Computer Aided Design / Computer Aided Manufacturing (CAD/CAM)
Computer Hided and Integrated Manufacturing Systems fl S-Volume Set Cornelius
T Leondes
Vol.4 Computer Aided Design / Computer Aided Manufacturing (CAD/CAM)
Compurer Hided m Integrated Monuficruring Siisfems fl S-Volume Ser
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Vol.4 Computer Aided Design / Computer Aided Manufacturing (CAD/CAM)
C o m p u t e r A i d e d and Integrated Manufacturing Systems H S-Volume Set
Cornelius T Leondes Umrnly of California, Los Angeks, USA
fj|)p World Scientific NEW JERSEY • LONDON • SINGAPORE • SHANGHAI • HONGKONG • TAIPEI * BANGALORE
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
COMPUTER AIDED AND INTEGRATED MANUFACTURING SYSTEMS A 5-Volume Set Volume 4: Computer Aided Design/Computer Aided Manufacturing (CAD/CAM) Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-238-339-5 (Set) ISBN 981-238-980-6 (Vol. 4)
Typeset by Stallion Press
Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore
Preface
Computer Technology This 5 volume MRW (Major Reference Work) is entitled "Computer Aided and Integrated Manufacturing Systems". A brief summary description of each of the 5 volumes will be noted in their respective PREFACES. An MRW is normally on a broad subject of major importance on the international scene. Because of the breadth of a major subject area, an MRW will normally consist of an integrated set of distinctly titled and well-integrated volumes each of which occupies a major role in the broad subject of the MRW. MRWs are normally required when a given major subject cannot be adequately treated in a single volume or, for that matter, by a single author or coauthors. Normally, the individual chapter authors for the respective volumes of an MRW will be among the leading contributors on the international scene in the subject area of their chapter. The great breadth and significance of the subject of this MRW evidently calls for treatment by means of an MRW. As will be noted later in this preface, the technology and techniques utilized in the methods of computer aided and integrated manufacturing systems have produced and will, no doubt, continue to produce significant annual improvement in productivity — the goods and services produced from each hour of work. In addition, as will be noted later in this preface, the positive economic implications of constant annual improvements in productivity have very positive implications for national economies as, in fact, might be expected. Before getting into these matters, it is perhaps interesting to briefly touch on Moore's Law for integrated circuits because, while Moore's Law is in an entirely different area, some significant and somewhat interesting parallels can be seen. In 1965, Gordon Moore, cofounder of INTEL made the observation that the number of transistors per square inch on integrated circuits could be expected to double every year for the foreseeable future. In subsequent years, the pace slowed down a bit, but density has doubled approximately every 18 months, and this is the current definition of Moore's Law. Currently, experts, including Moore himself, expect Moore's Law to hold for at least another decade and a half. This is impressive with many significant implications in technology and economies on the international scene. With these observations in mind, we now turn our attention to the greatly significant and broad subject area of this MRW.
VI
Preface
"The Magic Elixir of Productivity" is the title of a significant editorial which appeared in the Wall Street Journal. While the focus in this editorial was on productivity trends in the United States and the significant positive implications for the economy in the United States, the issues addressed apply, in general, to developed economies on the international scene. Economists split productivity growth into two components: Capital Deepening which refers to expenditures in capital equipment, particularly IT (Information Technology) equipment: and what is called Multifactor Productivity Growth, in which existing resources of capital and labor are utilized more effectively. It is observed by economists that Multifactor Productivity Growth is a better gauge of true productivity. In fact, computer aided and integrated manufacturing systems are, in essence, Multifactor Productivity Growth in the hugely important manufacturing sector of global economies. Finally, in the United States, although there are various estimates by economists on what the annual growth in productivity might be, Chairman of the Federal Reserve Board, Alan Greenspan — the one economist whose opinions actually count, remains an optimist that actual annual productivity gains can be expected to be close to 3% for the next 5 to 10 years. Further, the Treasure Secretary in the President's Cabinet is of the view that the potential for productivity gains in the US economy is higher than we realize. He observes that the penetration of good ideas suggests that we are still at the 20 to 30% level of what is possible. The economic implications of significant annual growth in productivity are huge. A half-percentage point rise in annual productivity adds $1.2 trillion to the federal budget revenues over a period of ten years. This means, of course, that an annual growth rate of 2.5 to 3% in productivity over 10 years would generate anywhere from $6 to $7 trillion in federal budget revenues over that time period and, of course, that is hugely significant. Further, the faster productivity rises, the faster wages climb. That is obviously good for workers, but it also means more taxes flowing into social security. This, of course, strengthens the social security program. Further, the annual productivity growth rate is a significant factor in controlling the growth rate of inflation. This continuing annual growth in productivity can be compared with Moore's Law, both with huge implications for the economy. The respective volumes of this MRW "Computer Aided and Integrated Manufacturing Systems" are entitled: Volume 1: Computer Techniques Volume 2: Intelligent Systems Technology Volume 3: Optimization Methods Volume 4: Computer Aided Design/Computer Aided Manufacturing (CAD/CAM) Volume 5: Manufacturing Process A description of the contents of each of the volumes is included in the PREFACE for that respective volume.
Preface
vn
There is really very little doubt that all future manufacturing systems and processes will utilize the methods of CAD/CAM (Computer Aided Design/Computer Aided Manufacturing), and this is the subject of Volume 4. Key to the processes of CAD/CAM is the generation of three dimensional shapes, a subject treated at the beginning of this volume, 2D assembly drawings are what are generally utilized for conversion to 3D part drawings in the CAD process in order to generate three dimensional shapes for the CAM process, and this is treated in depth and rather comprehensively in this volume. The evolution of a design process and product is often referred to as an adaptive growth representation in the CAD process and this receives necessary treatment in this volume. Fixture designs for the manufacturing process utilize modular elements, and the CAD methods for this essential process are treated rather comprehensively in this volume. Finite element techniques are becoming a way of life for CADS and CAE (Computer Aided Engineering) and rather powerful optimization techniques for processes involved here are also treated in depth in this volume. Rapid prototyping techniques are now a way of life in manufacturing systems, and CAD techniques for this are presented in this volume. These and numerous other techniques are treated rather comprehensively in this volume. As noted earlier, this MRW (Major Reference Work) on "Computer Aided and Integrated Manufacturing Systems" consists of 5 distinctly titled and well-integrated volumes. It is appropriate to mention that each of the volumes can be utilized individually. The significance and the potential pervasiveness of the very broad subject of this MRW certainly suggests the clear requirement of an MRW for a comprehensive treatment. All the contributors to this MRW are to be highly commended for their splendid contributions that will provide a significant and unique reference source for students, research workers, practitioners, computer scientists and others, as well as institutional libraries on the international scene for years to come.
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Contents
Preface Chapter 1 G e n e r a t i o n of T h r e e - D i m e n s i o n a l S h a p e s in C A D / C A M S y s t e m s using A r t - t o - P a r t Technique C. K. Chua and K. Y. Chow Chapter 2 C o m p u t e r Techniques a n d Applications of C o n v e r t i n g 2D Assembly Drawings into 3D P a r t Drawings in C o m p u t e r Aided Design Masaji Tanaka, Kenzo Iwama, Atsushi Hosoda and Tohru Watanabe Chapter 3 C o m p u t e r Techniques a n d Applications of A d a p t i v e - G r o w t h - T y p e R e p r e s e n t a t i o n in C o m p u t e r Aided Design ( C A D ) /. Nagasaka, K. Veda and T. Taura Chapter 4 C o m p u t e r - A i d e d M o d u l a r F i x t u r e Design Yiming (Kevin) Rong Chapter 5 O p t i m i z a t i o n in Finite E l e m e n t a n d Differential Q u a d r a t u r e E l e m e n t Analysis Techniques in C o m p u t e r Aided Design a n d Engineering C.-N. Chen
v
1
35
73
101
171
Chapter 6 C o m p u t e r Techniques a n d Applications in R a p i d P r o t o t y p i n g Gill Barequet
281
Index
297
CHAPTER 1 G E N E R A T I O N OF T H R E E - D I M E N S I O N A L S H A P E S IN C A D / C A M SYSTEMS U S I N G ART-TO-PART T E C H N I Q U E
CHUA C. K. and CHOW K. Y. School of Mechanical & Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 In some industries, products have elements of complex engraving or low relief on them. Traditionally, such work is carried out by skilled engravers working from 2D artwork manually. This process is costly, open to unwanted misinterpretations and lengthens the design cycle. This research presents the Art-to-Part technique which relies on computers and automation from the scanning of 2D artwork, to 3D surface and relief generation, and finally to the fabrication of the model by rapid prototyping. The technique links design to manufacturing stages together and reduces the whole production time. Furthermore, the quality is increased and reproducibility and reliability are ensured, as demonstrated in the 3 case studies. Keywords: 3D relief; Art-to-Part; CAD/CAM; rapid prototyping.
1.
Introduction
There are presently numerous commercially-available software for product design for a particular range of industries which include ceramics, glassware, bottle making, b o t h plastic and glass, jewelry, packaging and food processing for molded products and products produced from forming rolls, coins and badges, and embossing r o l l e r s . 1 - 3 All of these industries share a common problem: most of their products have elements of complex engraving or low relief on t h e m . 4 Traditionally, such work is carried out by skilled engravers either in-house or more often by a t h i r d - p a r t y sub-contractor, working from 2D artwork. This process is costly, open to unwanted misinterpretation of the design by t h e engraver and most importantly, lengthens the time of t h e design cycle. Advances in manufacturing technology allow many industries t o upgrade and change their usual production practices from labor-intensive to a u t o m a t e d and computerized methods. W i t h these changes, the production cycle time and cost 1
2
Chua C. K. and Chow K. Y.
could be reduced tremendously with an improvement in the quality of the product. In recent years, computer-aided design and computer-aided manufacturing (CAD/CAM) have become very popular, especially in the manufacturing industries. It links the designing and manufacturing stages together and thus reduces the whole production time. It is a significant step toward the design of the factory of the future. 5
2. Art-to-Part Process The use of CAD/CAM and Stereolithography Apparatus (SLA) reduces the time required for design modifications and improvement of prototypes. The steps involved in the art-to-part process include the following: 1. 2. 3. 4. 5. 6.
Scanning of artwork Generation of surfaces Generation of 3D relief Wrapping of relief on surfaces Converting triangular mesh files to STL file Building of model by the SLA.
The flow of this series of stages is illustrated using coin design as a case study. Figure 1 shows the steps involved in the art-to-part process.
2.1. Scanning
of
artwork
The function of scanning software is to create a 2D image from 2D artwork automatically or semi-automatically. It would normally be applied in cases where it would be too complicated and time consuming to model the part from a drawing using existing CAD techniques. The 2D artwork is first read into ArtCAM, the CAD/CAM system used for the project, using a Sharp JX A4 scanner. Figure 2 shows the 2D artwork of a series of Chinese characters and a roaring dragon. This combination of hardware and software allows the direct production of a standard image from the artwork, which can be read directly into ArtCAM. The 2D artwork in such instances represent the designs to be used on the face of the coin. In the ArtCAM environment, the scanned image is first reduced from a colour image to a monochrome image with the fully automatic "Gray Scale" function. Alternatively, the number of colours in the image can be reduced using the "Reduce Colour" function. A colour palette is provided for colour selection and the various areas of the images are coloured, either using different sizes or types of brushes or the automatic flood fill function. Figure 3 illustrates the touched-up image.
Generation
of 3D Shapes in CAD/CAM
Systems
Scanning of artwork
_^_ Generation of surfaces (eg.coin shape)
XL Relief generation using ArtCAM
v ^ w v ? •.
.5
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w^ % w. • • ~ w ~ J , » ^ ¥ vwvwi » ™ ^
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Fig. 80. Original photograph of the orchid.
^ ^ ^ ^ ^ ^ ^ ^ ^ A
Fig. 31. Edit image of the scanned-in orchid.
The number of hours spent on building the orchid model using the SLA is close to that for the human face model. The duration of model building depends largely on the number of layers required and the size of the model.
26
Chua C. K. and Chow K. Y.
Fig. 32. Resin prototype of the orchid.
The level of complexity in the orchid design is less than that of the human facial design. The reliefs produced are close to the pattern done by the craftsman. However, steps axe seen on the slope of the convex profile, resulting in a substantial amount of polishing work needed to be carried out to improve the surface finish. The convex curvature of the flower pattern appears to be slightly higher than the ideal size, however, this can be easily amended by redefining the angle of the relief in the Art CAM environment, which is actually the main advantage of using this technique in prototyping. A new model can be rebuilt within 10 hours. Texts of all font types can be added to the design by scanning the required texts together with the main design. The reliefs' of the texts are usually defined as a plane surface. The CAD/CAM system has the full capability of generating such reliefs. The resulting model built by the SLA (see Fig. 32) has the advantage of avoiding undesirable steps due to its plane surface nature.
6. Application of A r t - t o » P a r t Technique In M i n t I n d u s t r y The mint industry has traditionally been regarded as very labour-intensive and craft-based. It relies primarily on the skills of trained craftsmen. At present, automation in this industry has been restricted to the use of machines at certain individual stages of the manufacturing process. Several of these stages are not linked up and thus slow down the whole process. The flexibility of CAD/CAM enables the modelling and manufacturing of working dies needed for stamping coins. It is able to link up the design and die-making stages and to provide a common database.
