Mechanisms, Transmissions and Applications

Mechanisms, Transmissions and Applications MECHANISMS AND MACHINE SCIENCE Volume 3 Series Editor MARCO CECCARELLI Fo...

Author: Erwin-Christian Lovasz | Burkhard Corves

279 downloads 2796 Views 17MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Mechanisms, Transmissions and Applications

MECHANISMS AND MACHINE SCIENCE Volume 3 Series Editor MARCO CECCARELLI

For further volumes: http://www.springer.com/series/8779

Erwin-Christian Lovasz · Burkhard Corves Editors

Mechanisms, Transmissions and Applications

123

Editors Erwin-Christian Lovasz Politehnica University of Timi¸soara Mechanical Engineering Faculty Bv. Mihai Viteazul 1 RO-300222 Timi¸soara Romania [email protected]

Prof. Dr. Burkhard Corves RWTH Aachen University Aachen Eilfschornsteinstrasse 18 52056 Aachen Germany [email protected]

ISSN 2211-0984 e-ISSN 2211-0992 ISBN 978-94-007-2726-7 e-ISBN 978-94-007-2727-4 DOI 10.1007/978-94-007-2727-4 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011940322 © Springer Science+Business Media B.V. 2012 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Scientific Committees

Workshop Chairman Erwin-Christian Lovasz (Romania)

Workshop Co-Chairmans Burkhard Corves (Germany) G.K. Ananthasuresh (India) Marco Ceccarelli (Italy) Ion Visa (Romania)

International Scientific Committee Florin Breaban (France) Giuseppe Carbone (Italy) Nicolae Mircea Dehelean (Romania) Ionut Doroftei (Romania) Gheorghe Gheorghe (Romania) Grigore Gogu (France) Clement Gosselin (Canada) Antoni Gronowicz (Poland) Mikio Horie (Japan) Mathias Hüsing (Germany) Manfred Husty (Austria) Brian Jensen (USA) Dan Mandru (Romania) Dan Margineanu (Romania) Karl-Heinz Modler (Germany) Mircea Neagoe (Romania) Nenad Pavlovic (Serbia)

v

vi

Scientific Committees

Dan Perju (Romania) Victor Petuya (Spain) Doina Pisla (Romania) Anupa Saxena (India) Lin Song (China)

Local Organising Committee at Politehnica University of Timisoara Dan Perju Inocentiu Maniu Iosif Carabas Valer Dolga Corina Gruescu Valeria Vacarescu Mesaro¸s-Anghel Voicu Nicolae Mircea Dehelean Dan Margineanu Steliana Vatau

with the support of Romanian IFToMM National Committee ARoTMM Ion Visa Vistrian Maties Erwin-Christian Lovasz Catalin Alexandru

and the support of the Romanian IFToMM Association of Mechanical Transmissions - ROAMET George Dobre Vasile Bolos Gheorghe Moldovean

Preface

The first Workshop on Mechanisms, Transmissions and Applications - MeTrApp2011 is organized by the Mechatronics Department at the Mechanical Engineering Faculty, “Politehnica” University of Timisoara, under the patronage of the IFToMM Technical Committees Linkages and Mechanical Controls (old Mechanical Transmission) and Micromachines. The workshop aims to bring together researchers and students who develop their work in disciplines associated with Mechanisms Science in a friendly, colleague-like and collaboration environment. MeTrApp-2011 offers a great opportunity for scientists all over the world to present their achievements, exchange innovative ideas and create solid international links. The scientific event is meant to gather the novelty and originality in Mechanisms Science and its application and to depict the trend of this important and creative field. The topics proposed for the Workshop are Mechanisms and Machine Design, Mechanical Transmissions, Mechatronic and Biomechanic Applications, Computational and experimental methods, History of Mechanisms and Mechanisms Science and Teaching Methods. MeTrApp-2011 received 29 papers which were carefully reviewed by at least two prestigious reviewers chosen from the international academic or scientific communities. A number of 26 papers were accepted for publication and presentation during the workshop. We would like to express grateful thanks to IFToMM International Federation for the Promotion of Mechanism and Machine Science, to the Romanian IFToMM National Committee ARoTMM and the members of the International Scientific Committee of MeTrApp-2011. We appreciate the effort of the reviewers gathered in the International Scientific Committee of MeTrApp-2011. They spent time for a serious work of evaluation and improvement guidance, meant to assure a high quality of all papers. We thank to the authors who contributed valuable papers on different subjects, covering a wide scientific area of the Mechanisms and Machine Science. We thank to the “Politehnica” University of Timisoara, Mechanical Engineering Faculty and Mechatronics Department for hosting the scientific event and supporting all associated activities. The First Workshop on Mechanisms, Transmissions and Applications became a real event due to an enthusiastic and hard-working vii

viii

Preface

organizing team: Dan Perju, Inocentiu Maniu, Iosif Carabas, Valer Dolga, Corina Gruescu, Valeria Vacarescu, Mesaro¸s-Anghel Voicu, Nicolae Mircea Dehelean, Dan Margineanu, Steliana Vatau and Valentine Ciupe. Last, but not least we are grateful to the staff at Springer Publishers for their excellent technical and editorial support. Special thanks go to Mrs. Nathalie Jacobs, Senior Publishing Editor. Timi¸soara, Romania Aachen, Germany July 2011

Erwin-Christian Lovasz Burkhard Corves

Contents

Part I

History of Mechanisms and Machine Science and Teaching Methods

On the Evolution of Graphical Representation of Gears . . . . . . . . . M. Ceccarelli and M. Cigola Kurt Hain and Walther Meyer zur Capellen: A View from Aachen at Two Shapers of German Mechanism Theory . . . . . . B. Corves

3

15

thinkMOTION – DMG-Lib Goes Europeana . . . . . . . . . . . . . . . U. Döring, R. Brecht, and T. Brix

37

Multilingual Illustrated µ-Thesaurus in Mechanisms Science . . . . . . E.-C. Lovasz, D. Perju, B. Corves, T. Brix, K.-H. Modler, I. Maniu, C.M. Gruescu, A. Lovasz, and V. Ciupe

47

A General Classification for Mechanisms Regarding the Motion Task . B. Corves, S. Kurtenbach, M. Hüsing, and C. Schneider

59

Part II

Mechatronic and Biomechanic Applications

Developments of a Knee Motion Assist Mechanism for Wearable Robot with a Non-circular Gear and Grooved Cams . . . . . . . . . . . H. Terada, Y. Zhu, M. Suzuki, C. Cheng, and R. Takahashi

69

Force Distribution for a Walking Robot with Articulated Body . . . . . I. Doroftei, Ghe Plesu, and B. Stirbu

77

Numerical Root Finding from an Engineer’s Point of View . . . . . . . M. Färber and U. Döring

91

Kinetostatic Analysis of an Articulated Walking Mechanism . . . . . . F. Moldovan, V. Dolga, and C. Pop

103

A Mirror Tracking Mechanism . . . . . . . . . . . . . . . . . . . . . . . N.M. Dehelean and L.M. Dehelean

111

ix

x

Contents

Choppers: Optomechatronic Devices for the Controlled Modulation of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.-F. Duma, M.F. Nicolov, M. Kiss, T. Ilca, C. Mnerie, D. Demian, and L. Szantho Motion Laws of a Varifocal Objective Working with a 1/2” CCD Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.M. Gruescu Part III

125

133

Mechanisms and Machine Design

A New Principle of Adaptive Compliant Gripper . . . . . . . . . . . . . D. Petković and N.D. Pavlović

143

Optimum Design of Cam Mechanisms . . . . . . . . . . . . . . . . . . . I. Simionescu, M. Ene, and L. Ciupitu

151

Linkage Solutions for the Manufacture of Complex, Thermoplastic Lightweight Structures . . . . . . . . . . . . . . . . . . . M. Krahl, U. Hanke, and K.-H. Modler RBS Simulation of Integrated Piezo-Ceramic Actuation for Textile Reinforced Composite Compliant Mechanisms . . . . . . . . . . N. Modler, K.-H. Modler, W. Hufenbach, D. Margineanu, E.-C. Lovasz, D. Perju, and Ev. Ianosi Part IV

161

169

Computational and Experimental Methods and Dynamics of Mechanisms and Machines

Maximally Regular Planar Non Fully Parallel Manipulators . . . . . . G. Gogu

187

An Experimental Characterization of a Rickshaw Prototype . . . . . . T. Li and M. Ceccarelli

203

Considerations upon the Influence of Manufacturing and Assembly Errors on the Kinematic and Dynamic Behavior in a Flight Simulator Stewart-Gough Platform . . . . . . . . . . . . . . A. Pisla, T. Itul, D. Pisla, and A. Szilaghyi

215

Characterization of Flexure Hinges Using the Script Oriented Programming Within a FEM Software Application . . . . . . . . . . . . I. Ivanov and B. Corves

225

Basic Result on Type II DM Self-Motions of Planar Stewart Gough Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Nawratil

235

Contents

xi

Active Bearing for Vibration Damping of Roller Systems with Piezoelectric Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . M.-C. Voicu, R. Schmidt, B. Lammen, M. Mersch, and I. Maniu

245

On the Kinematics of a Hybrid Parallel Robot Used in Minimally Invasive Surgery . . . . . . . . . . . . . . . . . . . . . . . M. Suciu, B. Gherman, C. Vaida, N. Plitea, A. Stoica, and D. Pisla

255

Part V

Mechanical Transmissions

Synthesis of Toothed Continuously Variable Transmission (CVT) . . . . K.S. Ivanov Differential Planetary Gear Transmissions Usable in Renewable Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Saulescu, C. Jaliu, D. Ciobanu, and D. Diaconescu

265

273

Loading Devices for Closed-Loop Gear Test Stands . . . . . . . . . . . C.M. Gruescu, I. Nicoara, and E. Busa

281

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291

This is Blank Page Integra

xii

Contributors

R. Brecht Ilmenau University of Technology, Ilmenau, Germany, [email protected] T. Brix Ilmenau University of Technology, Ilmenau, Germany, [email protected] E. Busa “Politehnica” University of Timisoara, Timisoara, Romania, [email protected] M. Ceccarelli LARM: Laboratory of Robotics and Mechatronics, University of Cassino, Cassino, Italy, [email protected] C. Cheng University of Yamanashi, Yamanashi, Japan, [email protected] B. Corves RWTH Aachen University, Aachen, Germany, [email protected] M. Cigola DART, University of Cassino, Cassino Frosinone, Italy, [email protected] D. Ciobanu Transilvania University of Brasov, Brasov, Romania, [email protected] V. Ciupe Universitatea “Politehnica” din Timi¸soara, Timi¸soara, Romania, [email protected] L. Ciupitu “Politehnica” University of Bucharest, Bucharest, Romania, [email protected] L.M. Dehelean “Politehnica” University of Timisoara, Timisoara, Romania, [email protected] N.M. Dehelean “Politehnica” University of Timisoara, Timisoara, Romania, [email protected] D. Demian “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected]

xiii

xiv

Contributors

D. Diaconescu Transilvania University of Brasov, Brasov, Romania, [email protected] V. Dolga “Politehnica” University of Timisoara, Timisoara, Romania, [email protected] U. Döring Ilmenau University of Technology, D-98684 Ilmenau, Germany, [email protected] I. Doroftei “Gheorghe Asachi” Technical University of Iasi, Ia¸si, Romania, [email protected] V.-F. Duma “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected] M. Ene University of Quebec in Abitibi Témiscamingue, Rouyn-Noranda, QC, Canada, [email protected] M. Färber Ilmenau University of Technology, D-98684 Ilmenau, Germany, [email protected] B. Gherman Technical University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] G. Gogu LaMI, IFMA, Clermont University, Clermont, France, [email protected] C.M. Gruescu “Politehnica” University of Timi¸soara, Timi¸soara, Romania, [email protected] U. Hanke Dresden University of Technology, Dresden, Germany, [email protected] W. Hufenbach Technische Universität Dresden, Dresden, Germany, [email protected] M. Hüsing RWTH Aachen University, Aachen, Germany, [email protected] Ev. Ianosi Universitatea “Politehnica” Timi¸soara, Timi¸soara, Romania, [email protected] T. Ilca “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected] T. Itul “Technical” University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] K.S. Ivanov Almatinsky University of Power Engineering and Telecommunications, Almaty, Kazakhstan, [email protected] I. Ivanov RWTH Aachen University, Aachen, Germany, [email protected]

Contributors

xv

C. Jaliu Transilvania University of Brasov, Brasov, Romania, [email protected] M. Kiss “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected] M. Krahl Dresden University of Technology, Dresden, Germany, [email protected] S. Kurtenbach RWTH Aachen University, Aachen, Germany, [email protected] B. Lammen University of Applied Sciences Osnabrück, Osnabrück, Germany, [email protected] A. Lovasz Universitatea “Politehnica” din Timi¸soara, Timi¸soara, Romania, [email protected] E.-C. Lovasz Universitatea “Politehnica” din Timi¸soara, Timi¸soara, Romania, [email protected] T. Li LARM: Laboratory of Robotics and Mechatronics, University of Cassino, Cassino, Italy, [email protected] D. Margineanu Universitatea “Politehnica” Timi¸soara, Timi¸soara, Romania, [email protected] I. Maniu University “Politehnica” Timi¸soara, Timi¸soara, Romania, [email protected] M. Mersch University of Applied Sciences Osnabrück, Osnabrück, Germany, [email protected] C. Mnerie “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected] K.-H. Modler Technische Universität Dresden, Ilmenau, Germany, [email protected] N. Modler Technische Universität Dresden, Dresden, Germany, [email protected] F. Moldovan “Politehnica” University of Timisoara, Timisoara, Romania, [email protected] G. Nawratil Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Vienna, Austria, [email protected] I. Nicoara “Politehnica” University of Timisoara, Timisoara, Romania, [email protected] M.F. Nicolov “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected] N.D. Pavlović Faculty of Mechanical Engineering, University of Niš, Niš, Serbia, [email protected]

xvi

Contributors

D. Perju Universitatea “Politehnica” Timi¸soara, Timi¸soara, Romania, [email protected] D. Petković Faculty of Mechanical Engineering, University of Niš, Niš, Serbia, [email protected] Ghe Plesu “Gheorghe Asachi” Technical University, Ia¸si, Romania, [email protected] N. Plitea Technical University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] A. Pisla “Technical” University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] D. Pisla Technical University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] C. Pop “Politehnica” University of Timi¸soara, Timi¸soara, Romania, [email protected] R. Saulescu Transilvania University of Brasov, Brasov, Romania, [email protected] R. Schmidt University of Applied Sciences Osnabrück, Osnabrück, Germany, [email protected] C. Schneider RWTH Aachen University, Aachen, Germany, [email protected] I. Simionescu “Politehnica” University of Bucharest, Bucharest, Romania, [email protected] B. Stirbu “Gheorghe Asachi” Technical University, Ia¸si, Romania, [email protected] A. Stoica Technical University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] M. Suciu Technical University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] M. Suzuki University of Yamanashi, Yamanashi, Japan, [email protected] L. Szantho “Aurel Vlaicu” University of Arad, Arad, Romania, [email protected] A. Szilaghyi “Technical” University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] R. Takahashi Suncall Engineering Corporation, Japan, [email protected]

Contributors

xvii

H. Terada University of Yamanashi, Yamanashi, Japan, [email protected] C. Vaida Technical University of Cluj-Napoca, Cluj-Napoca, Romania, [email protected] M.-C. Voicu University “Politehnica” Timisoara, Timisoara, Romania; University of Applied Sciences Osnabrück, Osnabrück, Germany, [email protected] Y. Zhu University of Yamanashi, Yamanashi, Japan, [email protected]


xii

Part I

History of Mechanisms and Machine Science and Teaching Methods


xii

On the Evolution of Graphical Representation of Gears M. Ceccarelli and M. Cigola

Abstract In this paper, a historical evolution of graphical representation of gear designs is reported as a short survey by using significant drawings bath in the history of gear systems and design standards. A graphical representation of gear systems has evolved from a pictorial illustration towards a synthetic drawing of functionality and characteristics of gears that is today summarised in international standards. Those standards as from ISO and Italian UNI codes are still based on graphical schematic representation of gear characteristics with pictorial representation for an intuitive understanding of an intuitive understanding of main features of gear designs. A historical study is focused on the last century practical application of this gear representative. Keywords History of MMS · Gears · Drawing · Standards · History of technical drawing

1 Introduction Gears and gear systems have bean fundamental machine components and machinery yet that have been used and contributed to an evolution of machinery efficiency and capability as reported in many historical surveys like for example [1]. Technological evolution of gear systems has been supported also by suitable graphical representation of design features and functionalities both in practical applications and theoretical studies. In this paper attention is addressed to historical evolution of graphical representation of designs and features of gears and gear applications with a special focus on looking at recent practical views.

M. Ceccarelli (B) LARM: Laboratory of Robotics and Mechatronics, University of Cassino, Cassino, Italy e-mail: [email protected]

E.-C. Lovasz, B. Corves (eds.), Mechanisms, Transmissions and Applications, Mechanisms and Machine Science 3, DOI 10.1007/978-94-007-2727-4_1, C Springer Science+Business Media B.V. 2012

3

4

M. Ceccarelli and M. Cigola

2 Characteristics of Gears Characteristics of gear designs are today stressed by means of geometrical parameters through an universally accepted representation as in Fig. 1, [2]. Such a scheme is usually used to explain main design features and to develop suitable formulation both for analysis and synthesis purposes. These schemes are technical representations that nevertheless contain a certain direct physical interpretation of gear figures.

3 Drawing Standards for Gears Currently in Italy drawings of gears is codified in the standard UNI EN ISO 2203:1999’ Technical drawings-conventional representation of gear’, which is the official Italian version of European standard EN ISO 2203 of October 1997. This standard establishes conventional representation of teeth, geared wheels, and gears, as well as screws and wheels for chains, by indicating drawing rules both for design details and general assembly. The basic drawing rule gives a gear as a solid ungeared body with thin lines indicating primitive surfaces. This representation rule is not applied for axial sections. Contours and edges of geared wheels are represented as for a bulky wheel whose contour is given by the external limiting surface where as the primitive surfaces are indicated with thin dashed lines, Fig. 2. The bottom surface of teeth is drawn usually as a thin line only within sections or when necessary for outlining the gear nature of the element, Fig. 3. In schematic drawings with planar view the teeth orientation can be indicated by thee continuous lines delimiting the body of a tooth, Fig. 4. Those general rules círculo base círculo de dedendo

círculo de paso círculo de adendo

Fig. 1 A scheme for geometrical parameters of gears, (Ceccarelli and Lopez 2008), where the indicated circles are for the base, dedendeum, and addendum, repacively

On the Evolution of Graphical Representation of Gears

Fig. 2 Standard drawings of geared wheels for contours and edges

Fig. 3 Thin lines for bottom surfaces of geared elements

Fig. 4 Standard schemes for teeth orientation on geared wheels

5

6


Fig. 5 Standard graphical representation of bevel gears

Fig. 6 Standard representation of gears with their assembly

for planar gears are extended to bevel gears with indication of primitive surfaces by lines up to their characteristic intersection points, Fig. 5. In general, both gears of an assembly are fully represented unless one is in front of the other and this is a functionality aspect, Fig. 5. In Fig. 6 examples are reported to summarize the today standard graphical technical representations of gears.

4 A Historical Outline of Gear Designs A brief account of a historical evolution of gear representation can be summarise by using significant illustrative examples, as reported in Fig. 7 to 11.


a)

7

b)

Fig. 7 Examples of gear designs in Middle Ages: a irrigating machine by Al Jazari (1205 ca.) [3]; b hydraulic saw by Villard de Honnecourt (1225-1235) [4]

a)

b)

Fig. 8 Examples of gear designs during Renaissance: a rotating crane for construction of the dome of S, Maria del Fiore cathedral in Firenze by Filippo Brunelleschi (1420-1421) [5]; b epicloidal gears by Leonardo da Vinci (1445-1519) [6]

8


a)

b)

Fig. 9 Examples of gear designs in Theatrum Machinarum by: a punching machine for powder production by Giovanni Branca (1629) [7]; b pumping machine in Fig. II by Jacob Leupold (17241739) [8]

It is worth to note how such a representation evolved from pictorial shows with no technical details to synthetic schemes that are the bases of current standards through a slow process that has been accelerated during the Industrial Revolution.

5 Recent Evolution of Practical Representation of Gears An interesting historical insight of modern practical use of graphical representation can be achieved by looking at textbooks and handbooks that were used by practitioners during 20th century. Interesting aspects of evolution of graphical representation of gears can be discussed by referring to an example of a recent drawings with practical insights for gear use. Thus, an investigation is reported by looking at the use of gear designs in practical handbooks in the first half of 20th century, since it is still possible to appreciate evolutions within a modern approach with current formalism. Thus, the discussion has been based on analysis of three Italian handbooks for drawing purposes and machining applications by Goffi in 1932) [13], Bianco in 1947 [14], and Massero in 1958, [15], as shown in Fig. 12. It is to note the characters of those handbooks as directed to technicians for practical work and implementation of the gear manufacturing and technology.


9

a)

b) Fig. 10 Examples of gear designs from early handbooks: a gears for clocks in the Encyclopediè by J.B. D’Alembert and D. Diderot (1774) [9]; b gear assembly for a humanoid automata in handbook by Gian Antonio Borgnis in 1825 [10]

10


a)

b)

Fig. 11 Examples of gear designs during Industrial Revolution: a power machine by W. Rankine Mc Quorn (1869) [11]; b hypoid gears by F. Masi in 1897 [12]

In particular, book by Goffi was published in 1932 as ninth edition of a book that was also translated into Spanish in 1912 and 1926, [16]. The handbook by Bianco in 1947 was published also with the aim to promote products and gears by the company Elli & Zamboni who was the publisher yet1 . The third book by Massero in 1958 has been considered in this study to stress the changes with respect to the previous two books as related to the period after the World War II, although it was originally published in a first edition between the two World Wars. By considering the illustrations and drawings of those books, it can be noted how the gear representations evolved together with the drawing of machinery to a synthetic illustration stressing technical details that have been later coded in the standards summarized in 3. The graphics in book by Bianco in 1947, [14], is the most attractive example, since explanatory text is combined with red colour figures, Fig. 13. The number of pages is written within small gears which are different for odd and even pages. The illustrations are given with all the details as prescribed by the standards and even all the dimensions are reported. In Fig. 14 the evolution that can be appreciated within the three books, is stressed by looking at the illustrative representation of few teeth in Fig. 14 giving the size details on them as an attempt towards the current standard.

1 The Elli & Zerboni society, with center to Turin, has changed the name in ElliZerboni, and has become leader in the field of the production of cutting tools. For information see the english site of the company: http://www.ellizerboni.it/ellizerboni/site/eng/


a)

11

b)

c) Fig. 12 Cover pages of handbooks for an analysis of recent evolution of gear representation as published: a by Goffi in 1932, [13]; b by Bianco in 1947, [14]; c by Massero1958, [15]

Even for a representation of teeth set the drawing is reduced to essential elements (Fig. 15). In fact, in [14] by Bianco in 1947 a tooth profile is indicated with lines over the whole gear body and in [15] by Massero in 1958 the gear representation is limited to only half of the gear body as a preliminary version of the current standards in Fig. 3.

12


a)

b) Fig. 13 Examples from book by Bianco [14]: a page 264; b page 259


a)

13

b)

Fig. 14 Graphical representations of gears in: a book by Goffi [13]; b by Massero [15]

a)

b)

Fig. 15 Graphical representations of tooth profiles by: a Bianco [14]; b Massero [15]

6 Conclusions Graphical representation of gears and geared systems has evolved from a descriptive (even pictorial) representation to a synthetic technical scheme that is the base of today standards for technical drawings of gears. A practical representation has been defined for technicians in 19 century and it is still persistent as the core of standards. Acknowledgments The research work reported here was made possible by Grant n.250485 CIPICT-PSP2009.2.3 for Thinkmotion project by the European Community. References (Goffi 1932; Bianco 1947; Massero 1958), have been donated to the authors by Dr Sergio Cappellani from personal library of his father Erminio Cappellani.

14


References 1. Bautista Paz E., Ceccarelli M., Echavarri Otero J., Munoz Sanz, J.J., A brief illustrated history of machines and mechanisms, Science and Engineering, Book series on History of Machines and Machine Science, Vol.10, Springer, Dordrecht, 2010. DOI 10.1007/978-90-481-2512-8. ISBN: 978-90-481-2511-1. 2. Lopez-Cajùn C.S., Ceccarelli M., Mecanismos: Fundamentos cinematicos para el diseno y la optimizacion de la maquinaria, Trillas, Ciudad de Mexico, 2008 (ISBN 978-968-24-8181-9). 3. Al Jazari: Kitab al-Hiyal, 1205 ca. Istambul, Topkapy Museum. 4. Villard de Honnecourt: Livre de Portraiture, 1225-1235, Paris Bibliothèque Nationale. 5. Buonaccorso Ghiberti, “Zibaldone” (1420-1421), Firenze Biblioteca Nazionale, in E. Battisti: Brunelleschi, Electa 1975. 6. Mario Cianchi: Le macchine di Leonardo, Becocci Editore, Firenze 1984. 7. Giovanni Branca : Le machine, Roma 1629. 8. Jacob Leupold: . Theatrum machinarum, oder Schau-Platz des grundes mecanischer Wissenschaften / deutlich vor Augen gestellet von Jacob Leupold, Leipzig, 1724-1739. 9. Jean. Baptiste D’Alambert, Denis Diderot: Recueil de Planches, sur les Sciences, les Arts Liberaux, et les Arts Mechaniques, 3rd edition, Livourne 1774. 10. Giovanni Antonio Borgnis: Traitè complet de mecanique appliqée aux Arts, Bachelier, Paris, 1818-1821, 9 Volumes. 11. William John Macquorn Rankine : The cyclopaedia of Machine and Hands Tools, London 1869. 12. Francesco Masi: La Teoria dei Meccanismi, Bologna, 18. 13. Goffi, Valentino: Il disegnatore meccanico. Ulrico Hoepli Editore, Milano 1932, 9 edizione 14. Bianco, Emilio: Manuale pratico, ad uso dei tecnici addetti alle lavorazioni meccaniche, Società Elli Zerboni & c. s.l. 1947, 3 edizione. 15. Massero, Ferdinando: L’operaio tornitore e fresatore, manuale pratico con regole, tavole prontuarie, esempi, Ulrico Hoepli Editore, Milano 1958, 20 edizione. 16. Goffi, Valentino: Manual del ingeniero mecánico y proyectista industrial con nociones técnicas generales. Adrián Romo Editor, Madrid1912 e 1926.

Kurt Hain and Walther Meyer zur Capellen: A View from Aachen at Two Shapers of German Mechanism Theory B. Corves Abstract The occasion of the 100th Birthday of Dr.-Ing. e.h. Kurt Hain is the starting point of the author to shed a light on two shapers of mechanism theory in Germany: Kurt Hain and Walther Meyer zur Capellen. Considering the restricted time and space of this contribution this can only just tackle the full story. Two joint publications of the two protagonists will be taken as basis of this paper. It will be shown that for both the scientific career was affected by their education, their technical and mathematical interest (or better genius) as well as their work enthusiasm. Two different people have made quite an impact on the development of mechanism theory in the decades after World War II. Keywords Kurt Hain · Walter Meyer zur Capellen · FIAT Review · Graphical acceleration analysis · Dead center construction · Mechanism models · Literature reviews · History of mechanism and machines

1 Introduction This contribution about certain aspects related to the history of mechanism and machine science in Germany is based on a presentation during an honorary colloquium at the Department of Mechanism Theory at the TU Dresden, hosted by the IFToMM (International Federation for the Promotion of Mechanism and Machine Science) and the DMG-Lib (Digitale Mechanismen und Getriebebibliothek, Leistungszentrum für Forschungsinformation der DFG) on the occasion of the 100th birthday of Kurt Hain (Fig. 1). Therefore I would like to begin with Kurt Hain. Kurt Hain is certainly one of the most renowned representatives of mechanism theory in Germany. With great affection, he dedicated his graphical and mathematical knowledge to the synthesis and analysis of mechanisms. His almost 400 publications, stretching from the Thirties to the Nineties of the last century, are an eloquent testament to his achievement and his influence on mechanism theory, not only in Germany. With 13 books and over 380 B. Corves (B) RWTH Aachen University, Aachen, Germany e-mail: [email protected]


15

16

B. Corves

a)

b)

Fig. 1 a Kurt Hain (1908–1995) and b Walther Meyer zur Capellen (1902–1985)

conference and journal publications, he made an important contribution to almost every area of mechanism theory [1]. Another protagonist of German mechanism theory in the 19th centure is Walther Meyer zur Capellen. With his contributions to mechanism theory and his teaching and research activities, Walther Meyer zur Capellen also made a crucial contribution to shaping the development of mechanism theory in many areas, both nationally and internationally. He became renowned around the world as a leading academic regarding kinematics and dynamics of mechanisms. He was able to transmit his great specialist knowledge to his listeners with clear mathematical representations and a flair for teaching in numerous seminars and lectures. Alongside 32 research reports, he published 180 articles in publications at home and abroad [2].

2 Education and First Publications Both Kurt Hain and Walther Meyer zur Capellen are certainly outstanding representatives of mechanism theory in West Germany especially after the end of the Second World War. An examination of their lives shows both certain similarities and significant differences. Both Kurt Hain and Walther Meyer zur Capellen were born in the first decade of the twentieth century, when Germany was still a monarchy and ruled by an Emperor. Although Walther Meyer zur Capellen was six years older, almost to the day, since both were born in May [1,2], the educational path which followed for both also displays certain similarities, since both chose to study mechanical engineering upon completing school. While Walther Meyer zur Capellen joined the Faculty of Mechanical Engineering at TH Darmstadt in the summer semester of 1921, Kurt Hain completed a degree in mechanical engineering at the Höhere Maschinenbauschule (Higher Mechanical Engineering School) in Leipzig. After graduation, however, their paths took them in very different directions. Kurt Hain first worked as a design and development engineer in various industrial companies in Leipzig, Dessau and Dresden from 1931 to 1939, before moving to the aviation research institute in Braunschweig as a research and test engineer in 1939.

Kurt Hain and Walther Meyer zur Capellen

17

In contrast, after completing his Dipl.-Ing studies in 1926, Walther Meyer zur Capellen decided to continue his academic education by taking up an assistant post at the TH Darmstadt, as well as a second degree in mathematics, physics, philosophy, psychology and education, leading to a graduate teaching qualification in 1928. On 17th February 1932, his dissertation topic “Methods for the approximate solution of eigenvalue problems with application for oscillation problems” earned him his doctorate and the title Dr.-Ing. On 1st October 1932, he was employed as a lecturer in the position of “Baurat” (government building officer) at the Technische Staatslehranstalt für Maschinenwesen (state polytechnic institute in mechanical engineering) in Aachen, now known as Aachen University of Applied Science. These differing paths following the completion of their mechanical engineering education in the late 1920s and early 1930s already signalled the fundamentally differing academic focus of Kurt Hain and Walther Meyer zur Capellen. For Kurt Hain, as a trained designer, the knowledge one aims to achieve should always result from current practical problems and allow solutions to be found as directly and straightforwardly as possible, resulting in the development of useful applications for the everyday work of mechanical engineers active in the design of motion and force transmitting devices. This can already be seen even in his first publications in 1936 and 1937 on force transmission and velocity analysis [3,4] (Fig. 2). It should be noted that Kurt Hain himself never saw these two first publications as being especially worth drawing attention to, which is why his compilation of his own publications, which appeared in a VDI report [5] in 1979, quoted a publication from 1938 as his first publication [6]. This method of counting was also continued in [7] on the occasion of a colloquium for Kurt Hain’s 80th birthday, and it was only in [1], as the result of bibliographical research on the life’s work of Kurt Hain as part of the DMG-Lib research project [8], that the existence of the aforementioned publications [3,4] became known to a wider specialist audience once again. In contrast to these publications by Kurt Hain, which are strongly orientated towards practical application, the first publications by Walther Meyer zur Capellen are strongly influenced by mathematics. It is important to mention his work on the handling of simple kinematic problems in educational mathematics [9], since this publication unites all three of the typical elements which characterise almost all of his publications: mathematical precision, didactic standards and applications in mechanism theory (Fig. 3). It should also be emphasised that alongside publications on mechanism theory, Walther Meyer zur Capellen was working also on issues related to vibrations and machine dynamics as already indicated by the topic of his dissertation [10]. It must also not be forgotten that, in the 1930s and 1940s, Walther Meyer zur Capellen had also worked in detail on mathematic instruments as shown in publications [11,12]. Altogether, both Kurt Hain and Walther Meyer zur Capellen could already look back on a significant number of academic publications by the end of the Second World War. The somewhat lower number of 17 academic publications until 1945 by Kurt Hain compared to Walther Meyer zur Capellen’s 32 can be explained by the following circumstances. As a lecturer at the state polytechnic institute in mechanical engineering in Aachen, Walther Meyer zur Capellen held a position which allowed

18

B. Corves

Fig. 2 First publication by Kurt Hain [3]

him more opportunities and freedom to carry out his own academic research than Kurt Hain, who worked as a designer and development engineer for various industrial companies in Saxony until 1939, had little opportunity to carry out his own academic research. For Kurt Hain, this situation did not change until he moved to the aviation research institute in Braunschweig as a research and test engineer in 1939, giving him better opportunities to carry out his own academic investigations. This also explains, why the majority of the aforementioned academic publications


19

Fig. 3 Pair of cardan circles according to [9]

of Kurt Hain date from the period between 1939 and 1945. In this context, it must also be noted that Kurt Hain additionally published a large number of write-ups and reviews on general technical and engineering topics during this time, which are not included in the count according to [5]. In these works, Kurt Hain predominantly discusses the work of other authors which was of particular interest for general designers, for example [13].

3 Joint Publication Looking at the lists of publications for both Kurt Hain and Walther Meyer zur Capellen, the modern reader immediately notices that the majority of publications by both men lack co-authors [14]. In Walther Meyer zur Capellen’s case, over two thirds of all publications have no co-author, while that figure is above 90% for Kurt Hain. This is even more amazing when one considers that, in their most productive period both authors sometimes produced more than ten solo publications in a single year.

3.1 Joint Publication in the FIAT Review of German Science Indeed the first co-authored publication of the two is a joint publication from 1948 [15]. This publication, the table of contents of which is shown in Fig. 4, is a contribution to a multi-volume edition on “Naturforschung und Medizin in Deutschland

20

Fig. 4 Table of contents of [15]

B. Corves


21

in den Jahren 1939-1946”, an edition of the FIAT Review of German Science especially designed for Germany. The “Field Information Agency, Technical” (or: “... Agencies ...”) was an Allied institution which published multiple series of work (“reports”, “technical bulletin”, “review of German science 1939 - 1946”) up to 1947/1948, containing reports, sometimes from German academics, on the state of German research at the end of the war [16]. As the table of contents shows, the article is structured like a textbook in terms of the classification of topics handled, and begins with a chapter on the systematics and topology of mechanisms. Following brief reference to underlying considerations by Reuleaux and Grübler, only works by German mechanism designers published between 1938 and 1945 are quoted. Three publications by Walther Meyer zur Capellen [9,17,18] form an exception to this. These are mentioned as a reference for further publications by Walther Meyer zur Capellen in the report period. A total of 136 citations are quoted, 131 of which are referenced directly in the text. At the beginning of Chapter A, two mechanism theory textbooks published during the report period are mentioned, including the second volume of the textbook “Praktische Getriebelehre” by Kurt Rauh [19] issued in 1939. This is of particular interest to the compiler of this article, because Kurt Rauh’s “private lectureship” at RWTH Aachen University was transformed in the same year, being renamed an honorary professorship for “Mechanism Theory, Machine Design Theory and Patent Theory”, thus finally giving mechanism theory a fixed academic position at RWTH Aachen University. As the title of the textbook suggests, this book is aimed at designers looking for relevant practical solutions for real-life applications, which also explains the large number of illustrations in the book (Fig. 5). All in all, however, Kurt Rauh’s textbook is one of the few textbooks cited in the joint publication of Kurt Hain and Walther Meyer zur Capellen [15], since with a percentage above 96%, journal articles dominate the literature quoted. From a modern point of view, the article by R. Kraus on crank drives with stroke and time adjustment [20] appears the most interesting in Chapter A of the joint publication, since such mechanisms are today the subject of particular interest as so-called mechatronic systems with controlled adjusting drives. This seems to have been of particular interest to Kurt Hain and Walther Meyer zur Capellen too, since they dedicated over a page and four illustrations to it (Fig. 6). A certain personal connection, at least between Kurt Hain and the writer R. Kraus, certainly also played a role in this, as Kurt Hain makes clear in his dedication on the occasion of R. Kraus’ 65th birthday, which is shown in Fig. 7 [21]. The extract from the Konstruktion journal shown in Fig. 7 is also eloquent testimony to the ubiquity of Kurt Hain. Whereas in the left hand column Kurt Hain’s appreciation of R. Kraus can be found, the right-hand column commemorates the award of the Max-Eyth-Gedenkmünze, the highest prize awarded by the Max-Eyth-Gesellschaft in the VDI, to Kurt Hain. In Chapter B of [15] articles on the state of motion of mechanisms are discussed. It is certainly no coincidence that no fewer than five articles by Walther Meyer zur Capellen are quoted in this chapter. A Comparison with his bibliography according to [14] shows that only articles which are concerned purely with mathematical

22

Fig. 5 Illustrations from [19]

Fig. 6 Adjusting mechanism according to Kraus [20] as shown in [15]

B. Corves


23

Fig. 7 Reports from Konstruktion [21]

problems are omitted here. A further interesting aspect of this chapter is the first appearance of an article by another very renowned specialist in mechanism theory, Rudolf Beyer [22]. Figure 8 shows the geometrical construction of the inflection pole presented there. As [15] continues, it becomes clear that compared to the quotation of eleven articles by Kurt Hain and twelve by Walther Meyer zur Capellen after all nine references to work by Rudolf Beyer are made. This is certainly an additional sign of Rudolf Beyer’s expertise in mechanism theory, as well as a tribute to this by the authors of [15]. Later, it would fall to Kurt Hain to compile the obituary for Rudolf Beyer following his relatively early death in 1960 [23]. Chapter C of [15] is dedicated to the topic of velocity and acceleration analysis. The first mention of an article by Kurt Hain can be found here [24]. In this, it is interesting to note that, while the article by Hain mentioned in [15] is illustrated in the diagram shown in Fig. 9, this graphical depiction in this precise form is not found in [24]. The description of the velocity analysis for six-bar mechanisms with a four-bar mechanism as a replacement crank mechanism, suggested by Hain, is depicted in a way which is much easier to understand than in his original publication in [15] – many years after [24].

24 Fig. 8 Construction of the interpole according to [22]

Fig. 9 Substitute mechanism according to Hain [24]

B. Corves


25

Fig. 10 Acceleration structure according to Rosenauer in [15]

The articles quoted for acceleration analysis include a distinctively high number of articles on the graphical acceleration analysis developed by Nicolai Rosenauer. The procedure presented by Nicolai Rosenauer using the example of an eight-bar mechanism is described in detail through diagrams shown in Fig. 10. Interestingly, Kurt Hain and Walther Meyer zur Capellen would later undertake work on this topic in articles of their own [25,26], in part based on the work of Nicolai Rosenauer.

26

B. Corves

During the report period of [15], Nicolai Rosenauer was a Professor of Applied Mechanics at the University of Latvia. The occupation of the Baltic states by the then Soviet Union led Nicolai Rosenauer initially to West Germany. A little later, he immigrated to Australia [27], where, together with A.H. Willis of the University of New South Wales, he published according to Frank Erskine Crossley [27] the first English-language kinematics textbook [28] based on the previous German work on kinematics [31]. Interestingly, [14] contains a reference to a VDI research journal from 1957 regarding the generation of non-uniform rotational motions [29], which brings many of the authors referenced in [15], such as Rosenauer, Beyer and Hain, together with young kinematic engineers of the time such as Kiper, Roessner, Hagedorn and Volmer. According to the summary of this research journal “ the major part of this investigation is devoted to the quadric chain mechanism and its maximum and minimum transmission ratios. Other mechanisms dealt with in detail are: spatial genevas, six-link mechanisms, the turning-block linkage, cams, rope mechanisms, toothed gears with non-circular pitch profiles, and link mechanisms used in conjunction with spur gears.” Chapter D of [15] is concerned with publications on spatial gears. The field is dominated by five articles by Rudolf Beyer. Especially worth mentioning is his postdoctoral thesis on the synthesis of planar and spatial crank mechanisms, which he presented in 1939 at the TU Dresden [30]. As the author of the textbook “Technische Kinematik” [31] and publisher of the Reuleaux articles, and with his numerous publications since 1928, he was then a giant within the world of mechanism theory. Indeed, the memories of Frank Erskine Crossley on forty years of mechanism theory [27] speak of the three “German Missionaries” who visited the USA between 1957 and 1961: Kurt Hain, Rudolf Beyer and Walther Meyer zur Capellen. “Just as in the sixth century Irish saints brought Christianity to the Germans, so now the Germans brought their kinematic science to the U.S.” – with these words, Frank Erskine Crossley described the influence of the three on research and teaching in mechanism theory in the US. Figure 11 depicts a photo from [27], which shows Rudolf Beyer together with Jacques Denavit at the Purdue Conference in 1958. Chapter E of [15] is dedicated to linkages, especially transfer mechanisms. Both this and the following Chapter F, which is about publications on dimensional synthesis, are dominated by articles by Kurt Hain and Hermann Alt. A representative of the Dresden School of mechanism theory, Hermann Alt is of course known to every mechanism designer for Alt dead centre design procedure often connected with his name and which is shown in Fig. 21 of [15] (Fig. 12) [32]. Hain’s coordinate reduction [3] must not be forgotten at this point either. This is explained in detail in [15] with the help of a corresponding picture (Fig. 13). Altogether it can be said that the joint article [15] by Kurt Hain and Walther Meyer zur Capellen, although initially begun as a backward-looking analysis of German source material on kinematics between 1939 and 1946, had a great effect on the development of mechanism theory not only in Germany but worldwide, thanks to the expert knowledge and vision of the two authors [27]. The work of almost all the mechanism designers who shaped development in Germany in the 1950s is mentioned in it.


27

Fig. 11 Photograph from [27]

Fig. 12 Alt dead centre construction according to [15]

3.2 Joint Publication in the Industrieanzeiger It was not until almost twenty years after [15] that the second joint publication by Kurt Hain and Walther Meyer zur Capellen appeared in the Industrie-Anzeiger as a two-part article [34,35] on four-link mechanisms with a prismatic joint. If one examines the bibliography of this article, one notices that only four of the 29 references do not refer to Walther Meyer zur Capellen. No really topic-specific citations by Kurt Hain are quoted, only his textbook “Angewandte Getriebelehre” [36]. This begs the

28

B. Corves

Fig. 13 Coordinate reduction according to Hain [15,33]

question of what kind of cooperation between the two men led to this publication. Indeed, the article relates to a topic which Walther Meyer zur Capellen had tackled in previous publications, while Kurt Hain had not. Simplifications for the graphical design of the acceleration of coupler points are derived, based on the assumption that, in mechanisms with prismatic joints, the linear relative motion of two gear elements causes the disappearance of the associated relative angular velocity. The determination of angular velocities and angular accelerations through graphical design is used to show how the acceleration pole and its locus can be determined. This is presented for the first type of sliding crank in Part I [34] of the article, and for the second type in Part II [35] (Fig. 14). In addition to this, the corresponding mathematical determination of the acceleration pole for the centric sliding crank is presented. In this, a distinction is still made as to whether the acceleration pole and its locus are to be shown in the crank plane, coupler plane or fixed plane. Compared to the joint publication described above, fewer details and less additional information on the two protagonists of this article can be gleaned from the joint publication [34,35] from 1965.

4 Mechanism Models Both Kurt Hain and Walther Meyer zur Capellen always placed great emphasis on transmitting their own knowledge to others, a fact which is eloquently demonstrated by the number of their publications. In this, both were very aware that knowledge is shared on different levels. In publications on paper especially, the static nature


29

Fig. 14 Sliding crank, first and second type [34,35]

of a geometric drawing often greatly limits how clearly kinematics, i.e. the theory of movement, can be represented. For this reason, both put great effort into driving forward a collection of mechanism models in their respective areas of responsibility. One of Walther Meyer zur Capellen’s first articles on mechanism models appeared in a publication not normally focused on mechanism theory, namely the journal “Fette, Seifen, Anstrichmittel, Die Ernährungsindustrie” (grease, soap, coating material, the nutrition industry), [37]. Most of the models mentioned there still exist today in the IGM mechanism collection, [38]. These first mechanism models were made from astralon, a pressed composite material made from impact-resistant PVC. Astralon has low flammability and its properties made it suitable for use in items for technical applications, which had high requirements regarding dimensional stability and resistance to chemical influences. These models were later replaced by acrylic glass models, so that, as demonstration objects, they could be projected onto a screen

30

B. Corves

Fig. 15 Images from [37] (excluding bottom left)

using an overhead projector. Figure 15 shows mechanism models taken from [37]. Only the image on the bottom left shows today’s electronic CAD version of the acrylic glass model of a dwell mechanism shown on the bottom right. It is based on the left substitute mechanism according to Roberts, shown in the centre, for the cycloidal mechanism shown in the top left. Kurt Hain also saw the use of mechanism models as a teaching aid to bridge the gap between academic publications and mechanism theory in practice in the engineering design office [39]. In 1959, he began to compile a collection of mechanism models at the Forschungsanstalt für Landwirtschaft (FAL) in Braunschweig [1]. By 1962, the collection already stretched to 130 models, the majority of which referred to applications in agricultural mechanical engineering. A Cam mechanism used as a pick-up mechanism in a straw baler is shown as an example in Fig. 16. The mechanism model collection built up by Kurt Hain formed the foundation for the atlas of mechanism examples published by him in 1973 [40]. Figure 17 shows diagrams and additional mechanism information for the pick-up mechanism from Fig. 16. In Kurt Hain’s view, which was certainly also shared by Walther Meyer zur Capellen, it is “of particular importance for the design of mechanisms to have a


31

Fig. 16 Model of a pick-up mechanism in a straw baler [1]

reference work for mechanism examples at hand. Both for new designs and for the improvement of existing machines, such a collection can not only offer suggestions for immediate application, but especially for further development in the prescribed direction, defined by the purpose in question”. Today, the two mechanism model collections started by Walther Meyer zur Capellen and Kurt Hain, as well as further mechanism model collections in Dresden and Ilmenau, form an important corner stone of the digital library project DMG-Lib [8] supported by the DFG (German research funding organization) and the European project thinkMOTION [41].

5 Literature Reviews As the work on the joint publication [15] has already made clear, both Kurt Hain and Walther Meyer zur Capellen displayed great interest in systematically recording and classifying literature on mechanism theory in corresponding overviews. Kurt Hain presented the first review of this kind in 1959, for the period 1956 to 1958 [42]. While making reference to corresponding reviews on mechanism theory literature from the English-speaking world and on Russian literature, he limited himself to works in the German language. In total, his article contained almost 200 citations. This review was followed by two further overviews for 1959 and 1960 [43,443], before this task was passed on to Walther Meyer zur Capellen [45]. He continued the annual publication of the literature review until 1969, from which point the tradition was carried on by Günther Dittrich, who had already acted as co-author for some of Walther Meyer zur Capellen literature reviews and was also Walther Meyer zur Capellen’s successor as Director of the Institut für Getriebetechnik und Maschinendynamik at RWTH Aachen University from 1972 [46]. The last of these literature reviews appeared in 1998 [47]. Today, these annual reviews form yet another inportant cornerstone of the DMG-Lib project [8] and the European project thinkMOTION [41].

32

Fig. 17 Mechanism theory information on pick-up mechanisms from [40]

B. Corves


33

6 Final Remarks This article has presented a few examples to demonstrate differences and similarities between Kurt Hain and Walther Meyer zur Capellen as key shapers of German mechanism theory, especially in the 1950s and 1960s. This article cannot do justice to all aspects of their life’s work, but it is certainly intended as inspiration to gain ideas and impetus for future work in mechanism theory by also looking at the past. Not least, projects such as DMG-Lib and thinkMOTION draw their standards for the transfer of knowledge in mechanism theory from precisely such ideas. Without a simple and comprehensive access to literature and other sources in mechanism theory, it would have been difficult for the author to compile this article in a reasonable amount of time.

Literature 1. Kerle, H.; Brix, T.; Modler, K.-H.: Presentation of the lifework of Kurt Hain - pioneer of applied kinematics in Germany. Proceedings of DETC2006: ASME 2006 Int. Tech. Conf. & Comp. and Inform. in Eng. Conf., September 10–13, 2006, Philadelphia, Pa., USA. 2. Dittrich, G.: Prof. Meyer zur Capellen 80 Jahre. antriebstechnik 21 (1982) Nr. 4, S. 134. 3. Hain, K.: Kräfteübertragung in Kurbelgetrieben. FTV-Nachrichten 16 (1936) Nr. 9/10, S.13 14. 4. Hain, K.: Geschwindigkeitsermittlung in Kurbelgetrieben. FTV-Nachrichten 17 (1937) Nr. 4/5, S.9-12. 5. Hain, K.: Verzeichnis der Veröffentlichungen. VDI-Berichte Nr. 321, 1979, S. 153-167. 6. Hain, K.: Geschwindigkeitsverhältnisse sämtlicher Koppelpunkte eines gegebenen Gelenkvierecks. Betrieb – Reuleaux-Mitteilungen/Archiv für Getriebetechnik 6 (1938) Nr. 2, S. 97-99. 7. Kerle, H.: Das bisherige Lebenswerk Kurt Hains im Zeichen seiner Veröffentlichungen. VDI Fortschrittberichte. Nr.159, 1988, S. 5-16. 8. Brix T., Döring U., Corves B., Modler K.H.: “DMG-Lib: the” “Digital Mechanism and Gear Library” “ - Project”. In: Proceedings of the 12th World Congress in Mechanism and Machine Science, June 18-21, 2007, Besancon, France. 9. Meyer zur Capellen, W.: Einfache kinematische Probleme in schulmathematischer Behandlung. Z. mat. naturw. Unterr. 64 (1933) Nr., S. 268-275. 10. Meyer zur Capellen, W.: Methode zur angenäherten Lösung von Eigenwertproblemen mit Anwendungen auf Schwingungsprobleme. (Dissertation), Ann. d. Phys. 5. Folge) (1931) Nr. 6, S. 299-352. 11. Meyer zur Capellen, W.: Zur kinematischen Analyse einiger mathematischer Instrumente. Z. Instrkd. 53 (1933) Nr. 2, S.56-64 und S. 108-115. 12. Meyer zur Capellen, W.: Instrumente zum Integrieren. Eine Uebersicht. Z. Instrkd. 58 (1938) Nr. 3, S. 93-99. 13. Hain, K.: Leistungssteigerung durch Konstruktionsarbeit im Vorrichtungsbau, W. Lindner, Maschinenmarkt 49 (1943), Nr. 44, S. 4-6, in: Die Messtechnik: Zeitschr. für zeitgemäße Betriebskontrolle, 1944. 14. Groot, de J.: Bibliography on Kinematics. Eindhoven University of Technology, 1970, Eindhoven, Netherlands. 15. Hain, K., Meyer zur Capellen, W.: Kinematik, in Naturforschung und Medizin in Deutschland 1939-46 (FIAT REVIEW OF GERMAN SCIENCE) Bd.7, angewandte Mathematik, Teil V (herausgegeben von Prof. Dr. A. Walther, Darmstadt). 1948 Dieterichsche Verlagsbuchhandlung, Wiesbaden.

