Advanced Fixture Design Method and Its Application [1 ed.] 981334492X, 9789813344921

Table of contents :
Preface
Contents
1 Introduction
1.1 Fixturing Method
1.1.1 Alignment Fixturing
1.1.2 Fixture Fixturing
1.2 Fixture Structure
1.2.1 Types of Fixtures
1.2.2 Configuration of Fixtures
1.3 Fixture Design
References
2 Analysis of Locating Determination
2.1 Model of Theoretical DOF
2.1.1 Relationship Between Machining Requirements and DOFs
2.1.2 Establishment of Theoretical DOF Model
2.1.3 Undetermined Coefficient Method
2.1.4 Solving Rank Method
2.2 DOF Level Model of Position Sizes
2.2.1 Position Size of Plane Relative to Plane
2.2.2 Position Size of Line Relative to Plane
2.2.3 Position Size of Plane Relative to Line
2.2.4 Position Size of Line Relative to Line
2.3 DOF Level Model of Orientations
2.3.1 Relation Between Parallelism Tolerance and DOFs
2.3.2 Relation Between Perpendicularity/Inclination Tolerance and DOFs
2.4 DOF Level Model of Locations
2.4.1 Relation Between Coaxiality Tolerance and DOFs
2.4.2 Relation Between Symmetry Tolerance and DOFs
2.4.3 Relation Between Position Tolerance and DOFs
2.4.4 Relation Between Runout Tolerance and DOFs
2.5 Model of Locating Point Layout
2.5.1 Establishment of Locating Point Layout Model
2.5.2 Solution of Locating Point Layout Model
2.6 Judgment Criteria of Locating Determination
2.6.1 Theoretical Condition
2.6.2 Process Condition
2.6.3 Corollary and Flowchart
2.7 Application and Analysis
2.7.1 Verification of Locating Point’s Number
2.7.2 Verification of the Layout of Locating Points
References
3 Analysis of Workpiece Stability
3.1 Modeling of Workpiece Stability
3.1.1 Static Equilibrium Conditions
3.1.2 Friction Cone Constraints
3.1.3 Analysis Model
3.2 Solution Techniques
3.2.1 Linear Programming Techniques
3.2.2 Solution to the Model of Form-Closure
3.2.3 Solution to the Model of Force-Closure
3.3 Numerical Examples
3.3.1 Analysis of Force Existence
3.3.2 Analysis of Force Feasibility Without Friction
3.3.3 Analysis of Force Feasibility with Friction
References
4 Analysis of Clamping Reasonability
4.1 Local Deformation
4.1.1 Contact Deformation
4.1.2 Fixel Deformation
4.1.3 Local Deformation
4.2 Workpiece Position Error
4.2.1 Static Equilibrium Equation
4.2.2 Friction Cone
4.2.3 Relationship Between Local Deformation and Contact Force
4.2.4 Relationship Between Workpiece Position Error and Local Deformation
4.2.5 Solution Method
4.3 Workpiece Deformation
4.4 Modeling of Clamping Reasonability
4.5 Application and Analysis
4.5.1 Analysis of the Clamping Error Due to the Weak Stiffness Workpiece
4.5.2 Evaluation of the Clamping Error of High Stiffness Workpiece
References
5 Analysis of Workpiece Attachment/Detachment
5.1 Attachment and Detachment Model
5.2 Judgment Method of the Attachment and Detachment
5.3 Analysis Algorithm of the Attachment and Detachment Direction
5.3.1 Calculation Algorithm of the Generator Matrix of the Non-Positive Dual Matrix
5.3.2 Classification Method of Coefficient Matrix
5.3.3 Application of Pivot Algorithm
5.4 Analysis and Application of Attachment and Detachment
5.4.1 Three Dimensional Workpiece
5.4.2 Two Dimensional Workpiece
References
6 Analysis of Locating Accuracy
6.1 Analysis Model
6.1.1 Locating Source Error
6.1.2 Workpiece Position Error
6.1.3 Locating Error
6.2 Position Error Model of Contact Point
6.2.1 Directional Dimension Path
6.2.2 Directional Dimension Chain
6.3 Examples and Experiments
6.3.1 One Vee Bloke-One Supporting Pin Locating Scheme
6.3.2 Two Cylindrical Pins Locating Scheme
6.3.3 One Plane-Two Holes Locating Scheme
6.3.4 Experimental Validation
References
7 Selection Algorithm of Locating Datum
7.1 Hierarchical Structure Model
7.1.1 Surface Roughness
7.1.2 Surface Feature
7.1.3 Valid Locating Domain
7.1.4 Dimension Error
7.2 Judgement Matrix
7.3 Layer Weight Vector
7.4 Combination Weight Vector
7.5 Reconstruction of Judgment Matrix
7.6 Algorithm and Application
7.6.1 Algorithm and Flow
7.6.2 Practical Example
References
8 Planning Algorithm of Locating Point Layout
8.1 Locating Point Layout Scheme
8.2 Causes of Locating Non-determination
8.3 Generative Planning Algorithm
8.4 Application and Design Process
References
9 Planning Algorithm of Clamping Forces for Rigid Workpieces
9.1 Mechanical Model of Fixturing Scheme
9.1.1 Workpiece Static Equilibrium Constraints
9.1.2 Direction Constraints
9.2 Analysis of Force Existence
9.3 Analysis of Force Feasibility
9.4 Planning Algorithm of 1-clamping Force
9.5 Planning Algorithm of n-clamping Force
9.6 Calculation of Clamping Forces
9.6.1 Single Clamping Force of 2D Workpiece
9.6.2 Single Clamping Force of 3D Workpiece
9.6.3 Double Clamping Forces of 2D Workpiece
References
10 Determination Method of Clamping Point Layout for Rigid Workpieces
10.1 Workpiece Stability
10.1.1 Stability Index
10.1.2 Stability Measurement
10.2 Planning Algorithm of Application Region
10.3 Planning Method of Fixturing Layout
10.4 Numerical Tests of Application Region Planning
10.4.1 Planning of Application Region for One Clamping Force
10.4.2 Calculation of Stable Region in Dynamic Machining Process
10.4.3 Planning of Application Region for Multiple Clamping Forces
10.5 Examples of Fixturing Layout Optimization
10.5.1 Fixturing Layout Optimization of 2D Workpiece
10.5.2 Fixturing Layout Optimization of 3D Workpiece
References
11 Fixturing Layout Optimization of Thin-Walled Workpieces
11.1 Finite Element Analysis of Fixturing Deformation
11.1.1 Static Equilibrium Conditions
11.1.2 Friction Cone Constraints
11.1.3 Unilateral Contact Constraints
11.2 Prediction Method for Fixturing Deformation
11.2.1 Structure of Neural Network
11.2.2 Selection of Training Samples
11.2.3 Update of Weight Coefficients
11.3 Optimization Technology of Fixturing Layout
11.3.1 Optimal Model
11.3.2 Solution Technology
11.4 Numerical Tests and Experiments
11.4.1 Prediction of Workpiece Deformation
11.4.2 Optimization of Fixturing Layout
References
12 Matching Method of Fixture Elements
12.1 Analytical Hierarchy Process
12.2 Construction of Hierarchical Structure Model
12.2.1 Structural Model
12.2.2 Selection Factor
12.3 Determination of Layer Weight Vector for Criterion Layer
12.3.1 Construction of Judgement Matrix
12.3.2 Check of Consistency
12.4 Determination of Layer Weight Vector for Scheme Layer
12.4.1 Construction of Judgment Matrix
12.4.2 Check of Consistency
12.4.3 Decision Index of Locators
References

Citation preview

Guohua Qin

Advanced Fixture Design Method and Its Application

Advanced Fixture Design Method and Its Application

Guohua Qin

Advanced Fixture Design Method and Its Application

Guohua Qin Nanchang Hangkong University Nanchang, Jiangxi, China

ISBN 978-981-33-4492-1 ISBN 978-981-33-4493-8 (eBook) https://doi.org/10.1007/978-981-33-4493-8 Jointly published with Shanghai Jiao Tong University Press. The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Shanghai Jiao Tong University Press. © Shanghai Jiao Tong University Press 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Manufacturing technology is the foundation and key to the development of all industries. But in the manufacturing technology, fixtures are one kind of indispensable technical equipment that can directly influence the machining accuracy, production efficiency and manufacturing cost. Therefore, fixture design plays an important role in production technology preparation as well as product design and manufacturing. Fixture design is an important and complex technical work. In the traditional fixture design method, the determination of fixturing layout, the selection of fixture elements and the assembly of fixture elements are all completed by fixture designers. The traditional design method not only requires more manpower and a longer design cycle, but also relies on the rich experience of designers. With the wide application of computer technology in the manufacturing field, a new fixture design method is formed by using the integration of the computer with advanced manufacturing technologies including the feature technology, group technology and artificial intelligence. This is the so-called computer-aided fixture design technology. In the past, the key to develop the computer-aided fixture design system is to collect and express the knowledge from the experience of the fixture designers. However, it is either impossible or unrealistic to fully express all fixture design knowledge. Combined with kinematics, contact mechanics, elastic mechanics, mathematical modeling technology and optimization technology, the advanced fixture design method is systematically proposed. The essence of the proposed design method is to iteratively analyze the fixturing performance for the new fixturing layout until it is satisfied. The established analysis model of fixturing performance includes the locating determination, the workpiece stability, the clamping reasonability, the workpiece attachment/detachment, and the locating accuracy. By discretizing the value range of design variables, some planning algorithms are suggested including the selection algorithm of locating datum, the planning algorithm of clamping force, and so forth. In the proposed advanced fixture design method, the continuous fixture design problem is transformed into a discrete problem so that it can be easily realized by programming. Accordingly, it can enrich and develop the basic theory of computeraided fixture design and change the empirical method of fixture design. The combination of theoretical analysis and mathematical modeling technology can resolve v

vi

Preface

the key problems in the process of fixture design, which will play a certain role in promoting the progress of manufacturing technology, improving the precision and level of product manufacturing, and meeting the higher and higher requirements of the mechanical manufacturing industry. The contents covered in this book include major research outcomes of numerous research projects sponsored by the National Natural Science Foundation of China (51765047; 51465045; 51165039), Major Discipline Academic and Technical Leader Training Plan Project of Jiangxi Province (20172BCB22013), Aeronautical Science Foundation of China (2006ZE56006; 2010ZE56014), Natural Science Foundation of Jiangxi Province (2009GZC0104), Key Project of Science and Technology Support Plan of Jiangxi Province (2010BGB00300), and China Postdoctoral Science Foundation (20070411142). Moreover, several postgraduate students participated in relevant research work, including Haichao Ye, Huaping Huang, Meidan Zhou, Xiyuan Guo, Yue Cui, Huamin Wang, Zikun Wang, Shuo Sun, Xuiang Zhao, Yuanjun Hou, Zhe Huang, Jiamei Li, Xiuping Dai, Weida Lou, Feng Lin, Jianpeng Qiu, etc. I hereby express my sincere gratitude to them. Nanchang, China September 2020