Generation
6.1. Current
of 3D Shapes in CAD/CAM
Systems
27
practices
This section briefly outlines the various processes involved in the design and manufacturing cycle of coin items (circulating and noncirculating). Based on the discussion held with a coin manufacturer as well as some literature, 19 ' 20 it is found that advanced technologies have been implemented in upgrading the production speed and quality of the coin manufacturing processes, especially in the designing stage whereby artworks are prepared using computer graphics software. However, the prototyping of a coin model is still as traditional as one hundred years ago, which is done by hand-carving a plaster mould. This section focuses on the suitability of applying CAD/CAM techniques to replace the conventional practices in prototyping coin models. Therefore, the discussion on the coin manufacturing processes will concentrate on the initial stages up to the making of the working die. The design and manufacturing cycle can be broken down into the following stages: design, plaster mould engraving, making of rubber mould, making of epoxy mould, making of master die and making of working die. 6.1.1. Design At the first stage of the coin production cycle, the designer prepares a 2D artwork based on the Aldus Freehand 3.1 graphical software package which accepts scanned images as input. Majority of the work is spent on touching up the scanned-in picture and editing the text which may not be found in the original image. The output is directed to a laser printer, the resolution of the print-out can be improved by camera shooting it, developed into a photograph, rescanned into Aldus Freehand environment and edit to yield a better image. It is essential to do a few iterations to produce a piece of high resolution artwork so as to reduce the probability of the craftsman's misinterpretation of the artwork while making the plaster mould. The designer's concern is concentrated on the aesthetic issues of the design rather than the mundane issues of precise geometry and the dimensions of the design. 6.1.2. Plaster mould engraving The creation of coin prototypes from a circular plaster block using simple tools such as small chisels involves a high level of skill and experience. The designer's artwork of a coin piece is interpreted by the craftsman who builds a prototype from this interpretation. The craftsman is responsible for dimensioning the various parts of the design based on the proportions provided by the designer, who does not dimension his artwork. These interpretations are greatly influenced by his skills and experience, as a direct consequence, misinterpretations often result. Generally, a number of iterations is carried out within the first two stages (i.e. design and prototyping) before a design is approved for manufacturing. The designer assesses the prototype and suggests modifications to be made to the prototype if it does not turn out in the expected form. The quintessential aspect of prototype building is
Chua C. K. and Chow K. Y.
Fig. 33. Engraving of plaster mould by a craftsman.
the skill of the craftsman, which ultimately determines the quality of the prototype, and consequently, that of the finished product. It takes about one to two weeks for the craftsman to complete one piece of plaster mould prototype. Any crack or mistake on the mould will result in a great deal of amendment work and a more serious case will require the whole work-piece to be discarded. Figure 33 illustrates the engraving of plaster mould prototype by a craftsman.
6.1.3. Making of rubber mould This is a negative side of the plaster mould produced by pouring liquid rubber over the surface of the plaster mould. The rubber mould is removed by peeling off from the plaster mould when solidified. The time taken for a rubber mould to solidify varies from 5 to 8 hours, depending on the amount of hardener mixed in the solution.
6.1.4. Making of epoxy mould The epoxy mould is similar to the plaster mould in terms of its shape and size, except that the epoxy mould is made up of a harder material The epoxy mould is produced by solidifying liquid epoxy over the rubber mould for about one day. It will be used for making the master die.
Generation
of 3D Shapes in CAD/CAM
Systems
29
6.1.5. Making of master die Steel is employed as the working material and processed by an instrument called the pantograph. The pantograph, an instrument for the mechanical copying of a drawing or diagram on the same, an enlarged or a reduced scale, is used for the engraving of coin dies. It consists of an arm that is used to trace an enlarged design at one end and at the other end, a revolving engraving tool simultaneously cuts an exact reduction of the original, to produce a hub from which working dies are made. 19 The whole process takes about two days to complete, after which it undergoes a polishing and touching up treatment.
6.1.6. Making of working die The final working die, a negative face, is produced by the hobbing process using the master die. The hobbing process takes one day to complete, excluding the annealing process which requires another 8 hours of the time. Polishing of the working die is necessary between hobbing processes. The same procedures are carried when processing the reverse surfaces of the working die of a coin. Flow A of Fig. 34 illustrates the steps involved in making coin prototype by the traditional method.
6.2. Use of CAD modeling
and CNC
machining
Once the client has given the requirements (usually in the form of artwork or photographs such as in Fig. 35), the designer would scan in the design and store as an image file in the workstation called Silicon Graphics Iris. The surface modeller, DUCT5.2, is used to map this 2D pattern onto 3D surfaces. The method used to create surfaces is to form four boundaries for each surface needed using the scanned image in the background as a guide for outlining. By doing so, the whole design would be made of many surfaces but is still 2D. Flow B of Fig. 34 gives a brief outline of the process. Flow C of Fig. 34 outlines the art-to-part process as described in Sec. 2 for coin manufacturing. Using the same software, the surfaces could be manipulated in all directions so as to edit on the shape, size (as in Fig. 36) and most important of all, give them a third dimension. Each surface is made up of many meshes. The number of meshes is proportional to the flexibility in manipulating the surfaces. The intersections of the meshes are known as points. DUCT5.2 defines meshes using laterals and longitudes which are perpendicular to each other at the points. DUCT5.2 uses NURBS to define surfaces. Thus, the control of each point is local, that is, moving the position of a point would not affect the rest of the points, even those that are near to that point. After some manipulation of the 3D surfaces, the design on the top surface of the coin is then ready for addition of text. After the top surface has been completed, the shape of the coin is then modelled.
Chua C. K. and Chow K. Y.
30
B
Use scanned in image as background to outline the picture so as to create surface
Surface Generation using DUCT
Relief Genertion using ArtCAM Create 3D coin with surfaces (real time) using DUCT
WrapArtwork with reliefs created onto 3D Coin
Surface painting to visualise the actual model of the coin before producing the die
Surface apinting to visualise the actual model of the coin before producing the prototype Create prototype using Stereolithography
Create machining path
Making of Epoxy Mould Pentograph Machining of Master Die
CNC machining
Fine Touching up and Polishing of Master Die
(?)
Create Machining Path
(?)
(?)
CNC Machining
Hobbing Process to produce Working Die
(?)
Yes T ->f Final Polishing
(?)
= ^ _ Working Die Fig. 34.
->- Ready to Manufacture Coin
Current and proposed coining processes.
After the coin has been completely modelled, surface painting is done to visualise the actual model of the coin before producing the die. Lighting on the model could be done to enhance the appearance of the model. Up till this stage, the soft prototype has been produced on the workstation. Before investing substantial sum of money in making the mould, the manager could actually see the prototype of the coin and comment on it. Alterations could be made and money would not be spent unnecessarily in making moulds.
Generation
of 3D Shapes in CAD/CAM
Systems
Fig. 35.
Artwork of a Dendrobium Singa Snow orchid.
Fig. 36.
Manipulation of a petal surface of the orchid.
31
After the higher authority has approved the overall design, the designer would then create the machining path and send the NC codes to the CNC machine for machining. At this stage, two alternatives arise, either to machine the actual working die (Fig. 37) or the master die (Fig. 38), or both. By using the above methods of producing the dies, it reduces many existing processes and replaces the use of a copying machine. The main disadvantage of the copying method using pantograph is the time spent in producing the master die, which Is made without using computerised equipment. With NC and CNC, the master die is not required because a working die could be machined at anytime if the NC codes are available. 21
Chna C. K, and Chow K. Y.
32
Fig. 37.
Fig. 38.
Machined working die using plastic.
Machined master die using plastic.
7. Conclusion The CAD/CAM software allows the formation of complicated and time consuming reliefs on models such as jewellery, ceramics tableware, pewter ware, coin dies, etc. to be semi-automatically or automatically created. The software employs the technique of colour segmentation in generation of three-dimensional relief. The software provides realistic viewing function to see the colour shaded Inal model and permits amendments to be made easily. The art-to-part technique has facilitated the creation of complex surfaces. Experiments on building models using the SLA have been carried out to study the application of the relief generating software system. Three models, the Chinese
Generation of 3D Shapes in CAD/CAM Systems
33
Legend and Tradition, the human face and the orchid were built and examined. It was found that substantial amount of polishing work is needed to improve the surface finish of the resin models. Steps were observed on the slope of the relief profile which was caused mainly by the minimum layer size obtainable on the SLA, which is not small enough to produce a smooth surface finish. The major advantage of this prototyping technique is the ability to create more prototypes for less time and cost.
References 1. C. K. Chua, W. Hoheisel, G. Keller and E. Werling, Computing and Control Engineering Journal 4 (1993) 100-112. 2. H. B. Lee, S. H. K. Micheal, R. K. L. Gay, K. F. Leong and C. K. Chua, International Journal of Computer Applications in Technology 5 (1992) 72-80. 3. "2D to 3D Software (ArtCAM) 'Queen's award' case study", Carl Jury International Sales, Delcam International, UK (1992). 4. "ArtCAM 'Rose' case study", Carl Jury International Sales, Delcam International, UK (1992). 5. Y. Koren, Computer Control of Manufacturing Systems (McGraw Hill, Singapore, 1983). 6. J. C. Fidoora, 49th Annual Technical Conference - ANTEC '91. (May 1991) 5-9. 7. I. Mueller, Die Casting Engineer 36, 3 (1992) 28-33. 8. F. E. DeAngelis, Laser in Microelectronic Manufacturing (1991) 61-70. 9. H. B. Lee, Computer-aided and Manufacturing for Jewellery Industry, a thesis for Degree of Master of Engineering in Nanyang Technological University, Singapore, 1993. 10. P. L. Ulerich, Proceeding of the 1992 ASME International Computer in Engineering and Exposition, Computer in Engineering, ASME (1992) 275-281. 11. C. Bradley, G. W. Vickers, and J. Tlusty, CIRP Annals 4 1 , 1 (1992) 437-440. 12. D. Kochan, Computers in Industry, 20, 2 (1992) 133-140. 13. D. L. Bourell, H. L. Marcus, J. W. Barlow and J. J. Beaman, International Journal of Powder Metallurgy (Princeton, New Jersey), 28, 4 (1992) 369-381. 14. L. L. Kimble, Winter Annaul Meeting of the American Society of Mechanical Engineers (1991) 73-80. 15. C. K. Chua, K. F. Leong and C. S. Lim, Rapid Prototyping: Principles and Applications (World Scientific, 2003). 16. C. T. Lee, Implementation of Repair Algorithms for Faulty Stereolithography Files, Final Year Project School of Mechanical and Production Engineering, Nanyang Technological University, Singapore, 1996. 17. Stephen J. Rock and Michael J. Wozny, Solid Freeform Fabrication Symposium Proceedings, the University of Texas at Austin, Texas, 12-14 (1991) 28-36. 18. K. F. Leong, C. K. Chua, and Y. M. Ng, International Journal of Advanced Manufacturing Technology 12 (1996) 407-414. 19. G. Hoberman, The art of coins and their photography, Harry N. Abrams, New York, 1982. 20. W. J. Zimmetman, The coin collector's fact book (AMO, New York, 1974). 21. Y. Koren, Computer Control of Manufacturing Systems (McGraw Hill, Singapore, 1983).
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CHAPTER 2 C O M P U T E R T E C H N I Q U E S A N D A P P L I C A T I O N S OF C O N V E R T I N G 2D ASSEMBLY D R A W I N G S I N T O 3 D PART DRAWINGS IN C O M P U T E R AIDED DESIGN MASAJI TANAKA Okayama University of Science, Dept. of Mechanical Systems Engineering, Ridai-cho 1-1, Okayama 700-0005, Japan Tel. & Fax: +81-86-256-9594 E-mail:
[email protected] KENZO IWAMA Engicom Corp., Jimpo-cho 2-32, Kanda, Chiyoda, Tokyo 101-0051, Japan ATSUSHI HOSODA and TOHRU WATANABE Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Siga 525-0058, Japan Presently, though solid modelers are introduced to CAD systems, 2D CAD systems are often used for designing products that do not need complex curved surfaces. However, solid models will be needed for many cases such as CAM, CAPP, catalogs and manuals in the future. As the result, it is necessary to convert 2D drawings in to solid models. Many research efforts have been conducted to automatically reconstruct solid models from orthographic views of one solid. However, previously proposed algorithms require a large amount of computational time when the solid model becomes complex because many combinatorial searches of geometric elements such as faces and blocks are needed. In this chapter, a method is proposed that can minimize the combinatorial search of blocks called solid elements. The method can very fast generate solid models from orthographic views and also generate solid models of parts from 2D assembly drawings simply. Detailed explanation of the idea in the method and many examples are indicated in this chapter. Keywords: Orthographic view; assembly drawing; part drawing; solid element; solid model. 1. Introduction CAD (Computer Aided Design)/CAM (Computer Aided Manufacturing) systems have advanced automated design and manufacturing. In particular, solid modeling 35
36
Masaji Tanaka et al.