34

B. Corves

16. http://de.wikipedia.org/wiki/Field_Information_Agency,_Technical (visited May 6th 2011) 17. Meyer zur Capellen, W.: Getriebependel. Z. Instrkd. 55 (1935) Nr. 10, S. 393-407 und Nr. 11, S. 437-448. 18. Meyer zur Capellen, W.: Die Abbildung durch die Euler-Savarysche Formel. Z. angew. Math. Mech. 17 (1937) Nr. 5, S. 288-295. 19. Rauh, K.: Praktische Getriebelehre. Bd.2, J. Springer, Berlin 1939. 20. Kraus, R.: Kurbeltriebe mit Hub- und Zeitverstellung. Maschinenbau, Betrieb (1942) 21, S. 173-175. 21. Hain, K.: Professor Dr.-Ing. habil. Robert A. Kraus 65 Jahre alt. Konstruktion 15 (1963) 7, S. 288. 22. Beyer, R.: Zur Konstruktion des Wendepols. Maschinenbau, Betrieb 18 (1939) S. 469-471. 23. Hain, K.: Professor Dr. phil. habil. Rudolf Beyer VDI (gest. 27.11.1960). VDI-Nachrichten aus Naturwissenschaft, Technik, Industrie. - Düsseldorf: VDI-Verl., Bd. 14 (1960), 39, S. 12. 24. Hain, K.: Geschwindigkeitsermittlung mit Hilfe von Ersatzkurbeltrieben. Getriebetechnik : Reuleaux-Mitteilungen. - Berlin : VDI-Verl., Bd. 9 (1941), 4, S. 177-178. 25. Hain, K.: Beschleunigungsermittlung in viergliedrigen Getrieben mit Hilfe von Ersatzgetrieben. Werkstatt und Betrieb 96 (1963), 1, S. 31-35. 26. Meyer zur Capellen, W.: Instrumentelle Mathematik für den Ingenieur. Girardet, Essen, 1952. 27. Crossley, F. R. E.: Recollections From Forty Years of Teaching Mechanisms. Trans. ASME: Jl. of Mechanism, Transmissions, and Automation in Design, vol. 110 (1988), 3, pp. 232-242. 28. Rosenauer, N., Willis, A.H.: Kinematics of Mechanisms. Associated General Publications Pvt. Ltd., Sidney, Australia, 1953. 29. Rosenauer, N., Beyer, R., Hain, K., Freudenstein, F. Kiper, G. Roessner, W. Hagedorn L., Schnarbach, K., Volmer, J.: Erzeugung ungleichförmiger Umlaufbewegungen. VDIForschungsheft 23 (1957) 461. 30. Beyer, R.: Zur Synthese ebener und räumlicher Kurbelgetriebe,. Forsch. Gebiete Ingenieurwes. 10, VDI Forschungsheft 394, Habil.-Schr. Dresden 1939. 31. Beyer, R.: Technische Kinematik : Zwangslaufmechanik nebst Bewegungsgeometrie und Dynamik der Getriebe in Theorie und Praxis. Leipzig: Barth, 1931. 32. Alt, H.: Über die Totlagen von Getriebegliedern, in: Getriebetechnik: Reuleaux-Mitteilungen, Band 8.1940, Heft 4, S. 17 20. 33. Hain, K.: Punktlagenreduktion als getriebesynthetisches Hilfsmittel. Getriebetechnik : Reuleaux-Mitteilungen. - Berlin : VDI-Verl., Bd. 11 (1943), 1, S. 29-31. 34. Hain, K., Meyer zur Capellen, W.: Gleichförmige Übersetzungen in viergliedrigen Getrieben mit einem Schubgelenk : [Teil 1]. Industrie-Anzeiger. - Essen : Girardet, Bd. 87 (1965), 33, S. 631-633. 35. Hain, K., Meyer zur Capellen, W.: Gleichförmige Übersetzungen in viergliedrigen Getrieben mit einem Schubgelenk : [Teil 2]. Industrie-Anzeiger. - Essen : Girardet, Bd. 87 (1965), 41, S. 777-781. 36. Hain, K.: Angewandte Getriebelehre. Düsseldorf : VDI Verlag, 1961. 37. Meyer zur Capellen, W.: Ueber gleichwertige periodische Getriebe.Z.: Fette, Seifen, Anstrichmittel, Die Ernährungsindustrie 59 (1957) Nr. 4, S. 257-266. 38. Corves, B.; Kloppenburg, J.: History and Future of the IGM-Mechanism Collection. IFToMM Workshop-Lectures: IFToMM 2006 History of Machines and Mechanisms 2006. in Eng. Workshop, September 9–10, 2006, Ithaca, NY., USA. 39. Hain, K.: Farbige Getriebemodelle zur Darstellung schwieriger Bewegungsgesetze. Landbauforschung Völkenrode, Bd. 12 (1962), 4, S. 83-86. 40. Hain, K.: Getriebebeispiel-Atlas. Düsseldorf: VDI-Verl., 1973. 41. http://www.thinkmotion.eu/ (visited May 6th 2011). 42. Hain, K.: Ungleichförmig übersetzende Getriebe : Schrifttumsübersicht der deutschen Arbeiten 1956 bis 1958. VDI Zeitschrift. - Düsseldorf : VDI-Verl., Bd. 101 (1959), 6, S. 255-259


35

43. Hain, K.: Ungleichförmig übersetzende Getriebe (Schrifttumsübersicht 1959). VDIZeitschrift. - Düsseldorf : VDI-Verl., Bd. 102 (1960), 6, S. 245-247. 44. Hain, K.: Ungleichförmig übersetzende Getriebe (Schrifttumsübersicht 1960). VDIZeitschrift. - Düsseldorf : VDI-Verl., Bd. 103 (1961), 6, S. 272-274. 45. Meyer zur Capellen, W.: Ungleichförmig übersetzende Getriebe (Schrifttumsübersicht 1961). VDI-Zeitschrift. - Düsseldorf : VDI-Verl., Bd. 104 (1962) Nr. 6, S. 301-03. 46. Corves, B.: Wissensvermittlung in der Getriebetechnik am Institut für Getriebetechnik und Maschinendynamik der RWTH Aachen. In: 6. Kolloquium Getriebetechnik – Aachen 2005. Aachen: 2005. S.3-29. 47. Dittrich, G.; Stolle, G.: Ungleichmäßig übersetzende Getriebe (Jahresübersicht 1997). VDIZeitschrift 140 (1998), Nr. II (Special Antriebstechnik), S. 70-77


xii

thinkMOTION – DMG-Lib Goes Europeana U. Döring, R. Brecht, and T. Brix

Abstract In 2011 the thinkMOTION project started. It focuses on making knowledge in the field of motion systems accessible to a broad public. The collected material covers forgotten treasures as well as current day research results. Different kinds of representation are supported, most notably books, articles, patents, images, videos, interactive animations and formalized mechanism descriptions. From the technical point of view the work in thinkMOTION is based on the DMG-Lib which was started as a German national project. Now workflow and tools were adapted to fit the needs of the work on European level. The content is collected on the DMG-Lib platform and presented to the international audience in different ways. The most important way for thinkMOTION is to make all the content accessible via Europeana, a multi-lingual online collection of millions of digitized items. This article focuses on the thinkMOTION workflow and the support of multilingualism. Keywords Digital library · Mechanism and machine science · Workflow · Rights clarification · Multilingual access

1 Introduction The access to the material reflecting the current knowledge, experience and skills in the field of motion systems is often restricted by its non-digital form and its scattering. Therefore the digitization of the content (Fig. 1) and the open access via an online platform are necessary to make it available to the wide range of user groups like scientists, engineers, historians, lecturers, students, pupils and interested laymen.

U. Döring (B) Ilmenau University of Technology, D-98684 Ilmenau, Germany e-mail: [email protected]


37

38

U. Döring et al.

Fig. 1 Different types of non-digital sources which are to be located, indexed, legally clarified, digitized and integrated into the online portal for multilingual access

1.1 A Historical Overview of the Project In 2004 the collection, digitization and online presentation of material concerning mechanism and machine science started as a German national project. The results are presented in the DMG-Lib (Digital Mechanism and Gear Library [2]) which was funded by the DFG (Deutsche Forschungsgemeinschaft) as competence centre for research information. The core team consisted of contributors from Ilmenau University of Technology, RWTH Aachen University and Dresden University of Technology, but soon the community grew and people from other institutions as well as private persons joined the project. Nevertheless during this time the focus was on the collection of German material. Since 2010 the material is collected on European level. Partners from: • • • • • •

Ilmenau University of Technology (Germany), University of Basque Country (Spain), Politehnica University of Timisoara (Romania), RWTH Aachen University (Germany), French Institute of Advanced Mechanics (France) and University of Cassino (Italy)

collaborate within the thinkMOTION project [9], which is funded by the European Commission. An objective of the thinkMOTION consortium is to seek for further international cooperation partners (other institutions like libraries, museums, research institutions or private people etc.). The work on European level raised new challenges concerning the support of multilingualism, fulfillment of special national demands (e.g. concerning rights clarification) and the coordination of the work of more than 100 contributing persons. The growth of the number of provided items substantiates the increased workflow requirements. In 2009 ca. 3000 items were available. At the end of the thinkMOTION project in 2013 more than 60000 items are expected to be online available. Achieving this in a good quality, supporting multilingual access and respecting the rights of use is indeed a big challenge.

thinkMOTION – DMG-Lib Goes Europeana

39

1.2 The Role of DMG-Lib e.V. Presenting the collected material online implies that the presenting institution must have the simple rights of use for that material or the material must be in the public domain. The main task of DMG-Lib e.V. is to receive the simple rights of use from the rights holders of the single items (books, articles, videos etc.) and to provide the material via the web pages of the DMG-Lib [1]. These web pages are indexed and referenced in Europeana [5] (see Fig. 2). Thus DMG-Lib e.V. (English: Association for Promoting the Digital Mechanism and Gear Library) plays the role of a legal body which is responsible for providing the content online. New members are welcome to take part in the work of DMGLib e.V. [3]. DMG-Lib e.V. as well as IFToMM (International Federation for the Promotion of Mechanism and Machine Science [7]) are thinkMOTION partners.

2 Workflow and Work Packages The material collected in thinkMOTION has a time range which starts from the cradle of motion systems and ends with the current state of the art (Fig. 3). These different sources are also reflected in the different ways of collecting the material. Usually all the material stemming from the 1980s and before is to be digitized. Only sometimes digital versions of such material can be found. Starting with the 1990s the content was mostly generated in digital formats. That means for the thinkMOTION workflow, that the digitization process can be skipped - at least if the sources are still available and can be imported. Nevertheless the material must be converted into web compliant derivatives according to their format and size.

Fig. 2 The role of DMG-Lib e.V. in thinkMOTION

40

U. Döring et al.

Fig. 3 Similar aspects of motion systems reflected in material stemming from different times

Fig. 4 Overview of the used thinkMOTION workflow

Figure 4 gives a short overview about the workflow showing different processing steps as well as the role of the database and the portal within the workflow. This workflow is except for the digitization the same for originally digital and non-digital content. Necessary preproduction of the digital raw data as well as the conversion to web compliant processed data is done in the step “Processing”. Unfortunately the step “Enrichment” is often very time consuming and must therefore usually be skipped. Nevertheless thinkMOTION plans to do enrichment of the material e.g. by supplying videos or interactive animations to certain images of mechanisms, because that is very important for the understanding and thus a real additional benefit for the users.


41

Fig. 5 Overview of the work package structure

Figure 5 describes the work package structure implemented for the European thinkMOTION project.

3 Special Aspects of the Work 3.1 Rights Clarification As stated before rights clarification is essential for online publishing. Therefore the process of clarifying the rights of use is an important step in the thinkMOTION workflow. The workflow is based on the experiences made in the DMG-Lib project and can be described as follows: • Each partner identifies and contacts rights holders. • The use of provided sample letters (covering letters etc.) makes communication with the rights holders more effective. • Letters of agreement (signed by the rights holders to grant the simple rights of use to DMG-Lib e.V.) are supplied in different languages, according PDF-files are generated with respect to the various national use cases. • The documents which shall be covered by a certain letter of agreement are generated based on the metadata stored in the database. • The handling of returned letters (storage of digitized signed letters, activating referenced documents etc.) is also supported in the database. • Barcodes identify the lists as well as single documents in the lists, this increases efficiency and helps to avoid errors during list processing.

42

U. Döring et al.

Fig. 6 Editing an empty English text block (creator data) for the letter of agreement

The uniform text blocks of the generated letters of agreement ensure that different objectives of the rights holders can be represented as well as that the permissions given by the rights holders can be handled automatically. To be uniform the translation work must be done very carefully. Figure 6 shows a translation dialog.

3.2 Multilingualism in Portal and Internal Database To make the knowledge stored in the digital library accessible to the European users the different languages of the users as well as of the content must be taken into account. Because good translations are usually quite expensive, it is necessary to focus on the most important parts. For the portal there are three fields where translations are made: 1. static texts used in the web pages like texts for navigation elements (headlines, menus, buttons etc.) and help texts, 2. titles of stored items (books, articles, images, videos etc.), and 3. metadata fields of items where the set of possible values is small. This way users should be able to navigate in a language they understand as well as to find matching content containing texts in other languages. If the non-textual


43

parts of the contents (e.g. images, animations, formulas, diagrams) dominate, then the user will probably derive benefit from foreign-language search results too. But even the textual content could be understood based on a raw automatic translation (e.g. provided via web translation services) and the usage of special dictionaries like the IFToMM dictionary [6]. In exceptional cases users could involve professional translators where it seems to be important. Translations of the internal database (called ProDB) are handled very similar to the translations of the portal. Texts for metadata and the texts of some navigation elements can be reused. In the future even the workflow for maintaining the IFToMM dictionary shall be based on the translation technology implemented in the DMG-Lib platform [4]. From a software technical point of view the usage of Lucene (a Java based search engine [8]) is of interest, because texts in different languages are to be handled with the appropriate stemming, stop words etc. According to interaction the effectiveness and the reliability are important. Therefore among the different possible ways of entering translations (e.g. list based translations, etc.) the dialog based translation as shown in Fig. 6 should be preferred. Being in the edit mode the translator can open the dialog with one click onto a certain text in a web page. That is fast and it is important for the reliability because the editor can see the context of the text. Context information is often important for the correctness of translations. Figure 6 also shows the possibility to consider more than one language as source for translations. That takes into account, that good translators are usually proficient in more than one language.

4 Expected Results As described before the content collected in the digital library is very heterogeneous. Figure 7 shows how many items of textual documents (books, articles, patents etc.), images and other items will be accessible in 2013. Figures 8 to 10 give a more detailed view onto the expected results.

Fig. 7 Overview of expected results according to the main types of the items

44

U. Döring et al.

Fig. 8 Expected results for available textual documents according to the subtype of the items

Fig. 9 Expected results for available images according to the subtype of the items

Fig. 10 Expected results for available other items according to the subtype of the items

Fig. 11 A main goal of thinkMOTION: making the DMG-Lib content accessible in Europeana


45

5 Conclusions As shown in this article a combination of qualified laborious staff members and optimized tools enables to achieve the ambitious goals of thinkMOTION, especially the provision of at least 60000 items with multilingual access in the field of motion systems. The thinkMOTION partner institutions are looking continuously for relevant material. If possible please become a contributor and help to enlarge the collection which is online accessible via DMG-Lib or other connected internet portals like Europeana (Fig. 11). Acknowledgments thinkMOTION is funded under the Information and Communication Technologies Policy Support Programme; Area: CIP-ICT-PSP.2009.2.3 - Digital Libraries: European Digital Library Digitising content for Europeana; Funding period: 01.06.2010 31.05.2013.

References 1. Brecht R., Krömker H., Kühlewind A.: Multimedia E-Books for Digital Mechanism and Gear Library. International Journal of Human and Social Sciences 4:3 2009, pp. 193-197 (2009) Available via WASET. http://www.waset.org/journals/ijhss/v4/v4-3-25.pdf 2. Digital Mechanism and Gear Library web site: http://www.dmg-lib.org 3. DMG-Lib e.V. web site: http://www.dmg-lib.org/association 4. Döring U., Brix T., Corves B.: Suggestion for a More Productive Workflow and Infrastructure of the Permanent Commission on Standardization of Terminology. In: Proceedings of 23rd Working Meeting of the IFToMM Permanent Commission for Standardization of Terminology on MMS, Minsk/Gomel, Belarus, 21-26 June 2010, pp.49-57 (2010) Available via DMG-Lib. http://www.dmg-lib.org/dmglib/handler?docum=10677009 5. Europeana web site: http://www.europeana.eu 6. IFToMM dictionaries online, newest Version is maintained at TU Delft, Version 2.3, April 2010, Available via TU Delft (June 2011). http://www.iftomm.3me.tudelft.nl 7. IFToMM web site: http://130.15.85.212 8. Lucene web site: http://lucene.apache.org 9. thinkMOTION web site: http://www.thinkmotion.eu


xii

Multilingual Illustrated µ-Thesaurus in Mechanisms Science E.-C. Lovasz, D. Perju, B. Corves, T. Brix, K.-H. Modler, I. Maniu, C.M. Gruescu, A. Lovasz, and V. Ciupe

Abstract In order to ensure a uniform indexing of technical information and a coordinated search engine in the field of mechanisms science, it is useful to develop a multilingual illustrated μ-thesaurus of “mechanism” indexing terms. This paper proposes a complete database of multilingual illustrated mechanisms science indexing terms, which will overcome the language barrier and will be integrated into technical macro-thesauri. Keywords Mechanisms science · Coordinated indexing · Multilingual illustrated thesaurus

1 State-of-Art in Indexing Methodology Indexing is a complex process in which the following elements are present: indexer, indexing language, indexing term(s). Indexers are specialized entities in indexing of documents. They can be a human operator or a machine/computer. If a person is indexer, he/she is called intellectual indexer. If the indexing is performed by a computer, the operation is called automatic indexing. Both intellectual and automated indexing may be free or controlled. Indexing language is a controlled set of terms selected from natural language to represent in condensed form the contents of documents. A classification of indexing language is presented bellow: • indexing language with hierarchical structure (uses descriptors), • indexing language with combinatorial structure (uses subjects), • natural language indexing (uses keywords). An indexing term can be a keyword, a subject or a descriptor.

E.-C. Lovasz (B) Universitatea “Politehnica” din Timi¸soara, Timi¸soara, Romania e-mail: [email protected]


47

48

E.-C. Lovasz et al.

An indexing term may appear both as a term derived from a natural language (simple or compound word) or as belonging to a particular classification code. In the traditional library, an indexing numerical code named classification was used as indexing term for bookstore. The American librarian, Melville Louis Kossuth (1851-1931), known as Melvil Dewey developed in 1876 a classification table for the library fund publications, based on the model of William Torrey Harris classification. The principles of Dewey Decimal Classification [1] are the universality (as it intended to include all human knowledge worldwide), the decimal division of knowledge into ten classes and structured hierarchy (which means each of the ten classes are divided into other ten classes through progressive division, from general to particular, resulting in a series of ranked indexes). The advantages of Dewey Decimal Classification are good practical applicability, efficient information retrieval tool and capability to sort documents by their content. The drawbacks are rigid hierarchical structure of concepts, lack of facilities regarding the representation of relationships between subjects and interdisciplinary areas the linear form of achievement of indexes. They are grouped three and delimitated by a single point and a small space, which makes them difficult to read. The Universal Decimal Classification [3, 4] is a processed Dewey Decimal Classification, result of continuously completing and amending in order to follow the widening of knowledge in all areas and in all forms, especially in technique and sciences. This system is adopted worldwide as a standard one and it is used in most libraries and information - documentation centers. This classification system allows depiction of interdependencies among subjects and interdisciplinary areas. The inconvenience consists of a large number of classification indexes and a rigid hierarchic structure of concepts. Information which is indexed using classification indexes can be found in traditional catalogues. The era of computerized indexing requires the use of indexing terms such as simple or complex phrases and different indexing languages. Indexing effectiveness on subjects depends very much on the quality of indexing language. Such a language should be very familiar to library users. The modern indexing is defined as representation, using natural language or documentary data resulting from analysis of content of the document or part of the document, in order to retrieve it. It is a type of analysis of the document whose content is expressed by keywords, subjects or descriptors. Indexing by keywords is a method of indexing, which uses keyword indexing terms. They consist of one word or a phrase, especially chosen from the title or text document, which characterizes the content of the text and allows retrieval of that document. This type of indexing is both non-unitary and non-controlled. Controlled indexing uses a thesaurus with controlled vocabulary. The most known thesauri are: LCSH – Library of Congress Subject Headings (USA), RAMEAU – Repertoire d’Autorite-Matiere Encyclopedique et Alfabetique Unifie (France), CSH – Canadian Subject Headings (Canada), RVM – Repertoire de Vedette de Matiere (Canada), MeSH – Medical Subject Headings and AAT – Art and Architecture Thesaurus. LCSH [2, 6] contains 317.000 subjects and references. The last issue in 2010 is the largest thesaurus worldwide today. Among the indexing terms, there are the

Multilingual Illustrated μ-Thesaurus in Mechanisms Science

49

following types of relationships: equivalence, hierarchy, associability, general and specific kind. Each type of subject is entered into a certain field, which contains more subfields. Each field and its subfields get a certain identification code. RAMEAU [5] is an encyclopedic indexing language, which introduces descriptors and is used in France. It is conceived as an open system, which allows a permanent update of the content. The indexing language is a controlled type one, with hierarchical structure, comprising generic, specific and associated terms. The technical area of indexing terms is poorly developed. CSH [13] is also an encyclopedic indexing language based on LCSH and it is used in Canada. Indexing terms are subjects. The French equivalent of CSH is RVM [8]. MeSH [15] is a specialized multilingual thesaurus in the field of medicine and AAT [x] is also a specialized thesaurus in the fields of architecture and arts, which contain a vocabulary of indexing terms descriptor-type.

2 Structure of Multilingual Illustrated µ-Thesaurus Thesaurus is a controlled vocabulary, flexible in terms, which are bonded together by semantic and generic relationships and which refer to a certain field of knowledge [9, 10]. Thesauri can be general or specialized on different domains. Specialized thesauri on domains can be called μ-thesauri, and may be included in General Thesauri called macro-thesauri. If the indexing language used in indexing is thesaurus-type, the term of indexing is considered descriptor. Indexing terms may be simple or compound. For compound terms, the indexing morphological term can be decomposed into separate components, expressed by nouns which can serve independently as indexing terms. If indexing language is a thesaurus, specific terms are chosen to represent a particular concept. For instance, controlled indexing uses the following abbreviation elements as indexing terms based on a thesaurus: ID – Identification number of the indexing term, DES – Descriptor: the term or the group of terms, which are used for controlled intellectual indexing with indexed language, NE – Explanatory note: the note associated with a term to indicate its meaning within an indexing language, VG – Generic Heading: the term that follows this symbol expresses the general concept which subordinates the given term, UP – Non-descriptor: synonym of the word regarded as descriptive term, TG – Generic Term: the term that follows this symbol expresses a wider concept than the given term and is placed in terms of hierarchical ascent immediately afterwards, TA – Associated Term: is a term which designates several common properties of more indexing terms [9, 10].

50

E.-C. Lovasz et al.

Multilingual μ-thesaurus assumes the following logical steps: selection of indexing terms, establishing of structure of indexing language, establishing of descriptors and translation of thesaurus. Conceiving of the multilingual illustrated μ-thesaurus considered the following elements: ID, DES, NE, VG, TG, TA, UP (see above), LE - translation of the term in English, LG - translation of the term in German, LF translation of the term in French and IM – display of an image or equation of the descriptor. The translation of terms is effective in more languages. The elements LE, LG, LF and IM are not quite used in standard indexing abbreviation, but in this multilingual illustrated μ-thesaurus it is an important element to multilingual and visual communication [7].

3 Multilingual Illustrated µ-Thesaurus Database The structure of multilingual illustrated μ-thesaurus in the field of mechanisms science contains 11 elements and corresponding fields. The name of the fields is the same with the abbreviation of the thesaurus elements. The indexing terms are established according to IFToMM Dictionary [11, 12]. For the field IM is added the file path of the image or equation, if one of these exists for the descriptor. In addition to the established 11 fields, the following fields were defined: file_DIGI_DB - field for connection to digitize information, file_STAND_DB - field standing for the link to the scanned image, file_A500_DB - field for connection to the database of the Central Library of UPT (Universitatea “Politehnica” din Timi¸soara). The application used to develop the database of the multilingual illustrated μthesaurus is XAMPP. μ-thesaurus database structure is shown in Fig. 1. All fields with necessary information on each indexing term of the illustrated multilingual μ-thesaurus were created as shown in Fig. 2.

Fig. 1 Multilingual illustrated μ-thesaurus database structure


51

Fig. 2 Fields of the indexing terms in the multilingual illustrated μ-thesaurus database

In order to explain the μ-thesaurus elements an example is given according to the mechanisms science definitions. The IFToMM definition of the “mechanism” derived of kinematic chain from structure point of view is: “The mechanism is a kinematic chain with one of its components (links) taken as a frame” and “The kinematic chain is an assemblage of links (elements) and joints”. The definition allows establishing a hierarchy of the μ-thesaurus structure elements (Table 1). The field VG - Generic Headings - contains in the mechanisms μ-thesaurus three general concepts: Machine/Apparatus, Mechanism and Technique, which subordinate all specific mechanisms terms [7]. As hierarchical descendent term of the generic heading, TG – Generic Term, indicates the specific chapter of mechanisms science: structure, kinematics, synthesis, Table 1 Example of the elements hierarchy in the μ-thesaurus structure 1 2 3 4

Generic Heading Generic Term Associated Terms Descriptors

mechanisms structure element cam, cylindrical cam, slider, rocker, crank, gear, rack, annulus, frame, etc.

joint lower pair, higher pair, revolute pair, prismatic pair, cylindrical pair, cam pair, etc.

52

E.-C. Lovasz et al.

kinetostatics and dynamics. Adequate term was used in the database field. This field may have one or more generic terms. For example, the descriptor “cam mechanism” allows entering of all five generic terms, but the descriptor “degree of freedom” is linked only to the generic term “structure” [7]. The associated term – TA - defines a common property of more terms, for instance the associated term “element” leads to the descriptors cam, rocker, gear, etc. The field DES – Descriptor - contains the specific term or a group of terms, which are indexed in the database (in Romanian language). These terms may be for instance cam, cylindrical cam, slider, rocker, crank, gear, rack, annulus, frame, etc. All these terms are used as specific mechanisms science terminology as they are included in IFToMM Dictionary [11, 12]. In case that the descriptor has a synonym (non-descriptor), this is shown in the field UP. For instance, we mention the following synonyms: drag-link mechanism = double-crank mechanism, hinge = revolute pair, Cardan joint = universal joint = universal coupling. The fields LE, LG and LF contain the corresponding terms of the descriptor in English, German and French languages. For each indexing term of the μ-thesaurus, a definition or some explanations are given. They are entered in the field NE - Explanatory Note.

Fig. 3 Interface of the multilingual illustrated μ-thesaurus database


53

If the mechanism term has specific illustration or drawing symbol, the picture file is loaded in the field “file_path”. After completing the information about each indexing term, this term will be validated. In the multilingual illustrated μ-thesaurus database are entered 385 mechanism science indexing terms [7]. The interface of the multilingual illustrated μ-thesaurus database is shown in Fig. 3.

4 Examples of Multilingual Illustrated µ-Thesaurus Database Example 1 If the indexing term “cam” is to be found, then this term must be typed into the search field and the result is shown in Fig. 4. The entry fields for the advanced search engine are not activated. The search results reveal more indexing term or group of terms, which contain the word “cam”. In this example 10 terms are found: cam, cam mechanism, cam pair, camshaft, cylindrical cam, etc.

Fig. 4 Search in the multilingual illustrated μ-thesaurus for the indexing term “cam”

54

E.-C. Lovasz et al.

Each resulting descriptor is accompanied on the right side by its specific illustration or drawing symbol (Fig. 4). The user can select the desired descriptor among the resulting descriptors containing the word “cam”. If one chooses the descriptor “cam”, a new window opens (Fig. 5). The window of the descriptor “cam” displays all information about the specific indexed term “cam” in the database. The content of the fields in the database in this example of indexing term “cam” are the following: DES: Cam NE: Component with a curved profile or surface whereby it imparts a displacement either by point or line contact with a cam follower VG: Mechanisms TG: Structure TA: Element LE: Cam LG: Kurvenglied LF: Came File_Path: C:/thesaur/cama.gif Example 2 In order to illustrate the use of advanced search in the μ-thesaurus database, only the word “element” is entered in the search field of associated term TA (Fig. 6).

Fig. 5 Specific window of the descriptor - cam


55

Fig. 6 Advanced search in the μ-thesaurus database for the associated term “element”

There are 36 indexing terms as search results. The first five shown terms are: bar, belt, cam, camshaft, Cardan shaft. The user can now select one of the descriptors and for each one of them a specific window opens, as shown in Fig. 7 for the descriptor “cylindrical cam”. From the specific windows of the descriptor “cam” (Fig. 5) and “cylindrical cam” (Fig. 7) one can notice that they share the same generic term TG “structure”. Displayed information is correct as it follows the definition of the element in the field of mechanisms science – “each of the component parts constituting a mechanism” [11, 12]. Example 3 If one selects for example “kinetostatic” in the field of the generic term TG, he or she gets all the specific kinematic descriptors (68 indexing terms). These descriptors in this example are: active force, angular momentum, axial force, balanced mechanism, balancing, bearing force, bending moment, central force, etc. Searching results are shown in Fig. 8 in alphabetical order. It is important to emphasize, that advanced search for generic terms is subordinated to general headings. Example 4 Synonyms of descriptors are known as non-descriptors, UP. To ensure the retrieval of these terms too, the non-descriptors are also included in the database, if they exist.

56

E.-C. Lovasz et al.

Fig. 7 Specific window of the descriptor – “cylindrical cam”

Fig. 8 Advanced search in the μ-thesaurus database for the generic term “ kinetostatic”


57

Fig. 9 Specific window of the descriptor “coupler”

One example is the descriptor “coupler”, which admits the synonym “floating link”. The specific window for the descriptor “coupler” is shown in Fig. 9.

5 Conclusions The multilingual illustrated μ-thesaurus in the science of mechanisms addresses to all types of users: – librarians – regarded as indexer persons, specialized in indexing documents in a library. The indexer uses this software tool featuring coordinated, consistent and unambiguous indexing, – researchers – assumed as specialists in mechanisms science. Beside basic function of getting structured information, they get a powerful tool for translating specific terms in different languages (English, German, French and Romanian), – students – who may access online the μ-thesaurus and get the benefit of familiarization with the basic concepts in mechanisms science. Use of the μ-thesaurus for indexing of documents related to mechanisms science was conceived to facilitate the convergence of indexing language and user’s searching language. This is a very important trait, considering that the user tends

58

E.-C. Lovasz et al.

to retrieve information from a very extended knowledge database, as mechanisms science provides. The translation of 385 descriptors in the μ-thesaurus database of mechanisms science indexing terms in English, German, French and Romanian fulfils a very difficult task, which is the overcoming of language barrier. The possibility of multilingual translation facilitates knowledge exchange among scientists, academic or industrial institutions. An important novelty, which does not exist in ordinary dictionaries is the association of figures or pictures to each descriptor of the database. Quick and intuitive information is offered to the user. Coordinated indexing versus free indexing brings firm benefit to the user, who reaches faster precise information. The multilingual illustrated μ-thesaurus database described above will be used in framework of the European Project thinkMOTION [16] in order to support the retrieval of information in the Digital Mechanism and Gear Library [14].

References 1. Chan, L. M. and all, Dewey Decimal Classification, A practical guide, 2nd. Edition, New York, Forest Press, 1996. 2. Chan, L. M., Library of Congress Subject Headings: Principles and application, 3rd edition, Englewood, Libraries Unlimited, 1995. 3. Frâncu,V. (trad.), Clasificarea zecimal˘a universal˘a: Ghid de utilizare, Bucure¸sti, ABIR, 2006. 4. Gilchrist, Alan ; Strachan, David. The UDC: essays for a new decade. London : Aslib, 1990. 5. Guide d’indexation Rameau: Répertoire d’Autorité-Matière Encyclopédique et Alphabétique Unifié. Centre national Rameau, Paris, 2004. 6. Library of Congress Subject Headings, 22nd edition, Washigton, D.C., Library of Congress, Cataloging Distribution Service, 1999. 7. Lovasz, E.C., Perju, D., Corves, B., Brix, T., Maniu, I., Modler, K.-H., Lovasz, A., Ciupe, V.: Multilingual illustrated μ-thesaurus of “mechanism” indexing terms, 13th IFToMM World Congress, Guanajuato, México, 19-25 June, 2011, A27_404. 8. Répertoire de vedettes-matiˇcre de la Bibliothˇcque de l’Université Laval: RVM, Bibliotheque de l’Université Laval. Quebec. Mars 1999. 9. STAS 10711-76. Informare s¸i documentare. Reguli pentru elaborarea tezaurelor monolingve. Bucure¸sti, Institutul Român de Standardizare, 1986. 10. STAS 12598-87. Metode de analiz˘a a documentelor, de determinare a con¸tinutului lor s¸i de selectare a termenilor de indexare. Bucure¸sti, Institutul Român de Standardizare, 1987. 11. Terminology for the mechanism and machine science. Mechanism and machine theory, vol. 38 (2003), 597-605. 12. http://130.15.85.212/terminology/TerminologyWeb/index.html (last accessed July 2011). 13. http://www.collectionscanada.gc.ca/csh/index-e.html (last accessed July 2011). 14. http://www.dmg-lib.org (last accessed July 2011). 15. http://www.nlm.nih.gov/mesh (last accessed July 2011). 16. http://www.thinkmotion.org (last accessed July 2011).

A General Classification for Mechanisms Regarding the Motion Task B. Corves, S. Kurtenbach, M. Hüsing, and C. Schneider

Abstract This paper presents a general classification method for mechanisms regarding the motion task which is used for storing mechanisms in a database. This classification bases on a practical oriented concept. A motion task to be solved can be described by pre-defined characteristics in a search engine in the IGM-Mechanism encyclopaedia. Through a comparison with the saved data in the database possible mechanisms were offered as solution for the motion task. The user can choose among these possibilities to find the best mechanism. Keywords Classification · Motion tasks · Mechanism · Linkage · IGM-Mechanism encyclopaedia

1 Introduction In general mechanisms can consist of several links and joints which can take different forms with different degrees of freedom. A variation of those joints, the number and the kind of links within the step of structural or typology synthesis leads to a large amount of different mechanisms. This typology synthesis is well known and done in [1] for a planar linkage with hinges and sliders up to four linkages. According to spherical mechanisms a classification is derived in [2] for linkages with four hinges. During the centuries many persons were engaged in generating a classification for mechanisms [3]. Assur was the first who classified mechanisms on base of their kinematic chain and categorised them in the so called Assur groups [4]. Later, Artobolevsky created practical catalogues symbolizing an encyclopaedia which bases on structural and functional features. More recently, Tsai introduces a systematic methodology using both an analytical and an algorithmic approach [5]. But even this kind of classification is focussed on kinematic chains. Due to the fact that these methods primarily are used for finding all possible combinations for a mechanisms but not for relating a solution to a special motion task the presented classification is developed. This classification is based on already B. Corves (B) RWTH Aachen University, Aachen, Germany e-mail: [email protected]


59

60

B. Corves et al.

implemented mechanisms which are realized as functional model or experimental prototype. The aim of this classification is at one side to describe existing mechanisms regarding their typology and motion task and at the other side to enable a typology syntheses based on this mechanism data base. The user has to consider the general purpose of the mechanism before making any choices.

2 Classification A classification of mechanisms regarding their number and type of links and joints and their dimensions (planar, spherical and/or spatial) is reasonable for classification in a theoretical way. The advantage is a clear and demonstrative procedure for classifying mechanisms. Using this classification the typology syntheses provides a huge and perhaps overcharged number of possible solutions. Plenty of time and a good intuition and expertise of mechanism behaviour are needed to solve required motion tasks. Sometimes it is possible to solve motion tasks based on already known solutions respectively mechanisms. In this case either experience or a classification of realized mechanisms regarding to their motion tasks is demanded. Due to that fact a general classification method, developed in [6] at the department for mechanisms theory and dynamic of machines (IGM), is introduced. This classification is subdivided into four main parts: • • • •

mechanism typology transfer tasks guidance tasks other tasks

2.1 Mechanism Typology The mechanism typology defines the mechanism in a structured way in regarding to the configuration of links, joints and motion. This kind of classification is an analogue to the theoretical ones with an offset regarding to the motion task. It helps to decrease the solution set, which is caused by the classification according to the purpose of the mechanism. It is divided in seven separate characteristics: • • • • • • •

mechanism dimension input motions output motions full-cycle mobility total number of links number of input links number of output links

A General Classification for Mechanisms

61

The mechanism dimension defines the dimension of the motion of the mechanism and takes the orientation of the input and output axis into account. Is divided into the following sub-characteristics: ◦ planar mechanism – general planar mechanism – planar mechanism with coaxial input and output axis ◦ spherical mechanism – general spherical mechanism – spherical mechanism with coaxial input and output axis – spherical mechanism with orthogonal input and output axis ◦ spatial mechanism – – – – –

general spatial mechanism spatial mechanism with orthogonal crossing input and output axis spatial mechanism with orthogonal intersecting input and output axis spatial mechanism with coaxial crossing input and output axis spatial mechanism with coaxial intersecting input and output axis

The input motion and output motions, actually representing two different characteristics, can be specified identical with the characteristics: – rotation – translation – positioning – orientating – guiding = positioning + orientating – screwing – rotation and translation along the same axis – several in and output motions The full-cycle mobility just can exist or even not. Both parameters “yes” and “no” can be selected. In a special case where the full-cycle mobility is not necessary for the solution but anyway possible, the characteristic “possible” can be chosen as well. The last three characteristics deal with the total number of links, number of output links and number of input links, respectively. The total number of links can be changed from “one” to “twelve”. Just in case a mechanism with “more than twelve” links is required, this setting can be chosen as well. The number of input and output links can be selected from “one” to “nine” with respect to the fact that there cannot be more input and output links than links in total.

62

B. Corves et al.

2.2 Transfer Tasks The classification transfer task can be separated in the characteristics • transfer function • motion of output link The transfer function describes the dependence of output motion upon input motion and can be specified into: – – – – – – – – –

general function between output and input motion symmetric function between output and input motion output and input motion are fully or sectional proportional velocity increase/decrease sectional sectional dwell sectional pilgrim-step special mathematical function superposition/distribution of motion other special function

The motion of output link can be classified into: – revolving – angular reciprocating (rocking) or linear reciprocating – rocking: angular range = 100◦ – rocking: angular range = 180◦ – rocking: angular range = 360◦ – linear reciprocating with a long stroke/lift

2.3 Guidance Tasks The last part of the presented classification describes the guidance task. It can be divided into the following characteristics: • • • • •

path of a point of a link – motion around fixed point motion along path/coupler curve orientation of a link rotation of link reference to input

The path of a point of a link – motion around fixed point defines a certain path of a special point on a link. A possible classification of this path is: – general path – closed path


– – – – – – –

63

symmetric path circular path rectilinear path elliptical path cycloid/trochoid path fixed point other special path

The motion along path/coupler curve can be “in both directions” or “in the same direction”. The orientation of link can take the settings – – – –

general angular positions symmetric angular positions parallel angular positions other special angular positions

The rotation of the output link of the guidance mechanism can be characterized similar to the motion of the output link of a transfer mechanism: – – – – –

revolving (Fig. 1a) reciprocating (Fig. 1b) rocking: angular range = 100◦ rocking: angular range = 180◦ rocking: angular range = 360◦

In principle, the guidance tasks have to be solved by generating of an output motion, in which the guided link or the guided coupler point is moved through a prescribed sequence of positions. It may be necessary to implement a specific relation between the output and the input motion. This case describes an inputrelated guidance task. It is characterized in that the prescribed sequence of positions is assigned to input positions. If the guidance task is independent to input motion

Fig. 1 Orientation of the output link: a revolving b reciprocation

64

B. Corves et al.

this task describes a not input-related guidance task. This relation is described with reference to input. The reference to input just can exist or not. The settings to be made there are “yes” (input-related guidance task) or “no” (not input-related guidance task).

2.4 Other Tasks The classification other tasks can be separated in the characteristics • locking mechanism • differential mechanism/summation mechanism • adjustable mechanism

2.5 Application of the Classification The objective of the described classification is on the one hand to offer a general method to describe mechanisms and on the other hand to make this classification useful in order to search mechanisms for given motion tasks. So the aim is to find at least one solution for a given motion task. To this purpose the IGM-Mechanism encyclopaedia [7] is derived basing on the numerous models in the collection of mechanisms at the IGM [8] and their descriptions [9]. This encyclopaedia contains knowledge about every mechanism principle developed in other classifications like [1,2] and is integrated an interactive software [7]. Entering several characteristics in the search engine of that software for at least one general classification definitely will lead to a suggestion of possible mechanisms solving the moving task. Figure 2 presents the procedure for evaluating possible solutions. Due to the fact that every deposited mechanism principle in the software is classified in the same way (Input Form) as the above described classification, the entered characteristics in the search engine (Search Form) enable a comparison. All mechanisms owning the same characteristics as the input data in the search form were suggested as possible solution mechanism (Result). Characteristics where no data was entered will be treated as “arbitrary” and do not cause a limitation of the solution set [10]. The search engine for entering the desired characteristics is shown in Fig. 3. Furthermore there is shown a practical example for using the IGM-Mechanism encyclopaedia: For an application in automobile industry there is searched for a mechanism transforming a rectilinear translation at the input link into a rotating motion at the output link. The mechanism has to be in planar arrangement and should have as less links as possible. By knowing this information, the search can be defined. The mechanism arrangement must be “general planar mechanism”. The input motions needs to be “translation” and the output motions must be “rotation”. Entering several numbers of links leads to a minimum number of four links. For less than four links


Fig. 2 Comparison between Search Form and Input Form

Fig. 3 Search engine and result

65

66

B. Corves et al.

there is no mechanism conforming to the input data in the search form. This configuration of input data leads to exactly four mechanisms of which the “slider-crank mechanism” can be evaluated as kinematic structure used in the above described combustion engine.

3 Conclusions The introduced general classification for mechanisms shows a practical orientated way for classifying any kind of mechanism. By knowing the purpose of the mechanism or a certain motion task the user has to define some characteristics. The database automatically searches for possible mechanism solving the motion task and displays them. The user can choose among these solutions for the optimal one. The digital mechanism and gear library (DMG-Lib) [11] picks up that kind of classification. The search engine is made up in the same way as the introduced classification. The actual research project called thinkMOTION, which is the European counterpart of DMG-Lib [12], is to be built up in an analogue way.

References 1. Kerle, H., Corves, B., Pittschellis, R.: Einführung in die Getriebelehre. 3. Auflage, TeubnerVerlag, 2007 2. Jesús Cervantes-Sánchez, J., Medellín-Castillo, H.I.: A robust classification scheme for spherical 4R linkages, Mechanism and Machine Theory 37 (2002) 1145-1163. 3. Ceccarelli M.: Historical evolution of the classification of mechanisms, International Symposium of History of Machines and Mechanisms HMM2004, 285-302, Cassino, 2004 4. Assur, L.V.: Reserches sur la structure et la classification des mechanismes planes articules à paires cinematique inferieures, Annales de l’Institution Polytechnique de St. Petersburg, Vol. 21, 187-283, 1913 5. Tsai, L.W.: Mechanism Design: Enumeration of kinematic stuctures according to Function, CRC Press, 2001 6. Niemeyer, J.: Methodische Entwicklung von Prinziplösungen bei der Auslegung ungleichmäßig übersetzender Getriebe unter Verwendung eines praxisorientierten interaktiven Wissensspeichers. Dissertation RWTH Aachen: Shaker-Verlag, 2003 7. http://www.igm.rwth-aachen.de/getriebemodellsammlung/home/index.php 8. Corves, B., Niemeyer, J.: Das IGM-Getriebelexikon als Instrument der Wissensverarbeitung in der Getriebetechnik. Proceedings of the IX. International Conference on the Theory of Machines and Mechanism, August 31 – Sept. 2, 2004, Liberec, Czech Republic. 9. Corves, B., Niemeyer, J., Kloppenburg, J.: IGM-Mechanism Encyclopaedia and the Digital Mechanism Library as a Knowledge Base in Mechanism Theory. ASME 2006 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, September 10-13, 2006, Philadelphia, Pennsylva 10. Niemeyer, J.: Das IGM-Getriebelexikon Wissensverarbeitung in der Getriebetechnik mit Hilfe der Internet-Technologie. In: Dittrich, G. (Hrsg.): IGM-Kolloquium Getriebetechnik 2000, Forschung & Lehre 1972-2000. Aachen: Mainz 2000, S. 53-66. 11. http://www.dmg-lib.org 12. http://www.thinkmotion.eu/

Part II

Mechatronic and Biomechanic Applications


xii

Developments of a Knee Motion Assist Mechanism for Wearable Robot with a Non-circular Gear and Grooved Cams H. Terada, Y. Zhu, M. Suzuki, C. Cheng, and R. Takahashi

Abstract For wearable robot, the active knee joint assist mechanism has been developed. Especially, this mechanism is driven by a non-circular gear and grooved cams. This mechanism rotates and slides simultaneously, which has 2 degree-offreedom. And relation between a rotation and a sliding motion can approximate arbitrary mathematical function. And using this mechanism, the knee joint can flex until 130degrees. It is larger than the angle of a conventional joint. In this report, the motion of a knee flexion is analyzed. We propose the fundamental design method. And to move smoothly, we investigate the motion curves. And the optimized locus of a non-circular gear pitch and a grooved cam are shown. Also, the prototype knee joint assist mechanism is evaluated. It is clear that proposed mechanism is useful for the motion assist of a knee joint. Keywords Wearable robot · Knee joint · Motion assist · Non-circular gear · Grooved cam

1 Introduction An Osteoarthritis (OA) is the mechanical abnormalities of human body involving degradation of knee joints which includes articular cartilage and subchondral bone. The estimated number of Japanese patients with knee OA is approximately 25 million [8]. This is popular joint disease which is caused by aging for Japanese. On the other hand, the Japanese sitting “Seiza” is very important knee flexion motion at the Japanese life style. At this “Seiza”, the knee joints are almost flexed, and the legs themselves are underneath thighs. And this flexion angle is over 120 degrees. If we cannot sit at the “Seiza”, it is very difficult to live in Japan. Therefore, the recovery of the knee motion with high flexion is very important.

H. Terada (B) University of Yamanashi, Yamanashi, Japan e-mail: [email protected]


69

70

H. Terada et al.

The replacement surgery of an artificial knee joint is one of the recovery methods which are popular in Japan. And after the surgery, the walk rehabilitation is essential to recover the performance of walking. In Japan, many active walk assist robots which have some actuators have been developed [1] [3]. Nevertheless, these assist robots don’t almost consider the motion of that equipped person. Therefore, when the equipped person has the knee OA, it has been difficult that person walks smoothly using these robots. So, the walk assist mechanism for rehabilitation using a linkage and cam had been developed [5]. However, that has a large shape, and it is afraid to be caught in the linkage. In other words, the performance of a proposed mechanism was insufficient. Considering these points, the knee motion assist mechanism for wearable robot with a non-circular gear and grooved cams has been newly developed. In this report, at first, the motion of a knee flexion on the wearing position is analyzed by an image processing. And the calculation method of a rotation center position which changes with the knee flexion is proposed. And using this motion, design methods of a non-circular gear profile and a grooved cam profile are proposed. Based on these methods, the prototype of knee motion assist mechanism is designed and tested to verify the proposed approach.

2 Measurement of a Knee Motion In general, the knee joint rotates within 0 to about 130 degrees at the walking on a flat and a slope. And, during the 15 to 90 degrees, knee joint rotates with gliding to the back direction, at the bones contacts section, as shown in Fig. 1 [7]. In general,

Femur

Imaginary rotation center

Femur Patella

Roll-back

Thigh Contact points of each knee joint rotating angle

Leg

Tibia

Tibia Knee joint rotating

Fig. 1 Motion of a knee joint at the bones contacts section (Right leg; x-y view)

Development of a Knee Assist Mechanism for Wearable Robot

71

that motion at the bone center section is well known. So, this motion is defined as the “roll-back” motion. However, this roll-back motion length on the outside of leg is extended from that length at the bone center section, which is caused by a medial rotation. Therefore, the knee motion in which the assist robot is attached has been measured using an image processing system (Plexlogger PL1). Table 1 shows the motion lengths for 4 persons on the outside of the knee. These lengths are different from each other. Also, they don’t depend on the height. Therefore, to reproduce the knee motion precisely for each person wearing an assist robot, we have to make a new motion mechanism which can realize the arbitrary motion loci of that imaginary rotation center. And then, when the knee motion assist mechanism will be made, it is easy to make the structure using a relative motion; a coordinate on the thigh side will be changed to the coordinate on the leg side. So, to quantify the roll-back motion, the imaginary rotation center on the leg side is defined. Figure 2 shows the imaginary rotation center loci on the leg side coordinate. These loci don’t be generated with the constant velocities. Especially, the velocities will be changed, at the start and end angles of the roll-back motion. So, the acceleration curve motions have to be considered. Table 1 Motion length of the roll-back at horizontal and vertical directions Subject

Tall height (mm)

Maximum horizontal motion (mm)

Maximum vertical motion (mm)

A B C D

1660 1660 1700 1735

16.7 11.4 13.8 23.3

8.3 5.8 16.5 24.5

50

Motion at the vertical direction mm

Subject A

Fig. 2 Imaginary rotation center loci on the leg side coordinate (Right leg)

Subject B Subject C

25

Subject D

0 −50

−25

0

25

−25

−50 Motion at the horizontal direction mm

50

72

H. Terada et al.

3 Design of a Knee Motion Assist Mechanism Considering these points, a knee motion assist mechanism is proposed as shown in Fig. 3. This mechanism consists of two grooved cams and one non-circular gear [2]. Especially, this non-circular gear profile is generated using two circular curves and an arbitrary curve. And a grooved cam at the mechanism center is the imaginary rotation center on the leg side. In general, the rotation center of non-circular gear is almost fixed, however, in this case a rotation center of this gear moves along this groove. Then, the other grooved cam and a non-circular gear drive the leg with roll-back motion. So, each terminology is defined as Table 2. Frame of a thigh side Grooved cam of generating a roll-back motion

Non-Circular gear Grooved cam of an imaginary rotation center

Driving gear

Roller follower

Roller follower

Knee joint rotating Roll-back Frame of a leg side

Fig. 3 Structure of a knee motion assist mechanism (Right leg)

Table 2 Terminology of a knee motion-assisting mechanism Symbol

Terminology

Symbol

Terminology

hx hxmax

hy hymax

rp

Displacement to x-axis direction Maximum displacement to x-axis direction Non dimensional displacement Radius of a cam follower assignment Radius of a pinion assignment

θs θe θ pf θ max

Start angle of the roll-back motion End angle of the roll-back motion Offset angle of a pinion assignment Maximum knee rotating angle

Displacement to y-axis direction Maximum displacement to y-axis direction Non dimensional time Equivalent length of the imaginary rotational center Equivalent angle of the imaginary rotational center Rotation angle of the knee joint Offset angle of a cam follower Initial knee rotating angle

S rc

T rk θk θ θ cf θ min


73

25 Displacement to x-axis direction Displacement to y-axis direction

Displacement mm

20

Approximate curve to x-axis direction Approximate curve to y-axis direction

15

Roll-back motion 10

Rotation only

Rotation only

5

0

0

30

60 90 Knee rotating angle arc.degree

120

Fig. 4 Relations between the knee joint rotation angle and the displacements of the imaginary rotation center

In the previous section, we have shown the imaginary rotation center loci on the leg side coordinate. And it is clear that there are large individual differences of the locus shape. Considering this point, Fig. 4 shows the relations between the knee joint rotation angle and the displacements of the imaginary rotation center. In this figure, the roll-back motion occurs between 20 degrees to 100 degrees. And on other angle, a knee joint rotates without gliding. When we use the linear interpolation with a constant velocity to make the locus of a grooved cam, a theoretical acceleration value will be infinity, it causes a vibration. Therefore, to eliminate the vibration, the roll-back motion is resolved into x-axis and y-axis direction. And these values are approximated using the modified constant velocity motion curve [4]. This motion curve is shown as: hx = hx max · S, hy = hy max · S, T=

θ − θs θe − θ s

(θ ≥ θs ) ,

(1) (2)

where, at the acceleration section; 8.01 8.01 T− sin 8π T (0 ≤ T ≤ 1/16) , 8π 64π 2 8.01 1 8.01 2 72.09 (16T − 1) π + T − + 1 − 1 − cos S= 6 8π 16 128π π 64π 2 (1/16 ≤ T ≤ 1/4) , S=

and where, at the constant velocity section:

(3)

(4)

74

H. Terada et al.

1 S = 1.28 T − 4

8.01 + 128π

16 4+ π

(1/4 ≤ T ≤ 3/4) .