Guohua Qin

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fixturing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Alignment Fixturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Fixture Fixturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fixture Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Types of Fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Configuration of Fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fixture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4 4 11 13 16

2

Analysis of Locating Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Model of Theoretical DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Relationship Between Machining Requirements and DOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Establishment of Theoretical DOF Model . . . . . . . . . . . . 2.1.3 Undetermined Coefficient Method . . . . . . . . . . . . . . . . . . . 2.1.4 Solving Rank Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 DOF Level Model of Position Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Position Size of Plane Relative to Plane . . . . . . . . . . . . . . 2.2.2 Position Size of Line Relative to Plane . . . . . . . . . . . . . . . 2.2.3 Position Size of Plane Relative to Line . . . . . . . . . . . . . . . 2.2.4 Position Size of Line Relative to Line . . . . . . . . . . . . . . . . 2.3 DOF Level Model of Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Relation Between Parallelism Tolerance and DOFs . . . . 2.3.2 Relation Between Perpendicularity/Inclination Tolerance and DOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 DOF Level Model of Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Relation Between Coaxiality Tolerance and DOFs . . . . . 2.4.2 Relation Between Symmetry Tolerance and DOFs . . . . . 2.4.3 Relation Between Position Tolerance and DOFs . . . . . . . 2.4.4 Relation Between Runout Tolerance and DOFs . . . . . . . . 2.5 Model of Locating Point Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19 23 25 28 28 30 31 33 36 36 44 52 52 53 57 61 63 vii

viii

Contents

2.5.1 Establishment of Locating Point Layout Model . . . . . . . 2.5.2 Solution of Locating Point Layout Model . . . . . . . . . . . . 2.6 Judgment Criteria of Locating Determination . . . . . . . . . . . . . . . . . 2.6.1 Theoretical Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Process Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Corollary and Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Application and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Verification of Locating Point’s Number . . . . . . . . . . . . . 2.7.2 Verification of the Layout of Locating Points . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 67 70 70 72 72 74 75 77 80

3

Analysis of Workpiece Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Modeling of Workpiece Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Static Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Friction Cone Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Linear Programming Techniques . . . . . . . . . . . . . . . . . . . . 3.2.2 Solution to the Model of Form-Closure . . . . . . . . . . . . . . 3.2.3 Solution to the Model of Force-Closure . . . . . . . . . . . . . . 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Analysis of Force Existence . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Analysis of Force Feasibility Without Friction . . . . . . . . 3.3.3 Analysis of Force Feasibility with Friction . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 85 90 91 93 93 94 100 102 102 104 106 109

4

Analysis of Clamping Reasonability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Contact Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Fixel Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Workpiece Position Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Static Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Friction Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Relationship Between Local Deformation and Contact Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Relationship Between Workpiece Position Error and Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Workpiece Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Modeling of Clamping Reasonability . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Application and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Analysis of the Clamping Error Due to the Weak Stiffness Workpiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Evaluation of the Clamping Error of High Stiffness Workpiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 113 117 119 120 121 122 123 123 127 128 128 129 129 131

Contents

ix

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5

6

7

Analysis of Workpiece Attachment/Detachment . . . . . . . . . . . . . . . . . . 5.1 Attachment and Detachment Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Judgment Method of the Attachment and Detachment . . . . . . . . . 5.3 Analysis Algorithm of the Attachment and Detachment Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Calculation Algorithm of the Generator Matrix of the Non-Positive Dual Matrix . . . . . . . . . . . . . . . . . . . . 5.3.2 Classification Method of Coefficient Matrix . . . . . . . . . . 5.3.3 Application of Pivot Algorithm . . . . . . . . . . . . . . . . . . . . . 5.4 Analysis and Application of Attachment and Detachment . . . . . . 5.4.1 Three Dimensional Workpiece . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Two Dimensional Workpiece . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 142 145 146 148 149 159 159 166 173

Analysis of Locating Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Locating Source Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Workpiece Position Error . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Locating Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Position Error Model of Contact Point . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Directional Dimension Path . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Directional Dimension Chain . . . . . . . . . . . . . . . . . . . . . . . 6.3 Examples and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 One Vee Bloke-One Supporting Pin Locating Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Two Cylindrical Pins Locating Scheme . . . . . . . . . . . . . . 6.3.3 One Plane-Two Holes Locating Scheme . . . . . . . . . . . . . . 6.3.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 176 177 182 183 183 185 189

Selection Algorithm of Locating Datum . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Hierarchical Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Surface Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Valid Locating Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Dimension Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Judgement Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Layer Weight Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Combination Weight Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Reconstruction of Judgment Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Algorithm and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Algorithm and Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Practical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 209 210 211 211 212 212 214 215 217 218 218 220 228

189 192 197 205 208

x

Contents

8

Planning Algorithm of Locating Point Layout . . . . . . . . . . . . . . . . . . . . 8.1 Locating Point Layout Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Causes of Locating Non-determination . . . . . . . . . . . . . . . . . . . . . . 8.3 Generative Planning Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Application and Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229 232 237 239 243

9

Planning Algorithm of Clamping Forces for Rigid Workpieces . . . . 9.1 Mechanical Model of Fixturing Scheme . . . . . . . . . . . . . . . . . . . . . 9.1.1 Workpiece Static Equilibrium Constraints . . . . . . . . . . . . 9.1.2 Direction Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Analysis of Force Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Analysis of Force Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Planning Algorithm of 1-clamping Force . . . . . . . . . . . . . . . . . . . . 9.5 Planning Algorithm of n-clamping Force . . . . . . . . . . . . . . . . . . . . 9.6 Calculation of Clamping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Single Clamping Force of 2D Workpiece . . . . . . . . . . . . . 9.6.2 Single Clamping Force of 3D Workpiece . . . . . . . . . . . . . 9.6.3 Double Clamping Forces of 2D Workpiece . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 245 246 247 247 248 250 253 257 257 261 263 270

10 Determination Method of Clamping Point Layout for Rigid Workpieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Workpiece Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Stability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Stability Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Planning Algorithm of Application Region . . . . . . . . . . . . . . . . . . . 10.3 Planning Method of Fixturing Layout . . . . . . . . . . . . . . . . . . . . . . . 10.4 Numerical Tests of Application Region Planning . . . . . . . . . . . . . . 10.4.1 Planning of Application Region for One Clamping Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Calculation of Stable Region in Dynamic Machining Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Planning of Application Region for Multiple Clamping Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Examples of Fixturing Layout Optimization . . . . . . . . . . . . . . . . . . 10.5.1 Fixturing Layout Optimization of 2D Workpiece . . . . . . 10.5.2 Fixturing Layout Optimization of 3D Workpiece . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Fixturing Layout Optimization of Thin-Walled Workpieces . . . . . . . 11.1 Finite Element Analysis of Fixturing Deformation . . . . . . . . . . . . 11.1.1 Static Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Friction Cone Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Unilateral Contact Constraints . . . . . . . . . . . . . . . . . . . . . . 11.2 Prediction Method for Fixturing Deformation . . . . . . . . . . . . . . . .

271 271 271 275 276 280 283 283 286 290 294 294 297 301 303 303 303 304 305 305

Contents

xi

11.2.1 Structure of Neural Network . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Selection of Training Samples . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Update of Weight Coefficients . . . . . . . . . . . . . . . . . . . . . . 11.3 Optimization Technology of Fixturing Layout . . . . . . . . . . . . . . . . 11.3.1 Optimal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Solution Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Numerical Tests and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Prediction of Workpiece Deformation . . . . . . . . . . . . . . . . 11.4.2 Optimization of Fixturing Layout . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

305 306 306 307 308 308 311 312 317 324

12 Matching Method of Fixture Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Analytical Hierarchy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Construction of Hierarchical Structure Model . . . . . . . . . . . . . . . . 12.2.1 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Selection Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Determination of Layer Weight Vector for Criterion Layer . . . . . 12.3.1 Construction of Judgement Matrix . . . . . . . . . . . . . . . . . . 12.3.2 Check of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Determination of Layer Weight Vector for Scheme Layer . . . . . . 12.4.1 Construction of Judgment Matrix . . . . . . . . . . . . . . . . . . . 12.4.2 Check of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Decision Index of Locators . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325 325 327 328 328 331 331 332 333 333 335 336 337

Chapter 1

Introduction

Whether in traditional manufacturing industry or modern flexible manufacturing system, workpiece fixturing is the first faced key problem during the machining process (Cai et al. 2002; Chen et al. 2008; Shi et al. 2020). The size, shape and relative position of the workpiece surfaces depend on the position relationship between the workpiece and the cutting tool in the process of cutting movement. The position between the workpiece and the cutting tool is achieved by the workpiece fixturing. The workpiece fixturing is to locate and clamp the workpiece in a fixture/jig on the machine tool (Cai et al. 2010). The workpiece locating is to make the workpiece occupy an accurate position with respect to the cutting tool and its cutting movement. It is worth mentioning that the cutting movement is usually provided by the machine tools. Only in this way, it is possible to ensure that the machined surface satisfied the specified machining requirements. And, the workpiece clamping is to hold the workpiece in a position which has been determined in the stage of workpiece locating. The workpiece clamping can prevent the workpiece, which is subjected to the external forces including the gravity force and the machining load during the machining process, from the position variation to upset the workpiece locating.