enables the manipulation of 3D (three-dimensional) models of objects such as mechanical products. As a result, beginners who have little experiences about design and manufacturing can make solid models of various objects. 3D models become popular in CAM systems to generate NC (Numerical Control) programs and in other applications to make assembly planning and robot programs. However, 2D (two-dimensional) CAD systems are still preferred by expert designers for designing correct and precise products. Therefore various input devices provide engineers with choices in 2D drawings and improve flexibility for designers. Perhaps many designers will use both 2D CAD systems and solid modelers. They have desired automatic conversion programs of 2D drawings into solid models, and the automatic conversion is still an important research issue. Many research efforts have been conducted to automatically reconstruct solid models from orthographic views of one solid. Especially Idesawa,1 and Wesley and Markowsky2 led the studies at the first stage of the research. Idesawa constructed (3D) wireframe models and (3D) surface models from orthographic views. Wireframe models consist of edges of solid models. Surface models consist of faces each of which is a closed loop of the edges. Wesley and Markowsky constructed blocks each of which is a closed region of the faces. Those edges, faces and blocks are virtual geometric elements because all of them do not form the objects drawn as orthographic views. There can be ghost elements that do not actually exist in the solid models of the objects. Therefore, the solid models of the objects are obtained by removing the ghost elements. They take combinatorial search to remove the ghost elements from the reconstructed elements. However, the number of combinations of geometric elements is exponential of the number of the elements. For example, when the number of faces is n, the number of all combinations becomes 2 n . All of the combinations must be examined because plural solutions can exist in orthographic views. As a result, their methods were not practical. In addition, they limited the shapes of the objects to polyhedrons. Later Sakurai et al.3 constructed blocks that include cylindrical, conical, spherical and toroidal faces. In this paper, real elements of the objects are called true elements and ghost elements are called false elements, and the blocks are called solid elements. Solid models were represented as B-Reps (Boundary Representation) in the three methods. 1 " 3 On the other hand, several methods have reconstructed solid models as CSG (Constructive Solid Geometry) by recognition of elementary objects such as cube, cylinder and fillet from orthographic views. 4 " 6 However, the orthographic views that can be applied to the methods are very restricted, and it is difficult to apply the methods to plural solutions. At the next stage of the research, many methods were proposed to effectively search for the solutions. For example, Sasaki et al.7 applied pseudo-Boolean algebra to distinguish the truth of edges and faces. Nishihara et al.8 applied heuristics to distinguish the truth of faces. Gujar and Nagendra 9 ' 10 surveyed the study and developed a more systematic method than Wesley and Markowsky's to generate solid elements. Though those methods could effectively search for the solutions,
Converting
2D Assembly Drawings into 3D Part Drawings in CAD
37
they required a large amount of computational time when the shapes of solid models become complex. The authors proposed a method to realize a practical system that can very fast generate solid models from orthographic views. 11 ' 12 The method uses solid elements to reconstruct solid models from orthographic views. Generally there are two kinds of geometric elements that can reconstruct solid models from orthographic views. One is face and the other is solid element. Solid elements are superior to faces in the combination because the number of solid elements is much fewer than the number of faces in solid models. In addition, it is more difficult to convert surface models to solid models than to convert solid elements to solid models because each of the solid elements already forms a solid model. The authors' method established a set of equations that represent relationships of the truth among solid elements by comparing the solid elements with orthographic views. Since the method solves the equations, it does not employ a mechanism of searching suitable combinations of faces or solid elements. The method runs faster than those which employ search mechanisms. The computational time of the method can be linear to the complexity of a solid. This paper describes a method of automatic conversion of 2D assembly drawings into 3D part drawings. 13 The method is developed as an application of the authors' method described above. Though there are many methods to reconstruct solid models from orthographic views, nobody has attempted to reconstruct solid models of more than one part from orthographic views to the knowledge of the authors. In general, product design proceeds from conceptual to detailed design. Designers are observed first drawing 2D assembly drawings of products, and then drawing each 2D part drawing despite the fact that there are methods of designing products by composing designed parts in CAD systems by superimposing layers of 2D part drawings. The 2D part drawings are transformed into solid models for CAM systems and others. It is routine and takes much time to decompose 2D assembly drawings into 2D part drawings and transform them into solid models. Therefore, these transformations are very often processed by operators other than designers. The automatic decomposition system is important for designers for the following two reasons: • It takes much time to draw each 2D part drawing from a 2D assembly drawing in proportion to the number of parts. • If operators other than designers process the decomposition, the operators may fail to correctly recognize each part of the 2D assembly drawing. In this method, wireframe models, surface models and solid elements are constructed from 2D assembly drawings that are drawn as orthographic views. Solid elements become the components of solid models of parts of assemblies. The method generates solid models of each part by classifying solid elements into elements of some parts and false elements that do not actually exist in any parts. If there is more than one solution, this method can generate all of the solutions. The domain
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Masaji Tanaka et al.
y 1^ 0
Top
z
Z A
A
0
-" A
Front Fig. 1.1.
0
Side
Coordinate system of orthographic views.
of 2D assembly drawings is limited to orthographic views consisting of front, top and side views, and cross-sectional views. Also, the types of faces are limited to planar, cylindrical, conical and spherical faces. This paper firstly explains a method that reconstructs solid models from orthographic views of one part by using cubic elements. The cubic element is a particular solid element and forms a cube. This method conveys the basic idea in this study, but it is only applied to orthographic views that consist of only straight lines that are parallel to the axes of the coordinate system. Secondary, the method is extended to include oblique lines and curved lines in orthographic views, and to convert cubic elements to solid elements. Finally, this paper explains the method that decompose 2D assembly drawings into solid models of parts. Figure 1.1 illustrates the coordinate system of orthographic views of this study (front: x-z, top: x-y, side: y-z). 2. Conversion of Orthographic Views into Solid Models by Cubic Element Equations 2.1. Construction
of cubic elements
from lattice
faces
When the orthographic view of a part is drawn by only straight lines that are parallel to the axes of the coordinate system, it is possible to convert the view to that consisting of three lattice faces by extending the lines. Here, solid lines and dotted lines are not distinguished. For example, Example 2.1 illustrated as in Fig. 2.1 can be converted to the lattice faces as in Fig. 2.2. Lattice faces obtained from an orthographic view can be regarded as the projections of a set of cubes. In this case, each square of lattice faces corresponds to a face of a cube. When an orthographic view can be converted to three lattice faces, the solid models of the part are obtained by searching suitable combinations of the cubes. Because all of
Converting 2D Assembly
Drawings into 3D Part Drawings in CAD
Fig. 2.1.
Fig. 2.2.
39
Example 2.1.
The lattice faces in Example 2.1.
the faces of the solid models are parallel to x-z, x-y or y-z planes, the set of the cubes form a three dimensional matrix. However, all of the cubes are not elements of the solid models. Let cubes that are elements of the solid models be true cubic elements and let cubes that are not elements of the solid models be false cubic elements. Then our goal is to obtain the solid models by distinguishing the truth
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Masaji Tanaka et al.
Fig. 2.3.
The wireframe model of the cubic elements.
of all of the cubic elements. Figure 2.3 illustrates the wireframe model of the cubic elements constructing from Fig. 2.2. 2.2. Relationships
of the truth among cubic
elements
The relationships between cubic elements and squares of lattice faces can be recognized since all the cubic elements form 3D matrices. For example, when the cubic elements in Example 2.1 are numbered as in Fig. 2.4, the numbers of cubic elements corresponding to the squares of the lattice faces in Fig. 2.2 are illustrated as in Fig. 2.5. The relationships of the truth among the cubic elements are obtained by comparing original orthographic views and the lattice faces corresponding to the cubic elements. The relationships are classified into five conditions explained in the following. Conditions of Cubic Elements • Solid Line Condition If a line exists between two squares of a lattice face and this line exists as a solid line in an original view, two cubic elements corresponding to these squares are not both true elements. The reason for this is that the solid line in the original view cannot be there or must be a dotted line if both of the two cubic elements are true. When the two cubic elements are Sx and Sy, the relationship is expressed as a Solid Line Condition, Sx x Sy. For example, 5i x S2 is obtained from the front view in Example 2.1. • Temporary Line Condition If a line exists between two squares of a lattice face and this line does not exist in the original view, two cubic elements corresponding to these squares are both true elements or both false elements. In short their truth values are the same. The reason for this is that a new solid line appears in the original view if their
Converting
2D Assembly Drawings into 3D Part Drawings in CAD
Fig. 2.4.
Fig. 2.5.
41
The cubic element in Example 2.1.
S?
Ss
S9
S4
Ss
S6
Si
S2
S3
Si
S2
S3
S3
S6
S9
Sio
Su
Sl2
Sl2
Sis
Sis
The relationship between lattice faces and cubic elements in Example 2.1.
truth values are not the same. When the two cubic elements are Sx and Sy, the relationship is expressed as a Temporary Line Condition, Sx — Sy. For example, Su — S12 is obtained from the front view in Example 2.1 and also Su — S15, Si7 — Sis a r e obtained because these elements inherit the relation of S u — S12 in the direction of the front view.
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Masaji Tanaka et al.
• Dotted Line Condition If a line exists between two squares of a lattice face and this line exists as a dotted line in the original view, the truth values of two cubic elements corresponding to these squares are the same. The dotted line must be a solid line if their truth values are not the same. For a dotted line to exist, cubic elements must exist that make edges corresponding to the dotted line. When the two cubic elements are Sx and Sy, the relationship is expressed as a Dotted Line Condition, Sx — Sy. For example, S\o — Su is obtained from the front view in Example 2.1 and the dotted line must be formed by the combinations of the truth in set {S13, S14} or set {516,517}.
• Existence Condition For each line segment that is an edge of squares of lattice faces, there must be one true cubic element in all of the cubic elements that have edges corresponding to one line segment. For example, though the line segment exists that is horizontal and corresponds to the top edge of S 7 in the top lattice face in Example 2.1, the corresponding line segment does not exist in the original top view. Therefore, S7 and Si6 are false elements. Also, since the line segment exists that is vertical and corresponds to the left edge of Si in the front lattice face in Example 2.1 and the corresponding line segment exists in the original front view, all elements of {Si,S4,Sr} are not false. • Mass Condition Cubic elements that form a solid model cannot be separated into two or more solids, nor can the cubic elements be connected only through vertices or edges. 2.3. Search of solutions
by cubic element
equations
The relationships of cubic elements corresponding to lattice faces are organized as a system of equations that is called a cubic element equation. For example, three systems of equations are organized from Fig. 2.5 as in Fig. 2.6(a) and they can be combined as in Fig. 2.6(b). The cubic element equation in Example 2.1 is organized as in Fig. 2.6(c) by adding S10 — S13, S u — S14, Si7 — Sis that do not appear on the lattice faces in Fig. 2.5 and simplifying the equations. Cubic element equations simplify the relationships of all cubic elements. In Fig. 2.6(c), two sets of cubic elements A = {S 1 ; S 3 , . . . , Sis} and B = {S2, S 5 , S 6 } are made, and the truth values of the cubic elements of each set are equal. As a result, solid models are obtained in cubic element equations by searching not suitable combinations of all cubic elements but suitable combinations of several sets of cubic elements. Therefore, cubic element equations can minimize the amount of the combinatorial search of cubic elements. In Example 2.1, the solution is obtained as the following. When Existence Conditions are applied to the cubic elements, Si and S10 become true elements. Therefore set A becomes true, and S7 and set B becomes false. All of Dotted Line Conditions are satisfied in set A. Therefore, the solution is given as set A. Figure 2.7 illustrates the solution.
Converting
A
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living organism (phenotype)
environment
CAD representation
Fig. 1.
—- •
form
Shape generation in nature and CAD.
itself: the shape is generated through the interaction with the environment. This brings about diversity in the shapes of the living organisms (Fig. 1) along with the other characteristics. Shape — /(genetic information,
environment).
In this research, the adaptive-growth-type shape representation is adopted for configuration design. In analogy to biological development, in our representation, a shape is determined by the interaction between the rules of generation and the environment. This representation aims not so much at the precise representation of shape, as usual in current CAD systems, but at the generation of various design solutions, that satisfy the constraints in configuration design. The authors proposed a Shape Feature Generating Process Model (SFGP Model),10 based on a similar analogy to biological development. However, in SFGP model, to produce an independent shape representation, the internal environment of the shape was given as environment. In the case of configuration design, since each shape here is to adapt to the constraints given by two or more other function carriers, the mechanism of adaptation to the external environment should be reconsidered, too. Therefore, in our study, the SFGP model is extended as an adaptive-growthtype 3D representation to make it adaptable to an external environment. 3.1. Outline of SFGP
model
In the SFGP model, shape generation starts with a primary shape (sphere) and rules are selected and applied according to the position and local conditions, that is, the internal environment of the shape. After a number of generations, a final shape is
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primary
final Fig. 2.
Cell division model.
generated. The cell division model, since it is the basis for our SFGP model, and the new shape generation process are explained briefly below.
3.1.1. Cell division model In our model, the set of rules consists of rules of the division for a dot (which we call a cell in analogy to biological development) on a sphere. The shape feature generation process is the series for cell divisions. As shown in Fig. 2, in the beginning, there are few cells on the sphere. According to the rules, they divide into two or more cells and spread over the sphere. Consequently, after a number of generations, the cell division results in a distribution of cell density at the surface of the sphere. As Fig. 3 illustrates, the shape is derived by processing the density of cells: O is the center of the sphere and d is the density of cells near point A. Density d is converted to a distance from O to a point on the actual surface in the direction of OA. By proportioning the density of cells to the distance to the surface of the shape, the cell division model can display fairly complicated shapes. The positions of points on the sphere where the density is measured are arbitrary, but clearly if more points are set, the resolution of representation of the shape increases.