(5)

If the kinds of these curves are different, the imaginary rotation center locus has a curved shape; when the imaginary rotation center locus has a curved shape, the motion curve has to be different each other. And the deceleration section velocity has a symmetrical shape of the acceleration section. And then, the roll-back motion is defined as: rk =

h2x + h2y ,

θk = atn2 hx , hy .

(6)

Using imaginary center motion locus, a center locus of a groove cam which generates the roll-back motion of each rotating angle is defined as: Pc = rc ej(θ+θcf )

(θmin ≤ θ ≤ θs ) ,

P’c = rc ej(θ+θcf ) + rk ejθk

(θs < θ < θe ) ,

P”c = rc ej(θ+θcf ) + rk max ejθk max

(θe ≤ θ ≤ θmax ) .

(7) (8) (9)

And a grooved cam profile accords to the envelope curve locus which offsets the calculated locus. Also, a center locus of a non-circular gear is defined as: Pp = rp ej(θ+θpf )

(θmin ≤ θ ≤ θs ) ,

P’p = rp ej(θ+θpf ) + rk ejθk P”p = rp ej(θ+θpf ) + rk max ejθkmax

(θs < θ < θe ) , (θe ≤ θ ≤ θmax ) .

(10) (11) (12)

In this report, the wire-cut electrical discharge machining is used to generate the non-circular gear profile. So, at first, an envelope curve which offsets the center locus of a non-circular gear until the pitch circle radius of the gear mesh is calculated. That offset direction is the imaginary rotation center of a knee side. The envelope curve length on the pitch circle is calculated, and we divided that length in a standard tooth pitch. Then, the normal line of the arbitrary curve on each divided position is calculated. And the standard tooth profiles are assigned on the envelope locus. However, the number of the teeth can select only an integer value. So, to generate the continuous locus of tooth center position, the addendum modification coefficient of each gear needs to adjust.


75

4 Verification of the Knee Motion Assist Mechanism Based on a proposed method, the prototype knee motion assist mechanism is made as shown in Fig. 5. The specifications of the prototype are shown as Table 3. This motion changes to the roll-back motion from a circular motion, and changes into circular motion again. So, the profiles of the grooved cam have very complicated shapes which include the two inflection point. For the normal person, attached on the knee joint, we have tested about the usefulness. When the conventional structure which rotates without gliding is used, the flexion angle is short about 5 to 12 degrees to the required angle. On the other hand, it has been confirmed that the proposed mechanism can assist the knee motion smoothly to the maximum rotating angle.

Grooved cam of an imaginary rotation center

Frame of a thigh side Non-Circular gear Driving gear

Grooved cam of generating a roll-back motion

Roller follower

Frame of a leg side

Fig. 5 Prototype of a knee motion assist mechanism [6]

Table 3 Specifications of the prototype knee motion assist mechanism Items

Values

Driving gear module Teeth number of a driving gear Each roller follower diameter Maximum displacement to x-axis direction hxmax Maximum displacement to y-axis direction hymax Radius of a cam follower assignment rc Radius of a driving gear assignment rp Equivalent length of the imaginary rotational center rk at θ = θ e Equivalent angle of the imaginary rotational center θ k at θ = θ e Start angle of the roll-back motion θ s End angle of the roll-back motion θ e Offset angle of a driving gear assignment θ pf Offset angle of a cam follower assignment θ cf Initial knee rotating angle θ min Maximum knee rotating angle θ max

0.8 10 8.0 mm 7.5 mm 18.4 mm 35.0 mm 38.4 mm 19.9 mm 67.8 degree 20.0 degree 100.0 degree 10.0 degree 170.0 degree 0.0 degree 130.0 degree

76

H. Terada et al.

5 Conclusions In this report, the motion of a knee flexion on the wearing position is analyzed. And the calculation method of a rotation center position which changes with the knee flexion is proposed. And using this motion, design methods of a non-circular gear profile and grooved cams profile are proposed. Also, the prototype of knee motion assist mechanism is tested to verify the proposed approach. In future work, using the proposed the knee motion assist mechanism, the wearable walk assist robot will investigate the usefulness for the walk rehabilitation after the Total Knee Arthroplasty. Acknowledgments The author is grateful to Dr. Katori, Tecpha-Japan Co., for cooperating to make the non-circular gear.

References 1. Ishii M. et al.: Development of Power Assist Suit for Assisting Nurse Labor, Improvement of Sensing System and Mechanism”, Proc. of the Welfare Engineering Symposium 2004, (2004), 123-126. 2. Katori H. et. al.: A Simplified Synthetic Design Method of Pitch Curves based on Motion Specifications for Noncircular Gears, Transactions of the Japan Society of Mechanical Engineers. Part C, vol.60, 570 (1994), 668-674. 3. Kawamoto H., Kanbe S., Sankai Y.: Power Assist Method for HAL-3 Estimating Operator’s Intension Based on Motion Information”, Proc. of the 12th IEEE Workshop on Robot and Human Interactive Communication, (2003), 1A1-E04. 4. Makino H.: Automatic Assembly Machine Kinematics, Nikkan Kogyo Shimbun Ltd., (1976), 29-31. 5. Terada H. et al.: Study of a Walk-Assisting System for Rehabilitation (1st report) Development of a Knee-supporting Mechanism, Proc. of the First International Conference on Design Engineering and Science, (2005), 329-334. 6. Terada H. et.al.: Japanese patents, Tokugan 2011-027547, (2011). 7. The Japanese Orthopaedic Association: Check Point of the Prosthetist and Orthotist, Igakushoin, (1978), 262-267. 8. Yoshimura N.: Progress of research in osteoarthritis. Epidemiolgy of osteoarthritis in Japanese population. - The ROAD study -, Clinical Calcium, 19(11) (2009), 1572-1577.

Force Distribution for a Walking Robot with Articulated Body I. Doroftei, Ghe Plesu, and B. Stirbu

Abstract Legged vehicles have attractive capabilities in terms of agility and obstacle avoidance. Also, these platforms are superior for locomotion on soft ground comparing to wheeled or tracked robots, particularly in low gravity. The functional capabilities of a walking robot can be essentially expanded by changing of the body design. It means that climbing over obstacles equal to the geometrical size of the robot could be possible by using a legged platform with articulated body. In this paper, some aspects regarding the force distribution for a hexapod walking robot with articulated body and its active compliance are presented. Keywords Walking robot · Kinematics · Compliance · Support force

1 Introduction Legged vehicles have attractive capabilities in terms of agility and obstacle avoidance. Also, these platforms are superior for locomotion on soft ground comparing to wheeled or tracked robots, particularly in low gravity. As we can see from the technical literature, walking robots with one, two, three, four, six or eight legs have been built. The number of legs affects some characteristics of the walking robots, such as: the stability, the efficiency, the possibility of walking with fewer legs when some of these are out of order (redundancy), the quality of the robot control, the price, the weight, the gait, etc. [16]. The functional capabilities of a walking robot can be essentially expanded by changing of the body design. It means that climbing over obstacles equal to the geometrical size of the robot could be possible by using a legged platform with articulated body. Several walking machines with articulated body have been developed in the past years [2], [3], [5], [14], [15]. The control of walking machines involves the

I. Doroftei (B) “Gheorghe Asachi” Technical University of Iasi, Ia¸si, Romania e-mail: [email protected]


77

78

I. Doroftei et al.

solution of the force distribution problem. There are many researches for resolving the foot force distribution problem for different objectives [6], [9-13], [17]. In this paper, some aspects regarding the force distribution for a hexapod walking robot with articulated body and its active compliance are presented. The robot, called IOAN has been built to Free University of Brussels.

2 Robot Kinematics Six legs offer a good compromise between weight and electromechanical complexity, on one hand, and stability, velocity and the variety of gaits, on the other hand. Each of the six legs was equipped with only two active degrees of freedom, in order to minimize its complexity. In spite of the limited mobility of the legs, this architecture led to spectacular vehicle mobility. Although the trajectory can never be a straight line because of the kinematics, the slippage does not cause any particular mechanical problem because of the small weight of the vehicle. To reduce this effect without adding much to the weight or the complexity, a passive joint equipped with a spring has been introduced in the ankle of the leg (Fig. 1). In order to improve the terrain adaptability of the vehicle, the body was separated in three articulated bodies (each of them equipped with two legs) connected with two servo-controlled universal joints (Fig. 2). The objective is twofold: active suspension and recovery from roll-over. The active compliance of the chassis increases the vehicle capability to get over obstacles by adapting its shape to the terrain profile. The robot (Fig. 4) is equipped with universal joints (with pitch and roll movements, see Fig. 2). Each degree of freedom of the universal joint is actuated by a four bar mechanism. The driven links of the mechanisms (BC and B1 C1 ) consist of flexible elements, equipped with strain gauges for force measurement purposes.

Fig. 1 Leg architecture: a kinematics; b 3D design

(a)

(b)

Force Distribution for a Walking Robot with Articulated Body

79

Fig. 2 Universal joint: a kinematics; b constructive solution

(a)

(b)

This configuration has two distinctive advantages: • It is possible to program the joints to behave as active springs of arbitrary stiffness (including negative). This configuration improves considerably the agility of the vehicle by adapting the configuration of the main body to the ground profile. • The vehicle can walk on both sides and transfer autonomously from back to belly and vice-versa. This allows recovery from roll-over. In order to adapt the robot to the ground, no force sensors on the feet have been used but some strain gauges have been disposed on the common link of the two four bar mechanisms that actuate the universal joint (Fig. 3).

Fig. 3 Strain gauges, indicated by the squares

80

I. Doroftei et al.

Fig. 4 Robot architecture: a kinematics; b constructive solution

(a)

(b)

3 Active Compliance and Force Distribution The principle of the active compliance consists of measuring the error of the position and the force distribution with respect to steady state reference values and enforcing the following equilibrium condition xr − x + k (fr − f ) = 0

(1)


81

where: • f and x are the current values of the force and the position; • fr and xr are the reference values; • k is the compliance of the system. In other projects [1], [6], [8], [12], [18], the active compliance of the legs is used. In our study, the active compliance has been implemented in the universal joints connecting the articulated bodies (see Figs. 2 and 4). As advantages, we can mention: • a smaller number of actuators; • a smaller number of points where the force should be measured (it means a smaller number a force transducers), etc. For each degree of freedom of the universal joint, we have θr − θ − k(Tr − T) = 0

(2)

where θ and T refer to the angular position and torque, respectively. We shall now examine how the reference torque Tr can be computed. As a first step, the force distribution for a given gait state of the robot can be computed, by expressing the static equilibrium equations of the whole vehicle. In order to simplify these equations, we assume that: • the speed of the robot and its mass are small enough as well as dynamical effects could be neglected; • the ground support forces are vertical (Fig. 5); • the mass of the leg link is negligible comparing to the mass of its actuators and also comparing to the mass of robot body. In other studies, the mass of all the leg is neglected. In our case, the mass of all the six legs is comparable with the mass of the robot body; this is why we consider that the influence of the leg mass should be taken into account. Assuming that the mass of the robot body, Mb , its center of mass(xb , yb ), the mass of a leg, ml and the coordinates of their centers of mass xli , yli are known, and assuming that all the six legs have the same mass, we get: ⎧ G = (Mb + 6 · ml ) · g ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎨ G·x =m ·g·x +M ·x ·g G l li b b i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ yli + Mb · xb · g ⎩ G · yG = ml · g · i=1

(3)

82

I. Doroftei et al.

Fig. 5 Force distribution for a given gait state

where: • • • • • • •

G is the weight of the vehicle, in [N]; i = 1 . . . 6, expresses the leg indices; xli , yli are the coordinates of the center of mass for i leg; (xG , yG ) are the coordinates of the robot center of mass; (xb , yb ) are the coordinates of the robot body geometrical center; Mb is the mass of the robot body; ml is the mass of a leg.

Because the robot has a symmetrical design, the geometrical center of the robot body is in the referential origin, it means that: xb = yb = 0,

(4)

and ⎧ G = (Mb + 6 · ml ) · g ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎨ G·x =m ·g·x G l li i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ yli ⎩ G · yG = ml · g ·

(5)

i=1

If we know α(t) and β(t) (see Fig. 1), we can compute xli , yli , for i = 1 . . . 6, at any moment and then (xG , yG ):


⎧ ⎪ ⎪ ⎪ ⎪ ⎪ xG = ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yG =

83

6

x li i=1 Mb ml +6

(6) 6

y li i=1 Mb ml +6

Assuming that the ground support forces are vertical, we get: ⎧ Fzi = G ⎪ ⎪ ⎪ i∈I ⎪ ⎪ ⎪ ⎪ ⎨ Fzi · xi = G · xG i∈I ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Fzi · yi = G · yG

(7)

i∈I

or ⎧ Fzi = (Mb + 6 · ml ) · g ⎪ ⎪ ⎪ i∈I ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎨ Fzi ·xi = ml · g · xli i∈I i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ Fzi · yi = ml · g · yli ⎩ i∈I

(8)

i=1

where: • I is the set of supporting legs (we may have three to six legs on the ground, according to the gait and its phase); • xi , yi , for i = 1 . . . 6, are the coordinates of the tip feet with respect to the referential center (geometrical center of the robot body). For the initial position, when α = 0 and β has the same value for all the legs, the center of mass and the geometrical center of the robot are identical, it means that ⎧ Fzi = G ⎪ ⎪ ⎪ i∈I ⎪ ⎪ ⎪ ⎪ ⎨ Fzi · xi = 0 ⎪ ⎪ i∈I ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Fzi · yi = 0 i∈I

(9)

84

I. Doroftei et al.

The equations (8) can be rewritten as follow: [A] · [Fz ] = [G],

(10)

with ⎡

1 1 1 1 1 [A] = ⎣ x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 ⎤ ⎡ Fz 1 ⎢ Fz ⎥ ⎢ 2⎥ ⎢ Fz 3 ⎥ ⎥ Fz = ⎢ ⎢ Fz ⎥ ⎢ 4⎥ ⎣ Fz 5 ⎦ Fz 6 ⎡ G ⎢ ⎢ ⎢ 6 ⎢ m ·g· x li ⎢ l [G] = ⎢ i=1 ⎢ ⎢ ⎢ 6 ⎣ ml · g · yli

⎤ 1 x6 ⎦ y6

(11)

(12)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(13)

i=1

Whenever there are more than three legs on the ground, this system of equations (A Fz = G) is under determined (there are more unknown than equations); the minimum norm solution is given by Fz = A+ G A+

(14)

AT (AAT )−1

= is the pseudo-inverse of matrix A. where 2 This solution minimizes J = Fzi while satisfying the equality constraints (5) of the static equilibrium. Once the vertical force distribution has been obtained, the reference torques in the two degrees of freedom of the universal joints can be obtained by considering the rotation equilibrium about the corresponding axis. A precise knowledge of the reference torque is not necessary and the active compliance (2) can just as well be implemented with Tr = 0.

4 Theoretical Simulation and Experimental Results As we have mentioned before, no force sensors on the feet have been used but some strain gauges have been disposed on the common link of the two four bar mechanisms that actuate the universal joint (see Fig. 3). This is why we cannot get experimental data according to the feet support forces.


85

0.4 0.4 0.3

0.3

0.2

0.2

0.1 yG [mm]

xG [mm]

0.1 0 −0.1

0 −0.1

−0.2

−0.2

−0.3

−0.3

−0.4 0

−0.4

100 200 300 t*0.018 [s]

0

100 200 300 t*0.018 [s]

4

4

Fz2 [N]

Fz1 [N]

Fig. 6 Center of mass coordinates distribution for a wave gait with λ = 5/6

2

0

0 0

100 200 t*0.018 [s]

300

0

100 200 t*0.018 [s]

300

0

100 200 t*0.018 [s]

300

0

100 200 t*0.018 [s]

300

4 Fz4 [N]

Fz3 [N]

4 2 0

2 0

0

100 200 t*0.018 [s]

300

4 Fz6 [N]

4 Fz5 [N]

2

2 0

2 0

0

100 200 t*0.018 [s]

300

Fig. 7 Feet support force distribution for a wave gait with λ = 5/6

In Figs. 6–9, some numerical results of the feet support forces, for a tripod gait (λ = 3/6) and a wave gait with duty factor λ = 5/6, are presented. Also, theoretical simulation results and some experimental data concerning the pitch (θ movement in Fig. 3) and roll (ψ movement in Fig. 3) forces of the front and back universal joints are shown in Figs. 10–13.

86

I. Doroftei et al.

1

0.25

x 10−3

0.2 0

0.15

−1

0.05

yG [mm]

xG [mm]

0.1

0 −0.05

−2 −3

−0.1 −0.15

−4

−0.2 −0.25

0

−5

50 100 t*0.018 [s]

0

50 100 t*0.018 [s]

4

Fz2 [N]

Fz1 [N]

Fig. 8 Center of mass coordinates distribution for a tripod gait with λ = 3/6

2 0

50 t*0.018 [s]

Fz6 [N]

2 0 50 t*0.018 [s]

0

100

0

50 t*0.018 [s]

100

0

50 t*0.018 [s]

100

0

50 t*0.018 [s]

100

6 4 2 0

100

4

0

2

100

6 4 2 0 0

Fz5 [N]

50 t*0.018 [s]

Fz4 [N]

Fz3 [N]

0

4

4 2 0

Fig. 9 Feet support force distribution for a tripod gait with λ = 3/6

Based on the theoretical values of the feet support forces, Fzi , the forces on the strain gauges have been estimated (the black color in Figs. 10–13). In Figs. 10a and 11a, the Fpf force (front pitch joint) and Fpb force (back pitch joint), for a wave gait with λ = 5/6, are represented. Figures 10b and 11b show the forces Frf and Frb in the front and back strain gauges dedicated to the roll joints.


(a)

87

(b)

Fig. 10 Theoretical and experimental forces in the front universal joint, for a gait with λ = 5/6: a for pitch motion; b for roll motion

(a)

(b)

Fig. 11 Theoretical and experimental forces in the back universal joint, for a gait with λ = 5/6: a for pitch motion; b for roll motion

In the same figures, with red color the experimental forces in the mentioned strain gauges are represented. If we compare the theoretical and experimental forces, we may easily see that the forces in strain gauges for pitch joints are better estimated. Figures 12 and 13 show these forces for a tripod symmetrical gait. In this case, the estimation is good for roll joints too.

88

I. Doroftei et al.

(a)

(b)

Fig. 12 Theoretical and experimental forces in the front universal joint, for a gait with λ = 1/2: a for pitch motion; b for roll motion

(a)

(b)

Fig. 13 Theoretical and experimental forces in the bach universal joint, for a gait with λ = 1/2: a for pitch motion; b for roll motion

5 Conclusions Legged vehicles have attractive capabilities in terms of agility and obstacle avoidance. Also, these platforms are superior for locomotion on soft ground comparing to wheeled or tracked robots, particularly in low gravity. The functional capabilities of a walking robot can be essentially expanded by changing of the body design. It means that climbing over obstacles equal to the geometrical size of the robot could be possible by using a legged platform with articulated body. In this paper, some aspects regarding the force distribution for a hexapod walking robot with articulated body and its active compliance have been presented. Acknowledgments The research work reported here was made possible thanks to our stages to the Mechanical Engineering and Robotics Department, free University of Brussels.


89

References 1. Alexandre, P., “Le Controle Hierarchise d’un Robot Marcheur Hexapode”, These presentee en vue de l’obtention du grade de Docteur en Sciences Appliquees, Universite Libre de Bruxelles, Belgique, Nov. 1996. 2. Alexandre, P., Doroftei, I., Preumont, A., “An autonomous micro walking machine with articulated body”, Proceedings of the 3rd IFAC Symposium on Intelligent Autonomous Vehicles (IAV’98), Editor(s): Salichs, M.A., Halme, A., pp. 557-562. 3. Brazevic P., Iles A., Okhotsimski D., Platonov A., Pavlovsky V., Lensky A. “Development of multilegged walking robot with articulated body”, Proceedings of the 2nd International Conference on Climbing and Walking Robots, CLAWAR’99, Portsmouth, UK, pp. 205-212, 1999, ISBN 1-86058-207-9. 4. Colon, E., Hong, P., Habumuremyi, J.-C., Doroftei, I., Baudoin, Y., Shali, H., Milojevic, D., Weemaels, J., “An Integrated Robotic System for Antipersonnel Mines Detection”, Control Engineering Practice, Pergamon Press, Elsevier Science Ltd., Vol. 10, Issue 11, 2002, pp. 1283-1291, ISSN 0967-0661. 5. Doroftei I., Preumont A., “Development of an Autonomous Micro Walking Machine with Articulated Body”, Proceedings of the 2nd International Conference on Climbing and Walking Robots, CLAWAR ’99, Portsmouth, UK, 14-15 September, 1999, pp. 497-507, ISBN 1-86058-207-9. 6. Gorinevsky, D. M., Schneider, A. Yu., “Force Control in Locomotion of Legged Vehicles over Rigid and Soft Surfaces”, The International Journal of Robotics Research, Vol. 9, No. 2, April 1990. 7. Habumuremyi, J.-C., Doroftei, I., “Mechanical design and MANFIS control of a leg for a new demining walking robot” Proceedings of the 4th International Conference on Climbing and Walking Robots (CLAWAR 2001), Eds. Berns, K.; Dillmann, R., pp.: 457-464, 2001. 8. Halme, A., Hartikainen, K., Kärkkäinen, K., “Terrain adaptive motion and free gait of a six-legged walking machine”, 1st IFAC Workshop on Intelligent Autonomous Vehicles, Southampton, UK, 1993. 9. Kar, D. C., Kurien Issac, K., Jayarajan, K., “Minimum Energy Force Distribution for a Walking Robot”, Journal of Robotic Systems 18(2), pp. 47-54 (2001), ISSN 0741-2223. 10. Klein, C. A., Chung, T., “Force interaction and allocation for the legs of a walking vehicle”, IEEE J Robot Automat 3, 1987., pp. 546-555. 11. Klein, C. A., Kittivatcharapong, S., “Optimal force distribution for the legs of a walking machine with friction cone constraints, IEEE Trans Robot Automat 6, 1990., pp. 73-85. 12. Knoth, A., March of the Insectoids, International Defense Review, No.11, pp. 55-58, 1994. 13. Kumar, V., Waldron, K. J., “Force distribution in walking vehicles”, ASME J Mech Des 112, 1990., pp. 90-99. 14. Preumont, A., Alexandre, P., Doroftei, I., & Goffin, F., “A Conceptual Walking Vehicle for Planetary Exploration”, Mechatronics, Elsevier Science, Great Britain, 1997, Vol. 7, No. 3, 287-296, ISSN 0957-4158. 15. Schmucker, U., Rusin, V., Konyev, M., Contact Processing in the Simulation of the MultiBody Systems, EUROSIM 2007, 6th EUROSIM Congress on Modelling and Simulation : Ljubljana, Slovenia, 9-13 Sept., 2007, Proceedings, Vienna: ARGESIM, 2007, ISBN: 978-3901608-32-2, ISBN: 3-901608-32-X, pp.76. 16. Todd, D. J., “Walking Machines: an Introduction to Legged Robots”, Kogan page, London, 1985. 17. Waldron, K. J., “Force and motion management in legged locomotion”, IEEE J Robot Automat 2, 1986., pp. 214-220. 18. Weidemann, H., J., Pfeiffer, F., Eltze, J., “The Six-legged TUM Walking Robot”, International Workshop on Intelligent Robots and Systems IROS, 1994.


xii

Numerical Root Finding from an Engineer’s Point of View M. Färber and U. Döring

Abstract The article starts with a presentation of a straight-forward global root finding algorithm featuring Newton-Raphson-like local root finding to find all solutions. After an analysis of the causes why it may fail, strategies are given to improve root finding. The strategies are tested and statistically analyzed with respect to two practical mechanical engineering problems: determining the point positions of an Assur group of third order and determining the translation and orientation of a Stewart Gough platform. In conclusion, ten advices are given on how to do root finding successfully and quickly. Keywords Numerical root finding · Newton-Raphson method · Broyden’s method · BFGS method · Assur groups · Stewart Gough platform

1 Introduction Faced with the task of finding the solutions of a non-linear equation system derived from a practical problem, one usually remembers one of the well-known root finding algorithms, say the Newton-Raphson method, and uses it in the hope to get the solutions easily and quickly. Unfortunately, there are problems and much more effort is needed. What these problems are and how they can be dealt with shall be shown in this article with the help of examples from geometric kinematic simulation of planar and spatial mechanisms.

2 A Practical Global Method for Multivariate Root Finding A root finding problem is the task to find all solutions for a function y = f (x) = (Φ1 (x) Φ2 (x) . . . Φn (x))T = 0 where x = (x1 x2 . . . xn )T is a vector of n variables over some domain X ⊆ Rn and the Φ i represent the n non-linear equations to M. Färber (B) Ilmenau University of Technology, D-98684 Ilmenau, Germany e-mail: [email protected]


91

92

M. Färber and U. Döring

be solved. We assume that the domain X is bounded, which should be no serious condition for practical problems. Starting from some start point within X we now apply a root finding method like Newton-Raphson [4], Broyden [2] or even an optimization method like BFGS [1], assisted by some kind of step length control [1] if needed. After a given number of iterations the process is repeated starting from another point until a given time budget is exceeded. All roots found throughout the search are collected. In the next sections we will present some theoretical and practical considerations regarding the detailed embodiment of this apparently simple global root finding algorithm. A statistical analysis for some example problems from mechanical engineering follows before the article concludes with giving ten practical advices for everyday root finding.

3 A Note on Order of Convergence, Effectiveness and Efficiency To specify the speed of a root finding method, the literature usually provides the order of convergence. It indicates the theoretical minimum of how good a method iterates towards a root. It is actually a speed measure—higher orders mean faster progress—and it is only given for the case of convergence. This means, that we have no measure of how well the method finds a root at all. And we do not know the speed of the method for a practical problem, because the order of convergence only gives a theoretical minimum. Therefore we need measures of practical relevance: effectiveness and efficiency, which are to be determined by experiment. Effectiveness measures the percentage of roots found with respect to the number of existing roots while efficiency says how fast the method is in case of success. Unfortunately, mathematics so far produced only a few general theorems regarding convergence and speed of a root finding method. There is Miranda’s theorem [11] which investigates the arithmetic signs of the function values. It states that if the function values change their signs in a certain way within a given region, that region must contain a root. The theorem provides the foundation for the large number of univariate bracketing methods [12, 6], but cannot be exploited directly for multivariate root finding. And there is Kantorovich’s theorem [9] which says: If a root is known and we know some properties of the function (mainly Lipschitz continuity of the derivative) we can establish a region around the root and the Newton-Raphson method will always converge to this root when started inside that region. Again, the practical use of the theorem is limited: extensive knowledge of the derivative is seldom available and the resulting regions are often far to small.

4 Causes of Failure of Newton-Raphson-Like Root Finding Aside from failure due to numerical instability, wrong error thresholds and other software-related problems there are three main reasons for the failure of NewtonRaphson-like root finding: divergence, oscillation and spurious convergence [4].

Numerical Root Finding from an Engineer’s Point of View 0.35 0.30 0.25 0.20 0.15 0.10 0.05

1.0 0.5 f(x) = e−x X

−3

2 4 6 8 10 12 14 (a) Iteration runs into an asymptote

−2

−1 −0.5

f(x)

2

3

x

f(x) = arctan x

(b) Divergent oscillation 30

25

1

−1.0

x

30 20

93

f(x) = x2(x2−1) + 3

25

= −x2(x2−6) + 11

20

15

15

10

10 5

5 0 1 2 −1 (c) Exact oscillation between two points

x

0 1 2 3 −1 (d) Inexact oscillation within a region

x

Fig. 1 Typical failures of the Newton-Raphson method

Figure 1a shows a function f(x) with lim f (x) = 0 but no root for x > 0. When x→∞ starting from a point x0 > 1, the root at x = 0 will never be reached. Another case of failure is shown in Fig. 1b. The prognoses created during iteration oscillate around the root at x = 0 and depart more and more from the root. In Fig. 1c we see an exact eternal iteration between two prognoses, and in Fig. 1d there is an endless oscillation between four slightly differing points. Finally, Fig. 2 presents an example of spurious convergence into a local min√ √ imum. The root finding had been started from x0 = 1 + 2 + 2 2 + 2 .

25

f(x) = x2+1

20 15 10 5

Fig. 2 Spurious convergence of the Newton-Raphson method

0

1

2

3

4

5

x

94


5 Difficulties with Finding All Solutions Root finding problems can have multiple solutions. Figure 3 gives an example; it shows the four different solutions to the task of finding the point positions of an Assur group of 3rd order [17] for a given drive angle at point A0 . We would expect, that a root finding method would always converge to the solution closest to the start point. Generally, this is not the case, as shown in Fig. 4 where a small change of the start point results in convergence to another root. Small changes causing big effects is characteristic for chaotic behaviour, and as seen in Fig. 5, the Newton-Raphson method may indeed produce fractal structures [8]. In the figure, each root is coded in another color and each point is drawn in the color of the root that has been found by a Newton-Raphson search starting at that point.

D B0

A

C

B

B

C B0

A0 C0

B A

B0

A

C A 0

A0

C0

C0 D

D

B0 B

A

A0

C

C0 D

Fig. 3 Multiple solutions of an Assur group of 3rd order

1.0

1.0

0.5

0.5 0.5

−0.5 −1.0

1.0

1.5 x0

2.0 ξ2

2.5

3.0

f(x) = (x− 12)(x−2)

x

0.5 −0.5

(x−3)

(a) Start at x0 = 1,34 yields the root ξ2 = 2

−1.0

1.0

1.5 x0

2.0

2.5

3.0 ξ3

f(x) = (x− 12 )(x−2) (x−3)

(b) Start at x0 = 1,33 yields the root ξ3 = 3

Fig. 4 Dependence of the found root from the start point

Fig. 5 Chaotic dependence of the found root from the start point produces a Newton fractal [5]

x

Numerical Root Finding from an Engineer’s Point of View

95

6 Strategies to Improve Root Finding Having analyzed the causes for failure, we will now show a way to improve effectiveness and efficiency.

6.1 Reducing the Number of Variables Many root finding problems, especially practical problems, are not stated in an optimal way. Taken a problem in n variables, there exists often an implicit parameterization of lower dimensionality, in the sense that each variable xi of the root finding problem f (x1 x2 . . . xn )T = 0 can be expressed as x1 = p1 (t1 , t2 , ..., tm ) x2 = p2 (t1 , t2 , ..., tm ) .. . xn = pn (t1 , t2 , ..., tm ) for m parameters t1 , t2 , . . . , tm with m < n under the condition that all solutions of the original problem also exist in the parametric space and that there are no other solutions in the ti . This means that only an m-dimensional subspace of the n-dimensional function domain is of interest. Because all solutions must lie in this m-dimensional subspace, root finding in this lower-dimensional space is sufficient. Furthermore, the lower-dimensional search space lets us expect a reduction in the number of function evaluations needed. There is often no explicit automatic way to know whether an implicit lowerdimensional parameterization exists and how it can be derived mathematically. Of course, there are variable elimination techniques for polynoms such as Gröbner bases [3], the Wu-Ritt method [13] and resultant methods [16, 10], all of which have been used successfully to solve geometric kinematic simulation problems. Unfortunately, variable elimination is costly and often computationally unfeasible. We like to draw the attention to Döring’s constraint-based problem formulation [5] which also produces functions in a reduced number of variables. These functions represent sequences of basic geometric constructions, where some degrees of freedom are still unbound. These variables are determined by the constraint solver; they usually represent inherent geometric properties such as angles or distances. The resulting function values are error measures which express how much certain geometric properties differ from their required values. Variable reduction in general increases the degree of the function and can therefore cause computational problems. Nevertheless, the gain in reducing the search space is always promising and should be utilized in order to achieve more effective and efficient root finding.

96


6.2 Configuration Control Some of the multiple solutions can be excluded by default, for instance symmetric solutions to geometric problems. Constraint-based geometry representation as proposed in [5] provides an easy means to exclude unwanted solutions: The basic constructions that build up an iterative construction sequence often allow for explicit choice of a solution. The circle intersection constructor, for instance, determines the position of a point by intersecting two circles. It has a flag to choose exactly one of the two solutions.

7 Practical Evaluation After these theoretical reflections on root finding we will undertake some practical tests. We choose two selected geometric kinematic simulation problems, formulate them mathematically in two different ways and solve them numerically using the algorithm given in Section 2. As local root finding methods we use the Newton-Raphson method, the secant method and Broyden’s method. Alternatively the problems are also reformulated as optimization problems, and the BFGS method is used to find the solutions. Each n-dimensional root finding run is limited to 10(n + 1) iterations. Runs are repeated from random start points until a given budget of function evaluations has been used up. The calculation of the derivative (needed in the Newton-Raphson method and in the BFGS method) counts as a single function evaluation. Each method has also been tried in its damped variant featuring the Armijo Goldstein step length control algorithm [1].

7.1 Assur Group of 3rd Order Figure 6a shows a planar linkage consisting of three fixed points A0 , B0 , C0 and four free points A, B, C, D connected by seven bars of given constant lengths. It is called an Assur group of third order [17].

B0

A

A := intersect (ray (A0, δ), circle (A0, a))

α

α := guess (−π, π)

b B e

A0

δ

a

D := intersect (ray (A, α), circle (A, d)) d

f c C

g

B := intersect (circle (D, e), circle (B0, b), cfg1) C0

C := intersect (circle (B, f ), circle (D, g), true) ε := C−C0−c

D (a) Sketch

(b) Construction sequence

Fig. 6 Assur group of 3rd order with construction sequence [5]


97

Given a certain drive angle δ at A0 , the positions of the points A, B, C and D are to be determined. 7.1.1 8D Assur Group Function The first function is a naive approach where all four 2D positions of A, B, C and D, i. e. eight variables, are taken as input and the deviations from the specified drive angle and the seven point distances a, b, c, d, e, f and g are calculated in result. To correctly treat the stiffened triangle BCD, we introduce a check to determine the correct configuration of point C with respect to the line drawn through B and D. For positions of C which are in conflict with the wanted configuration, like C must always be right of BD, the calculation is aborted. 7.1.2 1D Assur Group Function The second representation of the Assur group of third order has been derived by the constraint solver from [5]. The constraint-based representation leads to the construction sequence given in Fig. 6b. In this scheme, point A is the first element to be determined, followed by point D depending on the angle parameter α. Construction of points B and C follows as intersection of two circles, where the location of C with respect to the line through B and D is given beforehand. For point B both possible intersection points are valid, which means, that each Assur group problem is represented by two different one-dimensional functions. The function value returned is an error measure ε. It indicates how much the distance between C0 and C differs from the wanted value c.

7.2 Stewart Gough Platform The Stewart Gough platform is a special type of parallel kinematics consisting of a fixed base and a free platform linked by six length-adjustable articulated legs [15]. The platform has six degrees of freedom, i. e., the platform can, within limits, freely translate and rotate. Our examples are chosen to have coplanar leg mounting points at the base as well as at the platform. The goal is to determine the platform transformation, i. e., its position and orientation in 3D space, for given lengths of the legs. Again we use two different mathematical problem representations. 7.2.1 6D Stewart Gough Platform Function The first representation is a root finding problem in six-dimensional space. The function’s variables describe the local coordinate system of the platform: three variables for the origin and three variables for the imaginary parts of a rotation quaternion [14] for the orientation. The function computes the deviation of each of the six legs from its required length.

98


7.2.2 3D Stewart Gough Platform Function The second representation is a function in three variables obtained using the techniques described in [5]. Figure 7b shows the resulting construction sequence, which can degenerate when the plane used to calculate PP2 is coplanar to the circle made by intersecting the spheres around PP1 and PB2 . The parameters in our examples are chosen to circumvent these degenerations. During determination of PP2 and PP3 there must be decided which of the two intersection points should be taken. This is done by two configuration variables cfg1 and cfg2 . Consequently, we get four different construction sequences for each Stewart Gough platform, and all existing roots for each of the four functions are to be found. The functions have three variables, which are the three angle parameters α1 , α2 and α3 , and compute three values, which are the deviations of legs 4, 5 and 6 from their required lengths. α 1 := guess (0,π) α 2 := guess (−π,π) PP1 :=PB1 + a (sinα1 cosα2, sinα1 sinα2, cosα1) α 3 := guess (−π,π) PP2 := intersect (sphere(PP1, m), sphere(PB2, b), plane ((cosα3, sinα3, 0), PB2)) PP3 := intersect (sphere(PP1, o), sphere(PP2, n), sphere(PB3, c)) M := makeTransform (PB1, PB2, PB3, PP1, PP2, PP3) PP4 :=M PB4 PP5 := M PB5 PP6 := M PB6 ε 1 := PP4−PB4−d ε 2 := PP5−PB5−e ε 3 := PP6−PB6−f

(a) Sketch

(b) Iterative construction sequence

Fig. 7 Stewart Gough platform with construction sequence

7.3 Experimental Results Now we come to analyze the experimental results. A total of 1,000 different problems had been generated for each of the four above-mentioned functions. For each problem we have conducted 1,000 root finding runs using the strategy described in Section 2. Each run was given a fixed budget of function evaluations, 5,000 for the Assur group and 20,000 for the Stewart Gough platform. As we have explicated in Section 3, we were mainly interested in effectiveness and efficiency measures. 7.3.1 Effectiveness The effectiveness indicates how many of the existing roots have been found by a certain method, see Tables 1 to 4.


99

Table 1 Effectiveness and efficiency for the 8D Assur group function effectiveness

efficiency (quantiles)

method

mean

std. dev. 5%

25%

50%

75%

95%

Newton-Raphson method Newton-Raphson method (Armijo) Broyden method Broyden method (Armijo) BFGS method BFGS method (Armijo)

45.9% 39.7% 80.7% 58.0% 10.7% 35.4%

32.7% 32.1% 27.1% 35.6% 19.3% 30.4%

19 20 63 159 929 767

23 24 100 369 1, 839 1, 483

28 30 173 773 3, 090 2, 632

37 90 349 1, 921 4, 538 4, 326

15 15 33 45 353 356

Table 2 Effectiveness and efficiency for the 1D Assur group function effectiveness


method

mean

std. dev.

5%

25%

50%

75%

95%

Broyden method Broyden method (Armijo) Secant method Secant method (Armijo) BFGS method BFGS method (Armijo)

96.2% 98.9% 94.8% 94.7% 71.5% 97.4%

15.7% 7.5% 18.1% 17.5% 38.1% 12.2%

8 8 7 9 17 15

10 10 9 12 29 18

14 15 12 34 71 28

33 44 32 92 260 73

122 311 221 556 1, 404 253

Table 3 Effectiveness and efficiency for the 6D Stewart Gough platform function effectiveness


method

mean

std. dev.

5%

25%

50%

75%

95%

Newton-Raphson method Newton-Raphson method (Armijo) Broyden method Broyden method (Armijo) BFGS method BFGS method (Armijo)

69.9% 91.7% 47.1% 29.1% 0.2% 88.2%

30.7% 16.7% 34.3% 28.9% 2.1% 19.1%

17 20 44 111 1.293 234

24 31 142 616 4,732 307

33 64 334 1,581 9,388 399

49 353 744 3,540 14,545 642

97 1,058 2,174 9,314 18,922 1,555

Table 4 Effectiveness and efficiency for the 3D Stewart Gough platform function effectiveness


method

mean

std. dev.

5%

25%

50%

Broyden method Broyden method (Armijo)

68.9% 74.5%

38.5% 36.1%

21 28

58 125

154 326

492 769

3,312 2,474

BFGS method BFGS method (Armijo)

6.1% 88.1%

20.1% 26.3%

92 77

243 163

599 342

1,697 779

6,617 2,813

75%

95%

100


The Broyden method is the most effective for the 8D Assur group problems. The achieved effectiveness of 80.7 ± 27.1% means that if we invest the budget of 5,000 function evaluations on an average problem we can expect to find four-fifth parts of the existing roots. The 1D representation of the same problems result in more effective root finding. Here the Broyden method with Armijo Goldstein step length control has an effectiveness of 98.9 ± 7.5%—a classic case for successful variable reduction. For the 1,000 Stewart Gough platform problems the situation is less obvious. The naive six-dimensional function can be solved effectively using the NewtonRaphson method with Armijo Goldstein step length control. 91.7 ± 16.7% of the existing roots had been found on a budget of 20,000 function evaluations. Unfortunately, the 3D problem formulation causes a loss in effectiveness—a hint for a non-optimal problem formulation. For the best performing method, which is the BFGS method with Armijo Goldstein step length control, the effectiveness of 88.1 ± 26.3% found roots is as good as in the 6D case, but cannot beat 3D NewtonRaphson search with step length control. 7.3.2 Efficiency The tables also provide an efficiency measure. Here we have only investigated the successful root finding runs. The statistics show the number of function evaluations needed to find exactly one root. Note, that therefore efficiency is only meaningful for effective methods. The most efficient method for the 8D Assur group problems is the NewtonRaphson method which needs 23 function evaluations on average. The secant method is still faster (12 function evaluations) when applied on the same problems in their univariate formulation. The six-dimensional root finding needed for the Stewart Gough platform solution is expectedly slower. Here we find the Newton-Raphson method to be the most efficient. An average of 33 function evaluations is necessary to find the first root of a problem. Interestingly, the 3D Stewart Gough platform problem is not faster to solve than the 6D problem (154 function evaluations using Broyden’s method), but Broyden and BFGS are still more efficient than (and at least as effective as) the 6D case.

8 Conclusions From the statistics presented above and from other tests undertaken by the authors, we are able to derive a compilation of advices concerning effective and efficient numerical root finding: 1. Find the mathematical problem description in the least number of variables. Investigate the computational properties of this description with respect to numerical stability. In some cases a more stable formulation in more variables is better.


101

2. Determine a search region where all the solutions are to be expected. Perform step length control for prognoses which fall outside the search region. 3. Use configuration control and similar techniques to exclude unwanted solutions. 4. Prefer functions that are defined over the whole variable domain over functions which have regions of undefined function values. 5. Root finding and optimization problems are complementary. Although optimization algorithms are generally less efficient than bespoke root finding algorithms, they can turn out to be more effective than root finding algorithms. 6. No method is guaranteed to find all roots. Always test several methods and determine their effectiveness statistically. 7. Efficient algorithms need start points close to an existing solution. In case that possible start points cannot be narrowed down (by approximative calculations for instance), a less efficient method, e. g. BFGS, is often the better choice. 8. The exact calculation of the Jacobian can be disproportionate and an approximate Jacobian would work just as well, especially for low-dimensional problems. In such cases Broyden’s method performs better than Newton-Raphson. 9. Step length control does not necessarily improve root finding. Generally, the need for step length control increases with the dimensionality of the problem. Optimization algorithms heavily rely on intelligent step length control. For these algorithms the respective function evaluations are usually well spent. 10. Limit the number of function evaluations spent for a single root finding run. Use the saved function evaluations instead for starting the search from many different start points (randomly chosen or grid-based). Alternatively, save all calculated points and find promising start points heuristically, like for instance in [7].

References 1. Bonnans, J. F., Gilbert, J. C., Lemaréchal, C., Sagastizábal, C. A.: Numerical Optimization. Theoretical and Practical Aspects. 2nd ed., Springer, Berlin, 2006. 2. Broyden, C. G.: A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, 19(92), 577-593 (1965). 3. Buchberger, B., Winkler, F. (eds): Gröbner Bases and Applications. Cambridge University Press, Cambridge, MA, 2000. 4. Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin, 2004. 5. Döring, U.: Ficucs - Ein Constraint-Solver für geometrische Constraints in 2D und 3D. PhD thesis, Ilmenau University of Technology, Ilmenau, Germany, 2011. 6. Engeln-Müllges, G., Uhlig, F: Numerical Algorithms with C. Springer, Berlin, 1996. 7. M. Färber, B. Brüderlin: Multivariate root finding with search space decomposition and randomisation. in: Proceedings of the Symposium on Applied Computing (SAC’09), Honululu, HI, pp. 1142-1143 (2009). 8. Hubbard, J. H., Schleicher, D., Sutherland, S.: How to find all roots of complex polynomials by Newton’s method. Inventiones Mathematicae, 146(1), 1-33 (2001). 9. Kantorowitsch, L. W., Akilow, G. P.: Funktionalanalysis in normierten Räumen. 2nd ed., Akademie-Verlag, Berlin, 1978.

102


10. Manocha, D., Canny, J. F.: Multipolynomial resultant algorithms. Journal of Symbolic Computation, 15(2), 99-122 (1993). 11. Miranda, C.: Un’osservazione su un teorema di Brouwer. Bollettino dell’Unione Matematica Italiana, II. Serie, 3, 5-7 (1940). 12. Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T.: Numerical Recipes in C: The Art of Scientific Computing. 2nd ed., Cambridge University Press, Cambridge, MA, 1992. 13. Ritt, J. F.: Differential Algebra. Dover Publications, New York, NY, 1950. 14. Shoemake, K.: Quaternions and 4 x 4 Matrices. In: Arvo, J. (ed.): Computer Graphics Gems II. Academic Press, San Diego, CA, 1991. 15. Stewart, D.: A platform with six degrees of freedom. In: UK Institution of Mechanical Engineers Proceedings, Volume 180, Part 1, Number 15, pp. 371-386 (1965). 16. Sturmfels, B.: Introduction to resultants. In: Proceedings of Symposia in Applied Mathematics, 53, pp. 25-39 (1997). 17. Volmer, J. (ed.): Grundlagen Getriebetechnik. Verlag Technik, Berlin, 1992.

Kinetostatic Analysis of an Articulated Walking Mechanism F. Moldovan, V. Dolga, and C. Pop

Abstract The mathematical model for kinetostatic analysis of Jansen’s type walking mechanism is presented in this paper. This mechanism will be implemented in the structure of a walking robot. Simulation results from CAD programs are also presented for validation of the mathematical model and for analyzing the 3D model of the mechanism. From a structural point of view the investigated mechanism consists in eight bar elements connected through revolute joints. The shape of the leg mechanism described is similar with the original one proposed by Theo Jansen. Keywords Walking mechanism · Structural analysis · Kinetostatics · Virtual prototype · Simulation

1 Introduction The main walking robots systems developed till now are destined to laboratory researches and sustain the studies for locomotion and development of new prototypes that are more efficient in terms of energy consumption and economical aspect. This prototypes are based on different and various types of walking mechanisms configurations [1], [2] and [3]. Articulated leg mechanisms are used by many researchers for building various types of walking robots such as bipeds, tripods, quadrupeds and many other forms that are inspired by nature [3]. An example of leg configuration used for building several types of walking structures and artificial life forms, consists in the twelve bar walking mechanism designed by Dutch kinetic sculptor Theo Jansen. This mechanism appears in studies of few researchers and was initially designed for walking over sand powered by the wind [4], [5] and [6]. Also it consists in the main object of study for present research.

F. Moldovan (B) “Politehnica” University of Timisoara, Timisoara, Romania e-mail: [email protected]


103

104

F. Moldovan et al.

The reason for choosing this configuration to be analyzed, simulated and in final implemented in a walking robot structure is based on several advantages that it possesses in comparison with other leg mechanisms. In the beginning, for studying the kinematics and kinetostatics of the investigated mechanism, a mathematical model it is important to be created. For validation of the results obtained from the analytical part, a 3D simulation of the mechanism model should also be developed [7], [8]. The article is divided in five parts. The first part presents a short introduction regarding the Jansen type of walking mechanisms that represents the main object of interest and study in this case. A general description of the mechanism and its main advantages are presented in the second part. The third part consists of mathematical models used for kinetostatic analysis of the mechanism. The simulation results for the reaction forces of the mechanism obtained from CAD are described in the fourth part. In the fifth part, which is the final one, are outlined the final conclusions.

2 General Description of Jansen Type of Mechanism A new type a leg configuration for a walking mechanism was introduced around 90’s by Dutch physicist and kinetic sculptor Theo Jansen similar to the one presented in Fig. 1. This mechanism is studied and designed in order to analyze the possibility of using it for building a walking robot structure [9], [10]. The proportions of the leg were established with help of evolutionary algorithms which perform well approximating solutions based upon an evolution process similar to natural one [11]. The virtual prototype of a new type of walking robot design

Fig. 1 The virtual prototype (at the left) and the kinematic sketch view of a variant of Jansen type of mechanism (at the right)

Kinetostatic Analysis of an Articulated Walking Mechanism

105

Table 1 Structural analysis of the leg mechanism [7], [8] Components

Symbol

Description

Number of elements Kinematic joints Joint angle First contour Second contour Third contour Termination of the foot Degree of freedom Number of nodes

n c5 ϕi O1 ABO2 O1 ADO2 O2 CED DEF M N

1. . .8 O1 , A, B,O2 , C, D,E,F; c5 = 10. i = 1. . .7 four bar mechanism reverse four bar mechanism irregular parallelogram triangle shape 1 20

is also presented in Fig. 1. In terms of structural analysis the mechanism described is analyzed in Table 1. This mechanism configuration offers certain advantages among other types of similar structures which consist in: minimal power consumption due to the single actuator that possess, deterministic foot trajectory described by the end point of the leg F, a scalable design because of the numerical values assigned to the proportions of the foot and the revolute joints that are connecting the elements [4].

3 Mathematical Models For further dynamic analysis of the investigated system during the trajectory path from p0 (starts point) to pf (final point) as shown in Fig. 1, it is important to establish the mathematical model that consists in structural, kinematics and kinetostatic analysis of the mechanism. In this case the first steps regarding structural and kinematic analysis were already done in previous work [9]. Forward it will be described the kinetostatic analysis for the leg mechanism proposed in Fig. 2. The kinetostatic analysis consists in calculation for the moment of inertia and reaction forces that appear in the mechanism joints with or without friction. In case of kinetostatic analysis without friction the mathematical relations are based on algebraic equations and the critical positions are defined through the singularities of the mathematical models [10], [12]. Reaction forces that appear between the kinematic elements of the mechanism can be calculated based upon few assumptions like neglecting the friction forces within the joints and consider the reaction forces normal to the contact surface. For plane mechanisms is preferable that applied and inertial forces to form a system parallel to the plane of movement. In these circumstances, the reaction forces acting on this plan became null from equilibrium equations. As a result the reaction forces for the kinematic revolute joints are reduced to a single force value Fij [N] applied in the center of the joint and equal to zero [12].

106

F. Moldovan et al.

Fig. 2 The geometry shape of the leg mechanism proposed for kinetostatic analysis on a current position designed in SolidWorks

The mathematical model proposed for determination of reaction forces and moments of the mechanism presented in Table 3 takes into account the parameters described in Table 2. Also it considers the fact that friction forces are neglected and the equations correspond to the equilibrium situation considered in a certain position of the leg. Table 2 Principal parameters for kinetostatic analysis Components

Symbol

Description

Force reactions

Fij

Resultant force applied in the center of joint [N]; i – element that determines the reaction force and j- the element that receives the force. Reaction force component across x and y direction [N]; i or j = 0. . .7.