1.1 Fixturing Method When the workpiece is machined on a machine tool, either the alignment fixturing or the fixture fixturing can ordinarily use to located and hold it according to machining requirements and machining batch of the workpiece (Bai 1997).

© Shanghai Jiao Tong University Press 2021 G. Qin, Advanced Fixture Design Method and Its Application, https://doi.org/10.1007/978-981-33-4493-8_1

1

2

1 Introduction

1.1.1 Alignment Fixturing The alignment fixturing is to use the indicator (e.g. micrometer, dial indicator, etc.) or the scriber to determine the correct position of the workpiece according to the relevant surface or line of the workpiece. And then, the workpiece is hold to be machining. As shown in Fig. 1.1, a radial hole with a diameter of d will be drilled on the shaft sleeve workpiece. If the number of workpieces is not large, the alignment fixturing can be adopted. The extension line of L can be firstly drawn for the radial hole on the workpiece. Secondly, it is placed in the vice to fasten. Thirdly, the vice along with the workpiece is moved against the drilling bit on the drilling machine to find the highest point on the dimension line of the radial hole. Finally, the machine can be started for drilling the radial hole. The alignment fixturing can better adapt to the variation of the machining process or the machining objects. The used fixture is of simple structure. However, this fixturing method has low productivity, high labor intensity and low machining accuracy. In Fig. 1.1, the error of size L is large, and the position accuracy of axis of the radial hole d relative to the shaft sleeve is poor. Therefore, the alignment fixturing is mostly used for single piece and small batch production. With the development of production, the requirement of product quantity and quality is increasing, which promotes the development of fixture structure. People have created a new technological device and fixturing method. Thus, the workpiece can directly be clamped without the need for alignment. This novel technological device and the corresponding fixturing method is the machine fixture and fixture fixturing.

Fig. 1.1 Alignment fixturing on the vice according to the line

1.1 Fixturing Method

3

1.1.2 Fixture Fixturing Fixture fixturing is to place the workpiece in a fixture to obtain the correct position by the contact of the locating reference surface with the fixture locators. And then, fixture clamps are used to press the workpiece firmly. Figure 1.2 is a drilling fixture for machining the radial hole with the diameter of d on the shaft sleeve workpiece. The inner hole and the end plane of the workpiece 5 are selected as the locating references to keep in contact with locator 4. Thus, the workpiece is located for the deterministic position. The through hole at the right of the locating pin 4 can be used for drilling bit crossing and chip removal. The opening washer 2 is employed to clamp the workpiece by tightening the nut 3. The drilling sleeve 1 is used to guide the drilling bit in case that the drilling bit will deviate. Dimension L between the axis of the drill sleeve and the shoulder end plane of the locating pin is determined to obtain the correct position between the workpiece and the bit. Because the movement of the fixture relative to the machine tool can be adjusted in advance as well as the position of the fixture relative to the cutting tool, it is not necessary to align a batch of workpieces one by one when they are machined by Fig. 1.2 Drilling fixture of shaft sleeve workpiece

1

2 3

4 L

5

4

1 Introduction

the fixture fixturing method. Therefore, it is time-saving and convenient. It can not only have a high repeat precision, but also ensure the machining requirements of the workpiece. Because the fixture needs a certain production cost and preparation cycle, fixture is widely used in batch production and mass production.

1.2 Fixture Structure Fixture is the main component of technological equipment. Fixture design plays an important role in manufacturing system, which can directly influence the machining quality, production efficiency and manufacturing cost of workpieces. Consequently, fixture is considered to be one of the most active factors in the technological process (Wang 2005). And the manufacturing industry attaches great importance to the research on fixture.

1.2.1 Types of Fixtures With the difference of structure and dimension, machining accuracy and production mode of the workpiece, the structure, type and generalization degree of the fixture are different.

1.2.1.1

General Fixture

When the alignment fixturing is selected for the workpiece, the technological equipment, such as vice, three jaw chuck, four jaw chuck, is frequently used, as shown in Fig. 1.3. Such technological equipment has generally been standardized and supplied to users as accessories to machine tools. Because they are used to hold the workpiece, it belongs to the category of fixture, and called general fixture. The general fixture is mainly used for the single piece and small batch production. The use of this kind of fixture to hold the workpiece is frequently time-consuming,

(a) Vice

Fig. 1.3 General fixture

(b) Three jaw chuck

(c) Four jaw chuck

1.2 Fixture Structure

5

complex operation, and low production efficiency, especially for the workpiece with complex shape or high machining precision. For the mass production, it is not economically feasible to use a general fixture to hold the workpiece.

1.2.1.2

Dedicated Fixture

According to the characteristics of mass production, the machining process is divided into many simple processes carried out on different machine tools. And then, a continuous workpiece transmission line is adopted to connect these simple processes. This is the process dispersion principle—based machining process. The dedicated fixture is designed and manufactured for each process of a workpiece. Figure 1.4 is the drill jig for machining three radial holes on a fan-shaped workpiece. The fanshaped workpiece is located on the rotation shaft 4 and the stop pin 11 based on its inner hole, bottom surface and side surface. The nut 2 and the open washer 3 are sued to press the workpiece on the dividing dial 8. If a radial hole has been drilled on the dividing dial 8, the operation position should be changed. Therefore, the dividing dial 8 must firstly be released and the dividing pin 1 must secondly be pulled out. Thus, the next operation position can be achieved by turning the dividing dial 8. When the dividing pin 1 is plugged in and the dividing dial 8 is locked, the drill operation for the next radial hole can be carried out. Moreover, the open washer 3 is used to quickly load and unload the workpiece.

Fig. 1.4 Drill jig

6

1 Introduction

As seen from above, without consideration of generality, the dedicated fixture can be designed with compact structure or easy operation. Again, the design of the dedicated fixture can be assisted by selecting the labor-saving mechanism or the power device. Therefore, the dedicated fixture can ensure high machining accuracy and production efficiency. However, the long cycle of design and manufacturing will be expended for a dedicated fixture. According to the statistics, the production preparation cycle of the product generally accounts for 50–70% of the whole development cycle of the product on the basis of the current level of China’s machinery industry. The design and manufacturing cycle of process equipment accounts for 50–70% of the product production preparation period. And fixture design and manufacturing accounted for 70–80% of the process equipment design and manufacturing period (Geng and Liu 2002; Zhang and Su 2015). Accordingly, the dedicated fixture design and manufacturing cycle greatly affects the development cycle of a new product. On the other hand, if the product is changed, the special fixture can not be re-used to be a waste. Consequently, this kind of fixture is suitable for large and mass production of fixed products. With the progress of science and technology and the development of production, the departments of national economy require modern manufacturing industries to continuously provide good product quality and develop new product varieties, in order to meet the needs of the sustainable development of the national economy and the continuous improvement of people’s life. It has led to significant changes of the production mode in the manufacturing industry. Thus, there are more and more multi—variety and small batch production between mass production and single piece production. Especially in recent years, with the application of advanced manufacturing technologies, such as computer numerical control machine tool (CNC), machining center (MC), and flexible manufacturing system (FMS), the dedicated fixture and the general fixture can no longer meet the needs of production (Zhu and Rong 2000; Yan and Liu 1996). Thus, a series of innovative fixtures appeared between the general fixture and the dedicated fixture.

1.2.1.3

Adjustable Fixture

Adjustable fixtures are a new kind of fixture developed for the defects of general fixture and dedicated fixture. The fixture can be used for different types and sizes of workpieces by only adjusting or replacing several locators and clamps. Adjustable fixtures are generally divided into general adjustable fixture and group fixture. The former has the larger use range than that of the general fixture. The latter is a special adjustable fixture which is designed for a family of workpiece with similar structures according to the group principle. Therefore, the group fixture has good economic benefit in multi—variety and small batch production. Figure 1.5a shows a general adjustable three-jaw chuck. The screw 1 is connected with the pneumatic device. The spring brake pin 3 in the nut 2 can prevent the screw 1 from loosening. The screw 1 can centering fasten the workpiece by the sleeve 4, the lever 5, the jaw seat 6 and the jaw 7. When the piston returns, the jaw 7 exits

1.2 Fixture Structure 5

7 6

7

4 3 2

1

8

(a) Chuck

(c) Adjustable jaw 2

(b) Adjustable jaw 1

(d) Adjustable jaw 3

Fig. 1.5 General adjustable three-jaw chuck

along the inclined surface of the sleeve 4 through the jaw seat 6 so that the workpiece can be loosed. The jaws shown in Fig. 1.5b, c, d are used for the outer circle of step, small diameter workpiece and large diameter workpiece, respectively. Group fixture is composed of the base part and the adjustment part in structure. The base part is the general part of the group fixture, which is fixed in use. But the components in the adjustment part must be adjusted or replaced for the machining of a new workpiece. Figure 1.6a shows an adjustable drill fixture which is used to

8

1 Introduction

1

2

3 4 5

(a) Adjustable drill fixture

(b) Workpiece family

Fig. 1.6 Sleeve-shaped workpiece and its adjustable drill fixture

machine a radial hole on a family of sleeved-shaped workpiece shown in Fig. 1.6b. The inner hole and the end face of the workpiece 3 is located on the locator 2. Three fixture components including the handle 1, the open washer 4 and the nut 5 are used to hold the workpiece 3. Here, the locator 2 belongs to the adjustment part of the fixture whereas the other elements are the base part of the fixture.