3.1.2. Shape generation process However, applying all rules to every single cell on the sphere surface is not only very inefficient, requiring an enormous amount of computational time, it also makes it very difficult to determine which rules are responsible for the generation of the various design features of the shape. Therefore, as shown in Fig. 4, the sphere is divided into a certain number of parts and, depending on the location (A, B, C, . . . ) of the cell on the sphere, a group of rules is selected (B). After the density around the cell is measured, a rule which matches the criterion of the density, that is, the condition in the rule, is applied (Rule B-2) to prompt cell division with parameters specified in the action side of
Adaptive-Growth-Type
Representation
in CAD
79
Point cm Surface
d = density of cells near point A Fig. 3.
Correspondence between shape and distribution of density in cell division model.
the rules. As shown in Fig. 5, actual sets of rules (classifiers) are binary codes with conditions referring to parts of the sphere (A, B, C, . . . ) . After a number of cell divisions, the distribution of the density of cells is measured and converted into a free form shape. Consequently, the shape is generated by applying the set of cell division rules through the classifier system. Figure 6 shows the correspondence between the genotype (set of cell division rules) and the phenotype (shape). 3.2. Extension
of SFGP
model
As previously mentioned, the SFGP model has been extended to make it adaptable to an external environment. With respect to the two issues of configuration design, the shape can be adapted to the environment of the given space in the following ways: (i) reshapes according to the environment, (ii) repositioning the shape, according to the environment. In our study, the SFGP model is first extended with taking (i) into consideration. To begin with, it is necessary to extend the model so that it adapts to the constraint of configuration design since this is one of the elements of the environment of the shape. The same process that determines the generation rule in the SFGP model can be directly applied to the algorithm that satisfies the constraints. In the SFGP model, the evaluation function is the degree of similarity to the shapes given
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Nagasaka et al.
set of rules
< Rule >
Fig. 4.
= cell density : cell division (condition) (action)
Rules applying to cell division.
by the designer. Here, in the extended model, the evaluation function is the degree of satisfaction of the constraints. Secondly, the SFGP model has been extended to adapt to the constraints among function carriers. Constraints between function carriers belong to the two categories as described in Sec. 2. Among the layout constraints, the most related constraint to (i) is the mutual interference among function carriers. For instance, an interference can be cancelled if it is possible to grow one of the shapes as shown by the interfering parts in the right side of Fig. 7. Therefore, the rule that is responsible for making interfered part is replaced: thanks for the characteristic of the SFGP model, it is possible to specify which rules are responsible for the generation of the various features of the shape. The following three techniques are sequentially applied, and the rule is selected if the interference is cancelled while the geometrical constraint is still satisfied. (i) Partial replacement of the rule is responsible for the interfering part with an other rules. For instance, as shown in Fig. 8, the rule which is responsible for making interfered part is specified in the set of rules of B, and exchanged with a same part of rule in the set of rules of other form with the same evaluation value for its geometric constraint.
Adaptive-Growth-T'ype
000010 000111 101101 001000
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^
Representation
001110 100001 001100 001100
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in CAD
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(action )
Set of rules in binary codes.
(ii) Replace the whole rule with another rule. For instance, as shown in Fig. 9, the whole set of rules of B is exchanged with the other set of rules that has the same evaluation value, (iii) Generate a new rule. For instance, as shown in Fig. 10, a new set of rules is generated from other rules with the same evaluation value as the interfered shape. The adaptation of shape is achieved by such strategies of modifying the constraint, that represent the environment. However, no mechanism of repositioning of the shapes have been implemented yet. As an alternative, we have adapted a method that repositions the shape as an external algorithm.
4. Methodology for Supporting Configuration Design 4.1. Implemented
system
A basic strategy is to generate shape gradually by using the adaptive-growth-type shape representation, and to search for the configuration design solution by adapting shapes to the situation of layout in a given space. There is a similarity between this
82
Nagasaka et at
Genotype
EJbenQtype
rules of cell division
Shape mmiBf^^^-
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)
) Cell Division
Fig. 6.
Genotype and phenotype.
Fig. 7.
Shape adaptation.
strategy and the work on skeleton-based techniques in which rays from a skeleton are used to determine shapes. 11 As shown in Fig. 11, the support system consists of three parts: a geometric constraint solver (GCS), a layout constraint solYer (LCS) and a configuration design unit (CDU). Each solver searches for configuration design solutions by interaction with the designer. The low of the constraint satisfaction process is in the GCS and the LCS, rough solutions are obtained, and the shape and layout are finally adapted by the CDU to each other within the same frame. This means that neither the 3-dimensional shape of the function carrier nor the layout are directly generated in parallel, but the outline of the solution of the layout problem is obtained by LCS using genetic algorithms (GAs). The outline shape is obtained by the GCS
Adaptive-Growth-Type
Representation
in CAD
•> •"-/ , • set of rules of other form
Fig. 8.
set of rules of B
Partial replacement of a rule.
set of rules of other form
Fig. 9.
set
of rules of B
t>
^-»*
Replacement of a rule as a whole.
•HUM set of rules of other form
Fig. 10.
set of rules of B
Generation of a new rule.
83
Nagasaka et al.
geometrical constraints
Spatial layout constraints
(2)
(1) Geometric Constraints Solver
Spatial Layout Constraints Solver
Spatial Design Unit
Fig. 11.
Flowchart of the system.
using adaptive-growth-type shape representation based on CS. Moreover, possible contradictions between the outlines of the solutions of the two constraint solvers are adjusted by exploiting the ability of generating a diversity of shapes in the same frame for the CDU. Such an approach is impossible with conventional shape representation. 4.2. Configuration
design solution
generation
algorithm
In this section, the algorithms, the specific role and solution generation methods of the three solvers are explained, mainly in terms of the CDU which plays the center role in the support system. 4.2.1. Geometric constraints solver The GCS searches for the outline of the solution for the geometrical constraints and obtains a rough shape before the CDU adjusts and generates the design solution. As mentioned previously, the process of finding the generation rule in the SFGP model is directly applied to satisfying the constraints.
Adaptive-Growth-Type
Representation
85
in CAD
4.2.2. Layout constraints solver The LCS searches for the outline layout which satisfies the layout constraints before obtaining the configuration design solution made by the CDU. Therefore, this solver only deals with the centerpoint of the shape of the function carrier, and from this centerpoint, the shape obtained by the GCS is generated. This solver is based on a similar technique in layout design of VLSI chips (e.g. Ref. 12) and utilizes GAs.
4.2.3. Configuration design unit After the outline shape of the function elements and the positions of the outline are obtained by the two solvers as discussed above, the configuration design solution is generated using this configuration design unit. The algorithm for the CDU is used not to achieve constraint satisfaction, but to integrate the outline of the solutions generated by the two solvers and to determine 3D shapes and their configuration layout in the same framework. Here, searching in the "same framework" means that a design solution is generated by searching for a point of compromise and correcting the contradictions — if there are any — between the outline of the solutions that were generated by the GCS and the LCS in one framework within the CDU. Each step is explained with reference to the numbers in Fig. 12 as follows. In Fig. 12-(1), the centerpoints of the function carrier generated by the LCS are obtained (Fig. 13-1).
0
input coordinates
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Flowchart of the configuration design unit.
86
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Next, a shape is generated by centering on coordinates obtained in (1), as shown in Fig. 13-2. The shape is gradually generated without any farther operations because there is no interference between function carriers in this step [Fig. 12-(2),(3)]. As shown in Fig. 13-3, as a shape is generated, the adjoining function carriers interfere and an overlapping region arises. In this case? as shown in Fig. 12-(4),(5),(6), the adjustment of the generation rules and the repositioning of the centerpoints of interfered shapes take place in the same framework. As shown in Fig. 14, the positions of centerpoints of interfered shapes A and B are adjusted by moving them in the directions of NA and NB (Fig. 14, left). The distance of movement is defined as,
\NA\ = \NB\ =
tfvi
(i)
where V* is the volume of the common region. The layout constraint is concerned with the positional relationship between all function carriers involved. Therefore, after moving the positions of the centerpoints of just two function carriers in the directions of NA and iVjg, in most cases, the overall positional relationship among the function carriers, including those two that were actually moved, may not satisfy some criteria of the layout constraint any
Fig. 13.
Growing shapes of function carriers. overall boundary
Fig. 14.
Movement of function carriers shape.
Adaptive-Growth-Type
Representation
87
in CAD
more. Thus, it is necessary to readjust the centerpoints of each function carrier again after moving some of them. Therefore, the center positions are readjusted so that they satisfy the constraints again (Fig. 14, right). At the same time, adjustment of the rules of the shape takes place, as described in the foregoing section (Fig. 12). After these processes, configuration design solutions are obtained in the CDU through the correction of the contradictions between the shape and its layout and they are presented the designer. The designer observes and examines the solutions, and if the designer accepts them, the configuration design process is finished. If the designer accept none of them, he/she modifies the constraints of the GCS or the LCS and the process repeats. In this way, the configuration design proceeds with interaction between the support system and the designer, and eventually, a configuration design solution can be obtained.
5. Prototype System for Supporting Configuration Design 5.1. Configuration
of the
system
The prototype of the supporting system is based on our method for configuration design as explained in the preceding sections. Figure 15 shows the overall configuration of the system. This system has been implemented using C + + on a Sun Workstation. It consists of five parts: the constraint specification unit, the configuration design unit, the constraint solver, the user interface, and the geometry engine.
5.2. Application
to configuration
design of a satellite
design
5.2.1. Configuration design in satellite design In most cases of configuration design, the task and the specification of the design object are already determined before this step of the design process. In satellite design, it is called the "mission" which is the purpose of the satellite (e.g. weather observation, communication, etc.). After the mission is given, the configuration design is made as follows. (1) Assume externals of the satellite according to the installed mission equipment. (2) Distribute functions necessary for accomplishing the mission, and determine the specifications of each component, and assume the size of the equipment. (3) Identify embodiment-determining main function carriers in each component. (4) Develop preliminary layout. (5) Select suitable preliminary layout as a result of the examination. After finishing the configuration design, the designer moves to a detailed embodiment design.
Nagasaka et at
88
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System configuration.
5.2.2. Example In configuration design of a satellite, since reliability is the most important issue, components are selected from among those in a catalog. However, despite the required reliability of the satellite, the fuel tank may have a comparatively high degree of freedom in the shape and is usually newly designed for each satellite, because it has little iniuence on reliability. Therefore, this example focuses on the application of our configuration design method to determining the outline shape of the fuel tank, as it is obtained by considering its volume, and the outline space of the installation and the layout of each component. External constraints of the satellite, as determined from the payload of the mission and the launch rocket, are shown in Fig. 16. Geometric constraints Geometric constraints in this example are the installation space necessary for the 11 main components. For the fuel tank, it is the volume of the fuel. Examples of these constraints for the fuel tank, battery charge control unit (BCCU) and battery are as follows. Fuel tank (volume) 10,000 cc BCCU 40.0 x 30.0 x 30.0 cm Battery 50.0 x 30.0 x 40.0 cm
Adaptive-Growth-Type
Representation
in CAD
89
Solar battery panel
,
Fig. 16.
1.000 m
.
Externals of a satellite.
Layout constraint The layout constraints concern the layout of the main components as follows: • Communication system ^••V-slots blind slots,
slot features
through keyways ^* blind keyways « + >
hole features
Manufacturing features
special holes edge cut boss features Protrusion features
fillet features rib features
Fig. 2. Manufacturing feature classification.
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Fig. 9. Overview of the Fix-Planning system.
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Fig. 10. An example of initialization of the system.
An example workpiece is shown in Fig. 6a where the step surface F46 is to be machined. Table 3 shows the results of the accessibility evaluation of candidate fixtaring surfaces and Fig. 11 shows a point accessibility distribution of a candidate bottom locating surface. The results of fixture planning in horizontal and vertical directions are shown in Fig. 12. The results may not be unique. Alternative planning is also provided when it is necessary. The fixture planning result can be
Computer-Aided Table 3.
Modular Fixture
Design
125
Results of accessibility analysis.
Face-id
Normal direction
Area
Function
Valid
OA
Fl F8 FIO Fll F12 F13 F14 F15 F16 F17 F18 F23 F28 F35 F36 F38 F40 F44 F59
(0, - 1 , 0) (1, 0, 0) (0, 1, 0) ( - 1 , 0, 0) (1, 0, 0) (0, 0, - 1 ) (0, 1, 0) (1, 0, 0) (0, - 1 , 0) ( - 1 , 0, 0) (0, 0, - 1 ) (0, 0, - 1 ) (0, 1, 0) (0, 1, 0) (0.707, 0.707, 0) (0, 0, 1) (0, - 1 , 0) (0, 1, 0) (0, - 1 , 0)
6095.04 1900 2322.58 6464.19 5126.26 3462.37 563.77 614.14 563.77 614.14 875.73 12342.47 3109.46 1008.58 1996.16 9942.9 3415.2 2322.58 1578.54
SL/SC SL/SC SL/SC SL/SC SL/SC BL SL/SC SL/SC SL/SC SL/SC BL BL SL/SC SL/SC SL/SC TC SL/SC SL/SC SL/SC
Yes No No Yes Yes Yes No No No No No Yes Yes Yes Yes Yes Yes Yes Yes
1.312395 N/A N/A 6.983632 4.819779 0.8750)0 N/A N/A N/A N/A N/A 16.962994 4.090943 0.967515 1.743387 19.599906 1.219500 0.579375 2.114299
BL-bottom-locating; SL-side-locating; SC-side-clamping; TG-top-clamping.