Fi,j,x,y Torque

Lix,y

Torque value around the position of center of mass [g∗ mm2 ]; i = 0. . .7.

Gravitational acceleration

g Al

9.81 mm/s2 Aluminum

Material characteristics

E c ρ

Young Modulus. E = 690+010 [N/m2 ] Poisson coefficient. c = 0.33[-] Density mass: ρ = 2700 kg/m3


107

Table 3 The geometry, reaction forces and moments of equilibrium on a current position of the leg mechanism Element Representation Element 1

Element 2

Element 3

Element 4

Mathematical Relationships

108

F. Moldovan et al. Table 3 (continued)

Element Representation

Mathematical Relationships

Element 5

Element 6

Element 7

4 Simulation Results The following sets of graphics are presented in order to compare the theoretical results, obtained after computation of the reaction forces and moments for the driving couple revolute joint of the leg mechanism, with the simulation results for the same objects made in Adams software. Each value is measured during a time period of 20 seconds corresponding to a walking cycle of the leg mechanism. In Fig. 3 the reaction forces for the fixed point throughout x direction presents more emphasized maximum points of the curve while the force along y direction


109

Fig. 3 The joint reaction forces considered in the fixed point of the leg O1

varies in a small range. Those maxims seem to repeat every 12 seconds during the walking cycle. The first maxim is located around the fourth second. Extreme minimal values are observed for the moment exerted around the fixed point as in Fig. 4. From the simulation results obtained it can be stated that the values for reaction forces and moments corresponding to the end point of the leg, like in Fig. 5, are minimal in this context where no contact platform was added.

Fig. 4 The moments across x and y direction considered in fixed point of the leg O1

Fig. 5 The joint reaction force and moments for the end point of the leg F

110

F. Moldovan et al.

5 Conclusions The kinetostatic analysis is developed in order to achieve a complete mathematical model for the investigated mechanism along with kinematics, dynamics and optimization process. The results from simulation method provide data for graphical interpretation for the measured values and also help to see the errors and inconveniences of the model before building a real one. Based on the known values for the masses and kinematic parameters, it can be calculated further the kinetic and potential energy developed by the leg mechanism during walking, the power consumption and the necessary torque value for the motor that acts one pair of legs. Also it is necessary to optimise the structure of the leg in order to achieve a reduced level of energy consumption during walking and of the impact at the contact with the ground. Acknowledgments This work was partially supported by the strategic grant POSDRU/ 88/1.5/S/50783, Project ID50783 (2009), co-financed by the European Social Fund – Investing in People, within the Sectoral Operational Programme Human Resources Development 2007 – 2013.

References 1. Maniu I., R˘adulescu C., Bogdanov I., Varga S., Dolga V.,Ciupe V.: Robotic˘a. Aplica¸tii robotizate ( Robotics. Robotical applications),Editura Politehnica, Timi¸soara (2009). 2. Vat˘au S., R˘adulescu C.: Robo¸ti Mobili. Robo¸ti Mobili P˘asitori (Robots. Walking Mobile Robots), Editura Politehnica, Timi¸soara (2008). 3. Angeles J.: Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 2nd Edition, Springer Verlag New York, 171 (2003). 4. Ingram A.J.: A new type of walking machine. thesis, University of Johannesburg (2006). Available at: http://152.106.6.200:8080dspace/handle/10210/598. Accessed 2010-03-10. 5. Giesbrecht D., Qiong Wu C.: Dynamics of Legged Walking Mechanism “Wind Beast”, Department of Mechanical and Manufacturing Engineering, University of Manitoba, California, (2010). 6. Jansen T.: http://www.strandbeest.com/. Accessed:2011-03-01. 7. Mesaro¸s A. V., C˘ar˘aba¸s I., Lovasz E. C.: Manual pentru proiectarea mecanismelor (Manual for the design of mechanisms), Editura Mirton, Timi¸soara (1997). 8. Gogu G.: Chebychev–Grübler–Kutzbach’s criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations. European Journal of Mechanics A/Solids,Volume 24, Issue 3, 427-441(2005). 9. Moldovan F., Dolga V.: Analysis of Jansen walking mechanism using CAD. Solid State Phenomena (Volumes), Robotics and Automation Systems (166 - 167), 297-302 (2010). 10. Pisla D., Ceccarelli M., Husty M., Corves B.: New Trends in Mechanism Science.Analysis and Design. Springer Dordrecht Heidelberg London New York, 29-39 (2010). 11. Tudose L., Pop D.: Proiectarea optimala cu algoritmi genetici (Optimal design using genetic algorithms),Mediamira, Cluj-Napoca, 19-25 (2002). 12. Duca C., Buium F., Paraoaru G.: Mecanisme (Mechanisms), Gh. Asachi, Iasi, 106 (2003).

A Mirror Tracking Mechanism N.M. Dehelean and L.M. Dehelean

Abstract Solar heliostat field use a tower and a set of mirrors spread over a precise area. The publications refer to solar energy name the mirror that tracks to the Sun as heliostat. There is yet a theory related to the heliostat tracking. Indeed, there is a set of methods of tracking control based on calculus algorithm. The paper tries to promote a mechanism that implement the reflection function to facilitate the heliostat target orientation. The mechanism should be equipped with a target line sensor. This one must target the Sun. Keywords Solar energy · Solar concentrator · Tracking mechanism · Heliostat

1 Introduction The common heliostat uses a spherical mirror and a universal joint azimuthelevation. The meridian focus moves along a circle with a diameter equal to the paraxial focal length. The sagittal focus moves along a straight line with the increase of incident angle. In [1], the authors have approached the tracking system as a mechatronic concept. The electronic control system integrates the mechanical structure of the solar tracker at the virtual prototype level. It implements an algorithm for determining the optimal actuating time for the “step-by-step” tracking. In [2] Anton I. proposes a method of mirror tracking by the meaning of a CCD camera. There are four places around the concentrator where to place CCD camera. The CCD camera acquires the image of the target. This image gives the stage information of the tracking and the errors. The image analyze gives the data about the errors of orientation. A general tracking formula is shown in [3]. The paper improves the sun-tracking accuracy by tracking the installation error of the solar collector. This is useful for azimuth-elevation and tilt-roll tracking mechanism.

N.M. Dehelean (B) “Politehnica” University of Timisoara, Timisoara, Romania e-mail: [email protected]


111

112

N.M. Dehelean and L.M. Dehelean

In [4] Chong K.K uses a CCD camera for recording the solar image. The signal from CCD has been used to fine tuning the tracking mechanism. It has obtained a tracking accuracy up to 0.96 mrad. In [5] the tracking system is view as an application of Machine Vision. The system uses a webcam as a sensor that provides a signal. An electric motor performs the rotation of the sensor. A shade plate sensor presented in [6] and in [2]. The paper [11] presents a similar sensor, in detail. The paper tries to establish the optimal stopping angle β in the morning or afternoon (about 50◦ ) from the solar noon position. In order to decrease the acting of the tracking mechanism in this paper is proposed one axis tracking mechanism adjusted at three fixed angles: morning, noon and afternoon. The optimal stopping angle β, in the morning or afternoon is about 50◦ from the solar noon position. The price reduction is between 20% and 30% relative to a common tracking mechanism. Aiuchi Kosuke uses for heliostat tracking two addition sensors, a cloud sensor and a primary sensor [7]. The paper shows the schema of the heliostat with the sun-tracking photo-sensors. The heliostat has two cloud sensors in order to prevent the miss tracking when the clouds appear on the sky. The sensor consists of two photocells placed in a box that has an aperture (Fig. 1). In this figure, a rack and pinion system is used for the rotation around the right ascension axis, and a screw– nut system is used for the rotation around the declination axis; in fact, this is a differential sensor. It obtains a high accuracy of the tracking even in a cloudy sky. In fine weather, it obtains an angular error of 2 mrad and in cloudy weather an error of 10 mrad.

Fig. 1 Picture and structure of the sensor-controlled heliostat

A Mirror Tracking Mechanism

113

The paper demonstrates the advantage of equatorial mount against the azimuth/elevation mount. It shows that the equatorial mount needs a constant rotation speed for the right-ascension axis. For the declination axis, the rotation speed is almost zero. It is useful to preserve the idea of the advantage of equatorial mount relating to azimuth/elevation mount. In [8] the authors propose a mirror drive mechanism to track the Sun and reflect the rays to stationary photovoltaic arrays. Incident radiation is measured using pyranometers located on the plane of the module. The Mirror Positioning Device is a five bar spherical mechanism. The mechanism has two degrees of freedom, which allows for tracking the Sun along its azimuth and altitude. In [9] the authors refer to a parabolic collector, whose geometry imposes one axis tracking system. The reflector should rotate 180◦ during the month of March. In [10] Guo Minghuan proposes a strategy to check the tracking accuracy and to adjust the tracking parameters. It uses a small flat mirror glued onto the center of the clear glass board attached on the heliostat center. It creates a temporary target and observes the spot of the attached small mirror when the heliostat tracks the Sun. The mini mirror should be parallel to the elevation axes. Mousazadeh Hossein, in [11] proposes a set of three sensors useful in sunny weather shade plate, prismatic sensor and box sensor (tube sensor). The paper presents a variant of shade plate sensor for horizontal tracker. Figure 2 presents a synthetic schema of orientated panel with tracking surfaces including passive or active trackers. The paper does not recommend using tracking system for small solar panels because a high energy loses in the driving systems. The power consumption by tracking device is 2 – 3 % of the increased energy. The idea of solar panel tracking using a signal of maximum irradiance is in [12]. For prevent the system idling in overcast or night some systems use a photo-switch and a clock-trigger-switch. In [13] Wei Xindong proposes a set of ray tracking equation for the target-aligned heliostat. It uses two axes and calculates the rotation angles of heliostat according to the solar time, the heliostat and the target location on Earth. The calculus refers to a toroidal surface. In [14] Zaibel R. proposes an asymmetric reflector in order to correct the astigmatism and to reduce the solar image size. This reflector should have different curvature radii along the meridian and sagittal direction. All of these improvements require a higher precision of the Sun tracking.

2 A New Tracking Mechanism Many solar systems perform an automatic mirror orientation. These act the mirror by a constant angular speed. This speed is about 15◦ /h for a solar panel and 7.5◦ /h for a mirror. The automatic orientation systems perform low accuracy of the orientation angle. This precision is good enough for the common solar panels. In contrast, the solar

114


Fig. 2 Types of sun trackers

mirrors (heliostats) need a high accuracy of the orientation angle because the reflected spot must target the power tower. A heliostat and the solar system it belongs to, need a high performance of orientation. The system must track the Sun and the spot must target the power tower. Any azimuth sensor is able to target the Sun. It is good to track directly, a common solar panel. In the case of mirror, the azimuth sensor must be integrated in a special mechanism. This mechanism has to perform a tracking function useful for a mirror. A solar tracker for a heliostat must perform θ angle for the Sun and θ/2 angle for the mirror normal direction. The paper solves this request and promotes a new mechanism to track the mirror to the Sun. The new mechanism performs the angular relation between the Sun, the target and the mirror. The paper recommends preserving (use in parallel) the automatic angular orientation useful in the cloudy period of the day. The mirror of a solar area should be correct track by the Sun in order to reflect all the time the ray to the target tower of the system.

A Mirror Tracking Mechanism Fig. 3 The equatorial mount of the mirror

115 1 – shaft 2 – azimuth sensor 3 – vertical fixture 4 – rigidity ribs 5 – hoops 6 – mirror 7 – axial ribs

Obviously real world solar trackers (such as the one in Fig. 3) are not so simple. A solar tracker must be able to reset itself at sunset so it is ready for sunrise, it must cope with heavy cloud, and it must work reliably 365 days a year. A mount for the solar panel must be constructed which cope with strong winds, and a suitable motor can found. The main constrain is to avoid the sophisticated sensors and actuating motors. The mechanism is dedicated to an equatorial mount (one axis north south – Fig. 3). A mechanism able to divide the θ angle azimuth increases (theoretically) the angular accuracy of the sensor. This expected increasing could be decrease in the real application by the clearance from the joints.

2.1 Four Bar Linkage The main idea of the new mechanism is to track the Sun directly by a shade sensor. Tracking the Sun is necessary to rotate the mirror by an adequate angle to respect the reflection function. Therefore, the mechanism has to divide the angle between Sun and the target. This mechanism is proper for equatorial mount of the mirror. The first idea for this request is a four bar linkage. The schema of the linkage is shown in Fig. 4. The angle from Sun and target is angle θ. The mirror must rotate with θ/2. The mirror plane and the mechanism plan must be orthogonal. The mirror fixed the oscillating bar. It is not necessary that the four bar linkage to be able to cross over the dead point when the angle θ = 0◦ . The mechanism has an actuator (4) and this elementary linkage becomes itself an actuator for the second mechanism (Fig. 5). The joint C (Fig. 4) acts the crank rocker in B (Fig. 5). Using this combination, it results the final mechanism shown in Fig. 6. The joints in final mechanism (Sun Tracking Mechanism 1) becomes (C, D).

116 Fig. 4 Four bar linkage that perform θ/2 angle

N.M. Dehelean and L.M. Dehelean 1,2,3,4,5 – the elements of the mechanism 2 – actuator 6 – Sun tracking sensor 7 – target 8 – Sun C 8

7

5 O1

4

1

θ θ/2

6

B

A

3 2

O O2

1

1

The position of the tracking sensor is along the axis of element (3). The element (1) axe is the direction of the target. The actuator rotates the element (3) and track the Sun. The mirror performs the angle position to reflect the spot on the target surface. The practical development of the mechanism needs to place elements in some parallel planes to avoid different intersections of the joints.

1,5,6 – the elements of the mechanism 7 – mirror 8 – target 9 – Sun 6

8

B

5

A

O1 θ/2

9 7 π/2 O

Fig. 5 Crank Rocker Mechanism

1 7

A Mirror Tracking Mechanism Fig. 6 Sun Tracking Mechanism 1

117 1,2,3,4,5,6 – the elements of the mechanism 7 – mirror 2 – actuator 9 8 – Sun tracking sensor 9 – target 6 10 – Sun D 5 C O1

8

1

4

10 A

B 3

O

2 O2

7

1

1

2.2 The Minimized Linkage If minimal dimensions are requested the Sun Tracking Mechanism 1 (Fig. 6) can be transformed in a similar other one (Sun Tracking Mechanism 2) presented in Fig. 7. This is possible because the tracking need not a working angle more then ± 50◦ . The Sun Tracking Mechanism 2 is able to perform the requested function, but the paper presents the kinematics analyze for the Sun Tracking Mechanism 1.

1,2,3,4,5,6 – the elements of the mechanism 7 – mirror 2 – actuator 8 – Sun tracking sensor 9 – target 10 – Sun

9

10 6

O1

1

8 5

4 C

D

B 2

Fig. 7 Sun Tracking Mechanism 2

1

O2

A 3

O 7 1

118


D∞ C l5 O1 l60+se

l4

θ

θ l32

B

A

y

l20

θ/2

l20+s x

O(x0, y0)

A’

A

θ+π/2

l 31 ϕ

O2

θ

l1

O2

μmax

l20+s

l31

μmin O

l10

a)

l31

b)

Fig. 8 Sun Tracking Mechanism 2 – Kinematics Analyses

The mechanism schema for kinematics analyze is shown in Fig. 8a and the extreme positions of the joint A are presented in Fig. 8b. The design process of the real mechanism needs to establish the fitting position of the joint A (joint from actuator and bar l3 ). The length l31 can be critical in respect with the actuator stroke and the actuator force rate. If the force rate is low then the length l31 becomes longer. It applies the cosine theorem in scheme Fig. 8b. The start position is represented by the equation (1) and the final position by the equation (2). l210 = l220 + l231 − 2 l20 l31 cos μmax

(1)

l210 = (l20 + s)2 + l231 − 2 (l20 + s) l31 cos μmin

(2)

It multiples the equation (1) by (-1) and it adds the equation (2). It results the equation (3). (l20 + s)2 − l220 − 2 (l20 + s) l31 · cos μmin + 2 l20 · l31 · cos μmax ) = 0

(3)

To solve the equation (3) it must fix some constrains: μmin = 15o μmax = 90o

(4)


119

The cylinder stoke, s, cannot exceed 80% from the cylinder length, l20 (5). s < 0.8 · l20

(5)

It needs, also, to eliminate the variable l31 , using a proportional factor, k (6). l31 = k · l20 ; ⇒

k ∈ {0.1; 0.15; . . . ;1.35} s ∈ [0, smax ]

(6) (7)

It transforms the equation (3) in (8). 2 l20 · l31 (cos μmax − cos μmin ) + 2 (l20 − l31 · cos μmin ) s + s2 = 0

(8)

To solve the equation (8) in s, it makes the substitutions (9) and it uses k as parameter. b = l20 − l31 · cos μmin ; c = 2 l20 · l31 (cos μmax − cos μmin ); a=1

(9)

Therefore, the roots for the equation (8) are: s1,2 =

−b ±

√ 2 b − ac a

(10)

Running a solving routine of the equation (8) with parameter k and fixing the constrain (5), it results the data in Table 1. The compatible values, for l31 , adopting l20 = 200 mm, are shown in Table 1. The actuator stroke diagram (Fig. 9) represents the shape of stroke and joint position A (l31 ) for an actuator length, l20 = 200 mm. The data diagram is from Table 1. The vertical bar marker splits the domain for l31 in a compatible sub domain on the left side and an incompatible one on the right. For the length l31 > 120 mm, the stroke s cannot be performed by the actuator l20 = 200 mm because the constrain (5). The next step of the design algorithm refers to check out of the extreme position point of crank rocker. In the crank rocker mechanism in O2 AO from Fig. 8a, it is wrights: π (l20 + s) · eiϕ = x0 + i · y0 + l31 · ei(θ+ 2 ) (l20 + s) · eiϕ = x0 + i · y0 + i · l31 · eiθ (l20 + s) · e−iϕ = x0 − i · y0 − i · l31 · e−iθ

(11) (12)

120

N.M. Dehelean and L.M. Dehelean Table 1 Stroke and joint position A values

No.

k

b

c

s

l31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

180.6815 171.0222 161.363 151.7037 142.0445 132.3852 122.7259 113.0667 103.4074 93.74816 84.0889 74.42964 64.77038 55.11113 45.45187 35.79261 26.13335 16.47409 6.814835 -2.84442 -12.5037 -22.1629 -31.8222 -41.4815 -51.1407 -60.8

-7727.41 -11591.1 -15454.8 -19318.5 -23182.2 -27045.9 -30909.6 -34773.3 -38637 -42500.7 -46364.4 -50228.1 -54091.8 -57955.5 -61819.3 -65683 -69546.7 -73410.4 -77274.1 -81137.8 -85001.5 -88865.2 -92728.9 -96592.6 -100456 -104320

20.25 31.07 42.33 54.05 66.18 78.74 91.68 105.01 118.70 132.72 147.07 161.72 176.66 191.86 207.30 222.98 238.88 254.97 271.25 287.71 304.32 321.09 337.99 355.03 372.19 389.46

20 30 40 50 60 70 80 90 100 110 120 incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible incompatible

stroke, s

Length [mm]

Stroke Diagram for 120 = 200 mm 450.00 400.00 350.00 300.00 250.00 200.00 150.00 100.00 50.00 0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k

Fig. 9 Actuator stroke diagram

1.1 1.2 1.3

l31


121

(l20 + s)2 = x20 + y20 + l231 + i · x0 l31 eiθ − e−iθ + y0 l31 eiθ + e−iθ 0 = 2y0 l31 · cos θ − 2x0 l31 · sin θ + x20 + y20 + l231 − (l20 + s)2

(13) (14)

If it considers the substitution (15) and (16), the equation (14) becomes (17). ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sin θ =

2y0 l31 = A (s) 2x0 l31 = B (s)

(15)

x20 + y20 + l231 − (l20 + s)2 = C (s) 2tg

θ 2

θ 2 ; cos θ = θ 2 1 + tg 2 1 − tg2

θ 1 + tg2 2

;

tg

θ =t 2

(16)

2t 1 − t2 +B +C 1 + t2 1 + t2 0 = A 1 − t2 + 2Bt + C 1 + t2

(17’)

0 = (C − A) · t2 + 2B · t + (C + A)

(17”)

0=A

(17)

The roots for equation (17’’) are:

t1,2 =

−B ±

B2 − C2 − A2 C−A

(18)

Solving the roots (18), it results θ angle as follows: θ = 2arctg

−B (s) ±

A2 (s) + B2 (s) − C2 (s) A (s) − C (s)

(19)

In OBO1 (Fig. 10) applies cosine theorem it results: (l4 + l5 )2 = l21 + (l31 + l32 )2 − 2l1 (l31 + l32 ) · cos θ

(20)

However l31 + l32 = ll , so the equation (20) becomes: (l4 + l5 ) = 2l21 (1 − cos θ)

(20’)

(l4 + l5 )2 − 2l21 = 2l1 · cos θ

(20”)

or

122


Fig. 10 The mechanism position for θmax angle

θ θmax

O1 l5 D

C l4

B

l32

l1 A

l31

O O2

However l4 = l5 , so the equation (20’’) becomes: (l4 + l5 )2 − 2l21 2l1 2l24 − l21 = arccos l1

θ = arccos θmaxpossible

(21) (22)

that satisfied the condition to eliminate the singularity position θ < θmaxpossible .

3 Conclusions The paper proposes a set of new mechanisms to act the mirrors on solar fields. These come to improve the Sun tracking function. The new mechanisms allow to target the Sun directly using the common orientation sensors. The sensor signals must be in relation with the computed decision, based on time process algorithm. The mechanisms could be developing as a master mechanism able to drive synchronous a large set of mirrors.

References 1. Alexandru, C., Pozna, C.: Simulation of a Dual-Axis Solar Tracker for Improving the Performance of a Photovoltaic Panel, Proceedings of the Institution of Mechanical Engineers, Part A – Journal of Power and Energy, volume 224, Issue A6, pages 797-811, (2010).


123

2. Anton, A., Pachon, D., Sala, G.: Characterization of Optical Collectors for Concentration Photovoltaic Applications, Progress in Photovoltaics: Research and Applications 11, 387-405, John Wiley & Sons Ltd., (2003). 3. Chong, K.K., Wong, C.W.: General Formula for On-Axis Sun Tracking System and Its Application in Improving Tracking Accuracy of Solar Collector, Solar Energy, 83, 298-305, Elsevier Ltd., (2009). 4. Chong, K.K., Wong, C.W.: Open-Loop Azimuth-Elevation Sun-Tracking System Using OnAxis General Sun-Tracking Formula for Achieving Tracking Accuracy of below 1 mrad, 35th IEEE Photovoltaic Specialists Conference, Book Series: IEEE Photovoltaic Specialists Conference, Honolulu, (2010). 5. Hossain, E., Muhida, R., Dzulkipli, A.F., Rahman, K.A.A.: Solar Cell Efficiency Improvement Using Compound Parabolic Concentrator and an Implementation of Sun Tracking System, 11th International Conference on Computer and Information Technology ICCIT 2008, vols1 and 2, pp.174-179, Khulna, Bangladesh, (2008). 6. Huang, B.J., Sun, F.S.: Feasibility Study of One Axis Three Positions Tracking Solar PV with Low Concentration Ratio Reflector, Energy Conversion and Management, 48, 1273-1280, Elsevier Ltd., (2007). 7. Kosuke, A., Kazuo, Y., Masaki, O., Yukuo, K., Minoru, N., Katsushige, N.: Sensor-Control Heliostat with an Equatorial Mount, Solar Energy 80, 1089-1097, Elsevier Ltd., (2006). 8. Kulkami, S., Tonapi, S., Larochelle, P.: Effect of Tracking Flat Reflector Using Novel Auxiliary Drive Mechanism on the Performance of Stationary Photovoltaic Module, ASME International Mechanical Engineering Congress and Exposition, Vol.6 – Energy Systems: Analysis, Thermodynamics and Sustainability, Pages 351-356, Seattle, (2007). 9. Lepadatescu, D., Mastorakis, N.E.: Virtual Prototyping of a New Tracking System, Proceedings of the 2nd International Conference on Environmental and Geological Science and Engineering, Book Series: Mathematics and Computers in Science and Engineering, pp.142-146, Brasov, Romania, (2009). 10. Minghuan, G., Zhifeng, W., Wenfeng, L., Xiliang, Z., Chuncheng, Z., Zhenwu, ., Xiudong, W.: Tracking Formulas and Strategies for a Receiver Oriented Dual-Axis Tracking Toroidal Heliostat, Solar Energy 84, 939-947, Elsevier Ltd., (2010). 11. Mousazadeh, H., Keyhani, A., Javadi, A., Mobli, H., Abrinia, K., Sharifi, A.: A Review of Principle and Sun-Tracking Methods for Maximizing Solar Systems Output, Renewable and Sustainable Energy Reviews, 13, 1800-1818, Elsevier Ltd., (2009). 12. Wei, W., Su, J.: One Axis Sun-Tracking System with Auto-Test and Control, 3rd International Symposium on Test Automation & Instrumentation, vols. 1-4, pages 1083-1085, Xiamen, (2010). 13. Xiudong, W., Zhenwu, L., Weixing, Y., Hongxin, Z., Zhifeng, W.: Tracking and Ray Tracking Equations for the Target-Aligned Heliostat for Solar Tower Power Plants, Renewable Energy, 1-7, Elsevier Ltd., (2011). 14. Zaibel, R., Dagan, E., Karni, J., Ries, H.: An Astigmatic Corrected target-Aligned Heliostat for High Concentration, Solar Energy Material and Solar Cells, 37(2), pages 191-202, (1995).


xii

Choppers: Optomechatronic Devices for the Controlled Modulation of Light V.-F. Duma, M.F. Nicolov, M. Kiss, T. Ilca, C. Mnerie, D. Demian, and L. Szantho

Abstract We present our main contributions and current work regarding one of the most used mechatronic devices for optical applications: the choppers. The issues of the macroscopic choppers with rotating wheels are presented, with a focus on the chopping frequency and on the profile of the transmitted flux (the modulation function) generated. The profiles we have investigated theoretically for the classical chopper wheels, with linear margins, and for top hat light beam distributions are discussed. The new configuration of the “eclipse choppers” (with windows with circular margins) we have introduced to produce approximate triangular series of impulses is presented, and the modulation functions produced are characterized. An insight in the experimental part of the study we are currently working on is provided, from the prototype wheels we have manufactured to the optical setup built for the testing of the different geometrical configuration of wheels, and to the first results. Keywords Optomechatronics · Devices · Optical choppers · Modulation of light · Analysis · Design

1 Introduction Choppers [1,16] are optomechatronic devices used for generating trains of light (usually laser) impulses, of adjustable frequency and specific profiles. This controlled modulation of light has a wide range of applications: (i) attenuation of light, by example in colorimeters [1]; (ii) elimination of undesired wavelengths, as in telescopes [4] or lidars [12]; (iii) lasers [3]; (iv) selection of specific wavelengths (e.g., for radiometers [11]); (vi) generation of light impulses, in spectral systems [14], as well as in a variety of chemical physics studies; (vii) selection of optical paths, as in certain biomedical systems (e.g., when simultaneous confocal and optical coherence tomography images are captured [6]). One may therefore conclude that, especially in optical setups where lasers are concerned, choppers are as common as lenses, mirrors and prisms. V.-F. Duma (B) “Aurel Vlaicu” University of Arad, Arad, Romania e-mail: [email protected]


125

126

V.-F. Duma et al.

The importance of their study comes both from their extensive use and from the fact that, although they are such an established device, their analysis and design has not been completed so far (except for approximate and practical formulas), and current studies are undergoing for achieving a complete theory that may describe them. In this paper we shall present a review of our main contributions to this domain and of our current researches, in relationship with the main issues the study of choppers is facing, to satisfy the requirements of the different classes of applications they are involved in.

2 Types of Choppers and State of the Art There are two main types of choppers: with rotating wheels (Fig. 1) and MEMS (Micro-Electrical-Mechanical Systems) [5,15]. The former involve a disc with several windows and “wings” (opaque portions) that regularly obscure the light (a laser beam, in Fig. 1), thus generating series of impulses of frequency f = n/T = nω/2π, where n is the number of windows, and T and ω the rotational period and speed, respectively. The MEMS used for the same purpose are somehow similar, but they generally consist of a cantilever, a resonant lamellar spring that has a larger part at the free end to chop the laser beam. Several other designs were developed through time, but they have not survived. Efforts were also made to develop solutions that avoid the use of mechanical choppers (especially macroscopic), because of their moving parts that imply motors, vibrations and mechanical wear. Despite these disadvantages, choppers could not be replaced but in certain, particular systems, and they have retained their importance as the one of the most used mechatronic devices for optical applications. Their main issues are nowadays: (1) the chopping frequency (f), which is around 1 kHz (and up to 3 kHz) for macroscopic (rotating wheel) devices and at the order of MHz for micro-choppers, which were developed especially [5] for increasing f; 2δ ω

Photo-detector ω

2r

R Ο

γ

α

α

Laser

γ a)

b)

Fig. 1 Optical choppers (classical configurations, with windows with linear margins): a with two wings; b with several windows

Choppers: Optomechatronic Devices for the Controlled Modulation of Light Fig. 2 Transmission function of optical choppers (classical configurations, with windows with linear margins), for finite diameter beams, and for: a large angle wings (γ ≥ 2δ); b small angle wings, with only one wing at the time in the beam section (γ < 2δ and α > 2δ + γ)

Φ

127

τt

τ

ΦS ΦS/2 t T/n

0

t0

a)

Φ

2π/nω = (α + γ)/ ω

Φs α−2δ 0

δ + γ/2 2δ + γ

t

Τ = 2π/nω

b)

(2) the profile of the transmitted [light] flux. This modulation function is a series of approximate rectangular impulses for a laser beam focused in the plane of the chopper wheel (when the transition phases of the process are so short that they can be neglected), but has non-linear transition portions of different shapes (Fig. 2) for a beam that has a finite diameter 2r in the plane of the wheel (Fig. 1b). This latter case is the general one that has to be approached in order to analyze the flux profiles provided by different types of wheels and to tackle with the inverse problem: to design the appropriate wheel to generate a specific, required profile of this function. Numerous studies, including early ones (since the late ‘50s) were made to solve this analysis and to develop novel types of wheels. We shall present our contributions in the former direction [7] in Section 3, and in the latter [8] in Section 4. Section 5 will be dedicated to our current experimental studies [13] to validate the theoretical studies mentioned above, although certain cases (for the classical type of wheel in Section 3) have already been validates by the experimental work of other groups [2].

3 Choppers with Windows with Linear Margins In Fig. 1 the two most used types of choppers with rotational wheels are shown. For the two wings device (Fig. 1a), the window angle α and the wing angle γ are usually equal, α = γ = π/2. The chopper with n windows may have α = γ , and also multiple rows of windows (with different angles) to allow for choosing different frequencies using the same wheel and value of ω. Several cases may be pointed out regarding the relationship between the diameter of the beam in the plane of the wheel (as seen from the pivot O, with the angle 2δ) – Fig. 1b) and the angles of the window and of the wing, respectively:

128

V.-F. Duma et al.

(i) The chopper wing is larger than the section of the beam (γ ≥ 2δ), thus the section can be completely obscured (Fig. 2a). The time intervals of the modulation function are presented in Table 1. The rectangular signal obtained when the beam can be considered perfectly focused in the plane of the wheel is a particular case. Trapezoidal and sinusoidal profiles of the signal were obtained for this geometry of the chopper (with large wings) [7], and our theoretical findings were confirmed for the trapezoidal profile by another group as well [2]; (ii) The chopper has a single wing in the section of the beam (for γ < 2δ and α > 2δ + γ ), thus the modulation function will have the profile in Fig. 2b; (iii) The wheel with several wings in front of the section of the beam, with α, γ < 2δ, where there are two cases, for α < δ + γ /2, and for δ + γ /2 < α < 2δ, for which more complicated sinusoidal-type profiles with non-null levels are obtained. The equations of the transition phases were studied in [7] for all these three cases. Table 1 Characteristic time intervals of the modulation functions of chopper wheels Type of optical choppers Device with windows with circular margins Classical device (with linear margins – Fig. 1)

Time interval

Outward margins (Fig. 3a) γ − 2 (ϕ + δ) /ω 2δ 1 /ω (α + 2(ϕ − δ) /ω

(γ − 2δ) /ω 2δ/ω τ = (α − 2δ) /ω

Obscuration time (t0 ) Transition time (τ t ) Total transmission time (τ )

2r

Inward margins (Fig. 3b) γ + 2 (ϕ − δ) /ω 2δ 1 /ω (α − 2(ϕ + δ) /ω

2δ1

ω

ω ρ

R

Ο

R Ο

α

α ρ

a)

γ

γ b)

Fig. 3 Optical choppers with windows with (a) outward and (b) inward circular margins

Choppers: Optomechatronic Devices for the Controlled Modulation of Light

129

4 Choppers with Windows with Circular Margins We have proposed [8] a new type of wheels that has an interesting and useful property: triangular profiles can be obtained by using it. The idea is simple: instead of the classical linear margins of the windows (Fig. 1b), circular margins, outward (Fig. 3a) or inward (Fig. 3b) are used. As the beam is circular, its chopping is an eclipse-like process, therefore the name of “eclipse choppers” we have proposed. One may notice that the classical choppers described in Section 2 are actually a particular case that may be obtained from this more general configuration of wheel for ρ→∞. Besides the triangular shapes of the transmission function, different durations of the light impulses (therefore different time intervals) are obtained, as presented in Table 1.

5 Experimental Part The scheme of the stall we have designed and made to carry on the experimental part of our studies is presented in Fig. 4. The He-Ne laser source is equipped with a beam expander to provide different diameters of the beam in the plane of the wheel. The motor has a potentiometer to adjust the rotating speed ω. The chopper module has the possibility to change the wheels to test the various configurations developed. A set of different wheels was designed and manufactured by our group [10]: with linear margins and with different number of windows (n equals 2 and 10) and with circular margins, as described in the previous sections. The main phase (the Photodetector

Chopper wheel Transmission belt He-Ne Laser

Driving engine Potentiometer

Fig. 4 Experimental stall for the study of optical choppers

130

V.-F. Duma et al.

electro-erosion process) of their manufacturing is shown in Fig. 5 for the former and in Fig. 6 for the latter type of choppers. The experimental stall is presented in Fig. 7. In Fig. 8 a set of modulation functions obtained for the ten windows chopper with linear margins is shown, (a) for a beam focused in the plane of the wheel (rectangular impulses), and (b) for a beam of finite diameter in the plane of the

a)

b)

c)

Fig. 5 Optical choppers with ten wings: a simulation program for the electro-erosion phase of the window, and b of the entire wheel; c electro-erosion process

a)

b)

Fig. 6 Electro-erosion process of the optical choppers wheels with windows with circular margins: a with outward margins; b with inward margins

Fig. 7 Experimental setup for the study of optical choppers – photo (Optomechatronics Laboratory, www.3omgroup. appspot.com/, “Aurel Vlaicu” University of Arad)

Choppers: Optomechatronic Devices for the Controlled Modulation of Light

131

Fig. 8 Modulation functions: a rectangular impulses, for the laser beam focused in the plane of the wheel; b trapezoidal impulses, for a beam of finite diameter in the plane of the wheel

a)

b)

wheel, with 2δ < γ = 18 degrees, as described in Section 3. One may see the good qualitative match of the theoretical and experimental profiles.

6 Conclusions We have presented the main issues of optical choppers with rotating wheels, one of the most used mechatronic devices in optics. Our contributions regarding the study of these devices for top-hat light beam distributions were pointed out, both for the classical choppers with linear margins and for the eclipse choppers which we have proposed and developed. The experimental stall we have developed was presented, as well as the first qualitative results. Future work comprises other types of beams, e.g. Gaussian, and other types of chopper wheels. Applications are envisaged, especially for colorimeters, in competition with other types of attenuators, with Risley prisms, that we have developed previously [9].

132

V.-F. Duma et al.

Acknowledgments The research work reported here was made possible by Grant 1896/2008 of National University Research Council (NURC/CNCSIS), Romania.

References 1. Bass, M., ed.: Handbook of Optics, Mc. Graw-Hill, New York, 1995. 2. Benjamin, K., Armitage, A., South, R.: Harmonic errors associated with the use of choppers in optical experiments. Measurement 39 (2006) 764-770. 3. Benmair, R. M. J., et al.: Solar-pumped Er,Tm,Ho:YAG laser. Optics Letters 15 (1990) 36-38. 4. Bittner, H., Erdmann, M., Herdt, B., and Steinacher, A.: Optical system of the SOFIA Telescope. In: Proc. SPIE 3356 (1998) 512-521. 5. Ching, M. T., Brennen, R. A., White, R. M.: Microfabricated optical chopper. Optical Engineering 33 (1994) 3634-3642. 6. Cucu R, G., Podoleanu, A.G., Rogers J. A., Pedro, J., Rosen, R. B.: Combined confocal/en face T-scan-based ultrahigh-resolution optical coherence tomography in vivo retinal imaging. Optics Letters 31(11) (2006) 1684-1686. 7. Duma, V.-F.: Theoretical approach on optical choppers for top-hat light beam distributions. Journal of Optics A: Pure and Applied Optics 10 (2008) 064008. 8. Duma, V.-F.: Optical choppers with circular-shaped windows: Modulation functions. Communications in Nonlinear Science and Numerical Simulation 16 (2011) 2218-2224. 9. Duma, V.-F., Nicolov, M.: Neutral density filters with Risley prisms: analysis and design. Applied Optics 48 (2009) 2678-2685. 10. Duma, V.-F., Nicolov, M., Kiss, M.: Optical choppers: Modulators and attenuators. In: Proc. SPIE, 7469 (2010) 74690 V. 11. He, Y., Jin W., Liu, G., Gao, Z., Wang, X., Wang, L.: Modulate chopper technique used in pyroelectric uncooled focal plane thermal imager. In: Proc. SPIE 4919 (2002) 283-288. 12. McDermid, I. S., Walsh, T. D., Deslis, A., White, M. L., Optical systems design for a stratospheric lidar system, Applied Optics 34 (1995) 6201-6210. 13. Mnerie, C., et al, Duma V.-F.: Study of optical modulators with rotating wheels: The experimental setup. Scientific Bulletin of the Politehnica University of Timisoara, Transactions on Mathematics-Physics 56 (2011) – in print; 14. Olkonnen H.: Chopper stabilized laser-Doppler skin blood velocimeter. Proc. SPIE 1922 (1993) 219-224. 15. Sarkisov, S. S., Curley, M. J., Huey L., Fields A., Sarkisov, S. S. II, Adamovsky, G.: Lightdrive actuators based on polymer films. Optical Engineering 45 (2006) 034002. 16. Thorlabs Catalog 18, 2007.

Motion Laws of a Varifocal Objective Working with a 1/2” CCD Camera C.M. Gruescu

Abstract Varifocal objectives for imaging systems integrating CCD sensors are useful in numerous applications, but there are no standards to design such systems. The present paper proposes a varifocal objective, suited for a 1/2” CCD camera, working at constant distance between the object plane and the image plane and providing a variable magnification within the range of (0.33. . .0.68). The motion laws of the mid, negative lens, respectively the last, positive lens are calculated. The optical system is diffraction limited over the entire range. Keywords Varifocal objective · Motion law · Optical design · CCD sensor

1 Introduction Nowadays, inspection and measurement make use of systems integrating CCD cameras. The digital image is watched by a human operator, performing different tasks, such as observing flaws regarding shape, dimension and other specific quality features. A quality control cell consists of five basic assemblies (Fig. 1). The optical system associated to an inspection or measuring stand is an objective, which projects the image of the object on the sensitive area of the CCD sensor. Technical assemblies contain many details require a certain flexibility regarding the magnification. If a flaw is suspected, a better observation of a smaller area can be achieved using varifocal objectives.

Fig. 1 Basic assemblies of an imaging system integrating visual sensors

C.M. Gruescu (B) “Politehnica” University of Timi¸soara, Timi¸soara, Romania e-mail: [email protected]


133

134

C.M. Gruescu

Such objectives are used in other different applications, but the optical schemes and design algorithms are not given as public information by the manufacturers. The specialty literature is very poor both in basic information and substantial solutions [1], [2], [3], [4].

2 Theoretical Approach An inspection and/or measuring stand integrating visual sensors and aiming to observe different scale details needs a varifocal objective. Figure 2 presents the optical scheme of such an objective, accomplishing the following conditions:

constant distance between the object plane and the sensor surface constant distance between the object and the objective variable magnification within a given range movable mid-lens movable end-lens fixed first-lens object abscissa equal to focal length of the first-lens, so that its optical power is chosen according to the required working distance. The first lens, actually, takes no optical effect positive first and end-lens negative mid-lens infinite object abscissa relative to the mid-lens, no matter what its current position is variable magnification and field of view resulting mainly from moving the midlens constant position of image plane ensured by slight movement of the end-lens.

invariable image plane

y'

invariable object plane

F1

F'2

F'3 L1

a1 = const

L2

d12var.

L3 d23var.

d = const.

Fig. 2 Basic scheme of a varifocal objective

a'3 = var.

CCD sensor width

Motion Laws of a Varifocal Objective Working with a 1/2” CCD Camera

135

Mathematical modeling of the above conditions leads to the following equations: a1 = f 1 ,

(1)

a1 = +∞ = −a2 ,

(2)

a2 = f2 ,

(3)

1 1 1 − = , a3 a3 f3

(4)

a3 = f2 − d23 ,

(5)

d = −a1 + d12 + d23 + a3 = const.

(6)

where f’1, 2, 3 are the effective focal lengths of L1, 2, 3 , a1, 2, 3 – object abscissas and a’1, 2, 3 – image abscissas. Considering the effective focal lengths of the lenses known, the equations (1), (2) and (3) get immediate solution. The system of equations (4). . .(6) contains four unknown variables: a3 , a’3 , d12 and d23 . The expression of the final image abscissa is: a3 =

2 −f2 + d + a1 − d12 ± −f2 + d + a1 − d12 + 4 f3 f2 − f3 d − f3 a1 + f3 d12 2

,

(7)

where a’3 and d12 are unknown. Different values of d12 should give final image abscissa and establish the magnification range, defined as image, respectively object height ratio. Practically, for a given set of focal lengths, f’1 , f’2 and f’3 , equation (7) in its “plus” form should be applied for different values of d12 . Size d12 represents the displacement of the lens L2 and abscissa a’3 is an indirect parameter of lens L3 movement, as: d23 = d + a1 − d12 − a 3 .

3 Numerical Modeling and Simulation Assuming the following known sizes: -

f1 = 60 mm f2 = -100 mm f3 = 30 mm d = 190 mm,

(8)

136

C.M. Gruescu

equation (7) was applied for nine values of d12 within the range [15. . .85] mm. The results of L2 movement simulation are given in Table 1. The real system, with finite center thicknesses is presented in Fig. 3, which shows the exact data of extreme positions of L2 and L3 . One can notice that the positive lenses are achromats and the negative component is a thin plano-concave lens. Use of diffraction limited achromats is highly significant for the image quality. A 3-D image of the varifocal system is given in Fig. 4. The real constant distance between the object plane and image plane is slightly bigger than the starting value. Size d is 204 because of center thickness of real lenses. Table 1 Values of image abscissa for different displacements of L2 Position

f’1

f’2

f’3

d

a1

d12_var

a’3

1 2 3 4 5 6 7 8 9

60 60 60 60 60 60 60 60 60

-100 -100 -100 -100 -100 -100 -100 -100 -100

30 30 30 30 30 30 30 30 30

190 190 190 190 190 190 190 190 190

-60 -60 -60 -60 -60 -60 -60 -60 -60

15 20 30 40 50 60 70 80 85

36.04 36.26 36.75 37.34 38.04 38.90 40.00 41.46 42.40

invariable image plane

invariable object plane

60

5

14.4

79

15

27.1

204

60

5

14.4

85 2.6 204

Fig. 3 Real varifocal objective with L2 and L3 in extreme positions

33.3


137

Fig. 4 3D representation of the varifocal objective in its extreme positions Table 2 Numerical parameters of the varifocal objective d12 [mm]

d23 [mm]

y’ [mm]

PMAG

s’3 [mm]

15 20 30 40 50 60 70 80 85

78.96 73.74 63.25 52.66 41.96 31.10 20.00 8.54 2.60

3.20 3.28 3.44 3.64 3.90 4.22 4.64 5.22 5.60

-0.33 -0.34 -0.37 -0.40 -0.44 -0.49 -0.55 -0.63 -0.68

27.15 27.39 27.91 28.50 29.20 30.05 31.15 32.50 33.36

FOV [mm]

d [mm]

∼16

204 204 204 204 204 204 204 204 204

Tables 2 provides full data of the varifocal objective. One can notice the following observations:

the field of view (FOV) is practically constant (FOV = ∼16 mm) as the object abscissa is constant, a1 = - 60 mm, for all the positions of the inner lenses the primary magnification, PMAG, varies within the range of [-0.33. . .-0.68] the image height, y’, is variable from 3.20 mm to 5.60 mm, which is convenient for a 1/2” CCD camera (size of sensing area – 5.95x4.76 mm) the negative lens, L2 , moves along a path of 70 mm between the extreme positions the end-lens, L3 , needs a slight displacement of about 6 mm in order to ensure the stability of the image plane.

The optical components were designed using the algorithms described in [5]. The analysis of the objective and collection of exact numerical data was performed using the software OSLO. Figures 5, 6 and 7 illustrate the use of the program in simulating the optical behavior of the objective. The figures refer to the scheme where d12 =

138

C.M. Gruescu

Fig. 5 Ray tracing through the objective in the position d12 = 70, d23 = 20

Fig. 6 Full data of the numerical simulation for the objective in position d12 = 70, d23 = 20

Fig. 7 MTF and PTF of the varifocal objective


139

70 mm and d23 = 20 mm. Figure 4 traces the rays through the system on axis (green rays), at 0.7 of the half – field (blue rays) and full half – field (red rays). Figure 6 reveals full data regarding geometry of components, the constant aperture and maximum half – field angle, the distances between components, the image abscissa and the half – height of the image. Figure 7 contains a representation of system’s modulation transfer function (MTF) and phase transfer function (PTF) up to the limit resolution of the human observer (∼32lp/mm). Both quality and quantity – wise the system is very performant. It features a diffraction limited system.

4 Motion Laws of Mid-Lens and End-Lens In order to get variable magnification the mid-lens L2 and end-lens L3 must displace with certain amounts along the optical axis. Numerical data in Table 2 allow the 3D representation in Fig. 8. The precision points verify the linear equation: PMAG = 0.05509d12 + 0.055085d23 − 5.50584 .

(9)

Motion laws of L2 and L3 are: (10)

d23 = 428.6PMAG2 + 643.7PMAG + 243.1 .

(11)

−0.3 −0.35 −0.4 −0.45 −0.5 −0.55 −0.6 −0.65 −0.7

−0.3 −0.35 −0.4 PMG

−0.45 −0.5 −0.55 −0.6 −0.65 −0.7

70 60 50 40 d23

30

20

10

Fig. 8 3D representation of function PMAG(d12 , d23 )

80 70 60 50 2 40 d1 30 20 0 10

PMG

d12 = −429.8PMAG2 − 626.8PMAG − 143.5 ,

140

C.M. Gruescu

Fig. 9 Motion laws of mid-lens and end-lens of the objective

Figure 9 shows the motion functions d12 (PMAG), d23 (PMAG) and variation of image abscissa a’3 (PMAG).

5 Conclusions Varifocal objectives are useful in imaging systems, but their design does not fit into a general algorithm. Specific features of each application (working distance, magnification range, field of view, dimensions of sensing area) impose personalized approaches. The present paper described the steps in design of optical scheme, simulation of nine precision position points and calculation of motion laws for two mobile lenses. The objective which resulted provides high image quality over the entire magnification range.

References 1. Nicoara, I.: Calculul si constructia aparatelor optice, Curs, vol. II, Centrul de multiplicare al Institutului Politehnic “Traian Vuia” Timisoara, 1988. 2. Tao Ma, Jingchi Yu, Pei L., Chinhua, W.: Design of a freeform varifocal panoramic optical system with specified annular center of field of view, Optics Express, Vol. 19, Issue 5, pp. 3843-3853, doi:10.1364/OE.19.003843, 2011. 3. Benjamin C. I.: Paraxial Zoom Lens Design, ECEN 5616 Spring 2005, March 10th, 2005, available on http://ecee.colorado.edu/~mcleod/teaching/oesd/projects/Zoom%20lens.pdfpDesign of a freeform varifocal panoramic optical system wit 4. http://www.mir.com.my/rb/photography/companies/nikon/nikkoresources/zoomsMF/index.htm 5. Gruescu, C., Nicoara, I.: Aparate optice. Analiza si sinteza sistemelor optice lenticulare, Ed. Politehnica, 2004.

Part III

Mechanisms and Machine Design


xii

A New Principle of Adaptive Compliant Gripper D. Petković and N.D. Pavlović

Abstract Gripping and holding of objects are key tasks for robotic manipulators. The development of universal grippers able to pick up unfamiliar objects of widely varying shapes and surfaces is a very challenging task. This paper presents an approach of the new principle of a universal gripper with adaptable shape morphing surfaces. The adaptive surfaces will have the controllability by a compliant system with embedded actuators and sensors. The main sensing system has to be made of a conductive silicone rubber or foam. These are carbon-black filled silicone materials with good sensing properties whose electrical resistance is changed by compression. The implemented controllable system will be able to morph shapes of the gripper to accommodate different objects. A methodology for design of the compliant adaptive gripper will be presented. The main advantage of this compliant gripper is the connection of controllability and observability in one system by a compliant mechanism. Keywords Adaptive gripper · Compliant mechanism · Conductive silicone rubber · Shape morphing

1 Introduction The handling of irregular and unpredictably shaped objects introduces demands on gripper flexibility. Simple impactive implementations will almost certainly result in damage to the prehended object. These impactive techniques must be controlled by controllable compliance. If the degree of compliance can be controlled by external influences, then a gripper can be designed which exhibits quite different prehension and retention characteristics. Such grippers are known as shape adaptive. The inflexibility of grippers available today is limiting the use of robots. Much research has therefore been done to develop more universal flexible grippers. A flexible gripper design based on the use of compliant materials and internal pressure was introduced in [1]. This type of grippers conforms to the shape of D. Petković (B) Faculty of Mechanical Engineering, University of Niš, Niš, Serbia e-mail: [email protected]


143

144

D. Petković and N.D. Pavlović

an object by means of elastic gripping elements and pressurization. Paper [2] presented a universal robotic gripper which uses magnetorheological fluid to utilize the increase in yield stress for gripping various shapes and sizes when a magnetic field’s applied. In [3] another robotic finger with soft tips was proposed that can realize dynamic stable grasping with the aid of sensory feedback of the difference between centers of area-contacts at both sides of the object. Further investigation led to a completely different approach to a universal gripper where individual fingers are replaced by a single mass of granular material that, when pressed onto a target object, flows around it and conforms to its shape [4]. All previously mentioned examples of universal robotic joints do not use compliant mechanisms for their main structure. The use of compliant or flexible mechanisms is a very promising option for the successful realization of shapeadaptable structures. The distributed compliance and smooth deformation field of compliant mechanisms provide a viable means to achieve shape morphing in many systems. A compliant mechanism is a class of mechanism that relies on elastic deformation of its constituent elements to transmit motion and/or force [5]. Many practical benefits can be realized by exploiting compliance in engineering design. These include: reduced complexity, zero backlash and wear, sub-micron accuracy, and embedded actuation/sensing. Because of these advantages, properly designed compliant mechanisms are well suited for shape morphing applications [6]. There are many studies of the use of compliant mechanisms for shape adaptation. The main investigations were established in [7, 8] where a systematic synthesis approach was developed to find the optimal compliant mechanism for desired shape morphing. The results showed that the compliant mechanisms can achieve the desired shape. Another systematic approach for synthesizing compliant mechanisms that can change an initial curve shape into a desired target shape in the same plane through structural deformation was proposed in [9, 10]. A compliant mechanism for morphing aircraft structures in order to exploit aerodynamic benefits was designed in [11, 12]. Although the abovementioned designs only focus on changing an initial shape to one target shape, changing through a set of different shapes could be investigated by extending the proposed methods. For multi-stage shape morphing or for continuous morphing through a set of different shapes, the compliant mechanism requires more than one actuator. However, there are many situations where the desired shape change might not be known ahead of time and could be a function of the environment. Until now, only [13] was concerned with multiple shape changes or with unknown target shape, where a compliant system with embedded actuators and sensors was presented. The main purpose of this paper is to investigate the application of adaptive multiple shape changes in a compliant robotic gripper. As tactile sensing element, conductive silicone rubber or foam will be used. This is an electro active material whose electrical resistance changes with deformation. The key goal of this study is to establish the application of this silicone rubber material in robotic grippers as distributed tactile sensing structure. In the beginning of testing the main application for this gripper should be in macro domain.