1.2.1.4

Modular Fixture

Modular fixtures are a kind of fixture with high standardization, serialization and generalization. They are assembled by a set of pre-manufactured standard components and their units with different shapes, different specifications, different sizes and complete interchangeability, high wear resistance and high precision according to the machining requirements of different workpieces (Wang et al. 2003). After the fixture is used, it can be disassembled, cleaned and sealed for re-assembly and reuse. Modular fixtures change the dedicated fixture from the one-way process of “design, manufacture, use and waste” to the cycle process of “assembly, use, disassembly, reassembly, reuse and re-disassembly” (Rong et al. 1997, 2002). However, compared with dedicated fixture, modular fixtures have larger volume, heavier weight and weaker rigidity.

1.2 Fixture Structure

9

Fig. 1.7 Modular fixture with hole-based system

Modular fixturing systems can be generally classified into slot-based system and hole-based system. Hole-based systems have accurately positioned holes on the baseplates which are adopted to locate and fasten fixture components, as shown in Fig. 1.7. Slot-based system have parallel and perpendicular tee-slots on the baseplates. Functionally both hole- and slot-based modular systems serve the same purpose i.e., to provide configurability. However in slot-based systems, the order of assembly of modular elements has to be considered carefully, especially when the elements are fastened in the same row of slots. A comparison between the slot and hole-based system is given in Table 1.1.

1.2.1.5

Phase-Change Fixture

Phase-change fixtures use the the concept of a material phase change. This technique makes use of certain class of materials such as a low melting point alloy which is capable of rapidly converting its phase from liquid to solid and vice versa. Typically, a fixture of this kind consists of a container filled with this material and a mechanism to initiate the phase change. The fixturing procedure is initiated when the bi-phase material is in the liquid or semi-liquid state. A workpiece is immersed and placed in a desirable orientation in the container. The material is then subjected to some external influence (catalysts or cooling) which solidifies the material and firmly secure the

10 Table. 1.1 Characteristic of modular fixturing systems

1 Introduction Items

Slot-based modular element systems

Hole-based modular element systems

Cost of fabrication

High

Low

Ease of assembly and flexibility

Good

Relatively restricted

Requirement of skill and assembly

High

Relatively low

Adjustment of relative position of locators

Convenient and adjustment is not limited

Not convenient and adjustment is limited

Accessories for mounting the elements

More

Relatively few

workpiece in the desired position for the machining operations. When the machining operations are completed, the material is once again subjected to catalyst actions to return to its liquid form and the workpiece is easily removed from the fixture (Nee et al. 1995, 2005). This type of flexible fixturing is appropriate for irregular workpieces which are very difficult to hold. However, the phase-change fixtures still have some disadvantages. These fixtures provide supporting but not locating and some additional mechanism is needed to align the workpiece when it is immersed in the liquid medium. The locating function is therefore transferred to a separate alignment jig or a robot to hold the workpiece until the material becomes solid and strong enough to hold and maintain the position of the workpiece. The cost and difficulty of reconfiguring the separate locating mechanism remains. Most available systems of phase-change fixtures work by the direct encapsulation. It has been developed specifically for machining the roots of turbine and compressor blades. The sequence of operations is demonstrated in Fig. 1.8. The blade is contained in a diecasting mould and precisely located with respect to the mould by an external alignment jig. Liquid low melting point alloy is injected into the mould and allowed to cool and solidify. The blade encapsulation block has standardised location features so that it can be held in a standard fixture on the machine tool. When machining is complete the block is cracked open to release the blade and the block material is re-cycled. Obviously, the system works very well in this application which would be very difficult to fixture in any other way. There are drawbacks to the system. The external alignment jigs are a restricting factor for reconfigurability and the encapsulation material is not as stiff as conventional materials employed in fixture construction. The process can only be used for small workpiece, 300 mm or less. The mass of the block material is a limiting factor if robot handling is required.

1.2 Fixture Structure

(a) Positioned and solidified in die

(c) Held and Machined

11

(b) Encapsulated

(d) Released

Fig. 1.8 Encapsulation phase-change fixture

1.2.1.6

Conformable Fixture

Conformable fixtures are fixtures with clamping elements that automatically conform to the shape of the workpiece. They are passive device, which can change shape and reach a stable configuration when the clamping force is applied or they may be programmable to change shape under active control. Figure 1.9 is a vice with multiple leaves (Nee et al. 1995). The multi-leaf vice has a solid movable jaw and a fixed jaw made up from multiple leaves pivoted on a rod. The leaves are free to pivot about the rod but all other movements are constrained. The leaves are spring-loaded to a neutral position. This device is able to clamp long uneven cross-sectioned components.

1.2.2 Configuration of Fixtures As stated above, although fixtures are various and different from each ohter, the common structural components of a fixture can be summarized from different fixture types (Nee et al. 2005; Hoffman 1991). Thus, fixture design method can be further concluded.

12

1 Introduction

Torsional spring (Fixed jaw)

Movable jaw

Workpiece

Fig. 1.9 Multi-leaf conformable vice

(1)

Locator

The purpose of locator is to determine the position of the workpiece in the fixture. Components 4 and 11 shown in Fig. 1.4 belong to locator. A theoretical locator will prevent movement of the workpiece in one direction of one DOF. (2)

Clamp

The function of clamp is to provide a holding force. The holding force may hold the workpiece being fixtured against a locator by preventing motion in the opposite direction or provide a moment preventing rotation about some instantaneous centers. For example, components 2 and 3 illustrated in Fig. 1.4 are clamps. (3)

Fixture body

Fixture body is generally a rigid structure, the purpose of which is to provide a base for mounting the locators, clamps and other fixture components. Fixture bodies are made in three general forms: cast, forge, and welded. As a rule, the size and shape of the fixture body is determined by the size of the workpiece and the operation to be performed. Component 9 in Fig. 1.4 is the fixture body.

1.3 Fixture Design

13

1.3 Fixture Design In the manufacturing industry, especially in the mechanical manufacturing industry, no matter how the production scale of workpieces is, a variety of technological equipment are needed. Widely used in manufacturing, fixtures have a direct impact upon product quality, productivity and cost. Generally, the costs associated with fixture design and manufacturing can account for 10–20% of the total cost of a manufacturing system (Bi and Zhang 2001). Approximately 40% of rejected workpieces are due to dimensioning errors that are attributed to poor fixturing design (Wang et al. 2003; Hashemi et al. 2014). Fixture design work is also tedious and time-consuming. It often heavily relies on fixture design engineers’ experience and knowledge and frequently requires over 10 years manufacturing practice to design quality fixtures. Traditionally, the design and manufacture of a fixture can take several days or even longer to complete when human experience in fixture design is utilized. And a good fixture design is often based on the designer’s experience, his understanding of the products, and a try-and-error process. Therefore, with the increasingly intense global competition which pushes every manufacturing in industry to make the best efforts to sharpen its competitiveness by enhancing the product’s quality, squeezing the production costs and reducing the lead time to bring new products to the market, there is an strong desire for the upgrading of fixture design methodology with the hope of making sound fixture design more efficiently and at a lower cost. With the development of computer technology, computer aided technologies (CAx) for all stages of engineering, including CAD, CAPP, CAM, CAE and CAT, play an important role in mechanical manufacturing system. Since the 1980s, the application of computer technology to fixture design has been one of the research topics in the mechanical field. Recently in various industrial developed countries in the world, various CAFD (Computer Aided Fixture Design) systems have been developed based on the traditional fixture design theory and advanced manufacturing technology to be a new fixture design method (Liu and Shen 2004). This is of great significance to the development of fixture design and manufacturing. CAFD is a new idea of fixture design by fixture designers through computer-aided technology and scientific methods. It is not only the extension and development of traditional fixture design method, but also the need of the rapid development of flexible manufacturing system (FMS) and computer integrated manufacturing system (CIMS). The process of CAFD can be divided into four stages (Zhang and Su 2015; Duan et al. 2004; Chou et al. 1994; Rong and Huang 2005): fixture requirement analysis, fixturing layout planning, fixture configuration design and fixturing performance evaluation, as shown in Fig. 1.10. (1)

Fixture requirement analysis

The task of fixture requirement analysis stage is mainly to determine the number of fixturing required to perform all the manufacturing processes, the machining surface

14

1 Introduction

Fig. 1.10 The process of fixture design

and its machining requirements for each fixturing. A fixturing represents the combination of processes that can be performed on the workpiece by a single machine tool without having to change the position and orientation of the workpiece. The data of fixture requirement analysis comes from the process. Therefore, it is generally regarded as a subset of CAPP. Obviously, the function of fixture requirement analysis is the interactive interface of CAFD and CAPP integration. Because this stage is more closely related to CAPP, most of the researches in CAFD field focus on three stages: fixturing layout planning, fixture configuration design and fixturing performance evaluation. (2)

Fixturing layout planning

Fixturing layout planning is the initial and most creative stage in fixture design process, the purpose of which is to ensure that the workpiece has locating determination, workpiece stability, clamping reasonability, and attachment/detachment. These performances are achieved by reasonably planning the fixturing layout. Therefore,

1.3 Fixture Design

15

the core task of the fixturing layout planning stage is to determine the surface, upon which the locating/clamping points and clamping forces must act, as well as the actual positions of the locating and clamping points on the workpiece. Fixturing layout planning is a complex and abstract conceptual design at the highest level. It is the most critical technology to realize the automation and flexibility of fixture design, and it is also a bottleneck problem. Therefore, the fixturing layout planning is the key stage to identify the technical and economic benefits in the whole process of fixture design. Its development will promote the automatic design of fixture structure scheme, and even the whole fixture design to the direction of automation, intelligence and flexibility. (3)