Z\° i
0
1
(a)The distribution of sample points on surface F23 (b) PA values of all sample points on F23 Fig. 11.
PA values of sample points on a bottom-locating candidate surface F23.
used in automated modular fixture design system. 32 Figure 13 shows the fixture configuration design using the fixture plan from Fix-Planning.
4» S e t u p P l a n n i n g Setup planning is to determine the number and sequence of setups as well as the number of operations performed in each setup. A recursive backward setup
126
Yiming (Kevin) Rong
Fig. 12a. An example of horizontal locating/clamping Locating surfaces Locating points Gravity of center of workpiece Clamping surface Clamping point.
planning system has been developed in this research where setups are generated and sequenced based on the information of feature accuracy, machining methods including heat treatment requirement, tool axis direction, etc. Although current work is limited to the planning for machining processes, the method can be generally expanded to other processes. Manufacturing accuracy is the major consideration in setup planning where features with lower accuracy requirement are usually machined in very i r s t setups while features with higher accuracy are machined in the very last setups. Although sometime rough and finish machining may be performed in a single setup when a machining center is utilized, in the case of high precision, they may need to be separated where the large machining force involved in rough machining may cause serious vibration and damage the product quality generated in finish machining.
4.1. Heat treatment
mnd feature
voiume
Besides feature accuracy, several other factors need to be considered. When a heat treatment is required in manufacturing processes, the setup groups are separated by the heat treatment. In our research, three setup groups are defined as: (1) rough machining setups, (2) semi-finish machining setups, and (3) finish machining setups. These setup groups are separated by either the heat treatment requirement or fine
Gompmter-Aided Modular Fixture Design
BIHII^HHHHBHiH^^HBBHH
Fig. 12b. An example of vertical locating.
Fig. 12c. An example of vertical clamping corresponding to the vertical locating.
127
128
Yiming (Kevin) Rong
(a) 2-D top view
(b) 3-D View after removing hidden lines
Fig. 13. The final result of fixture configuration design.
manufacturing accuracy requirement. One example is the case that a grinding operation is necessary for a feature processing, which is usually performed in setup group three. Examples of planning rules for carbon-steel material are designed as shown in Table 4. In practice, features with small volumes removed (such as screw-holes) are usually machined in the very last setup since the machining force involved is little and has no effect on the accuracy of other features, although the feature accuracy may not be high. Therefore the feature volume is considered as a factor in our setup planning. In order to count a relative feature volume in an equivalent scale with feature accuracy calculations, the relative feature volume (V) is first defined as: V = Vc/V0
(23)
Computer-Aided Table 4.
Modular Fixture Design
129
Example of rules for feature assignment t o setup groups.
IF
HB < 350 and heat-treatment = 0 T H E N group = 1 ELSE IF HB < 350 and heat-treatment=l (1-steel, normal) IF IT < 9 or Ra < 250 or V < 0.00253 THEN group = 2 ELSE group = 1 ELSE IF HB > 350 or heat-treatment = 2 ( 2 - carbon steel, quenching) IF IT < 6 or Ra < 32 or V < 0.00063 THEN group = 3 ELSE IF (IT < 9 and IT > 6 ) or (Ra < 250 and Ra > 32) or (V < 0.00253 and V > 0.00063) THEN group = 2 ELSE group = 1
where V c is a feature volume calculated based on dimensional parameters of the feature; Vo is the volume of the workpiece. In our setup planning system, the following assumptions are made to consider the feature volume effect. When V < 1 x 10~ 3 in 3 , V c is small and the machining force would not affect the machining accuracy. When V < 4 x 10~ 3 in 3 , the feature may be machined in semi-finish machining setups because the machining force may affect the machining accuracy in a certain extent. If a feature accuracy is high (e.g. IT < 7), it needs to be machined in two setups in different groups. Finally when V is greater than the critical value, the best feature accuracy in rough machining setups is IT 9 and up. By following a similar procedure presented before, the feature volume factor can be calculated by: ,„
T
VI = I n t
TlogfVx 10- 4 )1
[ log(1.585) J
„ +1
-
,
N
<M>
Therefore, when feature volume factor is considered in setup grouping, a priority feature selection in the backward setup planning becomes: F a c = w t T g + w v VI. 4.2. Backward
setup
(25)
planning
Once the tool approaching and feeding directions are given to manufacturing features and fixturing features are identified, feature groups can be formed and sequenced for setup planning based on the considerations of feature accuracy and other factors. In order to overcome the problems of multiple and/or unreasonable approaching directions under certain feature combinations, the tool axis is considered with machine tool information to make the feature grouping decision. Only a group of features are assigned to a single setup, which can be processed in the same workpiece orientation and a feasible fixture configuration design can be realized (e.g. interference free).
130
Yiming (Kevin)
Rong
Figure 14 shows a recursive backward setup planning algorithm where the planning starts from the last setup with a product model and ends up with the first setup and workpiece blank model. In one setup planning cycle, following procedure is implemented: (1) A critical manufacturing feature is identified usually with a highest feature accuracy; (2) According to the manufacturing method and machine tool required for the feature processing, the workpiece orientation is determined; (3) Based on the machining tool axis which is determined by feature processing requirement and machine tool information, other features are grouped into the current setup usually with a same tool direction; (4) Fixturing features are identified and locating surfaces/points are selected for the setup based on the feature accuracy relationship and geometric accessibility; (5) Fixture configuration design is conducted to verify the setup planning. If a fixture configuration design with quality and interference-free cannot be generated, modification information is feedback to feature grouping and locating datum selection; (6) Operation details are generated for each operation within the setup where the depth of cut is determined and CNC programming is worked out; (7) When the setup planning is carried out for the specific manufacturing features, the material volumes removed in these operations are calculated in terms of machining parameters. These volumes are "added" to
•
Product model in setup #1 Critical manufacturing feature identification and analysis Machine tool information workpiece orientation Manufacturing feature groupingl Fixturing feature identification locating/clamping design Fixture configuration design
.+
Operation planning CNC programming
i Feature recovering (add material)
Finish
Product model generation for setup #i-l
Fig. 14.
Backward reasoning algorithm for setup planning.
Computer-Aided
Modular Fixture
Design
131
the workpiece model to form a "product" model for the setup prior to the current setup; (8) If the setup planning is not finished (i.e. the blank model is not reached), another cycle of the setup planning starts. It should be noted that during the setup planning feedback exists in each step for the setup modification. A manufacturing feature database, a machining tool database, and a manufacturing process database are necessary for setup planning decision making. 4.3. Implementation
examples
Figure 15 shows a virtual workpiece used to illustrate the setup planning method presented in this chapter, which is similar to the one used for fixture planning (Fig. 7). Because of the high feature accuracy and heat treatment requirement, three setup groups are necessary, as shown in Table 5. Within the first and second setup groups, the recursive backward planning algorithm is applied to generate the setup plans where a horizontal machining center is assumed available. In each setup groups, locating surfaces for the manufacturing features with major tolerance requirements are machined in the very first setups. Machining tools and the number of fixtures required are also determined. In the third setup group, grinding operations are concerned where the planning rules are much different with machining operations. The planning for grinding operations is not included in this chapter. Figure 16 shows another example of workpieces with a relatively complex geometry. Table 6 shows the setup planning results. The framework of setup planning includes manufacturing feature description and feature-base development, fixturing feature definition and locating surface selection, recursive backward setup planning algorithm, and fixture design and verification. These functions are integrated into a single package in the environment of AutoCAD and C + + platform. The inter-feature accuracy relationship is taken into account so that the manufacturing accuracy can be ensured. Geometric constraints are analyzed in fixture planning and fixture design. Other factors such as heat treatment and feature volumes are also considered in the setup planning. Therefore, a feasible setup plan can be generated for complex workpieces. The recursive backward planning algorithm leads to an automated setup planning. When this system is implemented, the lead-time of manufacturing planning would be significantly reduced.
5. Analysis of Modular Fixture Structures In order to develop an automated modular fixture configuration design system, the fundamental structure of dowel-pin based modular fixture and fixturing characteristics of commonly used modular fixture elements are first investigated. Figure 17 sketches a dowel-pin type modular fixturing system which includes a library of a large number of standard fixture elements. 33 With combinations of the fixture elements, an experienced fixture designer can build fixtures for
Yiming (Kevin)
132
others
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MATERIAL- CARBON STEEL HEAT-TREATMENT:
QUENCHING
HARDNESS' HRc 38-45
Fig. 15.
An example workpiece for setup planning.
Computer-Aided Table 5.
Modular Fixture
Design
133
Setup Planning for the First Example Part.
Material: Carbon steel GROUP: 1 SETUP1: Tool axis direction: X + ; locating surface: Ra: 500 Feature 11: plane IT: 15 SETUP2: Tool axis direction: Z - ; locating surface: Feature 1 : plane IT: 12 Ra: 125 SETUP3: Tool axis direction: Y - ; locating surface: Feature 8 : plane IT: 12 Ra: 500 SETUP4: Tool axis direction: X - ; locating surface: Feature 9 : plane IT: 12 Ra: 125 Ra: 250 Feature 10: slot IT: 12 SETUP5: Tool axis direction: Y + ; locating surface: Feature 4 : slot IT: 10 Ra: 125 Ra: 1000 Feature 3 : hole IT: 12 SETUP6: Tool axis direction: Z+; locating surface: Feature 6 : plane IT: 12 Ra: 500 Feature 5 : step IT: 12 Ra: 500
I, (6), 9, 2 mill I I , (9), 6, 2 mill 1, (6), 2. 11 mill 1, 11, 8 mill mill 1, 11, 8 mill drill 8, 1, 11 mill mill
VISE F I X T U R E D: 2 L: 4 VISE FIXTURE D: 2 L: 4 VISE FIXTURE D: 2 L: 4 FIXTURE 2 D: 2 L: 4 D: 1/2 L: l _ l / 4 FIXTURE 2 D: 27/32 L: 1.5 D: 1-1/2 L: 4 - 7 / 8 FIXTURE 1 D: 2 L: 4 D: 2 L: 4
1, (6), 8, 9 mill 1, (6), 8, 9 mill 1, 11, (9), 8 mill 1, 8, 11 mill mill bore 8, 1, 11 mill drill
VISE FIXTURE D: 2 L: 4 VISE FIXTURE D: 2 L: 4 VISE FIXTURE D: 2 L: 4 FIXTURE 2 D: 2 L: 4 D: 63/64 L: 1.5
Heat treatment: Normal GROUP: 2 SETUP 1: Tool axis direction: X + ; locating surface: Feature 11: plane IT: 12 Ra: 125 SETUP2: Tool axis direction: Y—; locating surface: Feature 8 : plane IT: 9 Ra: 125 SETUP3: Tool axis direction: Z—; locating surface: Feature 1 : plane IT: 9 Ra: 250 SETUP4: Tool axis direction: Y + ; locating surface Feature 2 : plane IT: 9 Ra: 63 Feature 4 : slot IT: 7 Ra: 63 Feature 3 : hole IT: 9 Ra: 250 SETUP5: Tool axis direction: Z+; locating surface: Feature 5 : step IT: 9 Ra: 125 Feature 7 : hole IT: 8 Ra: 125
FIXTURE 1 D: 2 L: 4 D: 11/16 L: 3 _ l / 8 bore
Heat treatment: Quenching , Hrc 38-45 Feature Feature Feature Feature
GROUP: 3 1: plane 2 : plane 3 : hole 4 : step
GRINDER IT: 6 IT: 6 IT: 6 IT: 6
Ra: Ra: Ra: Ra:
32 16 32 32
grinder grinder grinder grinder
1 1 2 1
a variety of workpieces. In order to automatically generate a fixture configuration design, the issues for the following problems are presented in the following sections: (1) The selection of suitable fixture elements and combinations of these elements into desired functional units; (2) the methodology to mount (position) the fixture units (or elements) in appropriate positions and orientations on a baseplate without interference with the
134
Yiming (Kevin)
Fig. 16.
Rong
Second example workpiece for setup planning.
space already occupied by the workpiece, machining envelope or other fixture units mounted in advance. It should be noted that kinematic constraints, locating accuracy, fixturing stability, and fixturing deformation are also important in fixture planning and fixture configuration design. Once a fixture configuration design is finished, these design performances, which are not presented in this chapter need to be verified.34"37 Verification results are the feedback information to the fixture configuration design module for alternative designs, if necessary. Fixturing features of a workpiece have been analyzed, including geometric, operational, and fixturing surface information. 38 Once a fixture structure is decomposed into functional units, fixture elements and functional surfaces, the fixture design process becomes a search for a match between the fixturing features and fixture structure. 18 In application of modular fixtures, a fixture element assembly relationship database is built up based on the analysis of the fixture structure.
5.1. Decomposition
of modular
fixture
structure
The advantage of modular fixtures is its adaptability for various workpieces by changing the configuration combinations of fixture elements. Modular fixture structures can be decomposed into functional units, elements, and functional surfaces. By applying Set Theory, a fixture body can be defined as a set or an assembly of fixture elements. Let F denote a fixture and e; (i = 1,2,..., n e ) a fixture element
Computer-Aided Table 6.