A New Principle of Adaptable Compliant Gripper

145

2 Materials and Methods Most conventional tactile sensors need a large number of sensing elements and wires because every detection point needs one sensing element and wiring. A soft areal tactile sensing structure made of pressure-sensitive conductive rubber has to be developed without any wire or sensing element in the tactile region. Another design of the tactile sensing structure can be made of pressure-sensitive conductive foam with wires in the tactile region. The distribution of applied pressure, relating to the resistivity change of the pressure-sensitive rubber, can be estimated by using inverse problem theory. Electrical impedance tomography (EIT) [14] will be employed to reconstruct the resistivity distribution from information obtained by electrodes placed around the region. EIT is a non-invasive technique for measuring the internal impedance distribution of a conductive material using only measurements done on its boundary. The processing of the tactile sensor consists of two steps. The first step is the measurement of conductivities using electrodes placed around the rubber plate at equal intervals, as shown in Fig. 1c. An AC voltage is applied, and resistance is calculated using the four-electrode method from the differential voltage between neighboring electrodes, for every possible combination. In the figure, alternating voltage is supplied between the first and second electrodes and the differential voltage is measured between the 10th and 11th electrodes. The number of possible combinations in this example is 208. However, the number of the independent measured data is actually 104, because the measurements are not independent when the supply and the measuring electrode pairs are symmetrically positioned. In the second step, the distribution of resistivity change is reconstructed from the measurements. This resistivity also reveals the pressure distribution since the resistivity change is caused by the change of the pressure applied to the plate. To get a reconstruction image, a sensitivity matrix is used that is the Jacobian matrix between δV and δρ [15] δV(m,n) = Mm,n,x,y δρ(x,y) .

(a)

(b)

(1)

(c)

Fig. 1 a Conventional approach for tactile sensing measurement, b Electrodes only on boundary – EIT method c Conductivity measurement method for EIT [15]

146


Here, m and n indicate the position of the applying electrode pair and the measuring electrode pair, respectively, and x and y indicate the position coordinates of the discrete domain in the rubberplate. The sensitivity matrix can be obtained by numerically solving a forward problem. Then, by solving (1) using least squares or generalized inverse matrix method, the distribution of resistivity change δρ(x,y) is obtained. Equation (2) shows the least squares method with a non-negative constraint. The constraint is implemented because of the unstability of this method. In the case of EIT, A, x, b are corresponding to M, δρ, δV, minAx − b2 (x ≥ 0)

(2)

This method needs the sensitivity matrix in advance, so this matrix is calculated using (3) Mm,n,x,y =

∇ϕm · ∇ϕn da

(3)

a

Here, ∇ϕm and ∇ϕn are the electric potential gradient, which is derived from the electric supply from the electrode pair of m or n, respectively. In other words, each component of the sensitivity matrix can be calculated by integrating the inner products of the potential gradients that are caused by the electric current from m and n electrode pairs in the area indicated by x and y. this equation can be derived from electromagnetic theory [15].

2.1 Processing of the Conductive Silicone Rubber The carbon-black filled silicone rubber is electrically conductive and its resistance changes by deformation. These properties make this material eligible to develop force or deformation sensors. The characteristic of resistance and force as function of deformation and time was examined. Figure 2 depicts the conductive silicone rubber and forming process of the sensors, i.e. molding process. The tool for form ing these sensors has to be designed depending on the sensor functions and requirements. Figure 2 shows two different tools for the molding process. Zwick ProLine material-testing machine Z005 was used to measure the mechanical and electrical properties of the sensors. Throughout the tests, some diagrams

Fig. 2 Conductive silicone rubber and molding process for the sensors forming


147

were recorded graphically and numerically at the same time. NI BNC-2120 card, together with a voltage divider, was also used for measuring some characteristics and this result was monitored and visualized simultaneously in an automatic program in LabVIEW.

2.2 Adaptive Shape Morphing Gripper The compliant mechanism could achieve the known target shape by using an external actuator and it could also be adapted to any unknown shape via embedded actuators (Fig. 3). The proposed shape morphing compliant gripper could be optimized by two objective functions. The first objective function is to minimize the distance of particular points between the desired target surface and achieved surface. In that way, a structure of compliant mechanism is obtained. The second objective function is to achieve any unknown surface via embedded actuators. The combination of three actuators could be used to cover as broad a range of output combinations as possible. All actuators ought to have independent influence on the output surface, thus making the maximization of independence of the embedded actuators effect an objective function. The main advantage of the new gripper is the combination of actuators, sensors, and compliant mechanism in one compliant system. Figure 5 shows two states of the compliant gripper when there is no grasping object, i.e. the unloaded state, and the state with one grasped cylindrical object. The unloaded state of the gripper could be accommodated by an optimization procedure to achieve desired target surface depending on the construction requirements. The second step of grasping is the accommodation to any unknown shape via embedded actuators.

3 Results Through the compression test with the load and unload phase, the electrical resistance changing and resisting force changing were monitored. These results were recorded and the visualized diagrams are shown in Fig. 4. It can be seen that the resistance had the lowest value in the beginning of the test. Also, the maximum value

Fig. 3 Principle of adaptable compliant robotic gripper with embedded actuators (grey beams) and distributed sensing element above whole grasping surface

148


Fig. 4 Electrical resistance changing curve

Fig. 5 Gripper without any grasping object and with one cylindrical objects

of resisting force decreased during the tests but the stationary behavior was obtained after more loading-unloading phases. Electrical resistance decreased during loading, while it increased very quickly within the relaxation phase.

4 Conclusions This paper presents a new principle for adaptive compliant gripping mechanism which utilizes both shape changing from the initial shape to the target one and universal adaptation to different grasping objects via embedded actuators. With the introduction of multiple, optimally placed actuators, this work does not focus on a specified target shape, but rather the ability to achieve any shape required of the gripper. This method is superior to anything one has tried to apply before and can be developed much further. Electrical impedance tomography is a technique used to estimate the internal conductivity of an electrically conductive body by using only measurements from its boundary. The proposed method enables one to have not only flexible and thin, but also stretchable tactile sensors. These sensors can be implemented over


149

the movable joints, where strechability is required. This measurement method has some disadvantages, e.g. it is very sensitive, and thus it cannot be ignored during measurements. Further investigations will stimulate other researchers to come closer to understanding and solving the problems. The electrically conductive carbon-black filled silicone rubbers are suitable to develop force or deformation sensors considering the achievable accuracy and repeatability. The understanding of the exact mechanism of the behavior of the material awaits further investigation. The main quality of the new robotic gripper is the synthesis of compliant mechanism, actuators, and sensors in one system to achieve any unknown target shape changes. Although it was designated for macro domain, it is possible to make this gripper in micro domain too. In general, the results indicate that this new principle for a compliant robotic gripper is a good starting point for further development and investigation. Acknowledgements This paper is supported by Project Grant III44004 (2011-2014) financed by Ministry of Education and Science, Republic of Serbia.

References 1. Choi, H., Koc, M.: Design and feasibility tests of a flexible gripper based on inflatable rubber pockets. Intern. J. Mach.Tools 46 (2006) 1350-1361. 2. Pettersson, A., Davis, S., Gray, J.O., Dodd, T.J., Ohlsson, T.: Design of a magnetorheological robot gripper for handling of delicate food products with varying shapes. J. Food Eng. 98 (2010) 332-338. 3. Han, H.-Y., Arimoto, S., Tahara, K., Yamaguchi, M., Nguyen, P.R.A.: Robotic pinching by means of a pair of soft fingers with sensory feedback. In: Proceedings of the 2001 IEEE International Conference on Robotics & Automation, Seoul, Korea, pp. 97-102 (2001). 4. Brown, E., Rodenberg, N., Amend, J., Mozeika, A., Steltz, E., Zakin, R., Lipson, H., Jaeger, M.: Universal robotic gripper based on jamming of granular material. In: Proceedings of the National Academy of Sciences of the United Stated of America, pp. 18743-19132 (2010). 5. Shuib, S., Ridzwan, M.I.Z., Kadarman, A.K.: Methodology of compliant mechanisms and its current developments in applications: a review. America J. App. Sc. 4(3) (2007) 160-167. 6. Rediniotis, O.K., Wilson, L.N., Lagoudas, D.C., Khan, M.M.: Development of a shapememory-alloy actuated biomimetic hydrofoil. J. of Intelligent Material Syst. And Struct. 13(1) (2002) 35. 7. Lu, K.-J., Kota, S.: Parametrization strategy for optimization of shape morphing compliant mechanisms using load path representation. In: Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Chicago, Illinois USA, pp. 693-702 (2003). 8. Lu, K.-J., Kota, S.: An effective method of synthesizing compliant adaptive structures using load path representation. J. of Intelligent Material Syst. And Struct. 16 (2005) 307-317. 9. Lu, K.-J., Kota, S.: Compliant mechanism synthesis for shape-change applications: preliminary results. Smart Struct. And Materials. 4693 (2002) 161-172. 10. Lu, K.-J., Kota, S.: Design of compliant mechanisms for morphing structural shapes. J of Intelligent Material and Struct. 14 (2003) 379-391. 11. Kota, S., Hetrick, J., Osborn, R., Paul, D., Pendleton, E., Flick, P., Tilmann, C.: Design and application of compliant mechanisms for morphing aircraft structures. Smart Struct. And Materials. 5054 (2003) 24-33.

150


12. Campanile, L.F.: Model synthesis of flexible mechanisms for airfoil shape control. J of Intelligent Material and Struct. 19 (2008) 779-789. 13. Trease, B., Kota, S.: Adaptive and controllable compliant system with embedded actuators and sensors. Active and Passive Smart Struc. And Integrated Syst. 6525 (2007) 65251R. 14. Alirezaei, H., Nagahubo, A., Kuniyoshi, Y.: A highly stretchable tactile distribution sensor for smooth humanoids. In: 7th IEEE-RAS International Conference on Humanoid Robots, pp. 167-173 (2007). 15. Hou, T.-C.H., Lynch, J.P.: Tomographic imaging of crack damage in cementitous structural components. In: 4th International Conference on Earthquake Engineering Taipei, Taiwan, pp. 1-10 (2006).

Optimum Design of Cam Mechanisms I. Simionescu, M. Ene, and L. Ciupitu

Abstract The paper presents some considerations about the optimum design of cam mechanisms with reciprocal and oscillating roller followers. The objective function express the dimension of the cam, while the constraints limit the size of the pressure angle and / or the magnitude of the unit stress to the higher contact between cam and follower or cam and roller. Finally, the result of solving a numerical example is presented. Keywords Cam mechanisms · Optimization · Contact stress · Hertz

1 Introduction The relative motion of the follower with respect to the cam is a combination of a relative turning motion and a relative oscillating motion. In this relative motion, the knife edge follower plots the pitch curve of the cam [8, 9, 10]. The cam profile is obtained as the envelope of the circles family which represents the relative positions of the roller with respect to the cam. The centers of the circles are situated on the pitch curve. The optimum design of the cam implies the solving of a nonlinear programming problem [2, 4, 5]. The objective function expresses one of the following features: the maximum radius of the pitch curve of the cam, the overall size of the cam mechanism, the efficiency of the mechanism, the actuating moment etc., or a combination of these. The constraints which must to be considered are: the maximum value of the pressure angle, the maximum value of the unit stress at the cam – follower contact point, the maximum value of the negative curvature radius of the cam pitch curve, the maximum magnitude of follower acceleration, the minimum value of the positive curvature radius of the cam pitch curve, to avoid the cam undercutting, and the load capacity of the higher pair between cam and follower.

I. Simionescu (B) “Politehnica” University of Bucharest, Bucharest, Romania e-mail: [email protected]


151

152

I. Simionescu et al.

A very important constraint limits the maximum value of the stress to the high contact between cam and roller or cam and follower. This value is influenced by: the magnitude of the pressure angle, the value of positive curvature radius of the cam profile at the contact point, and the size of the resistance force which acts on the follower, which include the inertial force and elastic force of back spring. But, the last constraint does not eliminate the restriction which limits the magnitude of the pressure angle, which is a working kinematics condition.

2 Optimum Synthesis of Cam Mechanisms The objective function, which expresses the maximum radius of the pitch curve of a cam with reciprocating roller follower (Fig. 1), is: Rmax

= (S0 + h)2 + e2 ,

where: S0 is the initial space of the follower, h is the follower linear stroke, e is the offset of follower.

M3

h3

δ

_ N1 B1 _ T1

d1

2

Me

Fig. 1 Cam Mechanism with reciprocating roller follower

e ω1 A

S

D _ T12

t d

y1

t

3

_ N12

_ R23

r2

l3

l

_ _ N2 B2 _ T2

d2

E

Y

δ'

_ F3

β

e2

C e1

1 ϕ

x1

X

153 '

δ

δ

_ R23 M3

_ F3

l

y1

_ T12

d

M f03 B

Me

1

C Y

t

ω3

2

D

E 3

_ N12

t

Fig. 2 Cam mechanism with oscillating roller follower

r2

Optimum Design of Cam Mechanisms

A

x1 ω1

ϕ

X

For the cam with oscillating roller follower (Fig. 2), the objective function is: Rmax =

l2 + d 2 − 2ld cos(ψ0 + θ),

where: l is the length of follower, d is the distance between axes of the revolute pairs between cam and frame and follower and frame, ψ0 is the initial angle of the follower, θ is the follower angular stroke. The pressure angle, of movement transmission from cam to reciprocating follower, is determined by using very well known formula [10]: dS −e dϕ , δ = arctan S0 + S (ϕ) and to cam mechanism with oscillating roller follower: δ = arctan

dψ − 1 l + d cos(ψ0 + ψ (ϕ)) dϕ . d sin(ψ0 + ψ (ϕ))

The extreme of the function δ = δ(ϕ) - for reciprocating follower - is in a point defined by ϕ∗ as solution of the equation: dδ =0 dϕ or: d2 S dS 2 dϕ dϕ = = tan δmax . dS S0 + S (ϕ) dϕ The solving is performed both in the rise phase and return phase, and results:

154


A−B B tan δ1 max − A tan δ3mib ,e= , D D ∗ dS dS , B = S ϕ − , tan δ where: A = S ϕ1∗ tan δ1 max − 3 min 3 dγ ϕ=ϕ1 ∗ dγ ϕ=ϕ3 ∗ S0 =

D = tan δ1 max − tan δ3 min = 0. The magnitude of the admissible pressure angle is calculated in terms of dimensions of the follower and guiding [10], and the minim size of the curvature radius is calculated in terms of maximum load capacity imposed to the higher contact l between cam and roller. For oscillating follower cam mechanism, the ratio and d the initial angle ψ0 are calculated as solution of a nonlinear programming problem [2, 4, 5, 10], where the objective function is the maximum radius of the cam pitch curve Rmax , and the constraints are: d dψ1 − 1 + cos(ψ0 + ψ1 (ϕ)) dϕ l ≤ tan δ1 max , d sin(ψ0 + ψ1 (ϕ)) l dψ3 d − 1 + cos(ψ0 + ψ3 (ϕ)) dϕ l ≥ tan δ3 min , d sin(ψ0 + ψ3 (ϕ)) l For solving may be used one of specific methods, so as Lagrange method, penalty functions method etc. The other constraints which must to be considered are:

a) the maximum value of the negative curvature radius of the cam profile do not exceed the roller radius. b) the minimum value of the positive curvature radius of the cam profile not be less than a critical value, which is calculated with respect to the loading of the higher contact cam – roller. c) the maximum value of the positive curvature radius of the pitch curve of the cam not be less than the radius of the cutting tool. In certain cases, some constraints are excluded by another. For example, the reaction force which stresses the higher contact cam – roller depends to the pressure angle. As a result, all constraints must be of the equality – inequality type.

2.1 Cam Mechanism with Reciprocating Roller Follower The applied forces and moments, namely technological resistances, elastic forces of back spring, gravitational forces and inertial forces, have in the point E (Fig. 1) the resultant force F 3 and the resultant moment M 3 . The normal reaction force N 12 ,


155

between cam and roller is inclined with pressure angle δ with respect to the translation direction. If in the prismatic pair is a play between translating follower and frame, the contact between these two elements is done in two points (Fig. 1), namely B1 and B2 . The equilibrium equations of the forces which act on the follower in the raised phase are: d2 S + g m3 = 0; F3 cos β − R23 cos δ + T1 + T2 + dt2 h3 h3 e1 − e2 + R23 cos δ + d1 sin δ + F3 cos β − (l + d2 ) sin β 2 2 +T2h3 − − N2 l − R23ρD = 0; h3 h3 cos δ + (d1 + l) sin δ + F3 cos β − d2 sin β R23 e1 + e2 − 2 2 −T1 h3 − − N1 l − R23 ρD = 0; R23 cos δ − N12 cos δ + T12 sin δ = 0; N12 sin δ + T12 cos δ − R23 sin δ = 0, N12 s12 + R23ρD + IG2 ε2 − T12 r2 = 0, (1) where: T1 = μB N1 , T2 = μB N2 , d1 = d − S, d2 = l3 − l − d1 , s12 is the rolling friction coefficient between roller (2) and cam (1), μB is the sliding frictional coefficient from prismatic pair B. The angular acceleration of the roller (2) is: ε2 =

dx1 d2 x1 dy1 d2 y1 ω21 + dϕ dϕ2 dϕ dϕ2 , dx1 2 dy1 2 r2 + dϕ dϕ

where: dy1D dϕ x1 = x1D − r2 2 ; dx1D dy1D 2 + dϕ dϕ dx1D dϕ y1 = y1D + r2 ! ; ! dx1D 2 dy1D 2 " + dϕ dϕ

(2)

156


x1D = S cos ϕ + e sin ϕ; y1 D = S sin ϕ − e cos ϕ. The simultaneous solving of the equilibrium Equations (1) is made with respect to the following unknowns: N1 , N2 , δ , R23 , N12 and μB . The absence of the system solutions leads to the not working of cam mechanism, owing to self-locking phenomena. From a simplified calculus, in which are neglected mass of roller and frictional forces and moments between cam and roller and follower (m2 ) = IG2 = s12 = ρD = 0), results: (e2 + 0.5h3 ) cosβ − (l + d2 ) sinβ (e2 − 0.5h3 ) cosβ − d2 sinβ + )μB 1 − μB h3 1 + μB h3 . (e1 + 0.5h3 ) cosδ + (l + d1 ) sinδ (e1 − 0.5h3 ) cosδ + d1 sinδ cosδ − ( + )μB 1 + μB h3 1 − μB h3

F3 (cosβ + ( N12 =

Under these circumstances: R23 = N12 . The critic pressure angle in which the self-locking of follower arose in the rise phase is: α cr = arctan

1 − 2e1 μB . μB (2d1 + l − μB h3 )

The movement of the follower under action of the reaction force R23 is not l . possible if e1 > 2μB In the return phase, which is performed under action of the force F¯ 3 , the selflocking of follower arise if angle β is greater than the critical value:

βcr = arctan

If e2 >

1 − 2e2 μB . μB (2d2 + l − μB h3 )

l , the force F¯ 3 can not move the follower in the return phase. 2μB

2.2 Cam Mechanism with Oscillating Roller Follower The applied forces and moments, namely technological resistances, elastic forces of back spring, gravitational forces and inertial forces, have in the point E (Fig. 2) the resultant force F 3 and the resultant moment M 3 . The equilibrium equations of the forces and moments, which act on the follower in the rise phase, are:


157

R03X + F3X − R23 sin ψ + α = 0; R03Y + F3Y + R23 cos ψ + α = 0; XB ) − F3X Y3 + R23 AB cosα +M3 +ρA SIGN(ω23 ) F3Y (X3 −

− R203X + R203Y ρA SIGN(ω23 ) = 0; R23 sin ψ + α − N12 sin(ψ 2 + α) − T12 cos(ψ + α) ω +ABm cosψ + ε sinψ = 0; 2 3 3 + α) T sin(ψ + R23 cos ψ + α − N12 cos(ψ 12 α) − m3 g +AB m2 ω23 sin ψ − ε3 cos ψ = 0; M12 s12 + R23 ρB + IG2 ε2 = T12 r2 ,

(3)

and is solved with respect to unknowns: R03X , R03Y , R23 , α , N12 and T12 . The angular acceleration ε2 of the roller (2) is calculated with formula (2).

3 Stress at the Higher Contact Between Cam and Roller At the contact of two cylinders cu radius R1 and R2 , with parallel axes, the unit stress which is caused by the force F comprised in the cylinders axes plane, and which press one cylinder to another, is calculated with Hertz formula [3, 6, 7]: σ =

FE0 , 2πlρ0 1 − υ 2

2E1 E2 R 1 R2 is the equivalent Young module, ρ0 = is the equivE1 + E2 R1 + R 2 alent curvature radius, l is the contact generatrix length, υ is the Poisson coefficient. This formula is valid with following simplifying assumptions: the length of contact generatrix of the cylinders is very long with respect to radii R1 and R2 ; the axis of cylinders are strictly parallel; the surfaces of the cylinders are perfectly smooth; the pressing force F is statically applied and uniform distributed along the contact generatrix; the unit stresses do not exceed the limit for which the deformations are into proportional limits, in accordance with Hooke law; the contact surfaces, which results owing to action of force F, are included into the common tangent plane of cylinders; no relative motion between cylinders. In fact, these assumptions do not accomplished entirely to cam mechanisms, namely: the length of contact generatrix of the cam and roller is not more long comparative with curvature radii; the state of stress is not statically, because its intensity is variable during working, the cam and the follower being movable; the contact pressure is not uniform distributed along the contact generatrix, because of elastic deformation of the axes, of machine inaccuracy, misalignments and wears. In the contact point C, the curvature radius of the cam profile is calculated with formula: where: E0 =

# ρC =

dx dλ

2

+

dy dλ

2 $1.5 % 2 dy d2 x dx d y − − r2 . dλ dλ2 dλ dλ2

158


where: x=x(λ) and y=y(λ) are the parametrical equations of the cam profile r2 is the radius of roller. This curvature radius is positive if the curvature center and the center of the revolute pair between cam and frame are positioned on the same part of the common tangent to the profiles [1], and negative otherwise.

4 Example In order to exemplify the proposed synthesis method, a numerical example has been solved. The transmission functions for a cam mechanism with reciprocating roller follower are [10]: 4x3 - in the rise phase: u = 0.4; x ∈ 0, u2 , y = 3u 2; u 4(u−x)3 x ∈ 2 , u , y = 3u2 + 2x − u; 3 x ∈ u, 1+u , y = 1 − 4(x−u)2 + 1 − 2x + u; x ∈ 1+u 2 2 ,1 ,y = 1 − 3(1−u)

- in the return phase: x ∈ [0, 1] , y = 10x3 − 15x4 + 6x5 ,

4(1−x)3 , 3(1−u)2

and are characterized by: follower stroke h = 80 mm, rotation angle of cam are: ϕ1 = 5π/6, ϕ2 = π/9, ϕ3 = 2π/3, maxim pressure angle in the rise phase: δmax = π/4, minim pressure angle in the return phase: δmin = −5π/18, roller radius r2 = 8 mm, minimum positive curvature radius of the cam profile ρmin = 30 mm. Using the above method, from minimization of the maxim radius of cam and taking into consideration the kinematic constraints, which limit the pressure angles only, the following dimension has been obtained: S0 = 33.6566 mm, e = 3.19540 mm. The minimum curvature radius is 25.8079 mm. By adding the constraint which limits the positive curvature radius ρmin ≥ 30 mm, the dimensions of the cam are: S0 = 37.8681 mm, e = 3.208943 mm and the pressure angle are diminished: δmax = .74602, δmin = −.835504. In the last case, the first constraints are ignored.

5 Conclusions The problem of design of machines and mechanisms are mathematically described by systems of nonlinear equations. For this reason, these problems have not unique solutions. The choosing of the best solution, from a certain point of view, implies the solving a problem of nonlinear programming. Some design problems involve multiple conflicting objectives and often they are attempted to be solved by converting them into a simple composite objective. In this paper, a robust mathematical model for optimum design of cam mechanisms is presented.


159

References 1. Duca C., a.o.: Bazele proiect˘arii mecanismelor cu came, Editura Gh.Asachi, Ia¸si, 1999. 2. Fiacco, A. W. and McCormick G. P.: Nonlinear Programming, Sequential Unconstrained Minimization Techniques, John Willey and Sons, New York, 1968. 3. Flemming M. and Baraf L., Die Theorie der Hertz’schen Pressung, Institut für Konstruktion und Bauweisen, ETH, Zürich, 1988. 4. Fox R. L.: Optimization Methods for Engineering Design, Addison Wesley Publishing Company, 1973. 5. Goldberg D. Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley Publishing Comp, 1999. 6. Hertz H.: Ueber die Berührung fester elastischer Korper, Gesamelte Werke, Bd. 1. Leipzig, 1895. 7. Johnson K. L.: Contact Mechanics, Cambridge University Press, Cambridge, 1985. 8. Rothbart H.: Cam Design Handbook, Dynamics and Accuracy, McGraw-Hill Handbooks, 2008. 9. Rothbart H.: Cam Design Handbook, Cam size and profile determination, Numerical control and manufacturing, Computer aided design techniques, McGraw-Hill Handbooks, 2003. 10. Simionescu I. and Moise V.: Mecanisme, Editura Tehnic˘a, Bucure¸sti, 1999.


xii

Linkage Solutions for the Manufacture of Complex, Thermoplastic Lightweight Structures M. Krahl, U. Hanke, and K.-H. Modler

Abstract The manufacturing of novel, high-performance fibre composite components based on hybrid yarns necessitates the development of new process solutions. In the particular case of manufacturing open core sandwich panels, tools are required which facilitate automated manufacturing. Such automation can be achieved using specially adapted wedge kinematics, which in turn require handling systems that ensure a correct position of these tools. One mainpart of such handling system is a comb system, which is successfully approved by the manufacturing of planar sandwich panels. Single-bend sandwich panels requires the development of new solutions, which facilitate the oriantation of the consolidation tools with a minimum of drive effort. Thereby the focus is on the design of a comb system based an a scissor mechanism. Keywords Composite sandwich panel · Consolidation process · Handling systems · Scissor mechanisms

1 Introduction The development of future-oriented, competitive lightweight components for applications in areas such as vehicle manufacturing and mechanical engineering requires innovative design concepts and efficient manufacturing processes. Thermoplasticbased hybrid yarns offer significant potentials in these areas: production-ready processing methods for load-conform design and a good design flexibility [1]. As a result, TU Dresden and in particular scientists active in SFB 639 do extensive investigation into the design and technological implementation of heavily loaded, textile-reinforced thermoplastic composite structures. One of the generic prototypes developed within the framework of SFB 639 is a textile-reinforced, multi-wall panel structure with a loose cross section. A continuous development process from filament to component level sees specially adapted hybrid yarns based on materials such as glass fibre and polypropylene processed into woven or knitted 3D spacer fabrics or stitched 3D spacer preforms before being compressed to spacer structures within the framework of multi-phase processing [2, 3, 4]. M. Krahl (B) Dresden University of Technology, Dresden, Germany e-mail: [email protected] E.-C. Lovasz, B. Corves (eds.), Mechanisms, Transmissions and Applications, Mechanisms and Machine Science 3, DOI 10.1007/978-94-007-2727-4_15, C Springer Science+Business Media B.V. 2012

161

162

M. Krahl et al.

2 Process The swelling behaviour of the semi-finished textiles represents a significant challenge when processing thermoplastic hybrid yarns. The swelling factor k signifies that the volume VH (the sum of the fibre volume of the yarn and the spaces) is k-times larger than volume VV (the volume of the consolidated composite). Experimental tests on textile semi-finished hybrid yarns yield a swelling factor of k = 3 for the given glass-fibre/polypropylene composite [5]. This means that the textile preform is 3-times thicker than the consolidated composite (s. Fig. 1). The swelling factor k is the key parameter for the design of the consolidation tool for the manufacturing of spacer structures. A novel tool concept using a compression-based manufacturing process has been developed in order to facilitate the reproducible processing of complex semi-finished textiles. To guarantee a strechted fibre orientation, it is expedient to keep the circumference u and centre-tocentre separation distance t of each tool as constant as possible (s. PP 1 in Table 1). This can be realised by using the consolidation kinematics as shown in Table 1. In order to ensure the smoothness of the component geometry, the functional principle of wedge kinematics is applied to core tools. The relative movement of the kinematic components enables the linear power transmission from the vertical movement of the press stamp to the lateral crosspiece areas of the hollow sandwich structure. During the transformation process the tool contour changes from vertical format (PP 3) when the panel is in an unconsolidated state to horizontal format (PP 4) when the panel is in a consolidated state. The whole consolidation process with its tool mode can be divided into five process steps (s. Table 1): 1. At the beginning of the process the cover profiles are forced apart vertically by the horizontal pressure applied to the side profiles. The wedge angles are not yet used. 2. With use of the limpness of the unconsolidated textile, the preform is pulled over the consolidation kinematics during the loading process. 3. The wedge angles are now pushed into the kinematic components. Therefor the precise guidance of each individual element is guaranteed.

Fig. 1 Swelling behaviour of a hybrid yarn textile (left: unconsolidated, right: consolidated)

Linkage Solutions for the Manufacture of Complex, Thermoplastic Lightweight . . .

PP 1

Sketch

163

Table 1 Consolidation process Comment Tools in “vertical format”, tool-orientation on separation distance t

2

Loading of tools with textile preform

3

Loading of tools with demoulding wedge angles

4

Consolidation, circumference u is equal to process phase 1

5

Removal of demoulding wedge angles

PP: Process phase

4. During the consolidation process, the movement of the press stamp path is used to force the coupled compression of the cover and side profiles. This ensures the surface of the kinematic levels is well-finished in its final form as well as the maintained relative positions of the chambers. 5. During demoulding it is necessary that the circumference of the consolidation kinematics can be reduced. This can be achived by retracting the wedge angels in axial direction. To ensure the function of the consolidation tool it is necessary to keep a defined alignment of each kinematic component. This can be achieved by using pneumatically driven comb systems (s. Fig. 2), which make it possible to maintain a consistent tool-to-tool separation distance. It is the bringing together of the side profiles by the combs in process phases 1 and 2 that ensures the vertical format. The comb system is based on a two-piece concept, and consists of two opposing pin bars which can be moved horizontally towards and away from each other. The separation distance between the comb teeth is calibrated to the core tool. Horizontal comb movement facilitates to achieve the above-mentioned “vertical format”. During the loading of the system with semi-finished textiles it is necessary to lower the guide combs with an additional vertical drive unit. The horizontal

164

M. Krahl et al.

Fig. 2 Comb systems for the guidance of kinematic components during loading and demoulding

relative movement of the pin bars also facilitates the entry and exit between the wedge angles.

3 Comb Guidance System for Single-Curved Spacer Fabrics The manufacturing of single-curved spacer fabrics is similar to the manufacturing of flat spacer fabrics. This necessitates significant modifications of the handling system. The following section focuses on the comb system. As in the scenario already described, combs are used to ensure adherence to the tool separation distance. Comb teeth always point in the precise direction of the respective bar (s. Fig. 3). This is a particular problem at the bend in the panel, however this issue can be overcome if each comb tooth has a separate drive unit. With the aid of a transmission-based solution it is possible to reduce the number of drive units required to one. The basic idea for the comb system for curved sandwich panels is based on the nuremberg scissors (s. Fig. 4a). The scissor members ASC and BSD are jointed in S. In case of AS = BS = CS = DS, the figure formed by the joining together of points ABCD is a rectangle (regardless of angle ϕ). By rotating the sections CS and DS are with the angle α in references to S (s. Fig. 4b), the figure formed by

Fig. 3 Sketch illustrating the principle behind the comb tooth guidance system


165

Fig. 4 Scissor kinematics. a Nuremberg scissors, b angulated scissors, c comb system

the joining together of points ABCD is a symmetrical trapezium with an angle of α [6, 7]. The angle ratios within the trapezium are independent of angle ϕ. The application of this effect to the comb tooth guidance problem delivers the comb tooth guidance solution presented in Fig. 4c. Each comb tooth is guided by a radial shaft, determined by the composite webs. The comb teeth used to control the orientation of the five trapezoidal chamber tools are therefore guided by a set of five scissor elements, connected in series. These elements form a complete scissor structure driven by pivot joints A, B (s. Fig. 5a). The problem of this solution is the high number of links. In order to minimize the accompanying tolerance problems the comb system is driven by two drive units (s. Fig. 5). In this system, the tool orientation is controlled by membrane bellows (s. Fig. 6a). Pressurization of the membranes leads to a change in cross-section (s. Fig. 6b), which is in turn used to alter tool orientation.

Fig. 5 Comb system for single curved spacer fabrics. a Position of the comb system prior to loading and unloading of the composite, b Consolidation tools in extended position

166

M. Krahl et al.

Fig. 6 Comb tooth design featuring bellows. a Unpressurized, b Pressurized

4 Conclusion The manufacturing of high-performance fibre composite components on the basis of hybrid yarns places considerable requirements on consolidation tool design. In the case of the sandwich components manufactured within the framework of SFB 639 in particular, it is wedge kinematics that facilitates the synchronized compression of all component surfaces in one process phase. This enables the realization of shorter cycle times and the development of an automated process. The orientation of consolidation tools is of particular importance to automation, and in the case of flat sandwich components it is performed by comb systems. The manufacturing of single-bend sandwich panels requires the development of new solutions, which facilitate the orientation of consolidation tools with a minimum of drive effort. The use of component symmetry enabled the development of a scissor mechanism as basis for a novel comb system. Acknowledgement The authors would like to express their gratitude towards the Deutsche Forschungsgemeinschaft (DFG), which supports this research within the scope of the subproject D4 of the Collaborative Research Centre SFB 639 “Textile-Reinforced Composite Components in Function-Integrating Multi-Material Design for Complex Lightweight Applications”.

References [1] Wakeman, M.; Zingraff, L.; Bourban, P.; Manson, J.; Blanchard, P.: Stamp forming of carbon fibre/PA12 composites - A comparison of a reactive impregnation process and a commingled yarn system. Composites Science and Technology. 2006, 66(1) S.19-35 [2] Golzar, M.; Brünig, H.; Mäder, E.: Commingled Hybrid Yarn Diameter Ratio in Continuous Fiber-reinforced Thermoplastic Composites. Journal of Thermoplastic Composite Materials. Composites Science and Technology. 2006, 66(1) S.19-35 [3] Abounaim, M.; Diestel, O.; Hoffmann, G.; Cherif, C.: Thermoplastic composites from curvilinear 3D multi-layer spacer fabrics. Journal of Reinforced Plastics and Composites. 2010, 29 (24) S.3554-3565 [4] Herzberg, C.; Zhao, N.; Rödel, H.: Beanspruchungsgerechte Abstandstrukturen für komplexe Leichtbauanwendungen durch funktionsgerechte Nähtechnik. Verbundwerkstoffe. Deutsche Gesellschaft fÃ1/4r Materialkunde e. V. Wiley-VCh GmbH Co. KG aA Verlag, Weinheim 2009, S. 440-446


167

[5] Hufenbach, W.; Adam, F.; Beyer, J.; Zichner, M.; Lin, S.; Hanke, U.: Development of an adapted process technology for complex lightweight structures based on hybrid yarns. 17th International Conference on Composite Materials (ICCM-17), Edinburgh, 27th-31th July 2009 [6] Langbecker, T.: Kinematic Analysis of Deployable Scissor Structures, International Journal of Space Structures, Vol. 14, No. 1, 1999 [7] Hanke, U.; Lin, S.; Modler, K.-H.: Synthesis of folding mechanisms for the consolidation of thermoplastic composites in spacer form; EuCoMeS-1st European Conference on Mechanism Science; Obergurgl- Austria; February 21-26, 2006


xii

RBS Simulation of Integrated Piezo-Ceramic Actuation for Textile Reinforced Composite Compliant Mechanisms N. Modler, K.-H. Modler, W. Hufenbach, D. Margineanu, E.-C. Lovasz, D. Perju, and Ev. Ianosi Abstract Structure-integrated drives use compliant elements to transmit motion just by structural deformations, and also, integrate piezo-ceramic actuators into the composite’s laminate lay-up to command and control the motion. The transmission of motion sequences can be obtained with reduced number of parts and low structural weight. However, the actual motion of the system can be obtained with the desired accuracy only after tests and re-design. Thus, reliable procedures able to predict the actual motion of compliant mechanisms with integrated actuators are necessary. This paper makes a contribution to the compliant structure simulation by developing a RBS method to obtain the piezo-ceramic actuation simulation for bending actuators using kinematical simulation models with single layer and double layer actuating cells. Keywords Compliant mechanisms · Piezo-ceramic actuator · Textile reinforced composite materials · Motion simulation

1 Introduction Compliant mechanisms CM, consisting of elements with higher rigidity and compliant elements (with higher elasticity) [1] can fulfill desired functions with minimum number of elements and joints. By integrating sensors and actuators in the composite structure, active compliant mechanisms A-CM are obtained, further reducing the product complexity and costs. The required motion of the A-CM can be obtained with sufficient accuracy only after tests, re-design and adjustments, implying high costs and consuming precious time, [2; 15], because it’s transmission function and the trajectories depend both on the rigid and elastic elements geometry and on the loads’ types, placements and values, and are influenced heavily by the elastic and inertial properties of the materials. The increasing number of compliant mechanisms industrial applications led to research on design approaches for developing compliant mechanisms for required N. Modler (B) Technische Universität Dresden, Dresden, Germany e-mail: [email protected]


169

170

N. Modler et al.

motion and force-deflection characteristics. Using the Pseudo-Rigid-Body Model (PRBM), two parallel mechanisms: a bicycle derailleur and parallel-motion bicycle brakes [9] were developed. Compliance comparison between multi-spring model and finite-element analysis is conducted for multidimensional acceleration sensors based on fully decoupled compliant parallel mechanism [16]. Analysis and design of an under-actuated compliant variable stroke mechanism by employing PRBM is studied [17] for prescribed output loading and constant input torque. The dynamics of a four-bar crank-rocker mechanism, suitable as a flapping mechanism and consisting of two flexural hinges is analyzed in [18] and the peak driving torque is minimized by modifying the stiffness and the unstrained positions of the flexural hinges. A novel finger mechanism with 1 active DOF able to obtain a humanlike grasping operation and which can be embedded within the finger body with human-size is proposed [19] for an under-actuated operation. Active or sensing CMs with various functioning principles are also extending their application range. On-chip sensing of bi-stable mechanism state using the piezo-resistive properties of poly-silicon [10], is made by detection of changes in position by changes in resistance across the mechanism. A compliant miniature parallel manipulator made of super-elastic nitinol pipe and actuated by shape memory alloy (SMA) wires is developed [12]. Micro electro mechanical systems (cantilevers, bistable mechanisms, and thermo-mechanical actuators) from a variety of materials with desired properties such as high-temperature stability or robustness can be developed using this technique. The development of a piezoelectric driven compliant-based micro-gripper mechanism capable of delivering high precision and fidelity manipulation of micro objects [14] adopts a flexure based concept on its joints to address the inherent nonlinearities associated with the application of conventional rigid hinges. A combination PRBM and FEA was implemented to expedite the prototyping procedure which leads to the establishment of high performance mechanism. A sufficiently precise approach is developing FEM models. However, the great numbers of design variables, including, besides all of the geometrical dimensions, the physical properties of the materials used, must be somehow known before the FEM modeling step in the A-CM developing process. Thus some dimensioning steps are previously required. In the case of A-CM, analytical calculations can only give rough approaches because the calculus models must include partial differential equations or equation systems, the initial and limit condition depending on the parts geometry (yet unknown). Or, the calculus model is simplified thoroughly, thus the results are not accurate enough. A simulation procedure should be carried out between the analytic dimensioning and the FEM modeling step. This simulation should observe some basic requirements: – all the initial simulation parameters should be defined according to the system’s dimensions, materials and loads; – the simulation results should estimate with an accepted limit error the behavior of the real system;

RBS Simulation of Integrated Piezo-Ceramic Actuation

171

– the simulation parameters should be easily changed, also according to optimization procedures; – all the simulation procedure should be fast, reliable and repeatable. The method to simulate the integrated piezo-ceramic actuation for some kinematical simulation models with active joints with concentrated rigidity, able to predict the actual motion of compliant mechanisms is studied in this paper. An existing A-CM (Fig. 1) is used as reference to evaluate the simulation accuracy. A flaps mechanism as a transmission A-TCM was chosen [2, 6], as these linkages can be found in various domains of equipment, machine and automotive applications. The A-TCM building method with its high elastic potential based on the material offers, next to other advantages, a reduced part number and thus an assembly cost reduction, in particular the possibility to adjust on purpose the flaps system’s opening behavior and its energy consumption [7, 8]. Therefore new innovative available solutions were studied on the basis of the flaps mechanism demonstrator (Fig. 2). The demonstrator consists of a planar coupler active textile reinforced compliant mechanisms A-TCM that, by an piezoelectric actuator integrated in the textile reinforced material with work ability adjusted to the flaps’ weight, allows an opening angle up to 90◦ . The structural pre-choice is done on the analytic equivalent model basis, developed by N. Modler [3], [4, 5]. Considering the complex requirement profile [2, 6] the drive and motion modes of operation were realized using compliant four bar linkages (Fig. 2). The flap’s

a

z

z

Aopen Bopen

A0

Bclose

B0 x

B0 x

A0

Aclose close (drive zone undeformed)

b

open (drive zone deformed)

z

z

Aopen

Bopen

B0 x

A0 open (drive zone undeformed)

A0

Bclose

B0 x

Aclose close (drive zone deformed)

Fig. 1 Operating modes of flaps A-TCM with 90◦ opening angle. a Hybrid flaps A-TCM in differential design: compliant drive link and rigid coupler – flexible connected, b Monolithic flaps A-TCM in integrally design: compliant drive link and compliant coupler – rigid connected

172

N. Modler et al.

Fig. 2 The actuator element dimensions

mechanism is based on a driving link A0 A, implemented by a bending actuator and connected to a rigid (differential design, Fig. 2a) or by a compliant (integral design, Fig. 2b) coupler AB with the output link. The pure rotation without guiding function requires the flap tightly connected to the output link B0 B. The A-TCM contains a resilient double-layer drive element. The base layer enables a large non-linear elastic deformation of the active element without damage, due to the good compliance of textile-reinforced thermoplastics [39]. The active layer consists of piezo-actuators MFC-P1: M – 8557 –P1 [1, 41] offered by Smart Materials. Applying voltage to the active zones produces a bending moment to open or close the flap.


173

Table 1 Mechanical characteristics of materials and composite Characteristic Young’s modulus

Poisson’s ratio

Shear modulus

Density

Material/Layer

E1 [GPa]

υ12 [-]

υ21 [-]

G12 [GPa]

ρ [kg/m3 ]

CF/HT-EP-UD – Base layer MFC active layer

111 30.336

0.29 0.3

0.022 0.16

4.5 5.515

1750 4750

Actuator layer Passive base layer

Fig. 3 The active element layers

The simulated active element has the dimensions given in Fig. 2. The first Zone 1 is composed of a polyamide PI layer reinforced with a carbon fiber composite passive layer CFK. In the second actuator Zone 2, the carbon fiber composite layer is glued on a macro fiber composite actuator layer. The third reinforced Zone 3 has a thicker CFK layer. The elasticity modules for the actuator element materials are given in the Table 1. The challenge is to develop a simulation model for actuation by a drive element consisting of a passive base layer and a piezoelectric layer (Fig. 3).

2 Piezo-Electric Actuation Simulation Basics Piezoelectric energy converters are characterized by great forces, response times of a few milliseconds and a positioning accuracy of a few nanometers, but only at more than 1-2 thousand Volt. The linear equation (1) in vector form is, in Voigt’s notation describes the inverse piezoelectric effect used for actuation, as the mechanical strain S depends on the mechanical load T and electric field strength E. {S} = sE · {T} + dt · {E} where: [dt ] is the matrix for the reverse piezoelectric effect, εT - permitivity Matrix for T = 0, sE - compliance matrix for E = 0.

(1)

174

N. Modler et al. U(t)

Fig. 4 The 2 phase system model

F(t) S1

x(t) S2

So, an accurate simulation model for an A-TCM should ensure both adequate compliance and actuation depending on the input signal and on the actuator effort. The piezoelectric actuator model can be mathematically represented using the input (voltage U(t)) and output (displacement x(t)) - Fig. 4 -, because the piezoelectric effect can be practically divided in 2 phases. The mathematic model of the system S, composed of the subsystem S1 (with voltage input and force output) and the subsystem S2 (with force input and displacement output), serial connected is given by the composition of both. For the first system S1: F(t) = Z · U(t)

(2)

F(t) - actuation force (output), Z – force- voltage constant

z=− tIDE ·

d33 · U h21 2 · E1 · I

+

.

(3)

1 E2 · l2 · b2

The mathematic model of the second systems S2 can be written as: m

d2 x dt2

= F,

(4)

with: m - mass [kg]; F – force [N]; and x = displacement The equation (2.4) can be also written in the form: #

where: x1 = x; x2 =

dx1 dt dx2 dt

dx1 dt

=

$

&

' & ' & ' 0 01 x = · 1 + 1 · [F] , 00 x2 m

dx dt ;

and dxdt2 = U(t) =

(5)

d2 x ; dt2

m d2 x . · Z dt2

(6)

The whole system can be written with the help of the transfer functions: G(s) =

x(s) U(s) 1 x(s) = · = G1 (s) · G2 (s) = Z · , U(s) F(s) F(s) ms2

with: G(s) = System’s transfer function; x(s) = output; U(s) = input Thus:

(7)


x(s) = G1 (s) · G2 (s) =

175

Z · U(s) . ms2

(8)

With the help of the equations (6) and (8) the system state model can be built in a system simulation software (Matlab, etc).

2.1 Developing a Model for Actuating Current Calculus In the case of piezo-electric actuators the resistance and the capacity are connected in parallel. The corresponding equations are: IR =

Vin , R

IC = C ·

dVin , dT

(9) (10)

with: Vin – input voltage, R - resistance, C – capacity, IR,C – resistive and capactive current. With the help of equations (9) and (10) a calculation model for the actuating current is developed (Fig. 5). The resulting current is represented as the sum of the resistive and capactive currents. The model will be also integrated in the simulation model [9]. As input data, a capacity of 14x10-9 F [60] and a resistance of 4.048x109 , on the base of the geometric parameters and the resistivity of PZT-A5 at 100 ◦ C and 1 kHz [62], and a voltage of 942.6 V are used, and a current of 3.196 μA is obtained. For more accurate results, it is necessary to run tests for the resistance calculation procedure before the use of the model. The resistance is dependent on the temperature, humidity but also on the frequency.

Fig. 5 Model for current calculus

176

N. Modler et al.

The active compliant element can be considered as a mechanism composed of rigid elements, passive and active rotational and/or translational joints, and torsion and translational springs. The simulated active element is composed of some simulation cells, serial connected. In the following, single and double layer cells/models will be studied.

3 Simulation Models for A-TCM 3.1 Single Layer Models Single layer simulation models approximate the mechanical behavior of flexible links by model cells j (see Fig. 6) composed of a rigid bar Aj-1 Aj , with the length Lj : Lj =

l2 , n

(11)

where l2 is the length of the active element and n is the number of cells in which the active element is composed. The cells are joined by the rotational joints Aj , and the equivalent torsion spring with the rigidity: kθa =

Mj Iz = Eeq · , θj Lj

(12)

kθa

Aj

Aj –1

Ma Lj

a) 11 13

A0 kθa Ma

A1

kθa

A2

kθa

Ma

A3 Ma

kθa

kθa

A4 Ma

Ma

A5 = A B k l kθp kθp kθp l4

B0 12

b) Fig. 6 The kinematic schema of an active element cell with single layer (a) and the A-TCM with single layer active element (b)


177

where: Mj - resulting moment, θj - the bending angle in the joint Aj , Eeq – equivalent Young’s (elasticity modulus) for the two layers composite; and Iz – the area moment of inertia of the two layer composite cross-section; for the rectangular cross-section:

Iz =

B · H3tot , 12

(13)

with B, the width and Htot , the total thickness of the active element. The actuation is obtained by introducing in the active joint Aj the actuating moment Ma , which can be calculated knowing the actuating force Fa and the actuating layer thickness ha : Ma = −Fa · ha .

(14)

The actuating force can be calculated as follows: Fa = −

h21 2 · E 1 · Iz

+

,

(15)

1 E2 · Lj · b2

with: - Actuating deformation [mm]; E1,2 - base and active layers Young’s modulus (Table 1); h1,2 , l1,2 , b1,2 – thickness, length, width of the two layers, respectively. The actuating deformation is calculated with respect to the input voltage U: = d33 · E = d33 ·

U , tIDE

(16)

with the piezoelectric constant d33 - 4.6 102 pC/N, and distance between electrodes tIDE - 0,5 mm. Using the active elements in the Fig. 4, a model for the whole A-TCM can be built, based on the kinematic schema (Fig. 6), simulation models in ADAMS and Matlab Software were constructed. The active element is built of more single layer elements, linked together by rotational joints, as shown in the Figs. 6 and 7. In the joints (JOINT_2 (3, 4, etc.)) bending rigidities of 34 N m/◦ (calculated with Eq. 12), are concentrated as TORSION_SPRING_1 (2, 3, etc.). The model cell is actuated by the moment SFORCE_28 (29, 30, etc.). The first active element cell is locked on the frame element (JOINT_1), and the other active elements are joined in series, a cell base to the tip of the previous cell, in order to compose the active element, which is then connected to the rest of the A-TCM model (Fig. 6). The compliant active element is modeled as a subsystem with the Matlab software (Fig. 8), composed of 8 active cells, simulating the integrated MFC/CFK active zones 2 (Fig. 3) and 4 passive cells simulating the zones 1 and 3. At both ends, connections are prepared to link it to the rest of the mechanism. In Fig. 9, the structure of a simulation cell as element system composed of a rigid element, a custom joint and an actuator bloc is presented. The element system

Fig. 7 The A-TCM single layer active cell model in ADAMS

178 N. Modler et al.


179 Element system

1 Actuator bloc Base

B CS1

F

Body

R_ZDir Bending around Z

2

1

Typ

Base

CS2

Base

2

Tip

Tip

Flexible element

Custom Joint

Element

Joint

Fig. 8 Simulation cell as an Element system

actuator

Fig. 9 Actuator bloc of the passive element 1 R_ZDir

senzor

K–

ap av

+ +

K–

ZDir1

bending damping

Fig. 10 Actuator bloc of the active element

1 R_ZDir

K–

ap av

K–

+ ++

ZDir1 u^2.3*10e-5

input signal Step

Fcn

M(U)

is connected to two other elements or element systems, by the connection 1 (Base) and the connection 2 (Tip) to the previous and next element, respectively. The actuator bloc of the element system consists of a sensor measuring the joint’s rotation and an actuator imposing the rotation, under the effect of a bending moment resulting by adding the damping effect (Fig. 11). In the active elements, the input voltage imposes an additional rotation of the element (Fig. 10) under the effect of a calculated function M(U) (Eq. 8): M(U) = 2.3 · 10−5 · U.