Fixture configuration design

The main work of this stage is to select fixture elements (i.e., determine the structural shape and dimensions of fixture elements). Finally, the fixture structure scheme is achieved by assembling the selected fixture elements. (4)

Fixturing performance evaluation

During the performance evaluation stage, the design is tested to ensure that all manufacturing requirements of the workpiece can be satisfied which include the locating determination, workpiece stability, clamping reasonability, and so on. The design also has to be verified to ensure that it meets other design considerations that may include fixture cost, fixture weight, assembly time, and loading/unloading time of both the workpiece and fixture components/units. As stated above, fixture design essentially includes three stages of fixturing layout planning, fixture configuration design and fixturing performance evaluation. However, after the work in the fixture configuration design stage is in-depth studied, it will be found that the work in the fixturing layout planning stage is often intertwined with the work in the fixture configuration design stage. Thus, fixture design process can be further summarized into two aspects: structure design and performance analysis. The structure design includes the fixturing layout planning and the fixture configuration design. When the fixture structure is drawn up or designed, the performance that can be achieved should be analyzed and evaluated to measure whether it can guarantee the machining requirements of the workpiece, so as to determine the rationality of the fixture structure. More importantly, the performance analysis can guide the structure design, and make the fixture structure obtain more reasonable design results. Therefore, the structure design and the performance analysis are two complementary aspects. The performance analysis is the means of structure design, and the structure design is the purpose of performance analysis.

16

1 Introduction

References Bai CX. New principles of fixture design method [M]. China Machine Press, 1997. (in Chinese). Bi ZM, Zhang WJ. Flexible fixture design and automation: review, issues and future direction [J]. Int J Prod Res. 2001;39(13):2867–94. Cai J, Duan GL, Li CY, Li DH. Summary on the development of fixture designing technology [J]. J Hebei Univ Technol. 2002;31(5):35–40 (in Chinese). Cai J, Duan GL, Yao T, Xu HJ. Summary on the review and development trend of computer aided fixture designing technology [J]. J Mach Des. 2010;27(2):1–6 (in Chinese). Chen H, Chen WF, Zheng HL. FEM simulation for optimization fixture scheme of thin-walled workpiece [J]. Modul Mach Tool Automat Manufact Techn. 2008;3:63–7 (in Chinese). Chou YC, Srinivas RA, Saraf S. Automatic design of machining fixtures: conceptual design [J]. Int J Adv Manuf Technol. 1994;9(1):3–12. Duan GL, Lin JP, Zhang MD, Qi HW. The system of computer aided 3d modular fixture configuration design with intelligence [J]. J Hebei Univer Technol. 2004;33(2):104–9 (in Chinese). Geng YX, Liu X. Function analyzing in fixture conceptual design [J]. J Beijing Ins Light Ind. 2002;20(1):44–8 (in Chinese). Hashemi H, Shaharoun AM, Izman S, Kurniawan D. Recent Developments on computer aided fixture design: case based reasoning approaches [J]. Adv Mechan Engin; 2014:484928–1–15. Hoffman EG. Jig and fixture design [M]. New York: Delmar Publishers; 1991. Liu J, Shen XH. Application of advanced manufacturing technology in computer-aided fixture design [J]. J Beijing Technol Busin Uni. 2004;22(2):38–41 (in Chinese). Nee AYC, Whybrew K, Senthil KA. Advanced fixture design for FMS [M]. London: SpringerVerlag; 1995. Nee AYC, Tao ZJ, Kumar AS. An Advanced treatise on fixture design and planning [M]. Singapore: World Scientific; 2005. Rong YM, Huang S. Advanced computer aided fixture design [M]. Boston (MA): Elsevier Academic Press; 2005. Rong Y, Liu X, Zhou J, Wen A. Computer aided step-up planning and fixture design [J]. Intell Auto Soft Comput. 1997;3(3):191–206. Rong Y, Zhu Y, Luo Z. Computer-Aided Fixture Design [M]. China Machine Press, 2002. (in Chinese). Shi C, Lu YM, Ye W, Liu Y. Research on multi-method fusion fixture based on orthogonal design [J]. Manufact Technol Mach Tool. 2020;10:67–74. Wang YM. The tongs production preparation in CIMS [J]. Mechan Res Applic. 2005;18(3):47–8. Wang FQ, Xu HJ, Guo W. Overview of computer-aided fixture design [J]. Aeronaut Manufact Technol. 2003;11:38–40 (in Chinese). Wang H, Rong Y, Li H, Shaun P. Computer aided fixture design: recent research and trends [J]. Comput Aided Des. 2010;42(12):1085–94. Yan ZZ, Liu XM. Computer aided fixture design [J]. J Inner Mong Fores College. 1996;18(3):69–74 (in Chinese). Zhang SW, Su YH. Overview of computer-aided fixture design [J]. Manufact Technol Mach Tool. 2015;4:50–5 (in Chinese). Zhao HX, Xiong LS. The studies and development of computer-aided fixture deign [J]. Modul Mach Tool Auto Manufact Techn. 2007;2:1–4 (in Chinese). Zhu Y, Rong Y. The Development of flexible fixtures and computer-aided fixture designing technology [J]. Manufact Technol Machine Tool. 2000;8:5–8 (in Chinese).

Chapter 2

Analysis of Locating Determination

Locating determination is the precondition to design of locating point layout scheme. if the workpiece locating is unreasonable and non-deterministic, it is not significant to design the locating point layout scheme. According to the condition that the absolute velocity of the workpiece in the direction of machining requirement is zero, a machining requirement model is established which describes the theoretical DOFs (DOF, Degree of Freedom) as a function of machining requirements in this chapter. And then, in combination with rigid body kinematics, Taylor expansion is used to deduce the locating point layout model in which the practical DOFs are expressed as a function of the positions and the corresponding unit normal vectors of locating points. Finally, in light of the inclusion relationship between the theoretical DOFs and the practical DOFs, the theoretical DOFs can be equivalent to the solution of the locating point layout model. With consideration of the practical technological conditions of the locating point layout, the quantitative criterion of locating determination is proposed based on the existence of the solutions of homogeneous linear equations.

2.1 Model of Theoretical DOF In the locating theory during the machining process, the workpiece is generally regarded as a rigid body. According to the free rigid body motion theory of theoretical mechanics, the position of a free rigid body in a space rectangular coordinate system can be determined by six independent coordinate parameters, as shown in Fig. 2.1. The position variations of these independent coordinates are called DOFs. Here, denote Ow -X w Y w Z w to be the moving coordinate system that is consolidated with the rigid body whereas O-XYZ be the fixed Coordinate System. So the moving coordinate system is also called the WCS (WCS, Workpiece Coordinate System) whereas the fixed coordinate system is called the GCS (GCS, Global Coordinate System). Thus, the position coordinate of the origin of the WCS in the GCS is δrw = [δx w , δyw , δzw ]T . In addition, arbitrary direction of the rigid body can be achieved by first rotating the © Shanghai Jiao Tong University Press 2021 G. Qin, Advanced Fixture Design Method and Its Application, https://doi.org/10.1007/978-981-33-4493-8_2

17

18

2 Analysis of Locating Determination

Fig. 2.1 Diagram of the position of a rigid body in a rectangular coordinate system

angle δα w around the axis X w , then rotating the angle δβ w around the new axis Y w , and finally rotating the angle δγ w around the new axis Z w . δΘ w = [δα w , δβ w , δγ w ]T is the cardan angle. When the position coordinates δrw and the cardan angle δΘ w are known, the position of the rigid body is completely determined. According to this principle, six DOFs of the free rigid body in GCS are selected as three translation DOFs T x , T y , T z and three rotation DOFs Rx , Ry , Rz . Obviously, the six DOFs correspond to six position parameters of the rigid body, i.e., δx w , δyw , δzw , δα w , δβ w and δγ w .

2.1.1 Relationship Between Machining Requirements and DOFs A plane with process dimension h ± Δh will be milled on the top of the workpiece. The design dimension h ± Δh is in the Y direction, as shown in Fig. 2.2. However, in the free state, the workpiece has 6 DOFs including 3 translation DOFs along the coordinate axis and 3 rotation DOFs around the coordinate axis. These DOFs will certainly affect the machining accuracy of the workpiece (Qin et al. 2010). If the workpiece has the translation DOF δx w in the X direction, the procedure dimension h1 can be obtained after milling the top surface. Because DOF δx w does not affect the dimension h in the Y direction between the bottom surface and the milling tool, there exist h1 = h. In other words, δx w has no effect on the design dimension, as shown in Fig. 2.3a.

2.1 Model of Theoretical DOF

19

Fig. 2.2 Machining surface and its machining requirement

If the workpiece has the translation DOF δyw in the Y direction, the process dimension obtained after milling the top surface is h2 . Obviously, the process dimension h2 is not equal to the process dimension h. Therefore, there is the machining error δh = h2 -h so that the process dimension h can not be satisfied, as shown in Fig. 2.3b. By analogy, DOFs δzw and δβ w can impact the the process dimension h, as shown in Fig. 2.3c, e. However, DOFs δzw and δβ w cannot guarantee the specification of dimension h, as shown in Fig. 2.3d, f. The position of a free workpiece is uncertain in space. In order to determine the position of the workpiece in accordance with certain requirements (i.e., the machining requirements for each process), it is necessary to constrain some or all DOFs of the workpiece.