Modular Fixture
135
Design
Setup planning for the second example part
Material: Casting iron GROUP 1 SETUP1: Tool axis direction: Y - ; locating surface: 1, 25, 2 Feature 25: plane IT: 12 Ra: 1000 mill SETUP2: Tool axis direction: X - ; locating surface: 1, 25, 2 Feature 17: plane IT: 12 Ra: 1000 mill mill Feature 21: plane IT: 12 Ra: 1000 SETUP3: Tool axis direction: Z - ; locating surface: 25, 17, 12 Feature 1 : plane IT: 12 Ra: 250 mill SETUP4: Tool axis direction: X + , locating surface: 1, 25, 21 mill IT: 12 Ra: 500 Feature 2 : plane mill IT: 12 Ra: 500 Feature 6 : plane R_ bore IT: 9 Ra: 250 Feature 7 : hole SETUP5: Tool axis direction: Z + ; locating surface: 1, 25, 2 mill Feature 12: plane IT: 15 Ra: 500 mill Feature 3 : plane IT: 12 Ra: 500 mill Feature 18: plane IT: 12 Ra: 500 drill Feature 4 : step hole IT: 12 Ra: 250 drill Feature 5 : step hole IT: 12 Ra: 250 drill Feature 19: step hole IT: 12 Ra: 250 drill Feature 20: step hole IT: 12 Ra: 250 SETUP6: Tool axis direction: Y + ; locating surface: 1, 25, 2 F _ bore Feature 24: en_hole IT: 9 Ra: 500 drill Feature 23: step hole IT: 10 Ra: 250
FIXTURE D: 2 FIXTURE D: 2 D: 2 FIXTURE D: 2 FIXTURE D: 2 D: 2
1 L: 4 1 L: 4 L: 4 3 L: 4 1 L: 4 L: 4
FIXTURE 2 L: 4 D: 2 L: 4 D: 2 L: 4 D: 2 L: 2 _ l / 4 C_bore D: 1/2 D: 1/2 L: 2 - 1 / 4 C - b o r e D: 1/2 L: 2 - 1 / 4 C - b o r e D: 1/2 L: 2 _ l / 4 C_bore FIXTURE 1 D: 7/8
L: l _ 3 / 8 C_bore
Heat Treatment: Normalization GROUP: 2 SETUP 1: Tool axis direction: Z - ; locating surface: 25 , 12, 2 Feature 1 : plane IT: 8 mill Ra: 63 SETUP2: Tool axis direction: Z+; locating surface: 1, 25, 2 Feature 12: plane IT: 12 Ra: 125 mill IT: 12 Feature 13: hole Ra: 250 drill IT: 12 Feature 14: hole Ra: 250 drill Feature 15: hole IT: 12 Ra: 250 drill Feature 16: hole IT: 12 Ra: 250 drill SETUP3: Tool axis direction: X + ; locating surface: 1, 25, 2 IT: 12 Feature 6 : plane mill Ra: 125 Feature 7 : hole IT: 6 R_ bore Ra: 63 Feature 8 : hole IT: 12 Ra: 250 drill Feature 9 : hole IT: 12 Ra: 250 drill Feature 10: hole IT: 12 drill Ra: 250 Feature 11: hole IT: 12 Ra: 250 drill SETUP4: Tool axis direction: Y + ; locating surface: 1, 25, 2 Feature 24: en_hole IT: 6 Ra: 125 F_bore Feature 23: step-hole IT: 7 Ra: 63 drill ,
FIXTURE D: 2 FIXTURE D: 3/4 D: 5/16 D: 5/16 D: 5/16 D: 5/16 FIXTURE D: 2 D: 1-1/2 D: 5/16 D: 5/16 D: 5/16 D: 5/16 FIXTURE
3 L: 4 2 + ANGLE PLATE L: 5/16 t a p D: 3/8 L: 1 - 1 3 / 1 6 tap D: 3/8 L: 1 - 1 3 / 1 6 tap D: 3/8 L: 1 - 1 3 / 1 6 tap D: 3/8 L: 1 - 1 3 / 1 6 1 L:4 L:4_7/8 tap D 3/8 L:l_ 13/16 tap D 3/8 L:l_ 13/16 tap D 3/8 L:l_ 13/16 t a p D 3/8 L:l_ 13/16 1
-
-
C-bore
D:l L : l _ 3 / 8
in F, where n e is the number of fixture elements in F, i.e. F={ei|iene}. This is a representation of a fixture at the level of fixture elements.
(26)
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Yiming (Kevin)
Fig. 17.
Rong
A sketch of Bluco Technik modular fixturing system.
A fixture consists of several sub-assemblies. Each sub-assembly performs one or more fixturing functions (usually one). These kinds of sub-assemblies in a fixture are considered as fixture functional units. In a fixture unit, all elements are connected one with another directly where only one element is connected directly with the baseplate and one or more elements in the subset are contacted directly with workpiece serving as locator, clamp or support. Let Uj denote a fixture unit in a fixture. From above description we have: Ui = {eij\j
enei}
(27)
where n e j is the number of elements in unit U^. Therefore a representation of a fixture at the level of fixture units can be written in the following way: F = {Ul|i€n„},
(28)
F = {{eij\jenei}\ienu}
(29)
where nu is the number of units in fixture F. Dividing a fixture structure into functional units and giving detailed analyses on the functional units plays a key role in automated modular fixture designs. A fixture element consists of several surfaces which can either serve as a locating, clamping or supporting surface in contact directly with workpiece (which is named a fixturing-functional surface) or serve as supporting or supported surfaces in contact with other fixture elements (which are called assembly-functional surfaces). Therefore an element can be represented by: e» = { s i f c | k e n s i }
(30)
where Sik denotes the functional surface k on fixture element i and nsi is the number of functional surfaces the element i contains.
Computer-Aided
Modular Fixture Design
ci
Fixture Structure
y
\,
k\
I
\
6 6 \
i
4 6 Functional Surfaces
Fig. 18.
137
Functional Units
Fixt«re Elements
Fixture structure tree.
By combining Eqs. 29 and 30, a fixture can be represented at the level of fixture surfaces in the following form: F
= {{{Sijk\k
e nsij}\j
€ nei}\i
e nu}.
(31)
In this way, a fixture structure is decomposed into three levels, i.e. unit, element, and functional surface levels. A conceptual sketch of the fixture structure decomposition is shown in Fig. 18. Based on the investigation of various application examples of dowel-pin modular fixtures and also for the purpose of automated fixture configuration design, a fixture structure can be classified into seven types of units (sub-structures): Vertical Locating Unit (VLU), Horizontal Locating Unit (HLU), Vertical-Horizontal Combination Locating Unit (VHCLU), Vertical Claming Unit (VCU), Horizontal Clamping Unit (HCU), Vertical Supporting Unit (VSU) and Horizontal Supporting Unit (HSU). Fixture units are composed of modular fixture elements. The functional surfaces of a fixture element perform the task of locating, supporting and clamping. All the above units are mounted on a baseplate. Figure 19 shows the fixture structure decomposition for dowel-pin modular fixture systems. 5.2. Fixture
units
and
elements
In general, a fixture unit consists of several fixture elements where usually only one element is in contact with the workpiece by its fixturing-functional surface to serve as a locator, supporter, or clamp. All fixture elements in a fixture unit are connected together through their assembly-functional surfaces. This fixturingfunctional surface in a fixture unit is defined as an acting surface of the fixture unit. Each unit must have at least one acting surface which performs the fixturing function. Usually the acting surface is a plane or a cylindrical surface. The acting plane of a fixture unit can be described by a point on the plane and the external normal vector of the plane. The center of the fixturing plane is chosen as the point to describe the plane. The acting cylindrical surface of a fixture unit can be described by a point on the axis of the cylinder and the vector of axis. The center point of the acting surface is defined as an acting point of the unit and the distance between the surface of baseplate and the acting point is defined as an acting height of the fixture unit. The acting direction of a fixture unit can also be defined by the direction of the external normal vector of the acting surface.
Yiming (Kevin)
138
. Vertical Locating. Unit (VLU)
. Horizontal Locating Unit (HLU)
Fixture __ Structure
Vertical and Horizontal ' Cmbination Locating — Unit(VHCLU) , Vertical Clamping Unit (VCU)
Horizontal Clamping — Unit (HCU)
, Vertical Supporting Unit (VSU)
. Horizontal Supporting— Unit (HSU)
Unit Level
Fig. 19.
Rong
Surface and Edgd3ar — Adjustable LocatinjBar
' Top Surface Side Surface
• Adjustment Stop V-Pad
' Surface and Edge Bar Dual Surface and Edge Block
— Top Surface • Side Surface
Clamping Support Clamping Bar Speed Clamp with Adjustable Block Serrated Edge Clamp
' Adjustable Bar V-Pad
Adjustable Stop
Element Level
Surface Level
D e c o m p o s i t i o n of m o d u l a r f i x t u r e s t r u c t u r e s .
For fixture units, the most important parameter in fixture design is the acting height. Figure 20 shows the acting heights of different fixture units in a fixture design. In general cases, several fixture elements need to be assembled together to achieve the acting height. The acting heights of fixture units are the parameters to know before the suitable fixture elements can be selected. The fixture element selection to form a fixture unit is based on a fixture element assembly relationship analysis as shown in the next section. Fixture configuration design is a process of selecting fixture elements from a fixture element library and allocating them together in space according to a certain sequence. In AFCD, a fixture element database needs to be built up, in which the geometry information such as geometric profile, the edges and surfaces of a fixture element is represented in its own (local) coordinate system. To represent the position and orientation of a fixture element in the fixture system, global and local coordinate systems need to be defined. If the global coordinate system which is associated with the fixture baseplate is defined by 0(X, Y, Z), the local coordinate system of fixture
Computer-Aided
Modular Fixture
139
Design
Workpiece
Acting Height of VCU
^ ,
/
Acting Height of HLU
Acting Height of VLU Fig. 20.
Acting heights of fixture units.
• X
Fig. 21.
Coordinate systems in automated fixture configuration design.
element i can be denned by three orthogonal unit vectors (u^v^w,) with a local origin pi(x,y,z) as seen in Fig. 21. Once a fixture configuration is built up, the position and orientation of each fixture element needs to be determined. Parameters (Pi,ax,ay,az,bx,by,bz) are used to represent the position and orientation of the fixture element i in the global coordinate system, where Pi is the origin of the element local coordinate system and symbol ax,ay,az,bx,by,bz are the directional cosines of unit vector u^ and v, respectively. The unit vector w;(c x ,c y ,c z ) is not independent and can be determined by: Wi =
UiXVi.
(32)
140
Yiming (Kevin)
Rong
During AFCD, the bottom element of a fixture unit is first placed on the fixture baseplate, i.e. the position and orientation of the bottom element is first determined relative to the global coordinate system, although this relationship may be adjusted later on. Then other fixture elements in the fixture unit are, in turn, allocated until the acting height is reached. This bottom-up approach has been applied to the fixture unit mounting algorithm in the AFCD system. 5.3. Assembly
characteristics
of modular fixture
elements
The methodology of selecting fixture elements and assembling them together to form a fixture functional unit is the key issue in automated fixture configuration design. If a detailed examination is made on the level of fixture functional units from many practical application cases, it is found that there are some commonly used fundamental structures in various fixture bodies. These fundamental structures have the properties of adaptability, rigidity, simplicity, ease of loading, etc. Studying the assembly relationship between fixture elements and extracting basic combinations of the elements is a way to achieve automated fixture configuration design. In fact, the assembly relationships between modular fixture elements are not arbitrary but constrained. A fixture element can be only assembled with a fraction of other modular fixture elements and usually it can only be used in one or several units. Following are examples showing the fixturing characteristics of some commonly used modular fixture elements and their possible assembly relationship with other modular fixture elements. Figure 22(a) shows a console, which is usually used as a riser to raise other fixture elements up to the necessary acting height. Two adjacent sides have an alternating pattern of clearance and tapped holes for accurately mounting the console to baseplate or other support elements. The other two sides have bushed and tapped holes for mounting locating or clamping elements. A console can be mounted on the top of another console of its kind, which is named as a self-supportable fixture element. Because a console is relatively larger than other locating and clamping elements, many elements can be mounted on the top of a console. But, a console usually can be only mounted on a baseplate or another console. A console may be used in building up different kinds of fixture units and it is one of the most adaptable fixture elements. Surface/edge bar and dual surface/edge block shown in Fig. 22(b) are used as rises or locators either individually or in a combination with other fixture elements. The slot edge can serve as a vertical-horizontal combination locator. The surface/edge bar can be assembled on the top of dual surface/edge block and both of them can be stacked on the top of a console to achieve an appropriate height. Figure 22(c) shows several surface locator towers, including locating tower, multi-surface tower, ground spacer, and tipped screw. All of these towers are only used as locators in VLU or VHCLU. They may be mounted at the top level of fixture units and contact directly with surfaces of the workpiece. These towers cannot
Computer-Aided
Modular Fixture
141
Design
(b) Surface/edge bar and block
(a) Console
(c) Surface locating towers
(d) Adjustable locating bars
(e) Adjustable stop
(f) V-blocks
(h) Edge clamps (g) Clamping stop Fig. 22.