(17)

180

N. Modler et al.

As,j As,j–1

kl

Ai,j Ai,j–1

Xa

Fig. 11 Active element model with kinematical cells with 3 DOF 2 prismatic joints simulating the piezo-actuating effect

3.2 Double Layer Simulation Models for the Active Link In order to describe better the piezoelectric actuation (Eq. 1), which supposes the addition of actuating and elastic deformations, the model presented in Fig. 11 is introduced. The simulation cell is a 6-bar mechanism with 4 rotational joints and 2 prismatic joints. The Ai,j Ai,j+1 active equivalent layer is composed of 3 elements, linked by 2 prismatic joints. The actuation obtained by applying the voltage U will be: xa = · Lj ,

(18)

with given in Eq. 7. The elasticity of the actuator piezo-ceramic layer is simulated by the traction spring with the rigidity kl : kl = E2 ·

b2 · ha . Lj

(19)

The active layer is joined to the interface element in the rotational joints Ai,j and Ai,j+1 . In order to simulate the bending compliance of the two layer actuator, torsion springs are inserted in the upper and lower As,i;j rotational joints. As two springs are connected on each side of the interface element to ensure equal angles with the current and the previous layer elements, the rigidity of these springs is the double of the equivalent bending rigidity of the CFK and MFC layers:


kθs,i =

Iz Mj = E1,2 · 1,2 , θs,i;j Lj

181

(20)

where Iz1,2 – the area moment of inertia of each of the two layers cross-section: Iz1,2 =

b1,2 · h1,2 3 , 12

(21)

with b1,2 , the width and h1,2 , the thickness of the active and passive layers. In comparison with the simpler single layer model, all the simulation parameters have direct physical meaning. It is not necessary to calculate equivalent Young’s module or area moments of inertia.

4 Examples of Flaps Mechanisms Simulation This active element was simulated and included in flaps mechanisms operating in the modes described in Fig. 1. The simulations were performed using the ADAMS 2007 r1 software. The model is constructed at 1:1 scale. The torsion springs simulating the bending rigidities of the two layers of 13,05 N mm/◦ and 6,25 N mm/◦ were calculated with Eq. 21, and the piezo-ceramic layer rigidity (Eq. 19) is set to 27 517 N/mm. The actuation simulated by an imposed motion of the prismatic active joints (Fig. 11) with the maximal amplitude of 0,054 mm resulting from Eq. 18 for a 1000 V input signal induces the deformation of the active element. Simulations performed using this model showed the important influence of all the physical and dimensional factors, including the damping in the equivalent joints. In Fig. 13, the simulation results for a motion analysis of the 0.4 g flaps A-TCM are presented as motion sequences, for a maximal 1000 V (100%) step signal. The Fig. 14 shows the results obtained for the same mechanism actuated by a sine wave signal with the amplitude of 1380 V.

5 Conclusions The developed modeling concepts allow to asses in advance the behavior of A-TCM by multi-body kinematical simulation. However, this step must be followed by FEM simulations before the construction of the prototype. Thus, only with this continuous, synchronized approach, the high lightweight potential of active composite compliant mechanisms with integrated actuators can be fully exploited. The simulation models take into account the piezoelectric effect for the compliant active elements with high bending moment variation on their length. The double layer simulation model with 3 DOF modeling cells with one active prismatic joint and torsion springs in the rotational joints gives the best results, as it has actuating and compliance properties and its parameters have physical meaning.

182

N. Modler et al.

Fig. 12 Monolithic A-TCM design simulation model, with 3 DOF cells, 1 active and one compliant prismatic joint and 4 concentrated rigidities

Opening angle 0° at 0 V

Opening angle 6.36° at 500 V (30%) under own weight

Opening angle 10.93° at 700 V (70%)


Fig. 13 Moving sequences of the Flaps A-TCM with the weight of 0.4 g, actuated with step signal


Opening angle 0° at 0 V

Opening angle 5.25° at 500 V (36.23%) under own weight

Opening angle = 8.51° at 690 V (50%)


183

Fig. 14 Moving sequences of the Flaps A-TCM with the weight of 0.4 g, actuated with sine signal

Acknowledgement The authors would like to express their gratitude towards the Deutsche Forschungsgemeinschaft (DFG), which supports this research within the scope of the subproject D2 of the Collaborative Research Centre SFB 639 “Textile-Reinforced Composite Components in Function-Integrating Multi-Material Design for Complex Lightweight Applications”.

References [1] Howell, L. L.: Compliant Mechanisms. New York : Wiley & Sons, 2001 [2] Hufenbach, W.; Modler, K.-H.; Täger, O.; Modler, N. ; Lovasz, E.-C.: Design and manufacturing of smart textile compliant hinges. Acta Technica Napocensis, 2 (2006), p. 831–836 [3] Modler, N. Nachgiebigkeitsmechanismen aus Textilverbunden mit integrierten aktorischen Elementen. Dissertation Technische Universität Dresden, 2008 [4] Elspass, W. J.; Flemming, M.: Aktive Funktionsbauweisen: Eine Einführung in die Struktronik. Springer-Verlag Berlin Heidelberg New York, 1998 [5] Hufenbach, W.; Modler, K.-H.; Täger, O.; Modler, N.; Renner, O.: Contribution to the development of active compliant lightweight mechanism structures. In: Proceedings of the 4th IFAC-Symposium on Mechatronic Systems, Heidelberg, 12.-14. September 2006. – Preprints on CD-ROM [6] Modler, N.; Hufenbach, W.; Modler, K.-H.; Lovasz, E-Ch.; Perju, D.; Margineanu, D. A Design of Compliant Mechanisms with Integrated Actuators. SYROM 2009 - Proceedings of the 10th IFToMM Interntional Symposium on Science of Mechanisms and Machines, Bra¸sov (Romania), 12.-15. October 2009, p. 655–664 [7] Hufenbach, W.; Gude, M. Analysis and optimisation of multi-stable composites under residual stresses. Composite Structures, 55 (2002), p. 319–327 [8] Hufenbach, W.; Gude, M.; Modler, N.; Kirvel, Ch.: Novel function-integrated lightweight solutions based on thermoplastic composites and material-adapted piezoceramic actor modules. Advanced Materials and Technologies (AMT), Nr. 3-4 (2007), p. 254–260

184

N. Modler et al.

[9] Mattson, C. A., Howell L. L., Magleby S. P.: Development of Commercially Viable Compliant Mechanisms Using the Pseudo-Rigid-Body Model: Case Studies of Parallel Mechanisms, Journal of Intelligent Material Systems and Structures 2004: 195; DOI: 10.1177/1045389X04033256 [10] Anderson, J.K., Howell, L.L., Wittwer, J.W., McLain. T.W.: Piezoresistive sensing of bistable micro mechanism state, INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING J. Micromech. Microeng. 16 (2006) 943–950 doi:10.1088/0960-1317/16/5/010 30 March 2006 [12] Sreekumar, M., Singaperumal, M.: A generalized analytical approach to the coupled effect of SMA actuation and elastica deflection, IOP PUBLISHING SMART MATERIALS AND STRUCTURES Smart Mater. Struct. 18 (2009) 115026 (15 pp) doi:10.1088/09641726/18/11/115026, 16 October 2009 [14] Nashrul M., Zubir, M., Shirinzadeh B., Tian, Y: A new design of piezoelectric driven compliant-based microgripper for micromanipulation, Mechanism and Machine Theory 44 (2009) 2248–2264, Accepted 8 July 2009, Available online 18 August 2009 [15] Midha, A., Larry L. Howell, Tony W. Norton: Limit positions of compliant mechanisms using the pseudo-rigid-body model concept, Mechanism and Machine Theory 35 (2000) 99±115 [16] Gao, Zh., Zhang, D: Design, analysis and fabrication of a multidimensional acceleration sensor based on fully decoupled compliant parallel mechanism, Sensors and Actuators A 163 (2010) 418–427 [17] Tanık, E., Söylemez, E.: Analysis and design of a compliant variable stroke mechanism, Mechanism and Machine Theory 45 (2010) 1385–1394 [18] Khatait, J.P., Mukherjee, S., Seth, B.: Compliant design for flapping mechanism: A minimum torque approach, Mechanism and Machine Theory 41 (2006) 3–16

Part IV

Computational and Experimental Methods and Dynamics of Mechanisms and Machines


xii

Maximally Regular Planar Non Fully Parallel Manipulators G. Gogu

Abstract This paper presents a new family of maximally regular planar non fully parallel manipulators (PPMs). The moving platform has three planar degrees of freedom, which are two planar translations and one rotation around an axis perpendicular to the plane of translations. A one-to-one correspondence exists between the actuated joint velocity space and the external velocity space of the moving platform. The Jacobian matrix mapping the two vector spaces of the maximally regular PPMs is a 3×3 identity matrix throughout the entire workspace. The condition number and the determinant of the Jacobian matrix being equal to one, the maximally regular parallel robots perform very well with regard to force and motion transmission capabilities. The new kinematic criteria, used for structural synthesis, are based on the recent formulae proposed by the author for mobility, connectivity, redundancy and overconstraint of parallel robots. These kinematic criteria allow us to get new families of over constrained and non overconstrained maximally regular parallel robots with planar motion of the moving platform. The non fully-parallel solutions are presented for the first time in the literature. Keywords Planar parallel manipulators · Mobility · Redundancy · Overconstraint

1 Introduction Rigidity, accuracy, high speed, and high load-to-weight ratio are the main merits of parallel manipulators (PMs). With respect to serial manipulators, disadvantages include a smaller workspace, complex command and a lower dexterity due to a high motion coupling and multiple singularities inside their workspace. Maximally regular and fully-isotropic parallel manipulators can overcome these disadvantages. They have a very simple command and realize important energy-saving due to the fact that for a unidirectional motion only one motor works as in serial Cartesian manipulators. G. Gogu (B) LaMI, IFMA, Clermont University, Clermont, France e-mail: [email protected]


187

188

G. Gogu

It is known that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the actuated joint velocity space and the end-effector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robotic manipulator is fully-isotropic if its Jacobian matrix is isotropic throughout the entire workspace, i.e., the condition number of the Jacobian matrix is one. Thus, the condition number of the Jacobian matrix is an interesting performance index characterizing the distortion of a unit sphere under this linear mapping. The condition number of the Jacobian matrix was first used by Salisbury and Craig [23] to design mechanical fingers and developed by Angeles [1] as a kinetostatic performance index of the robotic mechanical systems. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator [5]. In an isotropic configuration, the sensitivity of a manipulator is minimal with regard to both velocity and force errors and the manipulator can be controlled equally well in all directions. The concept of kinematic isotropy has been used as a criterion in the design of various parallel manipulators [24], [26]. Five types of PMs are identified in [11]: (i) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fully-isotropic PMs, if J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission. The term maximally regular parallel robot was recently coined by Merlet [20] to define isotropic robots. In the previous systematization, this term is used to define just the particular case of fully-isotropic PMs, when the Jacobian matrix is an identity matrix throughout the entire workspace. Parallel manipulators with planar motion of the moving platform are used in applications that require positioning and orienting a body in a plane. They enable two independent translations in the plane and one rotation around an axis perpendicular to the plane of translations. Planar and/or spatial mechanisms can be used to obtain the planar motion of the moving platform. Various architectures are used in the literature to obtain three-legged PMs based on planar mechanisms [3], [19], [25]. The possible combinations of revolute, R, and prismatic, P, joints which connect the moving platform to the fixed base in a threelegged planar PM are as follows: RRR, PRR, RPR, RRP, PPR, RPP and PRP. The successions of the three joints start from the fixed base to the mobile platform. Since any one of the three joints in any of the seven kinematic chains may be actuated there are 21 possible limb architectures. The various combinations of these limb architectures give 1653 possible planar PMs with only lower pairs possessing three DoFs [14]. These solutions have coupled motions.

Maximally Regular Planar Non Fully Parallel Manipulators

189

The first solutions of fully isotropic and maximally regular PMs and unlimited rotation of the moving platform in planar motion have been presented in [7], [13]. Spatial mechanisms have been used in these non overconstrained solutions to obtain the planar motion of the moving platform. A systematic approach of structural synthesis of fully parallel planar and spatial mechanisms used in the parallel robots with planar motion of the moving platform and various associated solutions are extensively presented by the author in [12]. This paper represents a development of the approach presented in [12] by presenting new families of non fully parallel overconstrained and non-overconstrained maximally regular PPMs and unlimited rotational capabilities. These solutions are obtained from the fully parallel counterparts by superposing the last revolute joints of two limbs. In this way, the non fully parallel solutions presented in this paper combine less joints than the fully parallel counterparts. The various methods used for structural synthesis of PMs are based on: (i) screw algebra [6], [17], [19], (ii) displacement group theory [2], [15], [21], (iii) Lie algebra [16], [22], (iv) the method based on velocity loop equations [4] and (v) the method based on the theory of linear transformations [11]. The approach presented in this paper is founded on the theory of linear transformations and integrates the new formulae of mobility, connectivity, redundancy and overconstraint of parallel manipulators proposed in [10], [11]. The main aims of this paper are to present for the first time overconstrained and non overconstrained maximally regular non fully parallel planar manipulators, and to show the corresponding calculation of their structural parameters.

2 Kinematic Criteria for Structural Synthesis The main kinematic criteria used for structural synthesis are associated with mobility, connectivity, redundancy and overconstraint of parallel mechanisms. Mobility is the main structural parameter of a mechanism and also one of the most fundamental concepts in the kinematic and dynamic modelling of mechanisms. IFToMM terminology defines the mobility or the degree of freedom as the number of independent coordinates required to define the configuration of a kinematic chain or mechanism [18]. Mobility M is used to verify the existence of a mechanism (M>0), to indicate the number of independent parameters in robot modelling and to determine the number of inputs needed to drive the mechanism. The classical formulae for mobility calculation, known as Chebychev-GrüblerKutzbach formulae do not fit many classical mechanisms and recent parallel robots. These formulae have been reviewed in [8] and their limits have been set up in [9]. New formulae for quick calculation of the mobility have been proposed in [10] and demonstrated via the theory of linear transformations. A development of these contributions can be found in [11]. The connectivity between two links of a mechanism represents the number of independent finite and/or infinitesimal displacements allowed by the mechanism between the two links.

190

G. Gogu

The number of overconstraints of a mechanism is given by the difference between the maximum number of joint kinematic parameters that could lose their independence in the closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops. The case of a parallel mechanism F ← G1 -G2- . . . -Gk in which the end-effector n ≡ nGi is connected to the reference link 1 ≡ 1Gi by k simple or complex kinematic chains Gi (1Gi -2Gi - . . . -nGi ) , i = 1, 2, . . . , k is considered. In a simple kinematic chain, each link is connected by one or two joints. In a complex kinematic chain at least one link is connected by three or more joints. The parallel mechanism F ← G1 -G2 - . . . -Gk is characterized by: RGi - the vector space of relative velocities between the distal links nGi and 1Gi in the kinematic chain Gi disconnected from the parallel mechanism F, RF - the vector space of relative velocities between the distal links n ≡ nGi and 1 ≡ 1Gi in the parallel mechanism F ← G1 -G2 - . . . -Gk , SGi =dim(RGi ) –the connectivity between the distal links nGi and 1Gi in the kinematic chain Gi disconnected from the mechanism F, SF =dim(RF ) - the connectivity between the distal links n ≡ nGi and 1 ≡ 1Gi in the mechanism F ← G1 -G2 - . . . -Gk . The new formulae demonstrated in [11] for mobility MF , connectivity SF , overconstraint NF and redundancy TF of the parallel mechanism F ← G1 -G2 - . . . -Gk are used as kinematic criteria for structural synthesis of parallel robotic manipulators: MF =

p (

fi − rF

(1)

i=1

NF = 6q-rF

(2)

TF = MF -SF

(3)

SF = dim(RF ) = dim RG1 ∩ RG2 ∩ . . . ∩ RGk

(4)

where

rF =

k (

SGi − SF + rl

(5)

i=1

and rl =

k ( i=l

rlGi

(6)


p=

k (

pGi

191

(7)

i=l

In (1)-(7), p represents the total number of joints, and pGi the number of joints in Gi -limb, m is the total number of links including the fixed base, q = p-m + 1 is the total number of independent closed loops in the sense of graph theory, fi is the mobility of the ith joint, r - the number of joint parameters that lose their independence in mechanism F, rlGi - the number of joint parameters that lose their independence in the closed loops of limb Gi , rl - the total number of joint parameters that lose their independence in the closed loops that may exist in the k limbs of the mechanism F, k1 -the number of simple limbs, k2 -the number of complex limbs (k = k1 + k2 ). The number of limbs is determined by respecting (7) in such a way that each joint belongs to just one limb. In a simple limb no closed loops exist and rlGi = 0. More details about the application of these formulae can be found in [11]. The finite displacements and the velocities in the actuated joints are denoted by qi and q˙ i , the linear velocities of the characteristic point H situated on the moving platform by v1 = x˙ , v2 = y˙ and the angular velocity of the moving platform by ˙ ωδ = δ. With these notations, the linear mapping between the actuated joint velocity space and the end-effector velocity space for the PPMs is defined by: ⎡

⎤ ⎡ ⎤ v1 q˙ 1 ⎣ v2 ⎦ = [J] ⎣ q˙ 2 ⎦ ωδ q˙ 3

(8)

where J is the Jacobian matrix. The solutions of maximally regular PMs presented in this paper give two translational velocities v1 and v2 along with the rotational velocity ωδ in the basis of the operational velocity vector space (RF ) = (v1 , v2 , ωδ ). We consider xy-plane as the plane of motion of the moving platform. The PPMs presented in this paper have the connectivity between the moving and the fixed platforms SF =3. For the maximally regular solutions presented in this paper, the Jacobian matrix in (8) is 3×3 identity matrix throughout the entire workspace, and the following relations exist between the actuated joint velocities and the operational velocities v1 = q˙ 1 , v2 = q˙ 2 and ωδ = q˙ 3 . The actuators are mounted on the fixed base and so these solutions have the advantages of reducing the moving masses and large workspace and compactness. Basic and derived solutions can be generated. No idle mobilities exist in the basic solutions. The derived solutions are obtained from the basic solutions by combining various idle mobilities. They can be overconstrained or not overconstrained, fully parallel or not fully parallel solutions. We recall that an idle mobility is a joint mobility that does not affect a mechanism’s mobility.

192

G. Gogu

3 Fully and Non Fully Parallel Solutions of Maximally Regular PPMs Planar parallel robotic manipulators (PPMs) are closed-loop mechanisms composed of an end effector (moving platform) connected to the base (fixed platform) by at least two kinematic chains called limbs or legs. The basic kinematic structures of maximally regular fully parallel PPMs (Fig. 1) are obtained by concatenating ≡ 0- . . . three limbs G1 (1G1 -nG1 ≡ n) , G2 (1G2 ≡ 0- . . . -nG2 ≡ n), and G3 1G3 ≡ 0 - . . . -nG3 ≡ n . The first link 1Gi of each limb is the fixed base (platform) denoted by 0 and the final link is the moving platform denoted by n. To simplify the notations of links eGi of limbs Gi (i = 1, . . . , 3 and e = 1, . . . , n), by avoiding the double index in Figs. 3–10, the links belonging to limb G1 are denoted by eA (eA ≡ eG1 ) and the links of limbs G2 and G3 by eB eB ≡ eG2 and eC (eC ≡ eG3 ). The actuated joint of each leg is underlined. In a maximally regular fully parallel solution proposed in [12], G1 limb contributes to positioning the moving platform on x-axis, G2 on y-axis and G3 limb to orienting the moving platform in xy-plane. In general, in the fully parallel robot the number of limbs is equal to the robot mobility (k=MF ) and each limb integrate just one actuator [11]. Basic non fully parallel solutions with just two limbs G1=GA-B and G2=GC can be generated from the fully parallel solutions by superposing the last revolute joints of limbs G1=GA and G2=GB used for positioning the moving platform. In this way,

Fig. 1 Basic kinematic structure of fully parallel planar parallel manipulator of type 2PPR-RPP with three limbs, nine joints and six overconstraints (NF =6)


193

Fig. 2 Basic kinematic structure of non fully parallel planar parallel manipulator of type (PPPP)RRPP with two limbs, eight joints and seven overconstraints (NF =7)

the two simple limbs G1 =GA and G2 =GB in Fig. 1 are combined in a single complex limb G1=GA-B (Fig. 2) integrating one closed loop and two actuators. In Fig. 2, the links of G1 -limb are denoted with indices A and B and the links of G2 -limb with index C by conserving the notations used in Fig. 1. Limb G2=GC contributes to orienting the moving platform and is actuated by just one motor. It has the same structure as in the fully parallel solution. The three motors are mounted on the fixed base as in the fully parallel counterparts (Fig. 1). With respect to the fully parallel solution (Fig. 1), the non fully parallel solution (Fig. 2) is more compact and combines less joints. We recall that in a planar mechanism all the links have a motion parallel to the same plane (planar motion), the rotation axes are parallel and the directions of the translations are parallel to a plane perpendicular to the rotation axes. Idle mobilities with other directions of the axes can also exist in these planar mechanisms. Limbs G1 =GA-B and G2 =GC used in the maximally regular non fully parallel basic solutions presented in this paper are systematized in Figs. 3 and 4. One closed loop is combined in the complex planar limb G1 =GA-B . Simple or complex kinematic chains can be used as G2 =GC -limb. The complex planar kinematic chains associated with G2=GC -limb combine one (Fig. 4b) or two parallelogram loops Pa (Fig. 4c, d). Just revolute R and prismatic P joints are used in these kinematic chains. The notations ⊥ or between two joints indicates that the joints have orthogonal or parallel axes/directions. The notation ⊥⊥ in P⊥P⊥⊥ R indicates that the axis of the revolute joint is perpendicular on the directions of both prismatic joints.

194

G. Gogu

Fig. 3 Basic kinematic structure for G1 -limb of maximally regular non fully parallel solutions

Fig. 4 Basic kinematic structures for G2 -limb


195

Formulae (1-7) are applicable to both fully and non fully parallel solutions by taking into account that in fully parallel solutions of PPMs three legs exists (k=3) and just two legs in the non fully parallel counterparts (k=2).

4 Overconstrained Solutions Equations (1) and (2) indicate that the overconstrained solutions of PMs with mobilp ity M=3 and N overconstraints meet the condition 1 fi < 3 + 6q. The basic kinematic structures of overconstrained fully parallel PMs presented in Fig. 2 and Figs. 5–7 are obtained by concatenating in a planar mechanism a G1 (Fig. 3) and a G2 limb (Fig. 4). The structural parameters of the solutions presented in Fig. 2 and Figs. 5–7 are systematised in Table 1. The independent velocities v1 , v2 , v3 , ωα , ωβ and ωδ in the bases (RGi ) of the operational velocity space in Table 1 represent the linear and angular velocities of the moving platform in a limb isolated from the parallel mechanism. The joints combined in the closed loop of complex limb G1 are denoted in a bracket. Two joint parameters lose their independence in a closed loop of type PPPP

Fig. 5 Basic kinematic structure of non fully parallel planar parallel manipulator of type (PPPP)RRPPa with ten overconstraints (NF =10)

196

G. Gogu

Fig. 6 Basic kinematic structure of non fully parallel planar parallel manipulator of type (PPPP)RRPaPa with thirteen overconstraints (NF =13)

Fig. 7 Basic kinematic structure of non fully parallel planar parallel manipulator of type (PPPP)RRPaPa with thirteen overconstraints (NF =13) and unlimited rotation

combined in the complex limb G1 (Fig. 2 and Figs. 5–7) and three in each parallelogram loop Pa combined in the complex limb G2 (Figs. 5–7). The number of joint parameters that lost their independence in a closed loop integrated in a limb can be determined by Eq. (5). In this case, a parallel kinematic chain with just two legs is associated to each loop. For example, in the closed loop PPPP in Fig. 11, link 3A is


197

Table 1 Structural parameters of non fully-parallel solutions in Fig 2 and Figs. 5–7 No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Structural parameter

Solutions in Fig. 2

Fig. 5

Figs. 6 and 7

m p1 p2 p q k1 k2 k (RG1 ) (RG2 ) SG1 SG2 rG1 rG2 MG1 MG2 (RF ) SF rl rF MF NF T F

7 5 3 8 2 1 1 2 (v1 , v2 , ωδ ) (v1 , v2 , ωδ ) 3 3 2 0 3 3 (v1 , v2 , ωδ ) 3 2 5 3 7 0 5 3 8

9 5 6 11 3 0 2 2 (v1 , v2 , ωδ ) (v1 , v2 , ωδ ) 3 3 2 3 3 3 (v1 , v2 , ωδ ) 3 5 8 3 10 0 5 6 11

11 5 9 14 4 0 2 2 (v1 , v2 , ωδ ) (v1 , v2 , ωδ ) 3 3 2 6 3 3 (v1 , v2 , ωδ ) 3 8 11 3 13 0 5 9 14

p

1 fj j=1 p2 fj j=1 p j=1 fj

connected in parallel to the fixed base by two kinematic chains of type PP and Eq. (5) indicates that two joint parameters lost their independence in this closed loop.

5 Non Overconstrained Solutions The derived kinematic structures of non overconstrained PPMs in Figs. 8–11 are obtained by combining the required idle mobilities in the basic overconstrained solutions presented in Fig. 2 and Figs. 5–7. The structural parameters of the solutions presented in Figs. 8–11 are systematised in Table 2. The joints in which the idle mobilities are introduced are denoted by ∗ . One idle mobility is introduced in the cylindrical joint denoted by C∗ and two idle mobilities in the spherical joint denoted by S∗ . Three idle mobilities are introduced in each parallelogram loop denoted by Pass in which two spherical joint are combined. There is an internal mobility in this closed loop. This is a rotational motion of the link adjacent to the two spherical joints around an axis passing by the centers of these two

198

G. Gogu

Fig. 8 Kinematic structure of non fully parallel and non overconstrained planar parallel manipulator of type (PC∗ C∗ C∗ P)R∗ R∗ C∗ -RPP

Fig. 9 Kinematic structure of non fully parallel and non overconstrained planar parallel manipulator of type (PC∗ C∗ C∗ P)R∗ R∗ C∗ -RPPass

spherical joints. Six joint parameters lose their independence in each closed loop combined in the complex limbs G1 and G2 in Figs. 8–11. Note: Special attention must be paid when introducing idle mobilities so as not to modify the connectivity of the moving platform. After introducing idle mobilities, equations (1)-(5) must always give SF =3 and (RF ) = (v1 , v2 , ωδ ). Equations


199

Fig. 10 Kinematic structure of non fully parallel and non overconstrained planar parallel manipulator of type (PC∗ C∗ C∗ P)R∗ R∗ C∗ -RPass Pass

Fig. 11 Kinematic structure of non fully parallel and non overconstrained planar parallel manipulator of type (PC∗ C∗ C∗ P)R∗ R∗ C∗ -RPass Pass with unlimited rotation

(1)-(5) represent a valid guide to avoid hazy use of idle mobilities in parallel mechanisms. To the best of our knowledge, no previous formulae are known to solve this problem for parallel manipulators. Other derived overconstrained solutions with 0 < NF < 13 can be obtained from the basic solutions in Fig. 2 and Figs. 5–7 by using partially the idle mobilities introduced in the non overconstrained solutions in Figs. 8–11.

200

G. Gogu Table 2 Structural parameters of non fully-parallel Solutions in Figs. 8–11

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Structural parameter m p1 p2 p q k1 k2 k (RG1 ) (RG2 ) SG1 SG2 rG1 rG2 MG1 MG2 (RF ) SF rl rF MF NF T F p

1 fj j=1 p2 fj j=1 p j=1 fj

Solutions in Fig. 8

Fig. 9

Figs. 10 and 11

10 8 3 11 2 1 1 2 v1 , v2 , v3 , ωα , ωβ , ωδ (v1 , v2 , ωδ ) 6 3 6 0 6 3 (v1 , v2 , ωδ ) 3 6 12 3 0 0 12 3 15

12 8 6 14 3 0 2 2 v 1 , v2 , v3 , ω α , ω β , ω δ (v1 , v2 , ωδ ) 6 3 6 6 6 4 (v1 , v2 , ωδ ) 3 12 18 4 0 1 12 10 22

14 8 9 17 4 0 2 2 v1 , v2 , v3 , ωα , ωβ , ωδ (v1 , v2 , ωδ ) 6 3 6 12 6 5 (v1 , v2 , ωδ ) 3 18 24 5 0 2 12 17 29

6 Conclusions Mobility, connectivity, redundancy and overconstraint represent the main kinematic criteria for structural synthesis of parallel mechanisms. An approach has been proposed for structural synthesis of overconstrained and non overconstrained non fully parallel solutions of maximally regular PPMs. These solutions are obtained by an original approach of structural synthesis founded on the theory of linear transformations. The approach integrates the new formulae for mobility, connectivity, redundancy and overconstraint of parallel mechanisms recently proposed by the author. These formulae are applicable to fully and not fully parallel solutions. The non fully parallel solutions presented in this paper are derived from the fully parallel counterparts by superposing the last revolute joints of two limbs. In this way, the non fully parallel solutions presented in this paper combine less joints than the fully parallel counterparts. These solutions are presented for the first time in the literature. The method proposed in this paper can be easily extended to structural


201

synthesis of other types of non fully parallel mechanisms with various combinations of translational and rotational motions of the moving platform.

References 1. Angeles, J: Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 1st ed. Springer, 1997. 2. Angeles, J: The qualitative synthesis of parallel manipulators. ASME Journal of Mechanical Design 126 (2004) 617-624. 3. Ceccarelli, M: Fundamentals of Mechanics of Robotic Manipulation. Springer, 2004. 4. Di Gregorio, R. and Parenti-Castelli, V: A translational 3-dof parallel manipulator. In Advances in Robot Kinematics: analysis and control, J. Lenarˇciˇc, M. Husty (eds), Kluwer Academic Publishers (1998), 49-58. 5. Fatah, A., Hasan Ghasemi, A.M.: Isotropic design of spatial parallel manipulators. International Journal of Robotics Research 21, 9 (2002), 811-824. 6. Frisoli, A., Checcacci, D., Salsedo, F. and Bergamasco, M.: Synthesis by screw algebra of translating in-parallel actuated mechanisms. In: Lenarˇciˇc J, Stanišić MM (eds) Advances in robot kinematics, Kluwer Academic Publishers (2000) 433-440. 7. Gogu, G.: Fully-isotropic over-constrained planar parallel manipulators. In Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, Sendai (2004) 3519-3524. 8. Gogu, G.: Mobility of mechanisms: a critical review. Mechanism and Machine Theory 40 (2005) 1068-1097. 9. Gogu, G.: Chebychev-Grubler-Kutzbach’s criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations. European Journal of MechanicsA/Solids 24 (2005), 427-441. 10. Gogu, G.: Mobility and spatiality of parallel robots revisited via theory of linear transformations. European Journal of Mechanics-A/Solids 24 (2005), 690-711. 11. Gogu, G.: Structural synthesis of parallel robots. Part 1: Methodology. Springer, 2008. 12. Gogu, G.: Structural synthesis of parallel robots. Part 3: Topologies with planar motion of the moving platform. Springer, 2010. 13. Gogu, G.: Kinematic criteria for structural synthesis of maximally regular parallel robots with planar motion of the moving platform. In Interdisciplinary Applications of Kinematics, A. Kecskeméthy (Ed.), Lima (2008) 61-79. 14. Hayes, M.J.D., Zsombor-Murray, P.J. and Chen, C.: Unified kinematic analysis of general planar parallel manipulators. ASME Journal of Mechanical Design 126 (2004) 866-874. 15. Herve, J.M.: Design of parallel manipulators via the displacement group. In Proceedings of the 9th IFToMM World Congress, Milan (1995) 2079-2082. 16. Herve, J.M.: The Lie group of rigid body displacements, a fundamental tool for mechanism design. Mechanism and Machine Theory 34 (1999) 719-730. 17. Huang, Z., Li, Q.C.: Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method. International Journal of Robotics Research 22(1) (2003) 59-79. 18. Ionescu, T.G.: Terminology for mechanisms and machine science. Mechanism and Machine Theory 38 (2003) 597-901. 19. Kong, X., Gosselin, C.M.:Type Synthesis of Parallel Mechanisms, Springer, 2007. 20. Merlet, J.P.: Parallel robots, 2nd ed. Springer, 2006. 21. Rico Martinez, J.M. and Ravani, B.: On mobility analysis of linkages using group theory. ASME Journal of Mechanical Design 125 (2003) 70-80.

202

G. Gogu

22. Rico Martinez, J.M., Gallardo, J. and Ravani, B.: Lie algebra and the mobility of kinematic chains. Journal of Robotic Systems 20(8) (2003) 477-499. 23. Salsbury, J.K. and Craig, J.J.: Articulated hands: force and kinematic issues. International Journal of Robotics Research 1(1) (1982) 1-17. 24. Tsai, K.Y. and Huang, D.: The design of isotropic 6-DOF parallel manipulators using isotropy generators. Mechanism and Machine Theory 38 (2003) 1199-1214. 25. Tsai, L.-W. Robot analysis: the mechanics of serial and parallel manipulators. Willey, 1999. 26. Zanganeh, K.E. and Angeles, J.: Kinematic isotropy and the optimum design of parallel manipulators. International Journal of Robotics Research 16(2) (1997) 185-197.

An Experimental Characterization of a Rickshaw Prototype T. Li and M. Ceccarelli

Abstract In this paper, a rickshaw robot has been characterized from aspects of mechanical design and experimental tests. Experiments have been done to measure the contact forces between the feet and terrain, acceleration of the robot body, and torque of the motor during a walking process. Experimental results of walking operation of a built prototype have been reported to show characteristic features of a rickshaw efficient performance. Keywords Mobile robots · Legged robots · Experimental characterization · Robotics

1 Introduction Mobile robots can be classified by the device they use to move, mainly as legged robots, wheeled robots, and track robots. Normally, wheeled robots have better stability and higher efficiency than legged robots, but they are not suitable in moving in uneven environments. On the other hand, legged robots are more suitable in unstructured environments than wheeled robots. Hybrid systems have been developed as combination of wheeled and legged solutions with aim to exploit the advantages of both types of mobile systems [1-3]. At LARM: Laboratory of Robotics and Mechatronics in Cassino, design and research activities have been carried out to the design of legged walking machines [4, 5], and even specifically for rickshaw robots. A rickshaw walking robot has been presented by Ceccarelli et al in [6]. A hybrid walking robot has been proposed in [7]. Then a low-cost easy-operation biped mechanism has been proposed for a prototype of rickshaw robot [8]. A prototype has been built at LARM in Cassino (see LARM web page and Fig. 7). This hybrid system is designed as combination of wheeled and legged solutions with aim to exploit the advantages of both types of mobile systems. An application of this robot can be as transportation platform, not only for humanoid service robots, whose main design characteristics are low-cost and easy-operation features. T. Li (B) LARM: Laboratory of Robotics and Mechatronics, University of Cassino, Cassino, Italy e-mail: [email protected] E.-C. Lovasz, B. Corves (eds.), Mechanisms, Transmissions and Applications, Mechanisms and Machine Science 3, DOI 10.1007/978-94-007-2727-4_18, C Springer Science+Business Media B.V. 2012

203

204

T. Li and M. Ceccarelli

In this paper, the rickshaw solution has been analyzed in detail through the following aspects: mechanical design and kinematic modelling of the identical legs in section II; an experimental layout in section III; and in section IV experiments are reported to characterize the contact forces between the feet and terrain, the accelerations of the body, and also the torque of the motor during a walking process.

2 The Rickshaw Solution One basic consideration for a robotic leg design is that the leg should generate an approximately straight-line trajectory for the foot with respect to the body [9, 10]. A Chebyshev mechanism is used in LARM leg in the design of the biped mechanism with a pantograph mechanism. The pantograph is used to amplify the trajectory generated by the Chebyshev mechanism and to have a human-like operation. Besides the biped mechanism, this robot has also been provided with a passive wheeled system to increase its load capability and stability of walking on different types of terrains. Namely, the rickshaw robot consists of a biped robot with one DOF and a chariot [4, 8]. It gives high maneuverability, simple use, reduced size, and it does not require high power in operation. A mechanical design of the rickshaw robot is shown in Fig. 1 with its outside dimensions. The biped mechanism is actuated by a DC motor. A gear box is used to transmit the motion. The lengths of links have been determined by considering that the shape of the end leg point trajectory should be similar to the shape of human’s foot trajectory [4]. A sketch for the rickshaw leg design is shown in Fig. 2. The solution considers a fully rotative actuation at point L that gives a suitable trajectory of point B, which can be suitably modified by changing the design parameters shown in Fig. 2. In particular, it has been shown that good features for leg operation can be obtained if the transmission angles γ1 and γ2 have suitable values. A parametric study had been carried out to study the influence of design parameters on motion capabilities of the one-DOF leg mechanism in rickshaw solution [8].

Fig. 1 Design of LARM rickshaw robot: mechanical solution with its main dimensions (a-right leg; b-motor; c-T shape gear box; d-left leg; e-pantograph mechanism; f-Chebyshev mechanism; g-passive two-wheeled chariot)

An Experimental Characterization of a Rickshaw Prototype

205 y

Fig. 2 Kinematic scheme of LARM rickshaw robot for leg operation

α

a C

E

m

L

X

β

c

d D

θ f Z2

γ1

B

ϕ1

G

P

Z6

Z4

=Z 3

H

γ2

M =Z Z7

Z3

4

ϕ2

I

Z5

A

When a reference system Cxy is attached at point C as shown in Fig. 2, the position of point B can be evaluated as a function of the input crank angle α and kinematic parameters of the Chebyshev mechanism LEBDC in the form

xB = −a + m cos α + (c + f) cos θ yB = −m sin α − (c + f) sin θ

(1)

in which θ can be obtained by considering the closure equation of the four-bar linkage CLED. The position of point A with respect to the fixed frame can be obtained from the geometry in Fig. 2. The velocity of point A can be computed by differentiating the expressions of the coordinates of point A with respect to time. As well as the velocity of point B can be obtained by differentiating (1) with respect to time. While giving a clockwise rotation to the crank LE around point L, numerical simulations can be computed to check the curve generated by point A, and to check the motion properties.

206


3 An Experimental Layout In order to evaluate the motion performance of the LARM rickshaw robot, experiments have been designed and carried out to test several main performance characteristics. These characteristics are the contact forces between the feet and terrain, the acceleration of the moving body, and the torque of the motor during a walking process. Contact forces are measured by force sensors installed on the bottom side of the feet. Each foot is equipped with one Piezoresistive force sensor [11]. This type of sensor has been chosen because of its small size and low cost. A measured acceleration can be calculated by a=

a2x + a2y + a2z

(2)

in which ax , ay and az are the accelerations along x-axis, y-axis and z-axis, respectively. Components ax , ay and az can be measured by using a 3-axis accelerometer. Here a 3-axis low-g accelerometer MMA7260 [12] has been chosen and attached to the body of the biped mechanism as shown in Fig. 3b. As for measuring the torque of the motor, a resistance of 0.1 is used by measuring its input voltage. It forms a series circuit with the motor. The data acquisition and elaboration are executed by a NI USB 6009 data acquisition board and a virtual instrument program in DAQ LabVIEW SignalExpress environment [13]. A Scheme of the experimental system is shown in Fig. 3a; and its experimental layout is shown in Fig. 3b. Contact forces are measured to check if there is big impact during a walking. Accelerometers are measured to check if the rickshaw robot has a smooth walking performance. The torque is measured to check if this rickshaw robot really does not require high power in operation as proposed.

4 Experimental Results for Characterization Snapshots of an experiment are shown in Fig. 4. All experimental results are represented in forms of voltage output. They need to be converted into force, acceleration and torque by using corresponding converting operation respectively. Experimental results corresponding to forces on left and right foot are shown in Fig. 5a and Fig. 5b respectively. For a simple force-to-voltage conversion, the FSR (Force Sensitive Resistors) device is tied to a measuring resistor in a voltage divider configuration. The output is described by the equation VOUT = V/ [1 + RFSR/RM]

(3)


conditioning board

GND +5V

P0.0 GND P0.1 AI0 P0.2 AI4 P0.3 GND P0.4 AI1 P0.5 AI5 P0.6 GND NI P0.7 AI2 AI6 USB 6009 P1.0 P1.1 GND P1.2 AI3 P1.3 AI7 PF10 GND +2.5V AO0 +5V AO1 GND GND

Left foot

Right foot

Acceleration transducer +5V GND

207

Z Y X

USB

Resistance (0.1Ω)

– +

+ –

Motor

PC: LabVIEW SignalExpress

Power supply +15V

(a)

(b) Fig. 3 The experimental system: (a) A Scheme for the experimental system with LARM Rickshaw prototype; (b) The experimental layout with a LARM rickshaw prototype (a-motor; b-accelerometer; c-force sensors; d-conditioning board; e-NI USB 6009; f-resistance)

208


Fig. 4 Snapshots of an experimental walking process

In the configuration shown in Fig. 3b, the output voltage increases with increasing force. The measuring resistor, RM, is chosen to maximize the desired force sensitivity range and to limit current. The current through the FSR should be limited to less than 1 mA/cm2 of applied force. Suggested op-amps for single sided supply designs are LM358 and LM324. The low bias currents of these op-amps reduce the error due to the source impedance of the voltage divider. The conditioning board has been developed following this guidance. The RM resistor used in these experiments has the value 3 k. The corresponding curve used for converting voltage into contact force can be found in a family of


209

Fig. 5 Voltage output for the contact forces with respect to time during the test in Fig. 3: (a) between the left foot and terrain (F31Lv ); (b) between the right foot and terrain (F31Rv )

FORCE vs. VOUT curves for a standard FSR in a voltage divider configuration with various RM resistors in [14]. From the outputted Volt curves shown in Fig. 5a and Fig. 5b, both of the maximum voltages are less than 2 V. Thus the contact force between the feet and terrain are less than 10 N. This means there is no big impact existing during walking. Curves of left and right foot should have the same shape

210


Table 1 A calibration for accelerometer

Voltage

Gravity (-g)

Gravity (g)

Uax (V) Uay (V) Uaz (V)

2.44 2.41 2.43

0.88 0.82 0.84

theoretically. However, in the prototype, there are some factors which make them differ from each other, like backlash, unsymmetry and friction. As listed in Table 1, the used accelerometer is calibrated by logging its output voltages with positive and negative gravity. Therefore, coefficient Kai (i=x, y, z) can be determined as g −g Kai = 2g/ Uai − Uai , (i = x, y, z) (4) g

−g

where Uai and Uai (i = x, y, z) are the original voltages under positive and negative gravity, respectively. Measured acceleration ai (i = x, y, z) can be computed as g −g ai = Uai − Uai + Uai /2 Kai , (i = x,y,z)

(5)

where Uai are the outputted voltages corresponding to the measured components ax , ay and az when i equals to x, y, and z, respectively. If g is given the value 9.8m/s2 , from (4), (5), and Table 1, it can be computed approximately ai = −12.25Uai + 20.21, (i = x,y,z)

(6)

From (6), values of ax and az can be obtained as shown in Fig. 6a and Fig. 6b. From Fig. 6, it can be seen that the minimum and maximum value of ax are about 0.3m/s2 and 1.4m/s2 (the negative signs in Fig. 6 represent the directions) respectively. The minimum and maximum values of az are approximately to 0.05m/s2 and 0.4m/s2 . ay shown in Fig. 7a seems like having much bigger values than ax and az . In fact, it senses both the acceleration of gravity and the acceleration along y-axis of rickshaw. Thus the acceleration along y-axis of rickshaw should be obtained by subtracting the acceleration of gravity from the values shown in Fig. 7a. The actual result of the acceleration along y-axis of rickshaw is shown in Fig. 7b. The values are around 0.55m/s2 . Figures 6 and 7b show that the variances of ax , ay and az are small and indicate that the rickshaw robot has a smooth movement. The torque T of the DC motor can be computed by T = CIa

(13)

in which C is the torque constant of the motor. It equals to 0.1Nm/A. Ia is the current of the motor. It can be computed by


211

Fig. 6 The accelerations with respect to time during the test in Fig. 4: (a) along x-axis (ax ); (b) along z-axis (az )

Ia = Ut /R

(14)

in which Ut is the measured voltage of R. R is the resistance that has the value 0.1 and forms a series circuit with the motor as mentioned in section III. Finally, the computed T is shown in Fig. 8. The minimum and the maximum torque of the motor are approximately to 0.066Nm and 0.118Nm respectively. It verified that the torque needed for actuating the proposed rickshaw robot is very small, and the rickshaw robot really does not require high power in operation.

212


Fig. 7 The acceleration along y-axis of rickshaw with respect to time during the test in Fig. 4: (a) with acceleration of gravity; (b) without acceleration of gravity

5 Conclusions A rickshaw robot has been studied from aspects of mechanical design, kinematic modelling and experimental tests. Experiments have been carried out to measure the contact forces between the feet and terrain, the acceleration of the robot body and torque of the motor during a walking process. During the experiments, all


213

Fig. 8 Voltage output for the torque of the motor (T) with respect to time during the test in Fig. 3

experimental results are represented in forms of voltage. Relationships between voltages and force, acceleration, torque have been reported to obtain the real values of the forces, the accelerations and the torque. Experimental results showed that the LARM rickshaw robot has good performance, i.e. the outputted forces show that there is no big impact existing during a walking process; the measured accelerometers show that the robot has a smooth movement; and the measured torque verifies that the rickshaw robot really does not require high power in operation as proposed. Acknowledgments The first author likes to acknowledge Chinese Scholarship Council (CSC) for supporting his PhD study and research at LARM in the University of Cassino in Italy for the years 2010-2012.

References 1. Yuan, J., Hirose, S., Zero carrier: a novel eight leg wheels hybrid stair climbing mobile vehicle, Journal of Robotics and Mechatronics, 2005, 17:44–51. 2. Modianot-Fox, D., Rickshaws reinvented–The ancient transportation takes a modern turn, Smithsonian.com, March 01, 2007. 3. World news, Chinese rickshaw robot, http://wn.com/Chinese Rickshaw Robot, 2011. 4. Ceccarelli M., Carbone G., Ottaviano E., and Lanni. C., Leg Designs for Walking Machines at LARM in Cassino, ASI workshop on Robotics for moon exploration, Rome, July, 2009 5. Ceccarelli M., Carbone G., A Study of Feasibility for a Leg Design with Parallel Mechanism Architecture, IEEE/ASME Conference on Advanced Intelligent Mechatronics AIM’09, Singapore, 2009, paper no. 131. 6. Ceccarelli M., M., Figliolini, G., Lanni, C., Ottaviano, E., A Study of Feasibility for Rickshaw Type Mobile Robot, IEEE International Conference on Industrial Electronics, Control and Instrumentation, Nagoya, 2000, paper MT9-MRC-2.

214


7. Tavolieri, C., Ottaviano, E., Ceccarelli, M., Design and problems of a new leg-wheel walking robot., 10th CLAWAR International Conference on Climbing and Walking Robots, Singapore, 2007, pp. 319-328. 8. Ottaviano E., Grande S., Ceccarelli M., A Biped Walking Mechanism for a Rickshaw Robot, Mechanics based design of structures, 38: 227-242, 2010. 9. Rosheim, M. E., Robot Evaluation, New York: Wiley, 1994. 10. Morecki, A., Waldron, K. J., Human and Machine Locomotion, New York: Springer, 1997. 11. Konteck Comatel, Specification Sheet for Standard Lusense Sensors of PS3 Family, 2001. 12. Freescale Semiconductor Technical Data, ±1.5 g-6 g Three Axis Low-g Micromachined Accelerometer, Document Number: MMA7260QT, Rev 5, 03, 2008. 13. National Instruments Corporation, User guide and specifications NI USB-6008/6009, May 2008 Edition, Part Number 371303L-01, 2008. 14. Interlink Electronics Inc., FSR Integration Guide and Evaluation Parts Catalog with Suggested Electrical Interfaces, Version 1.0, 90-45632 Rev. D, 2007.

Considerations upon the Influence of Manufacturing and Assembly Errors on the Kinematic and Dynamic Behavior in a Flight Simulator Stewart-Gough Platform A. Pisla, T. Itul, D. Pisla, and A. Szilaghyi

Abstract The paper presents a new approach concerning the accuracy of a dynamic flight simulator behavior by considering the influence of the manufacturing and assembling errors on the kinematics and dynamics of a Stewart-Gough platform. The obtained models offer the possibility of a complex study of the parallel structure in order to evaluate the kinematic and dynamic behaviors and to generate the control algorithms for an appropriate digital immersion. Using a numerical simulation the diagrams due to the manufacturing errors are computed and represented. Keyword Flight simulator · Kinematics · Dynamics · Manufacturing and assembling errors analysis

1 Introduction The flight simulators are extensively used by the aviation and the military industry not only for pilots initial and updating training, but also for aircrafts development, equipment testing, the crew emergency operating procedures training, disaster simulation and for medical application in identifying phobias. In all cases dynamics plays an important role in the behavior of the used parallel structures. The different types of flight simulators range from video games up to full-size cockpit replicas [2]. In this study is considered the full simulation with electromechanical actuators. The commands of the system must provide the pilot with realistic motion cues [9]. The correct platform actuation must consider the combination between the real aircraft dynamic model, the mobile platform trajectory center point (TCP) control, and motion of the cockpit center of gravity and for the real “passengers” positions the induced motions, induced accelerations, visual effects and sounds representations, while remaining within the simulator’s motion limits. For

D. Pisla (B) “Technical” University of Cluj-Napoca, Cluj-Napoca, Romania e-mail: [email protected]


215

216

A. Pisla et al.

not obtaining a dizzy effect but a correct digital immersion in the virtual reality and a measurable effect of the actions must be also to be considered the influences of the manufacturing and assembly errors in order to obtain the final desired path (controlled kinematic behavior) and the desired dynamic behavior. The parallel robots have some advantages, over the serial ones, a higher stiffness, equal or higher precision, higher speeds and accelerations, a better payload over weight ratio. However, kinematic and dynamic analysis of the parallel structures is much more complicated due to the existing constraints and singularities. Sometimes these structures tend to be inaccurate and compliant compared with Cartesian serial systems by using a large number of multi-degree-of-freedom joints (spherical and universal joints). A similar problem arises for the overall stiffness: even though the forces are distributed over a certain number of legs, the universal/spherical joints within the legs are much more compliant than the massive guide ways of conventional serial machine tools [15]. The new approach for modeling and determine the influence of the manufacturing and assembly errors upon the kinematic and dynamic behavior starts from the considerations and conclusions presented in a selection of other studies that are analyzing the influences of the errors on the Stewart-Gough platform behavior. Most approaches to error analysis are based on direct kinematics [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 17, 18, 19]. Since generally is not possible to obtain closed form solutions for forward kinematics of mechanisms that involve closed kinematic, the analytical derivation with respect to the parameters cannot be calculated. Even if the closed form solution exists, or computer algebra to handle the large equations is used, it is complex and difficult to work with and hard to interpret the results.