2.1.2 Establishment of Theoretical DOF Model Assumed that vw = [vwx , vwy , vwz ]T and ωw = [ωwx , ωwy , ωwz ]T are respectively the linear velocity and the angular velocity of the free workpiece in GCS, as shown in Fig. 2.4. Denote the origin Ow of WCS to be the instantaneous center of workpiece, the velocity of Ow is v O W = vw

(2.1)

w w T Denote that rP = [x P , yP , zP ]T and r wP = [x w P , y P , z P ] are respectively the position of arbitrary point P on the workpiece in GCS and WCS. The velocity of P can be obtained according to the principle of velocity synthesis of particles, that is

v P = vOW + v P O

(2.2)

where vP is the absolute velocity, v O w is transport velocity, and vPO is the relative velocity.

20

2 Analysis of Locating Determination

Fig. 2.3 Effect of DOFs on machining requirement

By substituting Eq. (2.1) into Eq. (2.2), the absolute velocity of point P can be rewritten as v P = vw + r P × (ωw )

(2.3)

The translation and rotation of the workpiece will vary the size or position of the machined surface. According to Eq. (2.3), the position variation of point P in a very short time δt can be obtained as v P δt = (vw δt) + r P × (ωw δt)

(2.4)

2.1 Model of Theoretical DOF

21

Fig. 2.4 Motion state of workpiece

where δrP = vP δt = [δx P , δyP , δzP ]T is the position variation of P. Obviously, vw δt and ωw δt are the position variation and direction variation of the workpiece which are resulted from the linear velocity vw and the angular velocity ωw , respectively. They can be expressed as 

δr w = vw δt δΘ w = ωw δt

(2.5)

where δrw = [δx w , δyw , δzw ]T is the position variation of the workpiece which includes three translation DOFs of the workpiece. δΘ w = [δα w , δβ w , δγ w ]T is the direction variation of the workpiece which consists of three rotation DOFs of the workpiece. By substituting Eq. (2.5) into Eq. (2.4), the relationship between the DOFs of the workpiece and the size variation (or location variation) in three machining directions (namely, X, Y and Z) can be achieved as δr P = Pδq w

(2.6)

22

2 Analysis of Locating Determination

where δqw = [δrwT , δΘ Tw ]T is the 6 DOFs of the workpiece. δrP = [δx P , δyP , δzP ]T is the size variation or location variation of the workpiece. P = [I, Ω] with the unit matrix I and the skew symmetric matrix Ω whose expression is ⎡

⎤ 0 −z P y P Ω=⎣ z P 0 −x P ⎦ −y P x P 0

(2.7)

On the other hand, the angle variation or orientation variation is not be caused by the translation vw of the workpiece but the rotation ωw of the workpiece. It is known from Fig. 2.3 that, the relative velocity of point P caused by ωw is v P O = r P × ωw

(2.8)

Thus, the relationship between the DOFs of the workpiece and the angle variation or orientation variation in the three machining directions can be obtained as δr P O = Dδq w

(2.9)

where D = [O, Ω] with the zero matrix O. δrPO is the angle variation or orientation variation. In fact, the workpiece may not have machining requirements in all three directions (that is, X, Y, and Z) at the same time. Denote e to be the directional vector of machining requirement, the DOFs must be limited as follows according to Eqs. (2.6, 2.9) eT Pδq w = 0

(2.10)

eT Dδq w = 0

(2.11)

or

where e = eX when the workpiece has the machining requirement in the X direction, e = eY when the workpiece has the machining requirement in the Y direction, and e = eZ when the workpiece has the machining requirement in the Z direction. e = [eX , eY ]T when the workpiece has simultaneously the machining requirements in the X and Y directions, e = [eY , eZ ]T when the workpiece has simultaneously the machining requirements in the Y and Z directions, and e = [eX , eY ]T when the workpiece has simultaneously the machining requirements in the X and Y directions. If the workpiece has the machining requirements in the X, Y, Z directions, e = [eX , eY , eZ ]T with eX = [1, 0, 0]T , eY = [0, 1, 0]T and eZ = [0, 0, 1]T . (Qin et al. 2008) Thus, if point P is on the process datum, the theoretical DOFs δq ∗w can be obtained by either undetermined coefficient method or solving rank method according to Eqs. (2.10, 2.12).

2.1 Model of Theoretical DOF

23

2.1.3 Undetermined Coefficient Method As shown in Fig. 2.5a, now there are the upper and lower planes of the fork will be machined on the shifting fork, and the size h is required to be guarantee. Since the size h is in the Z direction, e = [0, 0, 1]T . The following equation can be obtained according to Eq. (2.10) δz w + δαw y P − δβw x P = 0

(2.12)

Here, both x P and yP are arbitrary values. To make Eq. (2.12) always hold, if and only if δz w =δαw +δβw = 0

(2.13)

That is to say, in order to guarantee the machining requirement of size h, the first condition is that there should constrain the δzw , δα w and δβ w DOFs. So the theoretical DOFs δq ∗w is δq ∗w = ζ ∗ λ∗ = λx ζ x +λ y ζ y +λγ ζ γ

(2.14)

where the constant vector λ* = [λx , λy , λγ ]T with the non-zero arbitrary numbers λx , λy , λz , λα , λβ and λγ . The base vector matrix ζ * = [ζ x ,ζ y ,ζ γ ] with the unit vectors ζ x = [1, 0, 0, 0, 0, 0]T , ζ y = [0, 1, 0, 0, 0, 0]T , ζ z = [0, 0, 1, 0, 0, 0]T , ζ α = [0, 0, 0, 1, 0, 0]T , ζ β = [0, 0, 0, 0, 1, 0]T and ζ γ = [0, 0, 0, 0, 0, 1]T (Qin et al. 2008). Figure 2.5b is a oil pump shell. the lathe is used to bore holes 3 and 4 as well as holes 5 and 6. On one hand, the machining requirements of holes 3 and 5 are as follows: size 85 in the X and Y directions, and perpendicularity 0.02 in the Z direction. On the other hand, the machining requirements of holes 4 and 6 include size 70 in the X direction and size 85 in the Y direction. It follows that the workpiece has the size requirements in the X and Y directions in addition to the directionalposition requirement. Thus, there is e = [ex , ey ]T . Therefore, in order to guarantee the sizes in the X and Y directions, it is known from Eq. (2.10) that 

δxw + δβw z P − δγw y P = 0 δyw + δγw x P − δαw z P = 0

(2.15)

Because x P , yP and zP are arbitrary numbers, the following condition must be achieved to ensure Eq. (2.15) be true δxw = δyw = δαw = δβw = δγw = 0

(2.16)

Therefore, five DOFs (δx w , δyw , δα w , δβ w and δγ w ) must be limited to ensure the size requirements in both X and Y directions.

24

Fig. 2.5 Workpieces and these machining requirements

2 Analysis of Locating Determination

2.1 Model of Theoretical DOF

25

Moreover, to guarantee the perpendicularity requirement in the Z direction, the directional vector of e = ez is substituted into Eq. (2.11) to obtain y P δαw − x P δβw = 0

(2.17)

Here, x P and yP belong to arbitrary numbers. Therefore, in order for Eq. (2.17) to be true, it must be δαw = δβw

(2.18)

According to Eq. (2.16) and Eq. (2.18), in order to satisfy all the machining requirements of hole 3, hole 4, hole 5 and hole 6, the following conditions must be met δxw = δyw = δαw = δβw = δγw = 0

(2.19)

In other words, only by limiting the five DOFs (δx w , δyw , δα w , δβ w and δγ w ) can the size requirements in the X and Y directions be guaranteed in addition to the perpendicularity requirement in the Z direction. Likewise, a key slot will be milled on the stepped shaft. The guaranteed dimensions are not only sizes 22 and 42 in the X direction, but also sizes 35 and 12 in the Y and Z directions. Therefore, e = [ex , ey , ez ]T . According to Eq. (2.10), there should be ⎧ ⎨ δxw + δβw z P − δγw y P = 0 δy + δγw x P − δαw z P = 0 ⎩ w δz w + δαw y P − δβw x P = 0

(2.20)

Because x P , yP and zP can take arbitrary values, Eq. (2.20) is valid if and only if δxw = δyw = δαw = δβw = δγw = 0

(2.21)

As stated above, in order to ensure the size requirements in the X, Y, and Z directions at the same time, the six DOFs, δx w , δyw , δzw , δα w , δβ w and δγ w , of the workpiece must be constrained.

2.1.4 Solving Rank Method In reality, the shafting fork shown in Fig. 2.5a has the machining requirement only in the Z direction. Accordingly, it is necessary to meet Eq. (2.12), that is, δz w + δαw y P − δβw x P = 0. Since x P and yP are arbitrary values, Eq. (2.12) can be further described as the non-homogeneous linear equation about x P and yP , i.e.,

26

2 Analysis of Locating Determination

δαw y P − δβw x P = −δz w

(2.22)

Obviously, the solution form of Eq. (2.22) can be written as

yP xP





 1 0 = k1 + k2 + η∗ 0 1

(2.23)

with the arbitrary numbers k 1 and k 2 (Ma and Zhang 2013). In order to clearly understand Eq. (2.23), it is described as the following matrix form,



yP = −δz w δαw −δβw xP

(2.24)

Thus, if and only if the following condition hold, rank([ δαw −δβw ]) = rank([ δαw −δβw −δz w ]) = 0

(2.25)

x P and yP in Eq. (2.24) have infinite solutions. It is known from Eq. (2.25), there is δz w = δαw = δβw = 0 which means that DOFs δzw , δα w and δβ w are constrained. Only in this way can the size requirement in the Z direction be guaranteed. If the workpiece has the machining requirements in both directions, more DOFs must be limited. As shown in Fig. 2.5b, the oil pump shell has machining requirements in X and Y directions. Therefore, the following equation can be achieved

⎡ ⎤  xP 

0 −δγw δβw ⎣ ⎦ −δxw = yP δγw 0 −δαw −δyw zP

(2.26)