Typical modular fixture elements.
support any other fixture elements. A ground spacer is a self-supportable fixture element, which may provide a precise establishment of the acting height. Adjustable surface bar and adjustable locating bar, as shown as in Fig. 22(d), can be used in VLU as locators. Since these adjustable bars are fixed by a screw
142
Yiming (Kevin)
Rong
along a T-slot, the actual locating positions can be adjusted to any desired positions and orientation in the range which can be reached. Therefore, they are very useful in the case of a strict locating point position required. Other commonly used fixture elements in the modular fixture system include adjustment stop (Fig. 22(e)), V-bar, V-block and adjustable V-tower (Fig. 22(f)), clamping support (Fig. 22(g)), and edge clamps (Fig. 22(h)). Assembly characteristics of these fixture elements are similar to those analyzed previously. In order to automatically select and generate fixturing units in fixture configuration designs, the assembly relationships between fixture elements need to be analyzed and represented in a computer compatible format, which is the foundation of forming fixturing units with elements. A Modular Fixture Element Assembly Relationship Graph(MFEARG) has been developed to represent the assembly relationships in building fixture units. Figure 23 is a partial MFEARG composed of real fixture elements, showing assembly relationships of the fixture elements for possibly building a VLU. It should be noted that for the purpose of explicitness, only a few typical fixture elements are shown in Fig. 23. A total MFEARG for assembling a VLU may contain more fixture elements and more assembly relationships. MFEARG can be further represented by an abstract graph. A mathematical model and computer implementation of MFEARG will be introduced in the next section.
Fig. 23.
Modular fixture element assembly relationship graph for a VLU.
Computer-Aided
5.4. Modular fixture element (MFEARG)
Modular Fixture
assembly
Design
relationship
143
graph
An MFEARG can be denned, without loss of generality, as a directed graph (digraph) G, as shown as in Fig. 24, i.e. G = (V,E) and
V = {v | v G fixture elements}; E — {e | P(VJ, Vj) A (v*, Vj € V)}; (33)
where V is a set of vertices representing fixture elements used in building a specific fixture unit; and E is a set of directed pairs of members of V and is an edge representing the assembly relationship between fixture elements (i and j). The edge e(vj -^> Vj) presents that fixture element v^, the start-vertex of edge e, can be mounted on the fixture element v^, the end-vertex of edge e. The number of edges going from other vertices to an end-vertex denotes an indegree of the vertex and the number of edges coming from a start-vertex to other vertices denotes an outdegree of the vertex. An edge e(vj —» Vj) is called a self-loop if fixture element Vj can be assembled with a fixture element of its own kind. Consoles and adopter blocks discussed before are such kinds of fixture elements. A directed-path is a sequence of edges v,i —^-> VJ2 —^ v,3 —^» . . . such that the end-vertex of ej_i is the start-vertex of e^, which represents the possible assembly relationship for building a fixture unit. If the indegree of a vertex in MFEARG (vi, V2, or V3, in Fig. 24) is zero, that means that no fixture element can be mounted on the fixture element. Locating tower, multi-surface tower, etc. are such kinds of fixture elements. Similarly the outdegree of vg is zero in Fig. 24, which means there is no other fixture elements can be mounted to it except the baseplate. Therefore, a complete directed-path represents a possible formation of a fixture unit. In the AFCD system, a modular fixture element assembly relationship database (MFEARDB) is established to represent the MFEARG information where the
Fig. 24.
A sketch of MFEARG models.
144
Yirning (Kevin)
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relative positions and orientations between any two fixture elements are specified according to their possible assembly relationships (e.g. Fig. 23). Once the MFEARDB is built up, it can be used in fixture configuration design.
6. Establishment of M F E A R D B The MFEARG is stored in an MFEARDB. Based on the MFEARG model, algorithms were implemented to choose all suitable fixturing unit candidates and mount fixture units on a fixture baseplate. Since different fixture systems have different modular fixture elements, the corresponding MFEARGs will be different. In order to generally implement the AFCD system, the MFEARDB should be automatically constructed for various fixture systems. Figure 25 outlines the approach to automatically construct MFEARDB. For a modular fixture system, all modular fixture elements are first represented by CAD models with specified assembly features. Then, modular fixture element assembly relationship reasoning engine is applied to find all the possible assembly relationship between any element pairs. The reasoning results are used to construct the MFEARDB, which is based on MFEARG model. The MFEARDB needs to update only when any fixture element is added to or canceled from the fixture system.
6.1. Modular fixture element
modeling
Geometric information of fixture elements is used when interference of two elements is checked in specific spatial positions and orientations. Since the geometry of fixture elements is relatively simple and pre-known, a primitive instancing scheme 39 is used to model the fixture element geometry. Some geometry simplifications are made when modeling fixture elements to avoid time-consuming in intersection checking for complex geometry. Geometric information of a fixture element includes the shape type of the element and dimensional parameters. Figure 26 shows some examples of fixture element shape geometry: block, cylinder, bracket. Block-type elements are defined by three parameters (/, w, h), cylinder-type elements are represented by two parameters (r,h), and bracket-type elements are described by five parameters (li,l2,w,hi,h2).
Modular Fixture Element Database
Fig. 25.
Modular Fixture Element Assembly Relationship Reasoning Engine
Modular Fixture Element Assembly Relationship Database
System for constructing modular fixture element assembly relationship database.
Computer-Aided
Modular Fixture
Design
145
Z
AZ
X Block ( 1, w, h )
Y Cylinder ( r, h)
Bracket ( 11 , 12 , w, h ,, h2 ) Fig. 26.
Three categories of modular fixture element.
To reason assembly relationship between fixture element, assembly features together with the geometric information need to be defined and used to represent modular fixture elements. Following functional surfaces are defined as assembly features of fixture elements: (1) Supporting faces; (2) Supported faces; (3) Locating holes; (4) Counterbore holes; (5) Screw holes; (6) Fixing slots; (7) Pins; and (8) Screw bolts. Figure 27 shows the fixture assembly features. A supporting face is the surface that can be used to support other fixture elements or workpiece. A supported face is the surface that is supported by other fixture elements in a fixture design. A locating hole is the hole machined to a certain accuracy level and can be used as a locating datum with locating pins. Counterbore holes and fixing slots are used to fasten two elements with screw bolts. In a modular fixture system, assembly features of elements such as locating hole, counterbore hole, screw hole, pin and screw are designed with standard dimensions. Other parameters of an assembly feature are the position and orientation of the feature in the element's local coordinate system. The homogeneous transformation is used in this research to describe the position and orientation of features.
146
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Let F denote the feature position and orientation of an element, which can be represented by: F = (V,P) 1
(34)
where V = (vx vy vz 0) is the homogeneous representation of the orientation vector V of feature F and vx,vy,vz are the directional cosines of V. P = (x y z 1) is the homogeneous coordinate of origin of feature F. If F is a face type feature, its origin P is a point on the face, and the orientation vector V is normal to the face and points out from it (Fig. 27a). If F is a hole type feature, its origin P is the center of the hole end circle, and V points outwards along the axis of the hole (Fig. 27b). If F is a pin type feature, its origin P is the center
Fig. 27a.
Face type assembly feature.
Fig. 27b.
Hole type assembly features.
V" p
V' I
I
f—
Fig. 27c.
Pin type assembly features.
Computer-Aided
Modular Fixture
Design
147
t I Fig. 27d.
Fixing slot assembly feature.
on the tip of the shaft and V points outwards along the axis of the shaft (Fig. 27c). In the case of fixing slots, the origin P and vector V are defined as shown in Fig. 27d. In modular fixture systems, locating holes, counterbore holes, screw holes and fixing slots are designed to perpendicular to the supporting or supported face of an element. The locating holes, counterbore holes and fixing slots of a supported element are used to locate and fix the supported element to a supporting element. They are defined as associate assembly features with the supported face. Because of the standard design, their relative positions and orientations are known in the local coordinate system of the fixture element and can be extracted from the vector of the supported face. Similarly, locating holes and screw holes of a supporting element are used to locate and fix a supported element on the supporting element. They are also defined as associate assembly features with the supporting face. Their positions and orientations can be extracted from the vector of the supporting face in database. It should be noted that a fixture element may serve as a supporting element to a supported element in a fixture and may serve as a supported element to another supporting element. Since the number of assembly features on a face may vary, a linked lists structure is used in MFEARDB to represent the fixture elements (Fig. 28). In the MFEARDB, fixture element information is organized into four levels, i.e. an element list, element records, functional surfaces, and associate assembly features. In an element record, a fixture element identification code and shape type is first defined. The geometric dimensions are retrieved from element parameters. Associate assembly features are represented in terms of their assembly features on a functional surface, which provides a convenient way to find all associate assembly feature information for a specific surface. This will benefit in reasoning assembly relationship, which is mainly carried out according to supporting-supported face pairs. In the data structure, if there are no more assembly features associated with a functional face, the pointer just points to a symbol NIL which represents the end of list. Therefore this approach has the advantage of saving memory space. Figure 29 shows an example of the data structure for an edge-bar element where two functional surfaces (supporting and supported faces) and three types of associate assembly features (two locating holes, two screw holes, and one counterbore
148
Yiming (Kevin) Element List
Element Record ID Name Shape Type # of Parameters Parameter 1 Parameter N # of Supported Face Supported Face 1 Pointer Supported Face M Pointer # of Supporting Face Supporting Face 1 Pointer Supporting Face P Pointer
Fig. 28.
Rong
Supported Face 1 Record Index/ID Vx
Associate Locating Hole 1 — • Next ID X
Associate Locating Hole Pointer Associate Counterbore Pointer Associate fixing Slot Pointer Supporting Face 1 Record ID Vx Vy Vz Associate Locating Hole Pointer Associate Screw Hole Pointer
y z Associate Counterbore Next ID
Associate Locating Hole 1 — • Next —• ID X
y z
Associate Fixing Slot Next ID
Associate Screw hole 1—* Next ID X
y z
A linked list data structure representing fixture.
hole) can be identified with position and orientation information. The dimensions of the assembly features are standardized with a specific series of modular fixture systems. 6.2. Mathematical
reasoning
of assembly
relationships
When a data structure is designed to represent fixture element and mating relationships are defined between fixture elements, the assembly relationships between fixture elements can be obtained through a reasoning or inference procedure. Actually, the fixture configuration design is similar to an assembly process. Some previous work in assembly area provides valuable information for analyzing assembly relationships between modular fixture elements. 40 ~ 42 6.2.1. Mating relationship between assembly features Mating relationships have been used to define assembly relationships between part components. Researchers defined their own mating assembly relationship according to the application area. In this research, five types of relationships are defined between assembly features for the purpose of reasoning the assembly relationship between modular fixture elements (Fig. 30).
Computer-Aided
Modular Fixture
Supported Face 1 Record 1 0 0 1 LHptr CBptr FSptr -» Nil
Element Record 310020 Surface and Edge Bar Block 3 90 30 20 1 SPDF 1 Ptr 1 SPGF 1 Ptr
Supporting Face 1 Record 1 ' 0 0 -1 LHptr SHptr
^ ^ ^^5S&% Ky0^^ \J>^ Fig. 29.
149
Design
Associate Locating Hole — i Next 1 15 15 20
Next 2 75 15 20
•Nil
Associate Locating Hole Next 1 15 15 0
Next 2 75 15 n
Nil
Associate Screw Hole N«rt 1 30 15 0
Next 2 60 15 0
•Nil
Associate Counterbore Next —»Nil 1 45 15 20
D a t a structure representing.
(1) Against. Face 1 is against face 2 when they are coplanar and with opposite normals. This is the assembly relationship between a sporting face of an element and a supported face of another. Let Fi = (Vi, P i ) T and F2 = (V2, P2) T denote the positions and orientations of face 1 and face 2 respectively. Against condition can be represented by following equations: V 2 * M = Vi, and Vi * (P 2 - Pi) = 0
(35)
where M is mirror transformation matrix. (2) Align. A hole aligns another hole when their vectors lie along the same line but in opposition. This is the assembly relationship between two holes. Similarly let Fi = (Vi, P i ) T and F 2 = (V 2 , P2) T denote the positions and orientations of hole 1 and hole 2 respectively. Align condition can be represented by: V 2 * M = Vi, and K * (P 2 - P i ) = Vi or Pi = P 2
(36)
where K is a constant. (3) Fit. A pin fits a hole when their vectors lie along the same line but in opposition. This is an assembly relationship between a pin and a hole. In the same way, let Fi = (Vi, P i ) T and F 2 = (V 2 , P 2 ) T denote the positions and orientations
150
Yiming
(Kevin)
Rong
V2A
ft
^ (a) Aginst
^
^
(b) Align
£s^ (c) Fit
fi
(d) Screw fit
vector V2 points to reader.
(e) Across Fig. 30.
F i v e basic r e l a t i o n s h i p b e t w e e n
fixture,
of the pin and the hole respectively. Fit condition can be represented by: V 2 * M = Vi, and K * (P 2 - P i ) = Vi or Pi = P 2
(37)
(4) Screw Fit. A screw blot fits a screw hole when their vectors lie along the same line but in opposition. Let Fi = (Vi, P i ) T and F 2 = (V 2 , P 2 ) T denote the positions and orientations of the screw blot and the hole respectively. Screw fit condition can be represented by: V 2 * M = Vi, and K * (P 2 - P x ) = V x or Pi = P 2
(38)
(5) Across. A fixing slot crosses a screw hole when the vector of the fixing slot and the vector of the screw hole are coplanar and perpendicular. Let F x = (Vi, P i ) T and F 2 = (V2, P2) T denote the positions and orientations of the fixing slot and the screw hole respectively. Across condition can be represented by: Vi * V 2 = 0, and Vi * (Pi - P 2 ) = 0
(39)
These five types of mating relationship may cover the assembly relationships between assembly features of fixture elements in most fixture designs.