2 The Flight Simulator Structure Error Analysis Most methods use numerical differentiation to calculate the derivatives [1]. It is a time consuming method, changes in all parameters must be considered and complex direct kinematics for each parameter must be computed. For the tolerance synthesis it is crucial to analyze the impact of geometrical errors that emerge from manufacturing and assembly, the nonlinear direct kinematics make hard to predict the displacements of the platform TCP, Fig. 1. The mathematical modeling of parallel kinematics machines - PKM is a challenging task; sophisticated methods are needed to calculate the influence of geometrical errors and to analyze the correlation between the tolerances of the mechanical components and the accuracy of the end-effector [15]. In order to find the length of the legs, the position of guiding points Ai (i=1,2, . . . ,6) are first computed, where [R] is the rotation matrix, Fig. 2. Pi = P + R · p i ;

i=1,2, . . . ,6

(1)

Manufacturing Errors influence on the Dynamic Behavior of a Flight Simulator

217

Fig. 1 The flight simulator mobile platform

Fig. 2 The single leg kinematic chain representation

The leg lengths Ai Bi are materialized as generalized coordinates qi qi = Pi − PBi

(2)

The inverse kinematic model yields through the differentiating the equation (2): q˙ i = vi · ei ;

q¨ i = ai · ei +

v2i − q˙ 2i qi

(3)

where vi and ai are the velocities, respectively the accelerations of points Ai, and ei is the unit vector for the direction of leg Bi Ai ei =

Pi − PBi qi

(4)

218

A. Pisla et al.

Full form of the kinematic and dynamic models is given in the paper [14]. ⎡

T ⎤ T e P − P × e 1 ⎢ 1 1 T ⎥ ⎢ ⎥ ⎢ eT P − P × e ⎥ 2 2 ⎢ ⎥ 2 ⎢ ⎥ ⎥ and B = I6 (5) . . A · X¨ = B · q˙ , where A = ⎢ ⎢ ⎥ ⎢ . ⎥ . ⎢ ⎥ ⎢ . ⎥ . T ⎦ ⎣ T e6 P6 − P × e6 T T X˙ = P˙ ω = X˙ Y˙ Z˙ ωX ωY ωZ

(6)

T q˙ = q˙ 1 q˙ 2 q˙ 3 q˙ 4 q˙ 5 q˙ 6

(7)

The dynamic model consider frictionless joints and neglect legs Bi Ai inertia. −T Q=− A τo G + τo F j

(8)

Q - isthe vector of generalized forces from legs τ0 G - is the torsor in the mobile platform center of gravity forces τ0 F j - is the torsor of inertia forces in the same point The analysis of manufacturing errors considers only the imperfections of the lower joints; the upper joints, connected to the mobile platform are considered error free and equivalent to a spherical joint. The spherical lower joints are replaced with universal joints. The influence of the manufacturing errors represents a sequence of two rotational joint, Fig. 3, whose relative position depends on two parameters: a, the distance between the joints axes and deviation α of the twist angle. Assembly errors are modeled by three parameters R, δ, ν leading to a different positioning of points B and influence the orientation of the O1 X1 Y1 Z1 , reference system, Fig. 4. In this case, the kinematic model is more complex involving both the computing of the real generalized coordinates qr and the θ1 and θ2 angles from joints. For each leg is necessary to solve a nonlinear system derived from the equation: PB + a · i2 + qr · i3 = PA

(9)

where i2 = i2 θ1 ; i3 = i3 θ1 , θ2 ; PB = PB R, δ, ν Differentiating expression (9) in respect to time and successively multiplying with unit vectors of the leg attached reference system O3 X3 Y3 Z3 , result the actuator generalized velocity q˙ r and the angle velocities θ˙ 1 , θ˙ 2 of the passive joints:


219

Fig. 3 A kinematic chain with execution errors

Fig. 4 Location of Bi points

θ˙ 1 = −

vA · k3

;

a + qr cos θ2 · cos α

q˙ r = vA · i3 + aθ˙ 1 sin α sin θ2 vA · j3 + θ˙ 1 a cos θ2 + qr sin α θ˙ 2 = qr

(10)

Similarly are determined the accelerations: q¨ r , θ¨ 1 , θ¨ 2 : aA · k3 + vA · i2 + q˙ r cos θ2 + qr θ˙ 2 sin θ2 θ˙ 1 cos α θ¨ 1 = −

a + qr cos θ2 cos α

(11)

220

A. Pisla et al.

q¨ r = aA · i3 + vA · θ¨ 2 =

aA · j3 + vA ·

k1 × i3 θ˙ 1 + j3 θ˙ 2 + θ¨ 1 sin θ2 + θ˙ 1 θ˙ 2 cos θ2 a sin α (12)

k1 × k3 θ˙ 1 − i3 θ˙ 2 + θ¨ 2 cos θ2 − θ˙ 1 θ˙ 2 sin θ2 a sin α qr

+

+

qr θ¨ 1 + q˙ r θ˙ 1 sin α (13)

qr

Corrections to be applied to kinematic parameters are:

q = qr − q;

˙q = q˙ r − q˙ ;

¨q = q¨ r − q¨

(14)

To determine the corrections in case of a dynamic command control, is computed the Jacobi matrix [Ar ] corresponding to the real mechanism, replacing in [A], the unit vector e with the unit vector i3 . Values of the new generalized forces within the actuators can be determined with relation (8): −T Qr = − Ar · τ 0 G + τ 0 Fj

(15)

And the correction with:

Q = Qr − Q

(16)

3 Numerical Results The modeling and simulation of the defined behavior leads to the errors estimation in order to introduce the necessary correction in the system control. In the next diagrams are presented the most significant results of the simulation. The developed model enables to use the direct manufacturing and assembly measured errors to be used as model inputs and for the friction free joints and the error free upper joints to obtain the correction values for the control. The next diagrams present the computed position and force deviation for an input errors data set. A circular translation motion is defined as: X = 0.5 cos(π t) m ; Y = 0.5 sin(π t) m ; Z = 1.6 m The manufacturing errors data set:

a1 = 0.002 m ; α1 = 0.035 ; a2 = 0.001 m ; α2 = 0.0 15 ;

a3 = 0.003 m ; α3 = 0 ; a4 = 0 ; α4 = 0 ;

a5 = 0.001 m ; α5 = −0.015 ; a6 = 0.002 m ; α6 = −0.035 ;

Manufacturing Errors influence on the Dynamic Behavior of a Flight Simulator Fig. 5 The error positioning curve and the determined error values

[m]

221

0.002 0.001

ΔYn

0 −0.001 −0.002 −0.003 −0.002 −0.001

[m] 0

0.001

1.5

2

Δ Xn

[m]

0.003 0.002

Dn 0.001 0

[s] 0

0.5

1 tn

The assembly errors data set:

R23 = 0.002 m ; R45 = −0.002 m ; R61 = 0.003 m

δ23 = 0.0 15 ; δ45 = 0.03 ; δ61 = 0.03

ν23 = 0.015 ; ν45 = 0.015 ; ν61 = 0.015 The curve of the error positioning and the errors values on the TCP trajectory are shown in Fig. 5. The curve of the error positioning catches at a certain moment the dispersion of the real points that can be attended for the considered trajectory computed position. The shape is similar to the accuracy ellipse of dispersion but the area boundary is differently defined. The errors values for the TCP trajectory represent the maximum error estimated for each position attained in time on the computed trajectory. It has to be mentioned that the positioning errors are of the same order with manufacturing errors and the assembly errors. That indicates on one hand that a better positioning accuracy may be obtained by increasing the accuracy on the production line and a much accurate assembling. On the other hand, it indicates that up to a certain limit the positioning accuracy may be compensated by the control system. Moreover is determined that the twist angle α is not influencing at all the position accuracy.

222

A. Pisla et al. [N] 400

ΔQ1, n

200

ΔQ2, n

0

ΔQ3, n

0.04 ΔQ2, n max(Q2)

ΔQ3, n max(Q3)

−200 −400

[s] 0

0.5

1 tn

1.5

2

0.02 0 −0.02 −0.04

[s] 0

0.5

1 tn

1.5

2

Fig. 6 The force deviation diagrams, expressed as nominal values and in percentage

The forces deviation can be expressed in [N] or as percentage from the maximum actuating force required in a standard situation. For the input data set both cases are represented in Fig. 6. The forces compensation levels depend on the trajectory. In the considered trajectory the highest compensation force is under 400 N that in the extra simulations for the leg 2 and leg 3 indicates about 2-3% over the computed actuating force. The level of the compensation force can be ensured from the control.

4 Conclusions The general solution, in order to avoid errors determined by the manufacturing or assembly process, is to increase the manufacturing and assembling precision, and that will increase the costs and the time for the manufacturing and the implementation. For the already existing systems this method is not applicable therefore valid solutions are: - to optimize the kinematic parameters in order to reduce PKM’s sensitivity with respect to errors in the mechanical components; - to determine corrections models that provide an accurate kinematic and dynamic behavior. The results of the researches shown an estimation of the manufacturing and positioning errors and the compensation methodology for the control in order to follow the imposed paths and induce the required accelerations. The simplifying considerations allow future improving of the model by introducing manufacturing errors also in the upper joints and the friction effect influence. The presented selected diagrams demonstrates the necessity and the value of the elaborated method by indicating the value and the location of the positioning and the kinematic corrections and in the same time the determination of the imposed actuating forces in order to ensure the required simulation accelerations.


223

Acknowledgments This research activity continues the initial one financed from the research grants awarded by the Romanian Ministry of Education and Research, 120/2006.

References 1. Agheli M.M., Nategh M.J.: Identifying the Kinematic Parameters of Hexapod Machine Tool, World Academy of Science, Engineering and Technology 52, 2009. 2. Andreev A. N., Danilov A. M.: Information models for designing, conceptual broad-profile flight simulators. In: Measurement Techniques, 43 (2000), No. 8. 3. Bara M., Bri¸san C., Stan St.: Manufacturing and assembly errors modeling for anthropomorphic pseudo – tetrobots, Machine Building 2003 (55) – Number 11-12. 4. Briot S., Bonev I. A.: Accuracy analysis of 3-DOF planar parallel robots, Mechanism and Machine Theory 43, 2008, pp. 445–458. 5. Han S., Gao Z., Wang B.: Constraint Error Analysis for Three DOF Pure Translational Parallel Robot, International Conference on Measuring Technology and Mechatronics Automation, 2010 Changsha, China, March 13-March 14, icmtma, vol. 1, pp.824-827, 2010. 6. Kotlarski J., Heimann B. and Ortmaier T.: Improving the Pose Accuracy of Planar Parallel Robots using Mechanisms of Variable Geometry, Advances in Robot Manipulators, InTech, ISBN: 978-953-307-070-4, April 2010. 7. Merlet J. P.: Jacobian, manipulability, condition number and accuracy of parallel robots, Springer Tracts in Advanced Robotics, 2007, Volume 28/2007, pp. 175-184. 8. Merlet J-P. and Daney D.: A new design for wire-driven parallel robot, In 2nd Int. Congress, Design and Modelling of mechanical systems, Monastir, 19-21 Mars 2007. 9. Nahon M.A., Gosseli: A comparison of flight simulator motion – base architectures. In: Journal of Mechanical Design, 122 (2000). 10. Null Li Ma, Rong W., Sun L., Li Z.: Error Compensation for a Parallel Robot Using Back Propagation Neural Networks, Robotics and Biomimetics, IEEE International Conference on Robotics and Biomimetics, pp. 1658-1663, 2006. 11. Paccot F., Andreff N., Martinet Ph.: Enhancing tracking performances of parallel kinematic machines, 12th IFToMM World Congress, Besancon, June 18-21, 2007. 12. Pashkevich A., Chablat D. and Wenger P.: Kinematic calibration of Orthoglide-type mechanisms from observation of parallel leg motions, Mechatronics, Vol. 19(4), pp.478-488, June 2009. 13. Pernkopf F. and Husty M.: Reachable Workspace and Manufacturig Errors of Stewart-Gough Manipulators, Proceedings of MUSME, the International Symposium onMultibody Systems and Mechatronics, Uberlândia, Brasil, 6-9 March, 2005, pp. 293-304. 14. Pisla D.L., Itul T.P., Pisla A., Gherman B.: Dynamics of a Parallel Platform for Helicopter Flight Simulation Considering Friction, SYROM 2009, Proceedings of the 10th IFToMM International Symposium on Science of Mechanisms and Machines, Brasov, Romania, October 12-15, 2009, Springer Verlag, pp. 365-378. 15. Pott A. and Hiller M.: A New Approach To Error Analysis In parallel Kinematic Structures, In ARK, Sestri-Levante, 28 Juin-1 Juillet, 2004 16. Rizk R., Andreff N., Fauroux J. C., Lavest J. M. and Gogu G.: Precision Study of a Decoupled Four Degrees of Freedom Parallel Robor Including Manufacturig and Assembling Errors, Advanced in Integrated Design Manufacturing in Mechanical Engineering, Part II, pp. 111127, 2007, Springer Verlag. 17. Szatmari Sz.: Geometrical Errors of Parallel Robots, Periodica Polytechnica Ser. Mech. Eng. Vol. 43, No. 2, pp. 155-162, 1999. 18. Szep C., Stan S. D., Csibi V., Manic M., B˘alan R.: Kinematics, Workspace, Design and Accuracy Analysis of RPRPR Medical Parallel Robot, 2nd International Conference on Human System Interaction (HSI ’09), University of Catania, Catania, Italy, May 21-23, 2009. 19. Wang H., Kuang-Chao F.: Identification of strut and assembly errors of a 3-PRS serial–parallel machine tool, International Journal of Machine Tools & Manufacture 44, 1171–1178, 2004.


xii

Characterization of Flexure Hinges Using the Script Oriented Programming Within a FEM Software Application I. Ivanov and B. Corves

Abstract Using the script oriented programming within a software application based on the finite element method, it is possible to simulate the behaviour of a large number of different flexure hinges making a minor modelling effort. After creating a program, the stiffness / compliance and motion characteristics are determined and analyzed for circular and elliptical flexure hinges in this paper. Keywords Flexure hinges · Stiffness / compliance and motion characterization · Finite element method · Script oriented programming · Circular and elliptical notch shapes

1 Introduction Conventional joints enable mobility (degrees of freedom – DoF) through a form or force fit of the coupled links, theoretically with a zero stiffness in moving direction(s) and an infinite stiffness in the other, so-called parasitic, directions. Compliant joints utilize own flexibility and thus exhibit a finite stiffness in all directions. Among different types of compliant joints [6], flexure hinges are the most popular ones featuring simplicity and compactness. They are prismatic or cylindric bodies being notched for the purpose of flexibility enhancement in one or more directions. Although flexure hinges can be designed with two or three rotational DoF, those with one rotational DoF are more feasible both for planar and spatial applications. Flexure hinges bring many benefits – an insignificant backlash and friction, a high miniaturization potential, but also some drawbacks – parasitic effects, deviations and limitations of motion. In order to evaluate the performance of flexure hinges, it is crucial to know their stiffness or compliance characteristics, which determine their motion characteristics in predominant quasi-static analyses. The motion capability (desirable and undesirable mobility, motion accuracy, motion range) is condensed into the stiffness (K) or compliance (C) matrix I. Ivanov (B) RWTH Aachen University, Aachen, Germany e-mail: [email protected]


225

226

I. Ivanov and B. Corves

⎞ Cux −Fx 0 0 0 0 0 ⎜ 0 Cuy −Fy 0 0 0 Cuy −Mz ⎟ ⎟ ⎜ ⎜ 0 0 C 0 C 0 ⎟ uz −Fz uz −My ⎟ ⎜ C=⎜ 0 0 ⎟ 0 0 Cθx −Mx ⎟ ⎜ 0 ⎝ 0 0 Cθy −My 0 ⎠ 0 Cθy −Fz 0 Cθz −Fy 0 0 0 Cθz −Mz ⎛

(1)

which defines the linear elastic relation =K·δ

or

δ=C·

(2)

between the load vector () and the deformation vector (δ) = Fx Fy Fz Mx My Mz

T

and

δ=

ux uy uz θx θy θz

T

(3)

For a flexure hinge being fixed at one end and subjected to forces (F) and moments (M) at the other end, corresponding translations (u) and rotations (θ) can be calculated by using an analytical method (the pioneer contribution in [5]), an experimental test (the first one of the very rare examples in [8]) or the finite element method, which are possible approaches to find out the compliance matrix (C), indirectly the stiffness matrix K = C-1 . The finite element method is already a well-established tool for the characterization of flexure hinges [7, 9, 10]. Failing the experimental test, credible results for all the elements of the stiffness or compliance matrix, including shearing effects as well as torsion being less successfully described by the analytical method, can be obtained by means of the finite element method. Its classical implementation is extensive and only suitable for verifying individual results of the analytical method. Using the script oriented programming within a software application based on the finite element method, it is possible to simulate the behaviour of a large number of different flexure hinges making a minor modelling effort. Such a possibility is realized and employed in this paper.

2 Program for the Characterization of Flexure Hinges The FEM software application Abaqus is used here. Instead of working in its computer-aided engineering (CAE) environment, all the activities concerning modelling, simulation and post-processing are defined in the form of a script written in the programming language Python [3]. So a program for the characterization of flexure hinges is created with the following features: – The geometry and material parameters of flexure hinges are variables, which are simply modified. An iterative parameter modification is also available. – An adaptive remeshing provides a very fine mesh inside the notch area and a coarse mesh outside the notch area (Fig. 1).

Characterization of Flexure Hinges Using the Script Oriented Programming

227

Fig. 1 Mesh model of flexure hinge

– There is a universal procedure to apply forces (F) and moments (M) as well as to calculate corresponding translations (u) and rotations (θ). – The compliance (C) and stiffness (K) matrix are found out for the reference points of flexure hinges. – The parameters and the results are automatically and systematically archived. – The program is customized to the known families of notch shapes. With the help of the program, the stiffness / compliance and motion characteristics are determined and analyzed for circular and elliptical flexure hinges as follows.

3 Analysis of Circular and Elliptical Flexure Hinges In order to achieve the best characteristics of flexure hinges, various notch shapes are investigated [2, 4]. Among all of them, because of the easiest manufacturability, circular and elliptical flexure hinges are particularly interesting for applications. They represent the special cases of a general elliptical flexure hinge (Fig. 3) [1]. Accordingly, it is possible to integrally perform an analysis of all circular and elliptical flexure hinges. Because of the linear elastic relations, the material parameters are not specified. The modulus of elasticity (E), the modulus of shear (G) and the allowable stress (σmax ) are incorporated in the characteristics of flexure hinges. The geometry parameters are only variables in the analysis. Although the width (w) shows an influence on the characteristics of flexure hinges (increasing the width causes a stiffness enhancement in all directions), it is rather a construction parameter being

228 Fig. 2 Maximum rotation of flexure hinge


y

z

x θz max uy max

σmax

Mz max

Fig. 3 a General elliptical flexure hinge. Special cases: b right circular, c circular, d right elliptical, e elliptical notch shape

constrained due to technological capabilities. For this reason, a typical width (w = 10 mm) is assumed for all flexure hinges in the analysis. The geometry parameters defining the general elliptical notch shape, i.e. the lengths of the axes (a, b) and the thickness (t), are varied and their influence on the behaviour of flexure hinges is analysed. The length (l) is also a construction parameter (l = 10 mm). In case of a flexure hinge, unlike an ideal revolute joint, one desirable (rotational) degree of freedom is accompanied by several undesirable (rotational and translational) degrees of freedom. An explanation for this can be found in the stiffness or

Characterization of Flexure Hinges Using the Script Oriented Programming

229

compliance matrix. The compliance matrix of the ideal revolute joint is composed of zero elements, except one infinite element. In the compliance matrix of the flexure hinge (Eq. (1)), this element Cθz - Mz is finite. It defines the rotation angle (θz ) that is generated by the moment (Mz ) for the desirable degree of freedom. The maxby the allowable stress (σmax ). The midpoint imum rotation angle (θz max ) is limited reaches the maximum deflection uy max thereby (Fig. 2). The other non-zero elements of the compliance matrix of the flexure hinge entail the undesirable degrees of freedom. The element Cθx - Mx indicates a particularly high torsion compliance, which has not to be neglected in spatial applications. Therefore, the bending compliance (Cθz - Mz ), the motion range (θz max ), the motion accuracy uy max and the torsion compliance (Cθx - Mx ) are observed in the analysis. These characteristics are illustrated for a few combinations of the geometry parameters (Fig. 4). For an acceptable thickness (t = 1 mm), the lengths of the axes (a = 1/2 . . . 1, b = 0 . . . 1) are simultaneously or separately varied to form circular or (right) elliptical notch shapes. For b = 0 in case of right elliptical flexure hinges (for a → ∞ in case of elliptical or circular flexure hinges as well), a simple prismatic beam having a constant rectangular cross section is obtained. Circular flexure hinges are stiffer than elliptical flexure hinges. Right elliptical flexure hinges quickly converge to the simple prismatic beam, which is very compliant. So right elliptical

l = 10 mm t = 1 mm w = 10 mm

l = 10 mm t = 1 mm w = 10 mm

PB (r −> ∞)

l = 10 mm t = 1 mm w = 10 mm

PB (a −> ∞)

PB (r −> ∞) PB (a −> ∞)

PB (r −> ∞)

l = 10 mm t = 1 mm w = 10 mm

PB (a −> ∞)

PB (r −> ∞)

PB (a −> ∞)

Fig. 4 Characteristics of flexure hinges as function of geometry parameters of notch shapes (RC – right circular, C – circular, RE – right elliptical, E – elliptical, PB – prismatic beam)

230


flexure hinges with b < l/4 and elliptical flexure hinges with a > 3 l/4 are characterized by a relative large motion range but a low motion accuracy. Consequently, the analysis is narrowed down to circular flexure hinges as well as (right) elliptical flexure hinges with stronger notch curvatures. The right circular flexure hinge represents a very special case of the general elliptical flexure hinge. In comparison to the other flexure hinges having the same width (w), thickness (t) and length (l), it is characterized by the minimum compliance or the maximum stiffness in all directions as well as the minimum motion range and the maximum motion accuracy. The right circular notch shape is defined by a less number of the geometry parameters, because the lengths of the axes (a, b) merge into the radius (r = l/2). It is reasonable to preselect the right circular notch shape as an initial situation in the design of flexure hinges. A nomogram for dimensioning the right circular notch shape with regard to the material properties and the motion range is depicted in Fig. 5. It covers the influence of the radius (r) and the thickness (t), while that of the width (w) is not considered. Putting the formulae for the maximum bending stress and the bending stiffness [5] together, an inequality well matching with the nomogram is obtained E σmax

· θz max
0), which is called self-motion. Note that such motions are also solutions to the famous Borel Bricard problem (cf. [1, 3, 4, 12]).

2 Types of Self-Motions In section 2 and 3 we give a very short review of the results and ideas stated in [10], where also more details and examples can be found. It is known that architecturally singular SG platforms, which are well studied, possess self-motions in each pose. Therefore, we are only interested in the computation of self-motions of non-architecturally singular SG platforms. A detailed review of self-motions of this type was given by the author in [10]. Moreover, it is known that if a planar SG platform with anchor points m1 , . . . , M6 is not architecturally singular, then at least a one-parametric set of legs exists, which can be attached to the given manipulator without changing the forward kinematics [5, 9] and the singularity set [2] of the manipulator. Moreover, it was shown that in general the base anchor points Mi as well as the corresponding platform anchor points mi are located on planar cubic curves C and c, respectively. Assumption 1. We assume that there exist such cubics c and C (which can also be reducible) in the Euclidean domain of the platform and the base, respectively. We consider the complex projective extension P3C of the Euclidean 3-space with (ai , bi , 0) → (wi : xi : yi : 0), (Ai , Bi , 0) → (Wi : Xi : Yi : 0) and wi , xi , yi , Wi , Xi , Yi ∈ C. Note that ideal points are characterized by wi = 0 and Wi = 0, respectively. Moreover, we consider the correspondence between the points of C and c, which is determined by the geometry of the manipulator m1 , . . . , M6 (cf. [2, 5, 9]). As this correspondence has not to be a bijection, a point ∈ P3C of c resp. C is in general mapped to a non-empty set of points ∈ P3C of C resp. c. We denote this set by the term corresponding location and indicate this fact by the usage of brackets { }. In P3C the cubic C has three ideal points U1 , U2 , U3 , where at least one of these points (e.g. U1 ) is real. The remaining points U2 and U3 are real or conjugate complex. Then we compute the corresponding locations {u1 } , {u2 } , {u3 } of c (⇒ {u1 } contains real points). We denote the ideal points of c by u4 , u5 , u6 , where again one (e.g. u4 ) has to be real. The remaining points u5 and u6 are again real or conjugate complex. Then we compute the corresponding locations {U4 } , {U5 } , {U6 } of C (⇒ {U4 } contains real points). Assumption 2. For guaranteeing a general case, we assume that each of the corresponding locations {u1 } , {u2 } , {u3 } , {U4 } , {U5 } , {U6 } consists of a single point. Moreover, we assume that no 4 collinear platform anchor points uj or base anchor points Uj (j = 1, . . . , 6) exist.

Basic Result on Type II DM Self-Motions of Planar Stewart Gough Platforms

237

Under consideration of Assumption 1 and 2, following theorem was proven [10]: Theorem 1. The resulting manipulator u1 , . . . , U6 is architecturally singular. Moreover, it was proven in [10] that there only exist type I and type II Darboux Mannheim (DM) self-motions, where the definition of types reads as follows: Definition 1. Assume M is a one-parametric self-motion of a non-architecturally singular SG platform m1 , . . . , M6 . Then M is of the type n DM if the corresponding architecturally singular manipulator u1 , . . . , U6 has an n-parametric self-motion.

3 Computation of Type II DM Self-Motions The only examples of type II DM self-motions known to the author are those constructed by Karger in [7, 8], which are characterized by e0 = 0. The computation of type II DM self-motions in [10] was based on Darboux and Mannheim constraints, which are repeated next. With this approach it seems for the first time possible to give a complete classification of type II DM self-motions: Darboux constraint: The constraint that the platform anchor point ui (i = 1, 2, 3) moves in a plane of the fixed system orthogonal to the direction of the ideal point Ui can be written as (cf. [10]) Ωi : X i (ai r11 + bi r12 + t1 ) + Y i (ai r21 + bi r22 + t2 ) + Li K = 0, with Xi , Yi , ai , bi , Li ∈ C. This is a homogeneous quadratic equation in the Study parameters where X i and Y i denote the conjugate complex of Xi and Yi , respectively. Mannheim constraint: The constraint that the plane orthogonal to ui (i = 4, 5, 6) through the platform point (gi , hi , 0) slides through the point Ui of the fixed system can be written as (cf. [10]) Πi : xi [Ai r11 + Bi r21 − gi K − 2(e0 f1 − e1 f0 − e2 f3 + e3 f2 )]+ yi [Ai r12 + Bi r22 − hi K − 2(e0 f2 + e1 f3 − e2 f0 − e3 f1 )] = 0, with xi , yi , Ai , Bi , gi , hi ∈ C. This is again a homogeneous quadratic equation in the Study parameters where xi and yi denote the conjugate complex of xi and yi . The content of the following lemma was also proven in [10]: Lemma 1. Without loss of generality (w.l.o.g.) we can assume that the variety of the two-parametric self-motion of u1 , . . . , U6 is spanned by Ψ , Ω1 , Ω2 , Ω3 , Π4 , Π5 . Moreover, we can choose following special coordinate systems in Σ 0 and Σ w.l.o.g.: X1 = Y2 = Y3 = x4 = y5 = 1, a1 = b1 = y4 = A4 = B4 = Y1 = h4 = g5 = 0 and X2 (X2 − X3 )x5 = 0.

238

G. Nawratil

We solve the linear system of equations Ψ , Ω1 , Ω2 , Π4 for f0 , . . . , f3 and plug the obtained expressions in the remaining two equations1. This yields in general two homogeneous polynomials Ω[40] and Π [96] in the Euler parameters of degree 2 and 4, respectively. The number in the square brackets gives the number of terms. Finally, we compute the resultant of Ω and Π with respect to one of the Euler parameters. Here we choose2 e0 . This yields a homogeneous polynomial Γ [117 652] of degree 8 in e1 , e2 , e3 . In the following we denote the coefficients j of ei1 , e2 , ek3 of Γ by Γ ijk . We get a set E of 24 equations Γijk = 0 in the 14 unknowns a2 , b2 , a3 , b3 , A5 , B5 , X2 , X3 , x5 , L1 , L2 , L3 , g4 , h5 . j Moreover, it should be noted that we denote the coefficients of ei0 e1 , ek2 , el3 of Ω and Π by Ω ijkl and Π ijkl , respectively.

4 The Basic Result An important step in direction of a complete classification of type II DM selfmotions is done by the basic result given in Theorem 2. As preparatory work for the formulation of this theorem we have to define the following two special cases: It can easily be seen, that Ω does not depend on e0 and e3 (upper signs) or e1 and e2 (lower signs) if the following three equations are fulfilled: L1 (X 2 − X 3 ) − L2 + L3 = 0,

(1)

a2 (X 2 − X 3 ) ± X 3 (X 2 b2 − X 3 b3 ) ± b2 ∓ b3 = 0,

(2)

a3 (X 2 − X 3 ) ± X 2 (X 2 b2 − X 3 b3 ) ± b2 ∓ b3 = 0.

(3)

Theorem 2. With exception of the above mentioned two special cases, the corresponding manipulator u1 , . . . , U6 of a planar SG platform (fulfilling Assumptions 1, 2 and Lemma 1) with a type II DM self-motion, has to have further 3 collinear anchor points in the base or in the platform beside the points U1 , U2 , U3 and u4 , u5 , u6 . The proof is done by contradiction, i.e. we stop the case study if 3 anchor points beside U1 , U2 , U3 and u4 , u5 , u6 are collinear or if we get one of the 2 special cases. Proof for the general case Ω2000 Π3000 = 0 We assume Ω200 Π3000 = 0, as only those solutions of E correspond to type II self-motions, which do not cause a vanishing of the coefficient of the highest power of e0 in Ω and Π , respectively. 1 2

For e0 e2 − e1 e3 = 0 this can be done w.l.o.g., as this factor belongs to the denominator of fi . Therefore we are looking for a common factor of Ω and Π , which depends on e0 .


239

Γ 800 can only vanish without contradiction (w.c.) for L1 = g4 or if F = X 2 (L1 − a2 ) − X 3 (L1 − a3 ) − L2 + L3 + b2 − b3 is fulfilled identically. We distinguish 3 parts: Part [A] Assuming L1 = g4 : Now F = 0 has to hold. W.l.o.g. we express L1 from F = 0. Then Γ710 = 0 implies a2 = a3 − X 2 b2 + X 3 b3 . Now Γ 620 cannot vanish w.c.. Part [B] L1 = g4 and F = 0: We express L1 from F = 0. W.l.o.g. we can compute h5 from the only non-contradicting factor of Γ 602 . Now Γ 530 can vanish w.c. for: 1. L3 = X 3 (L2 − b2 ) /X 2 + X 3 (a2 − a3 ) + b3 : W.l.o.g. we can express A5 from the only non-contradicting factor of Γ422 . Again we distinguish two cases: a. X 2 b2 − X 3 b3 + a2 − a3 = 0: Now Γ 350 has only one non-contradicting factor, which can be solved for L2 w.l.o.g.. Then we can solve the only non-contradicting factor of Γ 314 for x5 w.l.o.g.. Now the resultant of the only non-contradicting factors of Γ206 and Γ 242 with respect to B5 cannot vanish w.c.. 2 b. a3 = X 2 b2 − X 3 b3 + a2 : Then Γ314 = 0 implies L2 = 2X 2 b2 + X 2 a2 + b2 . i. X 3 (X 2 b2 − X 3 b3 ) + a2 (X 3 − X 2 ) + x25 (b3 − b2 ) = 0: Under this assumption we can express B5 from the only non-contradicting factor of Γ 242 . Then Γ 224 can only vanish w.c. for X i = −x5 with i = 2 or i = 3. As for x5 bj + X j bi = 0 with i = j and i, j ∈ {2, 3} the expression Γ 080 cannot vanish w.c., we can assume x5 bj +X j bi = 0. Under this assumption we can compute a2 from Γ080 = 0 w.l.o.g.. Then the linear-combination Γ044 − Γ026 − Γ062 equals b22 b23 (X j + x5 )2 (b2 − b3 ), a contradiction. ii. X 3 (X 2 b2 − X 3 b3 ) + a2 (X 3 − X 2 ) + x25 (b3 − b2 ) = 0: W.l.o.g. we can solve this equation for a2 . Then Γ242 can only vanish w.c. for X i = −x5 and Γ080 = 0 implies X j = x5 with i = j and i, j ∈ {2, 3}. Now x25 Γ206 − Γ026 equals bi b2j B5 x65 (b2 − b3 ), a contradiction. 2. a2 = X 3 b3 − X 2 b2 + a3 and X 2 X 3 (a2 − a3 ) + X 2 (b3 − L3 ) − X 3 (b2 − L2 ) = 0: In this case Γ440 = 0 already yields the contradiction. Part [C] Assuming F = 0: Now L1 = g4 has to hold. Then Γ080 factors into G[8]H[18]2 . We distinguish 3 cases: 1. G[8] = 0: W.l.o.g. we can express L1 from G[8] = 0. Now Γ 170 can only vanish w.c. for (X 3 b3 − X 2 b2 − a2 + a3 )V[18] = 0: a. a2 = X 3 b3 − X 2 b2 + a3 : We can solve the only non-contradicting factor of Γ 620 for h5 . Now we can express L3 from the only non-contradicting factor of Γ 602 . Moreover, we can compute A5 from the only non-contradicting factor of Γ 260 . Then we can solve the only non-contradicting factor of Γ 062 for L2 .

240

G. Nawratil

i. x25 (b2 − b3 ) + 2x5 (X 2 b2 − X 3 b3 ) + a3 (X 3 − X 2 ) + X 2 (X 2 b2 − X 3 b3 ) = 0: Now we can compute B5 from the only non-contradicting factor of Γ 404 w.l.o.g.. We distinguish two cases: • X 2 b3 − X 3 b2 = 0: Under this assumption we can express a3 from the only non-contradicting factor of Γ 026 . Then Γ 206 cannot vanish w.c.. • X 2 = X 3 b2 /b3 : Now Γ026 = 0 implies b2 = −b3 . Finally Γ206 = 0 yields the contradiction. ii. x25 (b2 − b3 ) + 2x5 (X 2 b2 − X 3 b3 ) + a3 (X 3 − X 2 ) + X 2 (X 2 b2 − X 3 b3 ) = 0: W.l.o.g. we can solve this equation for a3 . Then Γ206 can only vanish w.c. for the following two cases: • X 3 = −x5 : Now Γ404 = 0 implies b2 = −b3 and from Γ422 = 0 we get B5 = −b3 . Then Γ440 − Γ242 = 0 yields the contradiction. • X 2 = −x5 : Γ404 = 0 implies b2 = −b3 and Γ 422 cannot vanish w.c.. b. V[18] = 0, X 3 b3 − X 2 b2 − a2 + a3 = 0: W.l.o.g. we can solve this equation for A5 . We can solve the only non-contradicting factor of Γ620 for h5 . Then we can express L3 from the only non-contradicting factor of Γ 602 . Moreover, we can solve the only non-contradicting factor of Γ062 for L2 . i. x25 (b3 − b2 ) + x5 (a2 − a3 + X 3 b3 − X 2 b2 ) + X 2 a2 − X 3 a3 = 0: Under this assumption we can solve Γ404 = 0 for B5 . Then Γ026 = 0 implies a2 = X 3 a3 b2 /(X 2 b3 ) and Γ206 = 0 yields the contradiction. ii. x25 (b3 − b2 ) + x5 (a2 − a3 + X 3 b3 − X 2 b2 ) + X 2 a2 − X 3 a3 = 0: • a2 − x5 b2 = 0: In this case we can express X 2 from the above equation. Now Γ206 = 0 implies a3 = x5 b3 . From Γ404 = 0 we get a2 = −X 3 b2 and Γ422 = 0 yields B5 = b3 . Then Γ440 − Γ242 cannot vanish w.c.. • a2 = x5 b2 : Now Γ026 = 0 implies b3 = X 3 a3 /(X 2 x5 ) and from Γ422 = 0 we get X 3 = −x5 . Then Γ440 − Γ242 = 0 yields the contradiction. 2. H[18] = 0, G[8] = 0 and X 2 a2 − X 3 a3 = 0: Under this assumption we can compute h5 from H[18] = 0. Then we can express B5 from the only noncontradicting factor of Γ 620 . Moreover, from the only non-contradicting factor of Γ 602 we can compute L2 . Then we consider the only non-contradicting factor I[14] of Γ 260 . a. x25 (b2 − b3 ) + x5 (X 3 b3 − X 2 b2 − a2 + a3 ) + X 2 a2 − X 3 a3 = 0: Under this assumption we can express A5 from I[14] = 0.


241

i. a3 − x5 b3 = 0: Under this assumption we can express L3 from the only non-contradicting factor of Γ062 . Then Γ044 = 0 implies a2 = X 2 b2 − X 3 b3 + a3 . Then the resultant of Γ 404 and Γ422 with respect to x5 can only vanish w.c. for J[12](X 2 b3 − a3 )(X 3 b2 + X 3 b3 − X 2 b2 − a3 ) = 0: • J[12] = 0: W.l.o.g. we can solve this equation for a3 . Then Γ404 = 0 implies X 3 = x5 . Now Γ 026 cannot vanish w.c.. • a3 = X 3 b2 + X 3 b3 − X 2 b2 : Then Γ 404 cannot vanish w.c.. • a3 = X 2 b3 : Now Γ404 = 0 implies X 3 = x5 and Γ206 cannot vanish w.c.. ii. a3 = x5 b3 : From Γ062 = 0 we get L1 = 0. Then Γ404 = 0 implies a2 = X 2 b2 − X 3 b2 + x5 b2 . Now Γ 422 cannot vanish w.c.. b. x25 (b2 − b3 ) + x5 (X 3 b3 − X 2 b2 − a2 + a3 ) + X 2 a2 − X 3 a3 = 0: i. X 3 = x5 : Under this assumption we can express a3 from the above equation. Then I = 0 implies X 3 = x5 . Now we can solve the only non-contradicting factor of Γ 404 for A5 w.l.o.g.. Then Γ026 = 0 implies a2 = X 3 b2 . Now the difference of the only non-contradicting factors of Γ062 and Γ 422 can only vanish w.c. for: • X 3 = 1/x5 : Then we can solve Γ062 = 0 for L3 w.l.o.g.. Finally Γ044 = 0 yields the contradiction. • L1 = 4b2 b3 (x5 − X 3 )/(b2 − b3 ): Now Γ 062 cannot vanish w.c.. ii. X 3 = x5 : The equation of item b can only vanish w.c. for a2 = x5 b2 . Then we can express A5 w.l.o.g. from the only non-contradicting factor of Γ 404 . Now Γ026 = 0 implies a3 = X 2 b3 . Then the difference of the only non-contradicting factors of Γ062 and Γ 422 can only vanish w.c. for: • X 2 = 1/x5 : Now we can solve Γ062 = 0 for L3 w.l.o.g.. Then Γ044 = 0 implies b2 = −b3 . Finally Γ026 = 0 yields the contradiction. • L1 = 4b2 b3 (X 2 − x5 )/(b2 − b3 ): Now Γ 062 cannot vanish w.c.. 3. H[18] = 0, G[8] = 0 and a2 = X 3 a3 /X 2 : Now H = 0 implies A5 = −X 3 a3 /x5 . W.l.o.g. we can express h5 from the only non-contradicting factor of Γ 620 . Moreover, the only non-contradicting factor of Γ 602 can be solved w.l.o.g. for L2 . Then we consider the only non-contradicting factor E[12] of Γ260 . a. X 2 x5 (b3 − b2 ) − X 2 (X 3 b3 − X 2 b2 ) − a3 (X 2 − X 3 ) = 0: Under this assumption we can solve E[12] = 0 for B5 . Then Γ062 = 0 implies a3 = x5 b3 .

242

G. Nawratil

Now we can express x5 w.l.o.g. from the only non-contradicting factor of Γ 404 . Then Γ026 = 0 implies X 3 = 0. Finally Γ044 = 0 yields the contradiction. b. X 2 x5 (b3 −b2 )−X 2 (X 3 b3 −X 2 b2 )−a3 (X 2 −X 3 ) = 0: W.l.o.g. we can express a3 from this equation. Then E can only vanish w.c. for: i. X 3 = x5 : Now Γ062 = 0 already yields the contradiction. ii. X 2 = x5 : Now Γ062 = 0 implies B5 = −b3 . Then we can solve the only non-contradicting factor of Γ 404 for x5 . Γ026 = 0 yields the contradiction. Proof for the special case Ω2000 Π3000 = 0 If we set ei equal to zero for any i ∈ {0, . . . , 3}, then Ω and Π have to be fulfilled identically. It can be seen immediately, that the conditions implied by Ω = 0 already yield a contradiction. Therefore we can assume e0 e1 e2 e3 = 0 w.l.o.g. for this section of the proof. Due to the limitation of pages we can only sketch the remaining proof, which splits up into the following parts: Part [A] Ω2000 = 0, Ω1000 Π3000 = 0 Part [B] Ω2000 = Π3000 = 0, Ω1000 Π2000 = 0 Part [C] Ω2000 = Π3000 = Π2000 = 0, Ω1000 Π1000 = 0 Part [D] Π3000 = 0, Ω2000 Π2000 = 0 Part [E] Π3000 = Π2000 = 0, Ω2000 Π1000 = 0 The discussion of the parts [A-E] can be done similarly to the presented general case. The case study takes further 2.5 pages and can be looked up in the corresponding technical report [11]. Moreover, one has to discuss the following two parts: Part [F] Ω2000 = Ω1000 = 0 Part [G] Π3000 = Π2000 = Π1000 = 0 In the parts [F,G] we compute the resultant of Ω and Π with respect to e3 , as at least one of these equations does not depend on e0 . This results in a very exhausting study of cases. We encourage the interested reader to have a look at the corresponding 5 pages of the technical report [11], as this discussion is not trivial. Due to the structure3 of Ω it can easily be seen, that Ω and Π can only have a common factor, which does not depend on e0 (cf. footnote 2) if Ω = 0 has this property too. As this case was already treated in part [F] we remain with the discussion of those cases excluded by the assumption e0 e2 − e1 e3 = 0 (cf. footnote 1). 3 2 Ω : i=0 ci ei + c4 e0 e3 + c5 e1 e2 where c0 , . . . , c5 only depend on the geometry of the SG platform.

3


243

Proof for the case e0 e2 − e1 e3 = 0 As e0 = e1 = e2 = e3 = 0 does not correspond with an Euclidean motion, we only have to study the following special cases, which fulfill e0 e2 − e1 e3 = 0 identically: e0 = e1 = e2 = 0, e0 = e2 = e3 = 0,

e0 = e1 = e3 = 0, e1 = e2 = e3 = 0,

e0 = e1 = 0, e1 = e2 = 0,

e0 = e3 = 0, e2 = e3 = 0.

Moreover, in each of these cases we assume that the remaining ei ’s are different from zero. For the detailed one-sided discussion of these eight cases we refer again to the corresponding technical report [11]. Due to this study of special cases we can assume w.l.o.g. e0 e1 e2 e3 = 0. Therefore we can solve e0 e2 − e1 e3 = 0 for e2 . Moreover, we can solve Ψ , Ω1 , Π4 , Π5 w.l.o.g. for f0 , f1 , f2 , f3 . Now Ω 2 and Ω 3 have to vanish independently of the choice of the unknowns e0 , e1 , e3 . Therefore the coefficient of e60 of Ω 2 implies L1 = g4 . Then the coefficient of e50 e3 of Ω 2 yields an expression for L2 . Now we get g4 = 2a2 − 2X 2 b2 from the coefficient of e40 e23 of Ω 2 . Moreover, we get a2 = X 2 b2 from the coefficient of e21 e43 of Ω 2 . Finally the coefficient of e0 e21 e33 of Ω 2 cannot vanish w.c.. This finishes the proof of Theorem 2.

5 Conclusion and Future Research In this paper we presented the basic result (cf. Theorem 2) on type II Darboux Mannheim (DM) self-motions of planar SG platforms. Due to Lemma 2 of [6] and Theorem 1 we can replace the word “or” in Theorem 2 by the word “and”; i.e. with exception of the two special cases there always exist three collinear platform points ui , uj , uk and three collinear base points Ul , Um , Un beside the points U1 , U2 , U3 and u4 , u5 , u6 where (i, j, k, l, m, n) consists of all indices from 1 to 6. The presented basic result raises the hope of giving a complete classification of type II DM self-motions in the future, which would be an important step in solving the famous Borel Bricard problem. On base of Theorem 2 the work towards this goal is in progress. Acknowledgements This research is supported by Grant No. I 408-N13 of the Austrian Science Fund FWF within the project “Flexible polyhedra and frameworks in different spaces”, an international cooperation between FWF and RFBR, the Russian Foundation for Basic Research.

References 1. Borel, E.: Mémoire sur les déplacements à trajectoires sphériques, Mém. présenteés par divers savants, Paris(2), 33, 1–128 (1908). 2. Borras, J., Thomas, F., Torras, C.: Singularity-invariant leg rearrangements in doubly-planar Stewart-Gough platforms, In Proc. of Robotics Science and Systems, Zaragoza, Spain (2010). 3. Bricard, R.: Mémoire sur les déplacements à trajectoires sphériques, Journ. École Polyt.(2), 11, 1–96 (1906).

244

G. Nawratil

4. Husty, M.: E. Borel’s and R. Bricard’s Papers on Displacements with Spherical Paths and their Relevance to Self-Motions of Parallel Manipulators, Int. Symp. on History of Machines and Mechanisms (M. Ceccarelli ed.), 163–172, Kluwer (2000). 5. Husty, M., Mielczarek, S., Hiller, M.: A redundant spatial Stewart-Gough platform with a maximal forward kinematics solution set, Advances in Robot Kinematics: Theory and Applications (J. Lenarcic, F. Thomas eds.), 147–154, Kluwer (2002). 6. Karger, A.: Architecture singular planar parallel manipulators, Mechanism and Machine Theory 38 (11) 1149–1164 (2003). 7. Karger, A.: New Self-Motions of Parallel Manipulators, Advances in Robot Kinematics: Analysis and Design (J. Lenarcic, P. Wenger eds.), 275–282, Springer (2008). 8. Karger, A.: Self-motions of Stewart-Gough platforms, Computer Aided Geometric Design, Special Issue: Classical Techniques for Applied Geometry (B. Jüttler, O. Röschel, E. Zagar eds.) 25 (9) 775–783 (2008). 9. Mielczarek, S., Husty, M.L., Hiller, M.: Designing a redundant Stewart-Gough platform with a maximal forward kinematics solution set, In Proc. of the International Symposion of Multibody Simulation and Mechatronics (MUSME), Mexico City, Mexico, September 2002. 10. Nawratil, G.: Types of self-motions of planar Stewart Gough platforms, under review. 11. Nawratil, G.: Basic result on type II DM self-motions of planar Stewart Gough platforms, Technical Report No. 215, Geometry Preprint Series, TU Vienna (2011). 12. Vogler, H.: Bemerkungen zu einem Satz von W. Blaschke und zur Methode von Borel-Bricard, Grazer Mathematische Berichte 352 1–16 (2008).

Active Bearing for Vibration Damping of Roller Systems with Piezoelectric Actuators M.-C. Voicu, R. Schmidt, B. Lammen, M. Mersch, and I. Maniu

Abstract To improve the efficiency of printing or coating processes for paper products the velocity of the web and the roller width can be increased. However, these measures cause deformations of the rollers, heating effects and streak print defects due to undesirable oscillations. The vibration characteristic is significant for the quality and efficiency of the printing or coating processes. Approved methods like balancing of the rollers and maximizing the bending stiffness have come to technical limits. This seems to be promising technology for further optimization of printing and coating machines. This paper describes an approach with piezoelectric actuators in a flexographic printing machine by introducing active elements in the bearings of a roller to compensate undesired bending vibrations. Similar active components for vibration damping have been introduced successfully in grinding and milling machines [1, 2]. Keywords Active bearing · Piezoelectric actuator · Vibration damping · Roller system

1 Introduction This paper describes the integration of active vibration damping into a flexographic printing machine which has several color stations placed around a central impression cylinder. In Fig. 1, one color station is exemplified. The ink is coated on the anilox roll (2) by means of a doctor blade (1) which transfers it onto the plate cylinder (3). The surface of the flexible printing plate mounted on the plate cylinder defines the image to be printed. The elevated parts of the printing plate carry the ink to the web by pressing onto the impression cylinder (4). Due to excitations caused by the canal impact or to the uneven surface of the printing plate when passing the nip in positions (5) and (6), oscillations can occur. Approved methods like balancing of the M.-C. Voicu (B) University “Politehnica” Timisoara, Timisoara, Romania University of Applied Sciences Osnabrück, Osnabrück, Germany e-mail: [email protected]


245

246

M.-C. Voicu et al.

Fig. 1 Layout of a roller system in a flexographic printing machine

1

2

3

5

4

6

rollers and maximizing the bending stiffness have come to technical limits. In the research project presented in this paper, the undesired impact of the vibrations on the printing process shall be diminished by active vibration damping. Piezoelectric actuators are able to generate huge forces at high frequencies with small displacements as required for this purpose. Figure 2 shows the integration of piezoelectric actuators in the bearings of a roller. Similar active components for vibration damping have been introduced successfully in grinding and milling machines [1, 2]. For measuring nip forces and vibrations of a roller, different technologies can be considered as discussed in [3]. Sensors and signal lines must be applied without mechanical impact on the system. Figure 3 illustrates the over-all control structure. The bending vibration of the roller is measured by strain gauges (1) at different axial positions on the roller (2). For transmitting these signals, a slip ring is mounted on the roller. A sensor (3) for measuring the rotational angle of the roller gives the position and the acceleration of the axle bearing points. The signals are the feedback for the control unit (4) which activates the piezoelectric actuators (5) via high voltage direct current amplifiers (not depicted in Fig. 3). In a first approach the forces of the actuators are applied in a radial direction to compensate excitations in the nip.

piezo-electric actuators

piezo-electric actuators

Fig. 2 Piezoelectric actuators in the bearings

Active Bearing for Vibration Damping of Roller Systems with Piezoelectric Actuators

247

Fig. 3 Control structure

3

1 1 5

2

5 u1(t)

u2(t)

4

2 Integration of a Piezoelectric Actor into the Bearing of an Industrial Flexographic Printing Machine As described above, the active vibrations damping shall be realized using piezoelectric actuators in the bearings of the plate cylinder because these have a good dynamic and can work with higher frequencies as hydraulic actuators for example. The main challenges for the construction are given by • the fact that the piezoelectric actuators can generate high forces but only small displacements, • the need to avoid bending moments on the actuator and • the limited available space. In the following part of the paper, the construction of the active bearings for the industrial roller system (Fig. 4) will be presented with accent of the problematic driven side B (Fig. 5).

Fig. 4 Roller system

248

M.-C. Voicu et al.

Fig. 5 Test set-up

B

A

2.1 Active Bearing in General An active bearing contains two parts, the outer stator and the movable inner roller bearing. Because of the small deflection of piezoelectric actuators the stator has to be as stiff as possible and the clearance of the roller bearing should be as small as possible. Further on in the direction of motion there should be only little friction or elastic stiffness. This will reduce the actuator force on the roller. The leading in the direction, perpendicular to the moving direction should be free from clearance and stiff. In case of piezoelectric actuators, well designed flexible hinges fulfill these requirements and seem to be the best solution.

2.2 Boundary Conditions of the Bearing of the Industrial Flexographic Printing Device The weight of the plate cylinder roller is about 100 kg, the total length about 2,5m. The piezoelectric actuators create pressure forces up to 30 kN and displacements up to 200 μm. Two different active bearings for the roller system had to be designed as described in [8]. On the side A the actuator pressed directly on the plate cylinder as shown in Fig. 6. On the side B, the actuator has to be positioned above the roller bearing because of the small space between the rollers. This parallel offset produces moments and additional stress for the stator and the flexible hinges when applying the actuator forces. Tension and bending moments disturb the actuators and have to be constructively avoided. This technical solution is now under construction and is shown in Fig. 8. Figures 7 and 8 show the flexible hinge two flexible beams. The beams are integrated into the housing that is manufactured as one single part. There was no other possibility to transmit the high tension forces occurring in the beams when operating the actuator.