Since x P , yP , and zP are arbitrary numbers, then Eq. (2.26) has a solution of the form ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ xP 1 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ (2.27) ⎣ y P ⎦ = k 1 ⎣ 0 ⎦ + k 2 ⎣ 1⎦ + k 3 ⎣ 0 ⎦ + η zP 0 0 1 where k 1 , k 2 and k 3 are the arbitrary numbers. It is clear that the rank of coefficient matrix of Eq. (2.26) is not only equal to the rank of the augmented matrix, but also equal to zero, i.e., 

rank

0 −δγw δβw δγw 0 −δαw





= rank

0 −δγw δβw −δxw δγw 0 −δαw −δyw

 =0

(2.28)

2.1 Model of Theoretical DOF

27

Therefore, δxw = δyw = δαw = δβw = δγw = 0

(2.29)

In other words, in order to guarantee the size requirements in the X and Y directions, it is necessary to constrain five DOFs, δx w , δyw , δα w , δβ w and δγ w . Moreover, the workpiece has also the perpendicularity requirement in the Z direction which belongs to the directional-position tolerance. According to Eq. (2.11), it can be obtained as



yP =0 (2.30) δαw −δβw xP Because x P and yP is arbitrary numbers, the solution of Eq. (2.30) can be described as

yP xP





 1 0 = k1 + k2 0 1

(2.31)

In order to make the solution of Eq. (2.30) be the form of Eqs. (2.30, 2.31) must satisfy the following condition, i.e., rank([ δαw −δβw ]) = 0

(2.32)

It can obtained as δα w = 0 and δβ w = 0. In other words, only by constraining δα w and δβ w DOFs can the directional-position requirement (i.e., the perpendicularity requirement) in the Z direction be guaranteed. As shown in Fig. 2.5c, in order to obtain the specified linear sizes in the X, Y and Z directions for the key slot milled on the stepped shaft, the equation can be concluded according to Eq. (2.10). ⎤⎡ ⎤ ⎡ ⎤ xP −δxw 0 −δγw δβw ⎣ δγw 0 −δαw ⎦⎣ y P ⎦ = ⎣ −δyw ⎦ −δβw δαw 0 zP −δz w ⎡

(2.33)

The fact that x P , yP and zP are the arbitrary values shows that the variables (i.e., x P , yP and zP ) in Eq. (2.33) have the infinite solution, therefore its solution form can be expressed as ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ xP 1 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ⎣ y P ⎦ = k 1 ⎣ 0 ⎦ + k 2 ⎣ 1⎦ + k 3 ⎣ 0 ⎦ + η zP

0

0

1

Accordingly, the following equation can be further be achieved.

(2.34)

28

2 Analysis of Locating Determination

⎛⎡

⎞ ⎤⎞ ⎛ 0 −δγw δβw 0 −δγw δβw −δxw rank⎝⎣ δγw 0 −δαw ⎦⎠ = rank⎝ δγw 0 −δαw −δyw ⎠ = 0 (2.35) −δβw δαw 0 −δβw δαw 0 −δyw Thus, if and only if the following condition is true, Eq. (2.35) can be hold. δxw = δyw = δz w = δαw = δβw = δγw = 0

(2.36)

As discussed above, only by limiting all DOFs of the workpiece (i.e., δx w , δyw , δα w , δβ w and δγ w ) can the machining requirements of the key slot be guaranteed.

2.2 DOF Level Model of Position Sizes The linear sizes of the machined surface on the process drawing can be divided into two categories. The first category represents the dimensions of the geometric features of the machined surface itself, such as the diameter of the hole (or shaft), the width of the key slot, and so on. It is called the self size. The second category represents the position between the geometric features of the machined surface and other geometric features, which is called the position size (Sun and Zhu 2006). In order to ensure this kind of position dimension, the DOF to be limited must be analyzed before the workpiece is clamped (Wu et al. 2007).

2.2.1 Position Size of Plane Relative to Plane The known plane is taken as the datum feature whereas the plane to be machined is taken as the measured feature (Li and Liu 2005). The workpiece has the machining requirements in the Z direction. The machining requirements between the measured feature and the datum feature is called the the position size requirements of the plane relative to the plane. As shown in Fig. 2.6a, with consideration of the machining requirement in the Z direction, the following equation can obtained according to Eq. (2.10). δz w + δαw yP − δβw xP = 0

(2.37)

x P and yP in Eq. (2.37) are arbitrary numbers. Thus, to make Eq. (2.38) always hold, the following conditions must be met. δz w = δαw = δβw = 0

(2.38)

2.2 DOF Level Model of Position Sizes

29 Z

Z

Z

y

z

y

z

(b) In the Y and Z directions

z

X

X

X (a) In the Z direction

Y

Y

Y

(c) In the X, Y and Z directions

Fig. 2.6 Machining requirements of plane to plane

Equation (2.38) indicates that the three DOFs (δzw , δα w and δβ w ) must be constrained to ensure the dimension requirements in the Z direction. Similarly, it can also be known that in order to ensure the machining requirements in the Y direction, the DOFs δyw , δγ w and δα w should be restricted. And, in order to ensure the machining requirements in the X direction, the DOFs δx w , δβ w and δγ w should be restricted. If the workpiece has machining requirements in multiple directions, it muse be constrained more DOFs. As illustrated in Fig. 2.6b, the workpiece has the machining requirements in the X direction as well as the Z direction, the position variations in the Y and Z directions must be zero, i.e., 

δyw +δγw xP − δαw z P = 0 δz w +δαw yP − δβw xP = 0

(2.39)

Because x P , yP and zP are the arbitrary values, the following conditions must be satisfied to make Eq. (2.39) always hold. δyw = δz w = δαw = δβw = δγw = 0

(2.40)

Therefore, if the five DOFs, δyw , δzw , δα w , δβ w and δγ w are limited, the specified dimensions in the Y and Z directions can be achieved. Likewise, if the workpiece is constrained its five DOFs, δx w , δyw , δα w , δβ w and δγ w , the specified dimensions in the X and Y directions can be achieved. To obtain the specified dimensions in the X and Z directions, the workpiece must be constrained its DOFs including δx w , δzw , δα w , δβ w and δγ w . As show in Fig. 2.6c, when the key slot with the design sizes in the X, Y and Z directions milled on the workpiece, the following equation can be written according to Eq. (2.10). ⎧ ⎨ δxw + δβw z P − δγw yP = 0 δy + δγw xP − δαw z P = 0 ⎩ w δz w + δαw yP − δβw xP = 0

(2.41)

30

2 Analysis of Locating Determination

In order to make Eq. (2.41) be true, the following conditions must be met, namely δxw = δyw = δz w = δαw = δβw = δγw = 0

(2.42)

Therefore, only when all 6 DOFs are constrained, the machining dimensions in three directions can be guaranteed.

2.2.2 Position Size of Line Relative to Plane The known plane and the measured line are taken as the datum feature and the measured feature, respectively. The position between the two features is required to guarantee during the machining process. This position is called the position size of the line with respect to the plane. As shown in Fig. 2.7, the process datum of the machined through-hole is based on the bottom surface (i.e., XOY plane). The size of the hole axis relative to the bottom surface is required to be guaranteed. In other words, the workpiece has the machining requirement in the Z direction. Therefore, it is easy to obtain the following equation according to Eq. (2.10). δz w + δαw y P − δβw x P = 0

(2.43)

Because the point P, whose coordinate is (x P , yP , zP ), is on the plane XOY, then x P and yP are the arbitrary values and zP = 0. Thus, in order to make Eq. (2.43) be true, the following conditions must be met. δz w = δαw = δβw = 0

(2.44)

This means that only the three DOFs δzw , δα w and δβ w are constrained can the size requirement in the Z direction be guaranteed.

Fig. 2.7 Machining requirement of line to plane in a single direction

2.2 DOF Level Model of Position Sizes

31

Likewise, in order to ensure the machining requirement in the Y direction, the DOFs δyw , δα w and δγ w must be limited. In order to ensure the machining requirement in the X direction, it is necessary to limit the DOFs δx w , δβ w and δγ w . The above deduces the constrained DOFs under the condition that the machining requirement of the measured line with respect to the datum plane is in a single direction. If the machining requirements of the measured line relative to the datum plane is in two or three directions, the conclusions are given as follows: If the position size requirements of the measured line relative to the datum plane are in the X and Y directions, the workpiece must be constrained its DOFs δx w , δyw , δα w , δβ w and δγ w . If the position size requirements of the measured line relative to the datum plane are in the Y and Z directions, the workpiece must be constrained its DOFs δyw , δzw , δα w , δβ w and δγ w . If the position size requirements of the measured line relative to the datum plane are in the Z and X directions, the workpiece must be constrained its DOFs δzw , δx w , δα w , δβ w and δγ w . If the position size requirements of the measured line relative to the datum plane are in arbitrary directions, the measured line belongs to a spatial line. Therefore, six DOFs δx w , δzw , δα w , δβ w and δγ w of the workpiece must be constrained to obtain the specified dimensions in three directions.