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Modular Fixture
Design
151
6.2.2. Assembly criteria between fixture elements In order to establish the MFEARDB, possible assembly relationships between fixture elements need to be evaluated. By examining typical fixture assembly structures, the following criteria in four cases for assembling two fixture elements are employed in modular fixture configuration design (Fig. 31). Let Ei donate a supporting fixture element and E2 a supported element. Case 1. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) A supporting face of Ei is against a supported face of E2. The face on El covers most part of the face on E2; (2) At least two locating holes of Ei align with locating holes of E2 respectively; (3) One or more counterbore holes of E2 align with the screw holes of Ei; and (4) Body of El does not intersect body of E2. Second half of condition 1 is a fuzzy condition. It ensures a firm connection between elements. Condition 2 ensures high locating accuracy between two elements since locating pins can be inserted into accuracy locating holes. Condition 3 ensures that two elements can be fixed together by using screws. Condition 4 is obviously an important criterion for interference free. Once these conditions are satisfied, an assembly relationship between fixture element Ei and E2 is identified and can be added to the MFEARDB. Case 2. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) The same as condition 1 in case 1; (2) The same as condition 3 in case 1; and (3) The same as condition 4 in case 1. The case is the same as last one except the requirement of locating hole alignment. In this case, locating accuracy can be only ensured in the direction of vector of supporting or supported face. Case 3. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) The same as condition in case 1; (2) A fixing slot of E2 is across a screw hole of Ei; and (3) The same as condition 4 in case 1. This case is similar to case 2. Again, in this case, the locating accuracy can be only ensured in the direction of vector of supporting or supported face. Case 4. E2 can be assembled into a position on Ei if the following conditions are satisfied: (1) A screw of E2 fits screw hole of Ei when E 2 is a crew bolt; and (2) The same as condition 4 in case 1. This kind of assembly case is usually used in adjustable locating fixture unit. The relative position between two elements is fixed by a nut.
6.2.3. Inference assembly relationship between element pairs Suppose two fixture elements Ei and E2 are an assembly pair. Assembly features and geometry of the two fixture elements are retrieved from MFEDDB. Let Fi = (Vi, P i ) T denote a supporting face of Ei and F2 = (V2, P2) T a supported face of E 2 . Assume P 1 1 ( P i 2 are any two locating holes on the supporting face and P 2 i, P22 are any two locating holes on the supported face. Note that V 1 ; P i , P n , P12 and
152
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counterbore holes
locating holes I I
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Ej
counterbore hole
is*
~r
E,
L
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case 2: Locating tower on edge block
o case 1: Edge block on console
screw holes E2.
© o
w-^ z*-t- fixing slot
o
o
o
E2
sm
screw bolt
w case 4: Adjustable locating sto
E2
E.
case 3: Surface bar on console Fig. 31.
Four cases of assembling two fixture elements.
V2, P2, P2I) P22 a r e represented in the fixture element local coordinate systems. If we can find a position and orientation that satisfies the following conditions: (1) Fi against F 2 and (2) P n , P12 align P 2 i, P22 respectively, the assembly position and orientation of E2 on Ei can be obtained from solving assembly mating equations.
Computer-Aided
Modular Fixture
153
Design
Our purpose is to find the position and orientation of E 2 on Ei in Ei 's local coordinate system. The local coordinate systems of Ei and E 2 are first made coincidence. Then, after a series of transformations, E2 is to be translated and rotated to a position and orientation that the relationship between E2 and Ei satisfies above conditions. Based on the mating conditions, we have: V 2 * T * M = Vi, P 2 i * T = P n , and P 2 2 * T = P i 2
(40)
where T is a transformation matrix calculated from: T = ROT x (a) * ROTy{(3) * ROT z ( 7 ) * TRAN(x,
y, z)
including rotation transformation matrices about the x, y, and z axes and a translation transformation matrix. T is further represented as: cos (3 sin 7 sin (3 0\ cos (3 cos 7 — sin a sin (3 cos 7 — cos a sin 7 sin a sin (3 sin 7 + cos a cos 7 sin a cos (3 0 T = — cos a sin (3 cos 7 + sin a sin 7 cos a sin j3 sin 7 — sin a cos 7 cos a cos (3 0 y z \) V /
The solution of above equations implies a potential assembly relationship between Ei and E 2 . Solution (x, y, z) is the position coordinate of E 2 on Ei in Ej local coordinate system, and solution (a, (3,7) is the orientation coordinate of E 2 on Ei in E^ local coordinate system. Furthermore, we should check whether the conditions 3 and 4 in case 1 are satisfied for Ei and E 2 in above position and orientation (x,y,z,a,P,j). If the checking is pass, (x, y, z, a, (3,7) will store into the MFEARDB as an assembly relationship between Ei and E 2 . Similar approach can be used to test if other assembly criteria are satisfied. 6.3. Assembly
relationship
reasoning
system
and
examples
Figure 32 shows the architecture of automatically reasoning assembly relationship engine. Once the MFEDDB is available, the reasoning engine will examine all element pairs to find their assembly relationship. The results are store in an MFEARDB, which is based on MFEARG model discussed in Ref. 39. This information is used to automatically design modular fixture configuration. To illustrate the implementation of the method, an example is given where a console and a surface/edge block are chosen as Ei and E 2 (Fig. 33). Vi, the vector of the supporting face Fj of Ei, is (0, 0, 1,0) and V2, the vector of the supported face F 2 of E 2 is (0, 0, - 1 , 0). P n = (60,75,120,1) and P 1 2 = (30,45,120,1) are the two locating holes on F i . P 2 i = (45,15,0,1) and P 2 2 = (15,45,0,1) are the two locating holes on F 2 . According to Eq. 40, one solution can be identified: x = 75, y = 30, z = 120, a = 0, /? = 0, 7 = 90. The solution shows that there is a potential assembly relationship between Ei and E 2 , which satisfied the conditions: (1) Fj against F 2 , and (2) P n , P i 2 align
154
Yiming (Kevin)
Nfodular Fixture Element Database (MFEDB)
Select El as supporting Eerrent Select E2 as supported Element *
*
Rong
Select a Face Fl in El L* Select aFaceF2 in E2 *
Search Assembly Position Satisfin Gisel yes
Search Assembly Position Satisfin Case2
Store Assembly Reationship yes
Modular Fixture Element Assembly Relationship Database (MFEARDB)
Vj2 —>...—> v j m which satisfy the following acting height constraint: m
H = h(vi) + 2 h (vifc)
(42)
fc=i
where h(v) is the acting height of fixture element v and H is the acting height desired for the fixture unit. Fixture unit candidates are listed in three sequences according to: (1) the number of fixture elements used in the fixture unit; (2) the total weight of the unit; and (3) the volume of the unit. When a specially high accuracy or stiffness is required, the fixture unit with the least number of elements is chosen with priority. In case a light fixture body is desired, the lightest fixture unit is first selected. If the spatial restriction becomes a big problem in the process of fixture unit mounting, the fixture unit with the smallest volume is the one to be selected. Other optimization methods may also be applied with different criteria. 7.2. Fixture
unit mount
module
At the fixture unit generation stage, only fixture elements are selected and the topdown assembly relationships between fixture elements are determined. The exact positions and orientations between fixture elements need to be further determined
Yiming (Kevin)
160
Rong
at the fixture unit mounting stage. The mounting procedure can be conducted in two steps: (1) mounting the bottom element of fixture unit onto the baseplate; (2) determining the positions and orientations of other fixture elements, which is presented in next section. To mount the bottom element onto the baseplate, the following factors are taken into consideration: the position and orientation of workpiece, the suggested locating or clamping points, the machining envelope, the position possessed by other mounted fixture units, and the positions of bushed and tapped holes on the baseplate. The mounting requirements include a satisfaction of the acting point position of the unit to the desired fixturing point and the assembly relationship between the bottom element of the unit and the base plate. The algorithm for mounting bottom elements is similar to that presented in Refs. 23 and 43. When fixture units are mounted on a baseplate, the baseplate size is selected from the fixture component database based on the workpiece size and an estimation of the space required for fixture configuration design, although it may be changed later. Figure 37 shows a typical baseplate with locating holes and tapped holes. As it is discussed before, the global coordinate system is associated with the baseplate. Two parameters are used to indicate the positions of center of locating or tapped holes on the surface of baseplate, which are integers u and v in the ranges of (-N, N) and ( - M , M). For the modular fixture system, the screws and holes are alternatively and evenly distributed in two dimensions (X and Y). The center positions of tapped
1
( T 4
•
/
\
© O © O © C> © o © 6 © (M-DO © O © O C » O © O © O ••• © o © o © c ) © O © O © i O © O © O C» O © O © O f i i n Hi O A o © O © O © Cj1 %J> KJ %J) KJ *y • 1 O O O O O 0 O O O O O ... © 0 © 0 © 0 © 0 © 0 © -(M-DO © 0 © 0 © 0 © 0 © 0 M
-M© ..u -N
Fig. 37.
X
O © O © C) O O C 3> (D © -(N-l) ...
-2-1
0
1
2 ...
(N-l)
N
Representation of baseplate in dowel-pin based fixture system.
Computer-Aided
Modular Fixture
Design
161
holes on the baseplate can be represented parametrically as: xs = 2Tu + T((u + 1) mod 2), V. = Tv.
(43)
The center positions of locating holes on the baseplate can be represented as: a;h = 2Tu + T(umod2), Vh = Tv
(44)
where u = - N , . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . , N; and v = - M , . . . , - 3 , - 2 , - 1 , 0 , 1 , 2, 3 , . . . , M. T is a spacing increment between the tapped and locating holes in the row or column directions. In the modular fixture system there are three series of modular fixtures with a uniformed spacing increment, e.g. T = 30, 40 or 50 mm. In mounting a fixture unit onto the baseplate, a fixturing point (x*, y*, z*) and direction is the target to be approached by the acting point and acting direction of the unit. The acting height of the unit is designed to approach the target in z direction, which is presented by Eq. 42. Therefore the fixturing point is projected onto XOY plane with the target (x*, y*). The two parameters are determined for the center position of the tapped hole on the baseplate which is nearest to point (x*, y*)\ v* =div(y*/T + 0.5), u* = div((x* - T((div(i/*/T + 0.5) + l)mod 2))/2T + 0.5).
(45)
The coordinates of the nearest tapped hole can be calculated with Eq. 18 where u* and v* are the variables. The determination of the center position of the locating hole follows a similar procedure and sometimes is not necessary when standard modular fixture elements are utilized because these holes are evenly distributed. The mounting range of a fixture unit largely depends on the fixturing direction. Once the fixturing direction is specified, an acceptable mount range can be determined by considering the information of the fixture unit, mainly the bottom element of the unit. The geometric and assembly relationship information of the bottom element is recalled to match with the holes on the baseplate as represented above. 7.3. Position
of fixture unit acting
point
After the position and orientation of the bottom element of a fixture unit is decided, positions and orientations of other fixture elements in the fixture unit can be determined. Then the position of fixturing unit acting point can be determined, which is desired to be at the closest to the required locating/clamping position. There are usually a number of assembly positions between two fixture elements. For different fixture elements selected to build a fixture unit, there may be many combinations between those fixture elements. Figure 38 shows a sketch of all possible position assembly combinations between the fixture elements, where a series of matrices A
162
Yiming (Kevin)
No. t (bottom) element Fig. 38.
No. t-1 element
Rong
No. 1 (top) element
Position assembly combinations of fixture.
present position relationships between two fixture elements. Ai^ (i\ = 1, 2 , . . . , ni) are the position relationship matrices between the top element and the element supporting it. Atit (it = 1,2,..., n t ) are the position relationship matrices between baseplate and the bottom element which are obtained from bottom element mounting. The number ni and nt are the numbers of candidate mounting locations for the top and bottom elements. Therefore, we can see that the first subscript of the transformation matrix A is a sequential index of a fixture element from the top element and the second subscript indicates the possible assembly relationship of the current element with the element under it. In automated fixture design process, the assembly relationship between every two connecting elements in a fixture is retrieved from the fixture element assembly relationship database. The assembly relationship includes relative position and orientation between two fixture elements. Let us assume element i and i + 1 are two directly connecting elements in a fixture unit. Element i is supported element and element i + 1 is supporting element. If A; denotes the transformation matrix between element i and i + 1 local coordinate systems, it can be described by a 4 x 4 homogeneous matrix in the following form:
where U is a 3 x 3 matrix representing a rotation of the two coordinate systems; and d is 1 x 3 vector representing the translation of the assembly pair coordinate systems. By recalling the fixture element representation described previously, the relative position of element i in element i + 1 coordinate system (pi+i,ui+i,vi+i,Wi+i) is represented by the coordinates of its origin p^x^y^z^) in (pi+i,Ui+i,Vi+\,Wi+i) system and the orientation of element i in (pi+i,Ui+i,Vi+i,Wi+i) is represented by the directional cosines of the unit vectors of Uj, Vi and Wi in (pi+\,Ui+i,Vi+i,Wi)
Computer-Aided
Modular Fixture
Design
163
system as shown in Fig. 8. Aj can be expressed as: &
&
Ki T
K T Ki "r c' .
xi
Ai=
c' .
yi
^Xl
l