249

Fig. 6 Active bearing on the side A

Fig. 7 Active bearing on the side B

(a)

(b)

Fig. 8 a Load forces on the beam in the bearing; b Analysis of mechanical stress of the active bearing

250

M.-C. Voicu et al.

Figure 8a demonstrates the load forces of the beams. The design of the beams was very challenging because they had to fulfill conflicting requirements: On the one hand • The operating actuator causes high pressure and tension forces in the beams, which causes high stress in the junction between the connection of the beams • Buckling has to be avoided. On the other hand • Deflection of the movable bearing causes bending stress that gets larger with increasing thickness of the beams • The bending stiffness of the beam is too large - it should be smaller than 10% of the actor stiffness Finite element calculation and optimisation of the shape of the edges at the ends of the beams provided a practicable solution as shown in Fig. 8b.

2.3 Protection of the Actuator In order to protect the actuator from disturbance caused by bending moments, two perpendicular arranged cylindrical bearings are positioned at both ends of the actuator as shown in Fig. 9. The first solution with spherical contact has been abolished

Fig. 9 Cylindrical beams


251

because the Herzian stress would be too large or the radius of the spheres has to be so large that a small angular deviation of the actuator will cause an unallowable shift of the force application point causing bending moments. Tension on the actuator is not possible in this construction. In order to apply push-pull action to the roller, the actuator will be pre-loaded using saucer spring packages.

3 Small Scale Test Set-up A small scale test set-up of the flexographic printing plant was designed for testing active vibration damping with piezoelectric actuators in the bearings [7]. The downscale of the test bed was constructed so that the roller system has the same eigenfrequencies as the industrial test set-up and considering proportionality between the roller diameters. Figure 10a shows the test bed with the three rollers connected by the printing plate (3) applied on the plate cylinder (1). Strain gauges are applied in (2) and measure the mechanical strain at two axial positions.

1

2 (b)

3 2 (a)

5

4 (c)

(b)

(c)

Fig. 10 a Small scale test set-up; b Active bearing; c Stress analysis of the flexible part of the bearing

252

M.-C. Voicu et al.

Active bearing depicted in Fig. 9b (without electrical drive and coupling) has a piezoelectric actuator (4) integrated in the bearings in the direction to the nip and preloaded using saucer spring packages (5) to compensate the tensile forces. The roller bearings must be flexible to carry out small displacements without bearing shake. Figure 9c shows the stress analysis of the flexible part of the active bearing. The displacement of this part given by the piezoelectric actuator force is defined from the thin elastic beam which is bonded with the body housing of the bearing.

4 Simulation and Control The roller system can be described by the equations of motion as M¨x + D˙x + Kx = f

(1)

where M denotes the mass matrix, D the damping matrix, K the stiffness matrix, and f the excitation vector. In a roller system consisting of three rollers as shown in Fig. 1, each roller is considered as a continuum elastic beam. To obtain an infinite number of degrees of freedom, a roller of the length L is discretised into n elements with the length l=L/n by using finite element method. Each element has 2 translational and 2 rotational degrees of freedom. Given the roller geometry and material parameters such as Young’s modulus of elasticity and material density, the mass matrix M and stiffness matrix K can be derived [3, 5]. The rollers are coupled by the printing plate which is modeled as a discretised stiffness-damping system with the stiffness k and the Rayleigh-damping d for each translational degree of freedom. Furthermore, the stiffness of the roller bearings has to be considered. The final over-all stiffness matrix superimposes the contributions described. Considering the Rayleigh formula, the damping matrix D is calculated as being proportional with the mass matrix M and the stiffness matrix K. The equation of motion (1) was implemented in Matlab/Simulink to simulate the dynamic of the roller system. The open-loop simulation of the vibrations of the plate cylinder with n=20 discretisations and 42 DOF is presented in Fig. 11a. The excitation is a sinuswave force in the middle of the roller. The plots show the bending curve at a sequence of time steps. For control purposes, the curvature signals of the strain gauges placed on the roller are of special interest. At the real test bed described further below, the vibrations are measured with 8 strain-gauges connected in half-bridges. These are applied at 2 section areas on the roller. Figure 11b shows the curvature signals of the simulation model at the corresponding positions for two different amplitudes of disturbance forces. A control algorithm calculates the control signals for the actuators in the left and right bearing. Figure 12 shows good results of active vibration damping with piezoelectric actuators on the small scale test set-up for an excitation with a shaker connected with the roller.


253

−4 6 x 10

Fig. 11a Roller bending

Roller bending

4 2 0 −2 −4 −6

Fig. 11b Simulated strain gauges signals

0

0.2 0.4 0.6 0.8 1 Length/m

1

1.2 1.4 1.6

x 10−3

0.8 0.6

Curvature/m−1

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

50 100 150 200 250 300 350 400 450 500 Time/ms

0.06

Strain gauge signal / V

0.04 0.02 0 –0.02 –0.04 –0.06

0

0.2

0.4

0.6

0.8

1

time/s Fig. 12 Active vibration damping

1.2

1.4

1.6

1.8

2

254

M.-C. Voicu et al.

5 Conclusions and Outlook A new possibility to damp the vibration of roller systems in a printing machine was presented in this paper. The test-bed is currently reconstructed to enable active vibration damping under conditions close to industrial application. In particular, the bearings have to be modified to integrate the actuators and to enable small displacements. The active vibration damping was proved on a small scale test set-up which is currently stimulated by an electro-dynamical shaker. Further research activities will address active vibration damping on the rotational roller system and the results will be transferred on the industrial test-set-up. Acknowledgments The authors wish to thank EFRE with the Ministry for Science and Culture of the land Lower Saxony for the granted sponsorship.

References 1. Ehmann, C., Nordmann, R.: Comparison of Control Strategies for Active Vibration Control of Flexible Structures, Archives of Control Sciences, Vol. 13, 2003, No 3, pp 303-312 2. Ehmann, Ch.; Schönhoff, U.; Nordmann R.: Aktive Schwingungstilgung bei Portalfräsmaschinen mit Hilfe integrierter Piezoaktoren, VDI Tagung Schwingungen in Anlagen und Maschinen, Veitshöchheim, Mai 2001 3. Klein, B: FEM, Grundlagen und Anwendungen der Finite-Element-Methode, 3. Auflage, Vieweg Verlag, 1999 4. Voicu, M.C., Lammen, B., Schmidt, R., Hillbrand, H.-H., Maniu, I,: Messung der Anpressdrücke im Nip von Walzensystemen mit neuentwickelten piezoelektrischen Sensoren, VDI-Fachtagung Mechatronik 2011, Dresden, April 2011, pp 19-24 5. Waller, H., Schmidt, R: Schwingungslehre für Ingenieure, Theorie, Simulation, Anwendungen, Wissenschaftsverlag, 1989 6. Roloff/Matek: Maschinenelemente, Normung, Berechnung, Gestaltung, 18. Auflage, Vieweg Verlag, 2007 7. Sander, S.: Dimensionierung, Konstruktion und Inbetriebnahme eines Versuchsstandes zur aktiven Dämpfung von Walzenschwingungen, Bachelorarbeit, Hochschule Osnabrück, 2010 8. Mersch, M.: Konstruktion / Auslegung aktiver Lagerungen, Bachelorarbeit, Hochschule Osnabrück, 2010

On the Kinematics of a Hybrid Parallel Robot Used in Minimally Invasive Surgery M. Suciu, B. Gherman, C. Vaida, N. Plitea, A. Stoica, and D. Pisla

Abstract The paper presents some aspects about the kinematics of a hybrid parallel robot used in minimally invasive surgery: PARASURG 9M. The robotic structure PARASURG 9M consists of two modules: a hybrid parallel robotic arm with five DOF - PARASURG 5M, and a parallel robotic surgical instrument PARASIM with four DOF. After a short introduction in the approached field, the hybrid parallel structure PARASURG 9M is presented. Using the developed geometrical models, some numerical results of PARASURG 9M are presented. Keywords Kinematics · Hybrid parallel robot · Minimally invasive surgery (MIS) · Spherical parallel mechanism (SPM) · Simulation

1 Introduction Robots are being used in increasingly complex surgical procedures. However, these robots are not autonomous machines that carry out simple, pre-programmed instructions. Surgical robots are designed to supplement the surgeon’s abilities, translating human movements into incredibly steady and accurate robotic movements, which, in turn, manipulate instruments to aid delicate operations [7]. Robots are useful tools in MIS, due to their capabilities: reduction of the hand tremor, 3D visualization, workspace scaling, and 360 wrist rotations [16]. It has already been shown that the progress in engineering and medicine has opened the way for the use of robots in the operating rooms [9]. Intuitive Surgical Company designed a fully operational robot, da VinciTM Surgical System [4]. Jaspers et al. [5] have published a detailed presentation of camera and instrument holders for MIS. Despite their applicability in real surgeries, however, these systems are known to occupy a large space in the operating room and cause interference with surgeons and assistants. A solution for a surgical instrument is the RSPR robot, an innovative mechanism developed by N. Simaan in [13]. Other existing applications made for the D. Pisla (B) Technical University of Cluj-Napoca, Cluj-Napoca, Romania e-mail: [email protected]


255

256

M. Suciu et al.

orientation of the laparoscope are: PARMIS robot developed at CESTER [9, 15, 16], capable of positioning the laparoscopic camera in the surgical field by vocal commands; COVER Robot developed at Osaka University by K. Taniguchi [14]; KaLAR [6] developed at the Korea Advanced Institute of Science and Technology; LER endoscope holder [8]. A solution for a SPM is presented in [3]. As for the kinematics of SPM’s, Bai S. has developed a model of a spherical robotic wrist in [1] with Euler parameters. This wrist consists of three links moving on a circular guide, which allow the end-effector to rotate, in addition to a limited pitch and yaw movements. The paper is organized as follows: Section 2 presents the hybrid parallel robot. Section 3 deals with the description of the kinematical structure of the SPM and its CAD model. Section 3 presents the kinematics of the SPM. Some obtained numerical results are presented in section 4, followed by conclusions in Section 5.

2 The Hybrid Parallel Robot The PARASURG 9M robot is presented in Fig. 1, consisting of PARASURG 5M which was extensively discussed in [2, 10, 11, 12], and the surgical instrument PARASIM. The surgical system was conceived so that the connection of PARASURG-5M (which performs the positioning in the surgical field) and PARASIM (which performs the orientation) is possible. During the design process of the surgical instrument PARASIM, some requirements had to be considered: positioning accuracy of 0.1 mm, capable of speeds of 10 mm/s and accelerations of 5 mm/s2 at the tip of the active instrument;

PARASURG 5M

Detail A

Detail A

a)

b)

Fig. 1 a The surgical robot PARASURG 9M, b detail A: the robotic instrument PARASIM

On the Kinematics of a Hybrid Parallel Robot Used in Minimally Invasive Surgery

257

compactness in size and low weight; diameter of maximum 10 mm of the active instrument; modular and interchangeable construction. PARASURG 5M can also be an independent robot that can work as a laparoscope holder [2, 10], or in combination with PARASIM [11]. PARASURG 5M has a hybrid parallel structure with five DOF- three DOF given by a module with parallel kinematics, for instrument positioning in the surgical field, and two DOF given by a module based on a serial chain for its orientation with respect to point B, aiming to eliminate any pressure upon the abdominal wall in this point. PARASIM is a parallel robot, therefore the final structure PARASURG 9M is a hybrid parallel robot for MIS in abdominal procedures (e.g. cholecystectomy). The components of PARASIM can be observed in Fig. 1b): 1-actuation system of PARASIM included in a box, which is mounted on the mobile plate of PARASURG 5M; 2-the instrument outside surgical field; 3-the trocar; 4- surgical field; 5-the insertion point on patient (point B); 6- the instrument inside the patient; 7-the SPM, which for a better observation is scaled 1.5:1; 8-gripper actuation mechanism; 9-the end-effector; 10-section view trough the instrument. The studied SPM belongs to a class of spherical parallel mechanisms in which all the links rotation axes intersect in a single point located at the center of the mechanism. The actuation system 1 from Fig. 1b) is placed on the mobile plate of PARASURG 5M. It consists of four Ø6 mm, brushless, DC motors, and other transmission elements (belts and pulleys) in order to achieve the required 4 DOF: three rotations: q1, q2 and q3, corresponding to the orientation of the SPM and a translation q4, corresponding to the actuation of the gripper. The kinematic structure of the SPM (Fig. 2a) consists of three identical, closed kinematic chains. Each “i” kinematic chain (i=1,2,3) (Fig. 2b) has an active joint qi , the proximal link which is the active element (first one from the base up), a passive revolute joint located in Di , a distal link which is the passive element and the passive joint located in Ai on the mobile platform. The crucial condition for the mechanism

a)

Fig. 2 The kinematic scheme a of the SPM, b one chain

b)

258

M. Suciu et al.

is to maintain all its joints on a sphere while in movement. The actuation of the SPM and orientation of the end-effector are achieved through three concentric bores (component no. 10 in Fig. 1b), that start from the connection system (component no. 1 in Fig. 1b) and travel the entire length of the instrument, until they reach the SPM where each bore actuates one of the three kinematic chains.

3 Kinematics 3.1 Inverse Geometric Model The inputs will be the coordinates of the centre of the SPM, point G (XG , YG , ZG ) and the orientation angles of the segment GE (h1 in Fig. 3a) with respect to the instrument rod, where E represents the tip of the surgical instrument. In this way the positioning of the instrument is achieved strictly by PARASURG 5M while the orientation of the tip is achieved by PARASIM. The position and orientation coordinates XG , YG , ZG , ψ, θ and ϕ are given for the inverse geometric problem of the robot PARASURG 9M. The active joint coordinates of PARASURG 5M: q1 , q2 , q3 , and q4 , q5 , q6 for PARASIM have to be found. In order to reduce the computation time, the 4th and 5th active joint of the PARASURG -5M robot are not taken into consideration as part of the algorithm as their values do not influence the positioning coordinates, as it is shown in [2]. Attaching a mobile reference system with the Z axes oriented along the endeffector and the origin in the centre of the mobile platform formed by the triangle A1 A2 A3 , the coordinates of points Ai , E and G can be obtained. A second coordinate reference system GX’Y’Z’ is considered which has the Z’ axes along GA axes. In this case the input is the centre of the sphere: point G (XG , YG , ZG ), having to

a)

b)

Fig. 3 a The hybrid parallel robot structure, b Coordinate reference systems for E and G points


259

find the active coordinates: q1 , q2 , q3, q4 , q5 and q6 . There are two cases to be considered depending on the position of the robotic arm with respect to the fixed point: B (XB , YB , ZB ), the insertion point in the abdominal wall of the patient. Knowing that points G, B and A are collinear, the inverse geometric problem has two cases: Case 1: If XG=XB and YG=YB , the following is obtained: XA = XG , YA = YG , ZA = ZG − h, ψ = 0, θ = 0

(1)

Case 2: If XG=XB or YG =YB , it yields: XA = XG −h1 ·cos(ψGB )·sin(θGB ) ; YA = YG −h1 ·sin(ψGB )·cos(θGB ) (2) ZA = ZG + h1 · cos(θGB ) ⎧ ⎪ ⎪ ⎨ θGB ⎪ ⎪ ⎩

ψGB =⎛a tan 2 (YG − YB , XG − XB ) ⎞ 2 − Z ) − Z ) (Z (Z B G B G ⎠ = a tan 2 ⎝ 1 − , hh1 hh1

(3)

Where, hh1 =

(XG − XB )2 + (YG − YB )2 + (ZG − ZB )2

(4)

So for both cases, the coordinates of q1 , q2 , and q3 will be obtained: q1 = ZG + h1 · cos(θGB ) ; q2 = q1 +

(2 · b)2 − XA2 + YA2

(5)

q3 = a tan 2 (YA , XA ) The matrix of the guiding angles cosines between the mobile system oxyz and GX’Y’Z’ are obtained using Euler’s case of XYZ, matrix noted with R. The coordinates of points Ai , are: ⎡ ⎤ ⎤ xAi XAi ⎣ YAi ⎦ = R · ⎣ yAi ⎦ ZAi zAi + hs ⎡

(6)

Starting from the next geometrical relation: 2 2 2 XAi − XDi + YAi − YDi + ZAi − ZDi − l2 = 0 (7) But ZDi = r, XDi = r·cos qi+3 , YDi = r·sin qi+3 where, i=1, 2, 3 so it results:

260

M. Suciu et al. 2 2 XAi − 2rXAi cos(q4 ) + r2 + YAi − 2rYAi sin(q4 )+(ZAi − r)2 = l2

(8)

Knowing that cos2 (qi ) + sin2 (qi ) = 1, and using the notations: a = XA1 , b = YA1 and c = −

2 + Y2 (ZA1 − r)2 − l2 + r2 + XA1 A1 2r

(9)

The above equation (9) can be solved with the solutions for qi+3 : √ −b · c ± a · a2 + b2 − c2 ; sin(qi+3 )12 = a2 + b2 √ −a · c ∓ b · a2 + b2 − c2 cos(qi+3 )12 = a2 + b2

(10)

qi+3 = a tan 2(sin(qi+3) , cos(qi+3 )) , where, i=1, 2, 3.

(11)

So it results:

The coordinates of point E can be determined with the following relations: ⎧ ⎨ XE = XA + h·cos(ψGB )·sin(θGB ) − cos (α3 )·(ds + hs ) YE = YA + h·sin(ψGB )·sin(θGB ) − cos (β3 )·(ds + hs ) ⎩ ZE = q1 − h·cos(θGB ) − ds − hs + cos (γ3 )·(ds + hs )

(12)

3.2 Forward Geometric Model In this case, the coordinates of the active joints q1 , q2 , q3 , q4 , q5 , q6 , are known, and the pose XE , YE , ZE , ψ, θ and ϕ of the end-effector are unknowns to be found. In order to determine XE , YE , ZE , ψ, θ and ϕ, the orientation angles of the instrument ψ, θ and ϕ need to be determined. Knowing that the lengths of the arms Ai Di are constant, a system of three equations results: 2 2 2 Fi = XAi − XDi + YAi − YDi + ZAi − ZDi − l2 = 0

(13)

Knowing the coordinates of points Di , in the above equation, using NewtonRaphson method the values for ψ, θ and ϕ are determined. Knowing q3 , q4 , q5 , the following equations result: rA = (2·b)2 − (q2 − q1 )2 ; XA = rA ·cos (q3 ); YA = rA ·sin (q3 ); ZA = q1 And if XA = XB and YA = YB :

(14)


ψAB = θAB = 0; XG = XA ; YG = YA ; ZG = ZA − h1 ;

261

(15)

If not, ψAB = a tan 2(YB − YA , XB − XA ) ; θAB = a tan 2 (XA − XB )2 + (YA − YB )2 , (ZA − ZB ) ; (16) XG = XA + h1 · cos(ψAB ) · sin(θAB) ; YG = YA + h1 · sin (ψAB ) · sin (θAB ) ; ZG = ZA − h1 · cos(θAB ) ; The coordinates of point E will result from the equations: ⎧ ⎨ XE = XA + h·cos (ψGB )·sin (θGB ) − cos (α3 )·(ds + hs ) YE = YA + h · sin (ψGB )·sin (θGB ) − cos (β3 )·(ds + hs ) ⎩ ZE = q1 − h·cos (θGB ) − ds − hs + cos (γ3 )·(ds + hs )

(17)

4 Numerical Results R A program in Matlab [17] was developed to compute and validate the inverse and forward geometric models. The geometric parameters fitting to the experimental model which is going to be built are given: XB =950 mm; YB =0 mm; ZB =-10 mm; bs =4.4 mm; ds =80 mm; r=5 mm; hs =4.3 mm; h=450 mm; h1 =84.3 mm. The input data for the inverse geometrical model: XG =970 mm, YG =10 mm, ZG =-120 mm, ψ=4o , θ=8o , ϕ=10o . The obtained results for the coordinates of the active joints are: q1 =238.37 mm, q2 =721.51 mm, q3 =0o , q4 =-145.36o , q5 =67.13o , q6 =83.82o . This second set of results is used as input parameters for the forward geometric model. After running the forward geometric model with Newton-Raphson numerical method, the following results have been obtained: ψ=4o , θ=8o , ϕ=10o , and the coordinates of point G are: XG =970 mm, YG =10 mm, ZG =-120 mm, the same as the initial input values. In each model the coordinates of the instrument tip have the same values: XE =973.28 mm, YE =23.33 mm, ZE =-203.63 mm. So it can be concluded, that the inverse and direct geometric models are numerically verified. Using the developed inverse geometric algorithm, a Matlab program has been developed regarding the workspace generation of the robot, and the robot’s endefector covers a volume of approximately: 300 mm x 300 mm x 300 mm; enough to investigate an average human belly.

5 Conclusions A new hybrid parallel robot used in MIS was presented in this paper. The inverse and forward geometric models of PARASURG 9M are presented and the numerical verification of these models are also described. A further step in the research is to

262

M. Suciu et al.

identify the robot possible singularities and to develop the experimental model with the adequate control of the robot. Acknowledgements This paper was supported by the project “Development project for doctoral studies in advanced technologies – “PRODOC” contract no. POSDRU/88/1.5/S/61178, project cofunded by the European Social Fund through the Sectorial Operational Program Human Resources 2007-2013.The research work was financed by PNCDI-2 P4 Grant: “Innovative development of a virtual e-learning system for hepatic laparoscopic surgery - HEPSIM” No. 92080/2008.

References 1. Bai, S., and Hansen, M.: Modeling of a Spherical Robotic Wrist with Euler Parameters, 12th IFToMM World Congress, Besancon, (2007) 2. Gherman, B.: Contributions to the Development of Kinematic, Dynamic and Functional Models for a Innovative Parallel Hybrid Structure for MIS, Phd Thesis, Cluj-Napoca, (2011) 3. Gosselin, C., and Angeles, J.: The optimum kinematics design of a spherical 3 DOF parallel manipulator, ASME J. of Mechanical Design, pp 202-207, (1989) 4. http://www.intuitivesurgical.com, Intuitive Surgical, (2011) 5. Jaspers, J. et. al.: Camera and Instrument Holders and Their Clinical Value in Minimally Invasive Surgery. Surg Laparosc Endosc Percutan Tech, (2004) 6. Kim, J., et. al.: Compact Camera Assistant Robot for MIS : KaLAR, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and systems, pp. 2587-2592 Sendai, (2004) 7. Lobontiu, A. and Loisance, D.: Robotic Surgery and Tele-surgery: Basic Principles and Description of a Novel Concept, Surgery Journal, Volume 3, pp. 208-214 (2007) 8. Long, J., et. al.: Development of the Miniaturized Endoscope Holder LER (Light Endoscope Robot) for laparoscopic surgery, Journal of Endourology 21, (2007) 9. Pisla, D., et. al.: PARAMIS Parallel Robot for Laparoscopic Surgery, Chirurgia Journal, Chirurgia 105: Nr. 5, pp. 677-683, (2010) 10. Pisla, D., et. al.: Kinematics and Design of a 5-DOF Parallel Robot used in Minimally Invasive Surgery, ARK 2010, pp 99-106, (2010) 11. Pisla, D., et. al.: PARASURG Hybrid Parallel Robot for minimally invasive surgery, Chirurgia (Bucur), Vol. 106, No. 5, pp. 619–625, (2011) 12. Plitea , N., et. al.: Robot chirurgical. Patent no. a 00525/7.07.2009, Romania (2009) 13. Simaan, N. and Shoman, M. : Robot Construction for Surgical Applications, Second Israeli symposium on computer-aided surgery, Medical Robotics, and Medical Imaging, (1999) 14. Taniguchi, K., et. al.: COVER: Compact Oblique-viewing Endoscope Robot for Laparoscopic Surgery, Journal of Computer Assisted Radiology and Surgery, pp. 207-209, (2006) 15. Vaida et. al.: Graphical simulation of a new concept of low sized surgical parallel robot for camera guidance in minimally invasive surgery, in: PAMM, Vol. 7, published online (2008) 16. Vaida, C.: Contributions to the Development and Kinematic-Dynamic Modeling of Parallel Robots for MIS, Phd Thesis, Cluj-Napoca, (2009) 17. www.mathworks.com/products/matlab/, MathWorks Inc., Matlab, (2011)

Part V

Mechanical Transmissions


xii

Synthesis of Toothed Continuously Variable Transmission (CVT) K.S. Ivanov

Abstract Continuously variable transmission contains the closed gear differential and the controlled device. It is proved that closed gear differential in itself carry out a function of a controlling device. The elementary gear (toothed) continuously variable transmission has form of the gear closed differential with two degrees of freedom. The gear differential with the closed contour of toothed wheels creates additional constraint and supports a transition regime of motion from a start-up to bi-mobile regime with independent stepless regulating. The equations of interconnection of power, kinematic and geometrical parameters of transmission in statics, kinematics and dynamics are searched out. The synthesis of transmission consists in determination of teeth wheels numbers on a condition of overcoming the given maximum moment of resistance. The design of the gear regulated transmission is presented. Design of gear continuously variable transmission is presented. The found regularities allow synthesizing gear continuously variable transmission with constant engagement of toothed wheels on the given operational parameters of motion. Keywords Gear transmission · Closed contour · Equilibrium · Stepless regulating

1 Introduction At last time a continuously variable transmission (CVT) which contains the closed gear differential mechanism with two degrees of freedom and the controlling device has been extended and patented. The controlling device imposes an additional constraint leading to definability of motion and self-regulating. Transmission has only one input (one engine) and meets the requirement of theoretical mechanics and theory of mechanisms and machines: number of initial links (or input links) of a mechanism should be equal to number of degrees of freedom [1, 2].

K.S. Ivanov (B) Almatinsky University of Power Engineering and Telecommunications, Almaty, Kazakhstan e-mail: [email protected]


265

266

K.S. Ivanov

In designs [3, 4] the engine transmits a motion to fluid converter which drive two links of gear differential with two degrees of freedom. The fluid converter creates an additional hydrodynamics constraint which realizes a self-regulating. An output link of the differential has definability of motion with stepless self-regulating. However the closed contour of the gear differential mechanism in itself is capable to impose additional constraint on motion of links. This property is used in the patent [5]. The transmission has self-regulation in operating regime with two degrees of freedom. However in this transmission the brake of one of wheels carries out an additional constraint for the start. The elementary designs of transmission in form gear differential mechanism [6, 7] create additional constraint by way of self-braking and inertial properties of the mechanism. In all patented designs of continuously variable transmission the closed contour is used. However the theoretical description of interacting parameters of transmission [3, 4, and 5] is executed without taking into account presence of two degrees of freedom. In works [6, 7] an interacting of parameters of gear differential is presented on the basis of a principle of virtual works. However in them there is no theoretical substantiation of a technique of research. Earlier in works [8, 9, 10, 11] the opportunity of transfer motion in a gear differential by means of the closed contour has been researched. The analytical description of transfer motion in the gear mechanism with two degrees of freedom has been executed on the basis of a principle of virtual works. It has been shown, that the closed contour imposes additional constraint on motion of links that leads to definability of motion of the kinematic chain with two degrees of freedom at presence only one input. However in these works there is no theoretical substantiation of an opportunity of use of a principle of virtual works. The purpose of the present work is to prove theoretically a definability of motion of gear differential with two degrees of freedom at presence only one input and to create analytical dependences of interconnection parameters of motion of gear continuous variable transmission with stepless self-regulation. Researches are executed on the basis of laws of theoretical mechanics and theory of mechanisms and machines.

2 Transfer of Motion in the Closed Differential Mechanism with Two Degrees of Freedom Subject of research is the closed toothed differential mechanism with two degrees of freedom which represents a continuously variable transmission. The toothed continuously variable transmission (Fig. 1) has form of the closed differential mechanism. Transmission contains frame 0, input carrier H1 , input satellite 2, block of central toothed wheels with external teeth 1 - 4, block of central toothed wheels (ring wheels) with internal teeth 3 - 6, output satellite 5 and output carrier H2 .

Synthesis of Toothed Continuously Variable Transmission (CVT)

267

6 3 5 2 H2

H1 0

1

4

Fig. 1 Toothed continuously variable transmission

Toothed wheels 4-1, 2, 3-6, 5 form the closed contour. On the input carrier H1 the external active motive moment MH1 acts. On an output link H2 the external active moment of resistance MH2 acts. In engagement of wheels we shall consider only horizontal components of reactions as vertical components are perceived by a frame. Forces and displacements (speeds) are parallel a horizontal axis Ox. Let’s accept following designations: rH1 , rH2 - radiuses of input and output carriers, ri (i = 1, 2...6) - radiuses of toothed wheels, MH1 , MH2 - moments on input and output carriers, Mi (i = 1, 2...6) - moments on toothed wheels, FH1 = MH1 /rH1 - input motive force, RH2 = MH2 /rH2 - output force of resistance, R12 = M1 /r1 , R32 = M3 /r3 , R45 = M4 /r4 , R65 = M6 /r6 - reactions in kinematic pairs. If a rest friction takes place in kinematic pairs of a closed contour with toothed wheels 1, 2, 3, 6, 5, 4 then all wheels rotate round the central axis as rigid system with one degree of freedom.

3 Interconnection of Parameters of the Kinematic Chain in a Condition with Two Degrees of Freedom First we shall consider a particular case of motion. We shall consider equilibrium of the all kinematic chain with two degrees of freedom on conditions of statics. Equilibrium occurs at equality of the moments of superposed forces MH2 = MH1 .

(1)

We shall consider equilibrium of the closed contour under act of the superimposed forces FH1 = MH1 /rH1 and RH2 = MH2 /rH2 .

268

K.S. Ivanov

From the input satellite 2 on intermediate links 1-4 and 3-6 reactions R21 = FH1 /2, R23 = FH1 /2 are transferred. From the output satellite 5 on intermediate links 1-4 and 3-6 reactions R54 = RH2 /2, R56 = RH2 /2 are transferred. Let’s express force of resistance RH2 through motive force FH1 . From the formula (1) follows RH2 = FH1 rH1 /rH2 . Then R54 = FH1 rH1 /2rH2 , R56 = FH1 rH1 /2rH2 . As a result the link 1-4 will appear under act of the unbalanced moment M1−4 = R21 r1 − R54 r4 =

FH1 2

rH2 r1 − rH1 r4 rH2

.

(2)

.

(3)

The link 3-6 will appear under act of the unbalanced moment M3−6 = R23 r3 − R56 r6 =

FH1 2

rH2 r3 − rH1 r6 rH2

Let’s substitute in formulas (2), (3) values rH1 = (r1 + r3 ) /2, rH2 = (r4 + r6 ) /2 we shall gain M1−4

FH1 = 2

r1 r6 − r3 r4 r4 + r 6

, M3−6

FH1 = 2

r3 r4 − r1 r6 r4 + r6

.

From here follows M1−4 = −M3−6 .

(4)

Thus internal forces are led to unbalanced forces and moments on links 1-4 and 3-6. The condition of equilibrium of internal forces by a principle of virtual works in view of a time is expressed as M1−4 ω1 + M3−6 ω3 = 0.

(5)

As M1−4 = −M3−6 , ω1 = ω3 . In this case angular speeds of all links are equal, and the kinematic chain is twirled as a single whole without internal mobility of links. Thus, the kinematic chain is in the equilibrium representing a special case when there is no motion of links inside of a contour. Thus equilibrium is carried out both on conditions of statics, and by a principle of virtual works. Further we shall consider a common case of motion. It is possible to assume, that in common case at presence of two degrees of freedom in the kinematic chain with the closed mobile contour equilibrium by a principle of virtual works also will occur.


269

4 Theorem If the kinematic chain with two degrees of freedom with the closed four-bar contour moves under action of the arbitrary driving moment and the arbitrary moment of resistance for these external moments of forces the condition of equilibrium by a principle of virtual works is carried out MH1 ωH1 + MH2 ωH2 = 0.

(6)

The main property closed four-bar contour is: all internal reactions in a contour can be expressed through superposed forces. (For comparison we shall note: it is impossible to express all internal reactions through the superposed forces applied to two links of a contour if the contour contains, for example, five mobile links). Let’s make for each satellite an equilibrium equation by a principle of virtual works. In view of a time we shall gain M1 ω1 + M3 ω3 = MH1 ωH1 , M4 ω4 + M6 ω6 = MH2 ωH2 .

(7) (8)

As satellites are a links of the mechanism, we shall sum the made expressions for satellites. We shall gain a condition of interacting of parameters of all mechanism as a whole M1 ω1 + M3 ω3 + M4 ω4 + M6 ω6 = MH1 ωH1 + MH2 ωH2 .

(9)

In the left side of an equation the sum of powers (works) of internal forces of a contour takes place. In the right side of an equation the sum of powers (works) of superposed forces of a contour takes place. All internal forces are single-valued functions of external forces. As constraints are ideal and stationary then work of constraint reactions is equal to null [1] M1 ω1 + M3 ω3 + M4 ω4 + M6 ω6 = 0.

(10)

Let’s substitute this value in the equation (9), we shall gain MH1 ωH1 + MH2 ωH2 = 0.

(11)

Thus, for the external moments of forces the condition of equilibrium by a principle of virtual works, as was to be shown is carried out. From the formula (11) in view of a moments signs we shall gain ωH2 = M

H1 ωH1 /M H2 .

(12)

According to the formula (12) at a constant input power PH1 = MH1 ωH1 the output angular speed ω H2 is in return proportional dependence on a variable output

270

K.S. Ivanov MH2

H2

Fig. 2 Dependence of torque from angular speed of output shaft

moment of resistance MH2 . This dependence expresses effect of force adaptation of an output link to a variable load. The closed contour with tooth gears has surprising property to supply definability of motion and power adaptation to variable technological load. There is no necessity to use any control system for change a transfer ratio. On Fig. 2 the diagram of dependence of torque from angular speed which corresponds to the formula (22) is presented. The output angular speed of output shaft ωH2 is inverse to drag torque MH2 on the output shaft at constant parameters of input power MH1 , ωH1 . That is ωH2 = MH1 ωH1 /MH2 . This relation is ideal towing performance for the machine with variable technological resistance In the closed contour circulation of energy takes place. The equation (10) contains positive and negative members and characterizes equilibrium of works (powers) on intermediate links of a contour. As ω1 = ω4 , ω3 = ω6 , M1 > M4 , M6 > M3 , from the equation (10) we shall gain (M1 − M4 )ω1 − (M6 − M3 )ω3 = 0.

(13)

Let’s designate (M1 − M4 ) = M1−4 , M6 − M3 = M6−3 - The moments on intermediate links 1-4 and 3-6. Then the equilibrium of works (powers) on intermediate links of a contour will become M1−4 ω1 = M6−3 ω3 . The equation (13) reflects unknown earlier the phenomenon of circulation of energy inside of a contour during its motion.

5 Synthesis of Mechanism Gear CVT Problem of a synthesis: at the set maximum transfer ratio umax = MH2 max /MH1 to define radiuses and numbers of teeth of wheels. Selection of numbers of t is carried out on a condition of a stop of the output shaft of the mechanism under act of the maximum moment of resistance MH2 max . The maximum moment of resistance matches to emersion in the closed contour of the maximum internal moment of resistance on the block of wheels 1-4 or 3-6. It is definable the transfer ratio of the closed gear differential without taking into account a friction as u = MH2 /MH1 . Or throw geometrical parameters


rH2 u= 2rH1

r1 r3 + r4 r6

271

.

(14)

In view of r1 = rH1 − r2 , r3 = rH1 + r2 we shall gain rH2 1 − a 1 + a . + 2 r4 r6 1 2ur4 r6 . Here a = r2 /rH1 . Then a = r6 + r4 − rH2 r6 − r4 From here in view of r6 + r4 = 2rH2 , r6 − r4 = 2r5 we shall gain u=

a=

2 − ur r rH2 4 6 . rH2 r5

(15)

(16)

Formula (16) allows determining relative parameter a of first planetary row on the set parameters r4 , r6 , r5 , rH2 of second planetary row and the set maximum transfer ratio. On the found parameter a of the first planetary row it is possible to determine all parameters of the first planetary row r1 , r2 , r3 , having set radius of the entrance carrier rH1 . The number of teeth of a wheel is determined through the module m and radius under the known formula z = 2r/m.

6 Conclusions The creation of gear continuously variable transmission in the form of the toothed closed differential mechanism with two degrees of freedom is theoretically proved. It is proved, that the mobile closed mechanical contour of transmission creates additional constraint and provides the transition regime of motion translating the mechanism of transfer from an one-mobile condition at start-up in bi- mobile condition in operating regime of motion. It is proved, that in operating regime of motion the equilibrium by a principle of virtual works, providing stepless regulating of transmission takes place. The equations of interconnection of force parameters, kinematic parameters and geometrical parameters of transmission in kinematics and dynamics are developed. These equations are mustered by numerical instances, confirmed by skilled acting specimens and the developed computer animation model. The assembly drawing of the gear continuously variable transmission is presented. The found regularities allow synthesizing gear continuously variable transmission on the given operational parameters of motion, to execute the kinematic and dynamic analysis of transmission and to determine design data of transmission. Gear continuously variable transmission in the form of the gear closed differential mechanism with constant engagement of wheels is the elementary transmission of such type and it has the reliability matching reliability of a gear mechanism.

272

K.S. Ivanov

The specified properties allow using the transmission both in easy local drives of manipulators, and in heavy drives of transport machines, including in the motor-wheels. The gear closed differential mechanism of transmission possesses effect of force adaptation to variable technological loading. Force adaptation allows creating the easy and hard loaded adaptive drives of machines with the variable transfer ratio depending on technological resistance (for velocipede, motorcycle, car, drilling rig, bulldozer, lorry, etc.).

References 1. Markeev A.P. Theoretical mechanics. Moscow. Science. 1990. 414 p. 2. Levitskiy N.I . Theory of mechanisms and machines. Moscow. Science. 1979. 576 p. 3. Samuel J. Crockett. Shiftless continuously-aligning transmission. Patent of USA 4,932,928, Cl. F16H 47/08, U.S. Cl. 475/51; 475/47.1990, 9 p. 4. Volkov I.V. Way’s of automatic and continuous change of a twisting moment and speed of twirl of the output shaft depending on a tractive resistance and the device for its realization. The description of the invention to the patent of Russia RU 2 234 626 from 3.27.2004. 26 p. 5. Harries John. Power transmission system comprising two sets of epicyclic gears. Patent of Great Britain GB2238090 (A.) 1991, 11 p. 6. Ivanov K.S. Transfer with automatically adjustable speed. The preliminary patent of republic Kazakhstan 3208 from 3/15/1996. 12 p. 7. Ivanov K.S., Yaroslavtseva E.K. Way of automatic and continuous change of a twisting moment and speed of twirl of the output shaft depending on a tractive resistance and the device for its realization. The description of the invention to the patent application of Russia RU 2007139258/11 (042965), the solution on output of the patent from March, 25th, 2010. 28 p. 8. Ivanov K.S. The Question of the Synthesis of Mechanical Automatic Variable Speed Drives. Proceedings of the Ninth World Congress on the Theory of Machines and Mechanisms, Vol.1, Politechnico di Milano, Italy, August 29/Sept 2, 1995. P. 580 - 584. 9. Ivanov K.S. Discovery of the Force Adaptation Effect. Proceedings of the 11th World Congress in Mechanism and Machine Science. V. 2. April 1 - 4, 2004, Tianjin, China. P. 581 - 585. 10. Ivanov K.S. Gear Automatic Adaptive Variator with Constant Engagement of Gears. Proceedings of the 12th World Congress in Mechanism and Machine Science. Besancon. France. 2007, Vol. 2. P. 182-188. 11. Ivanov K.S. The simplest automatic transfer box. WCE 2010. World Congress on Engineering. 201 (ICME). London, UK. 2010. P. 1179 – 1184.

Differential Planetary Gear Transmissions Usable in Renewable Energy Systems R. Saulescu, C. Jaliu, D. Ciobanu, and D. Diaconescu

Abstract Many of the renewable energy systems (RES) use planetary gears for the speed increase or reduction. The paper presents a 2 DOF planetary gear transmission to be used as “speed increaser” in the counter-rotating wind turbines, hydropower plants, and, also as “speed reducer”, in the tracking systems that equip the solar photovoltaic or solar thermal systems. The transmission modelling, simulation and examples of applications in the field of renewable energy systems is presented in the paper. Keywords Renewable energy system (RES) · Planetary gear · Counter-rotating

1 Introduction Increased fossil fuel prices, reduced fossil fuel stocks and concern for specialists to minimize emissions of greenhouse effect gas have led to increased interest for using renewable energy sources. The main renewable energy source is the sun. The direct conversion of solar radiation into electricity can be done by using photovoltaic panels [2]. Thermal energy can be obtained from solar radiation by using solar collectors. Using the indirect conversion, solar radiation can be transformed into electricity be means of wind turbines or hydropower plants [1,9,16,17]. In case of solar collectors or photovoltaic panels, because of the sun position change on the sky, these systems use tracking mechanisms to increase the amount of solar radiation. Tracking systems are classified by their motions: rotation around one axis or around two axes; they are composed of linkages or gear mechanisms driven by linear actuators or gear-motor. In case of wind turbines and hydropower plants, a gearbox can be used to transmit motion from rotor / turbine to generator [6,7,8,10,11,12,13]. Three basic types of gearboxes are used: belt transmissions [9], parallel-shaft gears [15,17] and 1 DOF planetary gears [1,6]. In the first two cases, the dimensions and cost increase

R. Saulescu (B) Transilvania University of Brasov, Brasov, Romania e-mail: [email protected]


273

274

R. Saulescu et al.

with the multiplication ratio, while in case of planetary transmissions, they are light and compact, but with a higher complexity than the other transmissions. The paper presents and analyzes a new solution of speed reducer / increaser for RES, which overcomes some of the previous problems. The 2 DOF planetary transmission, with two inputs and one output, connects 2 motors / rotors / turbines (coaxial or parallel) to a reducer / generator. The counter-rotating system runs based on the differential gear property of summing two input motions into an output motion. The efficiency and compact design are the most important advantages of these systems vs. the classical solutions. So far, the counter-rotating systems have been proposed mainly for the marine propulsion and aircraft (Figs. 1 and 2); thus, the transmission from Fig. 1 [17] is used in transmission of power from a drive shaft to propellers. The motion from the motor is transmitted through the two rows of paddles for propulsion. The running values of transmission speeds and torques depend on the mechanical characteristics of the engine and propellers, and, also, on the internal gear ratio of the planetary set. Firstly, the kinematical and dynamic parameters of the planetary transmission are established. Then, the synthesis of the number of teeth is made for particular conditions of RES. The models of the counter-rotating systems are built using modern CAD/CAE software (Dassault Systemes CATIA, AutoDesk Inventor). The numerical simulation results are analyzed and recommendations concerning the use of counter-rotating systems are settled up.

Propeller

Motor

2 Motor H 1

Propeller

Fig. 1 Planetary transmission used to propel boats [17]

Fig. 2 Exploded view of a planetary gear motor used to propel boats

3

Differential Planetary Gear Transmissions Usable in Renewable Energy Systems

275

2 Speed and Efficiency Analysis The block diagrams and structural schemes of the planetary gears to be used in RES are presented in Fig. 3, representing the starting point for the kinematical and dynamic analysis. Thus, the schemes from Fig. 3a belong to the gear transmission that has the following input elements: the ring gear -3 and the carrier H, and as output - the sun gear 1, while the planetary gear from Fig. 3b has as input elements: the sun gear 1 and carrier H, and output - the external ring gear 3 [3,4]. The structural schemes for two planetary gears which meet the above block diagrams are presented in Fig. 4. Based on these variants, Table 1 presents the area where the speed ratio takes values for different inputs – output combinations. The following notations are used in Table 1: i13 is the speed ratio between the gears 1 and 3; i0 is the interior speed ratio (when H is considered blocked). The version, where the gears 1 and 3 are input elements, is not possible because the ratio i13 is considered negative and the powers ω1 T1 and ω3 T3 can be not concomitantly positive for the negative values of the ratios i0 and i13 .

ω3

3

H

T3

ωH

ω1

TH

T1

1

ωH TH

1

ω1

H

3 ω3 T3

T1

3 2 H 1

a)

b)

Fig. 3 Block diagrams and structural schemes – front views, for the planetary transmission with: 3 and H (ωH TH > 0, ω3 T3 > 0) input shafts (a) and 1 and H (ωH TH > 0, ω1 T1 > 0) input shafts (b)

Table 1 The value of the ratio between the sun gears speeds for the planetary transmission from Fig. 4 i0 < 0

i13 < 0

H i 0 = i 13 , i 13 =

Input shafts

3&H

1&H

1&3

Output shaft

1

3

H

If i0 > 1 then 1 is sun gear and 3 is the ring gear; Obs.

If i0 < 1 then 3 is sun gear and 1 is the ring gear.

ω1 ω3

H ωH

1 ω1 3 ω 3

276

R. Saulescu et al. 2

2

3 or 1

2’

H 1 or 3

3 or 1 1 or 3 a)

H b)

Fig. 4 The structural schemes for the planetary transmission with: 1 planet gear (a) and 2 planet gears (b)

Considering the kinematical schemes from Fig. 3a and the running cases from Table 1, the following relations can be written [5,11, 14]: ω1 = (1 − i0 )ωH + i0 ω3 , T1 + T3 + TH = 0, (ω1 − ωH ) T1 η0x + (ω3 − ωH )T3 = 0. ω1 T1 + (ω3 T3 + ωH TH )η = 0;

ω3 T3 + (ω1 T1 + ωH TH )η = 0

(1) (2), (3) (4), (4’)

Eq. (1) represents the transmission function for angular speeds, eq. (2) is the torque equilibrium equation, eq. (3) is the power equilibrium equation for the fixed axes transmission that is associated to the 2 DOF planetary gear (the transmission with a fixed carrier) and equations (4), (4’) are the power equilibrium equations of the planetary transmission with 3&H inputs and respectively 1&H inputs. In relation (3), the exponent “x”, which is associated to the internal efficiency η0 (the efficiency of the gear mechanism with fixed axes derived from the planetary gear) specifies the role of shaft 1 (driving or driven) in the fixed axes mechanism; “x” can take the values ±1, as follows: x = + 1 when (ω1 -ωH )T1 > 0 (1 - input), or x = -1 when (ω1 -ωH )T1 < 0 (1 - output)[14]. In eq. (4) and eq. (4’), η represents the efficiency of the planetary transmission in the considered cases. The relations for the planetary gear output speed and efficiency in the two functioning cases from Table 1 are obtained from relations (1). . .(3), being centralized in Table 2, [3,4,10]. In Table 2, i0 represents the internal torque ratio and it is equal to the product between the internal efficiency η0 (usually, η0 ≈ 0.99 for a gear pair) and internal gear ratio i0 . Based on the equations from Table 2, the diagrams from Fig. 5 represent the variation of the speed ratio between the input elements vs. the speed ratio between the gears 1 and 3 (i13 ), for both running cases (Fig. 3a, b) and different values of the interior transmission ratio i0 (the values used in the example presented in the paper, |i0 | = 1 . . . 9, correspond to the range in which the speed is increased in renewable energy systems [9,12]).

Differential Planetary Gear Transmissions Usable in Renewable Energy Systems

277

Table 2 The output motion and the efficiency for the planetary transmission, in 2 running cases Input shafts

3 and H

1 and H

Output shaft

1

3

Motions

ω1 = (1 − i0 )ωH + i0 ω3

ω3 =

ω1 − i0

x

−sgn

−sgn

i13 − 1 1 − i0

Efficiency

i13 (1 − i0 ) i0 (1 − i0 ) + 1 − i0 (i13 − i0 )

–1

–3

–5

–7

–9

i0 (1 − i13 ) i13 (1 − i0 )

–11 –13 –15 –17 –19

0

i13

i0 (1 − i0 ) i13 (1 − i0 ) − 1 − i0 (i13 − i0 )

0

–0.2

–0.4

–0.6

0

–0.8

–1 i13

–0.25

–0.25 –0.5

–0.5

–0.75

–0.75 –1

–1

i1H (i0= –9)

–1.25

–1.25 –1.5 –1.75

(1−i0 )ωH i0

i3H (i0= –9)

i3H (i0= –7)

i3H (i0= –3)

i3H (i0= –1)

i3H

i3H (i0= –5)

i1H (i0= –7)

–1.5 –1.75

i1H (i0= –5) i1H

i1H (i0= –3) i1H (i0= –1)

–2

–2

a)

b)

Fig. 5 Speed increaser case: variations of the speed ratio between the input elements vs. the speed ratio i13 for different values of i0 when: 3 (ring gear) and H are inputs and 1 (sun gear) is output (a) and 1 (ring gear) and H are inputs and 3 (sun gear) is output (b)

According to the observation from Table 1, the two diagrams from Fig. 5 represent the same physical phenomenon: 1 is the sun gear and 3 the ring gear in Fig. 5a (|i0 |>1) while in Fig. 5b 3 is the sun gear and 1 – the ring gear (|i0 |

Mechanisms, Transmissions and Applications

Transmissions

Cryptography and Security Services: Mechanisms and Applications

Cryptography and Security Services. Mechanisms and Applications

Cryptography and Security Services: Mechanisms and Applications

Transmissions et reseaux

Homogeneous Catalysis: Mechanisms and Industrial Applications

Homogeneous catalysis: mechanisms and industrial applications

Networks and Devices Using Planar Transmissions Lines

The Starseed Transmissions

The Starseed Transmissions

Hypothermia and Cerebral Ischemia: Mechanisms and Clinical Applications

Automotive Transmissions: Fundamentals, Selection, Design and Application

Automotive Transmissions: Fundamentals, Selection, Design and Application

Neuroinflammation: Mechanisms and Management

Nanomaterials: Mechanics and Mechanisms

Neuroinflammation: Mechanisms and Management

Syncope: Mechanisms and Management

Hypertension and Hormone Mechanisms

Nanomaterials: Mechanics and mechanisms

Principles of Osteoimmunology: Molecular Mechanisms and Clinical Applications

The Marlowe Transmissions- Scavenger's War

Controlled and Living Polymerizations: From Mechanisms to Applications

Coatings Tribology Properties Mechanisms Techniques and Applications in

Chiral Recognition in Separation Methods: Mechanisms and Applications

Hot Thought: Mechanisms and Applications of Emotional Cognition (Bradford Books)

Towards a Theory of Educational Transmissions: Class, Codes and Control

Sensory Mechanisms

Photographic Sensitivity: Theory and Mechanisms

Ocular Disease: Mechanisms and Management

Mechanisms, Transmissions and Applications

Transmissions

Cryptography and Security Services: Mechanisms and Applications

Cryptography and Security Services. Mechanisms and Applications

Cryptography and Security Services: Mechanisms and Applications

Transmissions et reseaux

Homogeneous Catalysis: Mechanisms and Industrial Applications

Homogeneous catalysis: mechanisms and industrial applications

Networks and Devices Using Planar Transmissions Lines

The Starseed Transmissions

The Starseed Transmissions

Hypothermia and Cerebral Ischemia: Mechanisms and Clinical Applications

Automotive Transmissions: Fundamentals, Selection, Design and Application

Automotive Transmissions: Fundamentals, Selection, Design and Application

Neuroinflammation: Mechanisms and Management

Nanomaterials: Mechanics and Mechanisms

Neuroinflammation: Mechanisms and Management

Syncope: Mechanisms and Management

Hypertension and Hormone Mechanisms

Nanomaterials: Mechanics and mechanisms

Principles of Osteoimmunology: Molecular Mechanisms and Clinical Applications

The Marlowe Transmissions- Scavenger's War

Controlled and Living Polymerizations: From Mechanisms to Applications

Coatings Tribology Properties Mechanisms Techniques and Applications in

Chiral Recognition in Separation Methods: Mechanisms and Applications

Hot Thought: Mechanisms and Applications of Emotional Cognition (Bradford Books)

Towards a Theory of Educational Transmissions: Class, Codes and Control

Sensory Mechanisms

Photographic Sensitivity: Theory and Mechanisms

Ocular Disease: Mechanisms and Management

Recommend Documents