2.2.3 Position Size of Plane Relative to Line The known line is taken as the datum feature whereas the measured plane is taken as the measured feature. This is so-called the plane relative to line, such as the position size z shown in Fig. 2.8. Now, take the datum feature as the X axis for example. The measured feature (i.e., the machined plane) is parallel to the XOY plane. Because the machining size z is in the Z direction, the following condition can be obtained according to Eq. (2.10). δz w + δαw y P − δβw x P = 0

Z z

Fig. 2.8 One size of plane to line in the Z direction

(2.45)

Z X Y

32

2 Analysis of Locating Determination

Y

Y

y X

Z

Fig. 2.9 One size of plane to line in the Y direction

Because the point P with coordinate of (x P , yP , zP ) is in the Y axis, there are yP = 0, zP = 0, and x P is arbitrary number. Obviously, Eq. (2.45) can always be hold under the condition of δz w = δβw = 0

(2.46)

In other words, the size requirement in the Z direction can be guaranteed when DOFs δzw and δβ w are constrained. The X axis is still taken as the datum feature, but the measured feature is parallel to the XOZ plane, as shown in Fig. 2.9. Therefore, the required size y is specified in the Y direction. Thus, it is necessary to meet the following equation, that is δyw + δγw x P − δαw z P = 0

(2.47)

Under the condition that yP = zP = 0 and x P is arbitrary number, for Eq. (2.47) to be true, it must satisfy δyw = δγw = 0

(2.48)

In conclusion, if the rotation axis of the workpiece is on the X axis, the corresponding DOFs must be limited for the machined plane with different positions. If the measured plane is in a position parallel to the XOY plane, DOFs δzw and δβ w must be limited. If the measured plane is in a position parallel to the XOZ plane, DOFs δyw and δγ w must be limited. By analogy, if the Y axis is selected as the datum feature, DOFs δzw and δα w must be constrained when the machined plane is parallel to the XOY plane. In case the machined plane is parallel to the YOZ plane, DOFs δx w and δγ w should be limited. Again, the Z axis is assumed to be the datum feature, DOFs δyw and δα w must be constrained when the machined plane is parallel to the XOZ plane. But if the machined plane is parallel to the plane YOZ, DOFs δx w and δβ w must be limited. Provided that the workpiece has the machining requirements in two directions, then more DOFs must be limited than in one direction. If the workpiece is assumed to have the machining requirements in the X and Z directions, there is

2.2 DOF Level Model of Position Sizes



δxw + δβw z P − δγw y P = 0 δz w + δαw y P − δβw x P = 0

33

(2.49)

Because the Y axis is taken as the datum feature, yP is arbitrary value. Hence, if and only if the following condition is hold can Eq. (2.49) be hold. δxw = δz w = δαw = δγw = 0

(2.50)

Therefore, if DOFs δx w , δzw , δα w and δγ w are constrained, the size requirements in the X and Z directions can be guaranteed. Likewise, in order to ensure size requirements in the X and Y directions, the four DOFs to be constrained are δx w , δyw , δα w and δβ w . However, if the size requirements is in the Y and Z directions, the four DOFs to be constrained are δyw , δzw , δβ w and δγ w. Of course, if there are machining requirements in X, Y and Z directions and the datum axis is the X axis, the DOFs should be limited to δx w , δyw , δzw , δβ w and δγ w . If the datum axis is the Y axis, the workpiece must be limited DOFs δx w , δyw , δzw , δα w and δγ w . If the datum axis is the Z axis, the DOFs to be limited are δx w , δyw , δzw , δα w and δβ w.

2.2.4 Position Size of Line Relative to Line This kind of size takes the line as the datum feature and the line to be machined as the measured feature. It can be referred to as “line to line”. As shown in Fig. 2.10, the center line of the cylinder workpiece is selected as the X axis of the coordinate system. If a through hole will be bored on the cylinder workpiece with the size y relative to the X axis in the Y direction (or the size z in the Fig. 2.10 One size of line to line in the Y direction

34

2 Analysis of Locating Determination

Z direction), the following equation can be obtained according to Eq. (2.10). δyw + δγw x P − δαw z P = 0

(2.51)

δz w + δαw y P − δβw x P = 0

(2.52)

or

If the size y is required to guarantee for the through hole, yP = zP = 0 and x P is arbitrary number in Eq. (2.51). But if the machining requirement of the through hole is size z, Eq. (2.52) has also the same parameters as Eq. (2.51). Consequently, Eq. (2.51) or Eq. (2.52) is valid if and only if δyw = δγw = 0

(2.53)

δz w = δβw = 0

(2.54)

or

In other words, two DOFs, δyw and δγ w , must be constrained to ensure the size requirement in the Y direction. Two DOFs, δzw and δβ w , must be constrained to ensure the size requirement in the Z direction. Similarly, if the size x of the center line of the machined hole to the Y axis is in the X direction, DOFs δx w and δγ w must be constrained. If the size z of the center line of the machined hole to the Y axis is in the Z axis, DOFs δzw and δα w must be constrained. However, if the size x of the center line of the machined hole to the Z axis is in the X axis, DOFs δx w and δβ w must be constrained. If the size y of the center line of the machined hole to the Z axis is in the Y axis, DOFs δyw and δα w must be constrained. If the sizes of the center line of the through-hole to be machined relative to the datum feature is required to change from one direction to two directions, then more DOFs must be limited, as shown in Fig. 2.11. Because the specified machining requirements include the size y in the Y direction and size z in the Z direction, the following equation can easily be obtained 

δyw + δγw x P − δαw z P = 0 δz w + δαw y P − δβw x P = 0

(2.55)

Since point P is in the X axis, yP = zP = 0 and x P is an arbitrary number. By substituting them into Eq. (2.55), the DOFs to constrained can be obtained as δyw = δz w = δβw = δγw = 0

(2.56)

2.2 DOF Level Model of Position Sizes

35

Fig. 2.11 Two size of line to line in both directions

Accordingly, in order to guarantee the linear sizes in the Y and Z directions, DOFs, δyw , δzw , δβ w and δγ w must be constrained. Likewise, if the datum of the machining sizes is in coincidence with the Y axis, DOFs, δx w , δzw , δα w and δγ w must be constrained to satisfy the linear sizes in the X and Z directions. If the datum of the machining size is in coincidence with the Z axis, DOFs, δx w , δyw , δα w and δβ w must be constrained to satisfy the linear sizes in the X and Y directions. If the machined surface in Fig. 2.12 is not a through hole but a blind hole, its machining requirements have three linear sizes including the size x in the X direction, size y in the Y direction and size z in the Z direction. It is worth mentioning that the diameter d of the blind hole which belongs to the self size is nothing to do with fixture because it is determined by the selected drilling bit. Thus, the following equation can be achieved according to Eq. (2.10).

Fig. 2.12 Size requirements of line to line in three directions

36

2 Analysis of Locating Determination

⎧ ⎨ δxw + δβw z P − δγw y P = 0 δy + δγw x P − δαw z P = 0 ⎩ w δz w + δαw y P − δβw x P = 0

(2.57)

Because yP = zP = 0 and x P is arbitrary number, it is easy to get δxw = δyw = δz w = δβw = δγw = 0. Therefore, five DOFs, δx w , δyw , δzw , δβ w and δγ w must be constrained. Similarly, if the process datum is in the Y axis, the constrained DOFs are δx w , δyw , δzw , δα w and δγ w . If the process datum is in the Z axis, the constrained DOFs are δx w , δyw , δzw , δα w and δβ w . In order to understand clearly the relationship between the size and the theoretical DOFs, the level model (Wu 2003) can be established, as listed in Table 2.1.

2.3 DOF Level Model of Orientations Usually, the form and position tolerance requirements of the machined surface are marked on the process drawing. The form tolerance is to control the form accuracy of the geometric features of the machined surface, which is guaranteed by the geometric accuracy of the cutting forming movement. The position tolerance is the accuracy requirement for controlling the orientation and location (here, the run-out tolerance is grouped into the location tolerance) and between the geometric features of the machined surface and other geometric features of the workpiece. According to the working conditions of the workpiece, some features on the workpiece will have directional accuracy requirements relative to the datum. At this time, the orientation tolerance is used to limit the direction with respect to the associated features. The orientation tolerance refers to the total amount of allowable variation in the direction of the associated feature relative to the datum. The direction of the orientation tolerance zone is fixed, and determined by the datum. The position of the the orientation tolerance zone can float within the dimensional tolerance zone. The orientation tolerance includes parallelism, perpendicularity and inclination.

2.3.1 Relation Between Parallelism Tolerance and DOFs 2.3.1.1

Parallelism of Plane Relative to Plane

If the required parallelism tolerance is that the plane to be machined is relative to the datum plane, such a parallelism tolerance is called “plane to plane” parallelism tolerance requirement.

2.3 DOF Level Model of Orientations

37

Table 2.1 DOF level model of position size Type

Theoretical Theoretical DOF δq ∗w constrains

Base vector ζ*

YOZ

δx w , δβ w , δγ w

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z , ζ α]

ZOX

δyw , δα w , δγ w

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z , ζβ]

XOY

δzw , δα w , δβ w

λx ζ x + λy ζ y + λγ ζ γ

[ζ x , ζ y , ζγ ]

X, Z

YOZ, XOY

δx w , δzw , δα w , δβ w , δγ w

λy ζ y

ζy

X, Y

YOZ, ZOX

δx w , δyw , δα w , δβ w , δγ w

λz ζ z

ζz

Y, Z

ZOX, XOY

δyw , δzw , δα w , δβ w , δγ w

λx ζ x

ζx

X, Y, Z

XOY, YOZ, ZOX

δx w , δyw , δzw , δα w , δβ w , δγ w

0

0

X

Y

δx w , δγ w

λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ y , ζ z , ζ α, ζ β]

Z

δx w , δβ w

λy ζ y + λz ζ z + λα ζ α + λγ ζ γ

[ζ y , ζ z , ζ α, ζ γ ]

X

δyw , δγ w

λz ζ z + λx ζ x + λα ζ α + λβ ζ β

[ζ z , ζ x , ζ α, ζ β]

Z

δyw , δα w

λz ζ z + λx ζ x + λα ζ α + λγ ζ γ

[ζ z , ζ x , ζ α, ζ γ ]

X

δzw , δβ w

λx ζ x + λy ζ y + λα ζ α + λγ ζ γ

[ζ x , ζ y , ζ α, ζ γ ]

Y

δzw , δα w

λx ζ x + λy ζ y + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζβ, ζγ ]

X, Z

Y

δx w , δzw , δα w , δγ w

λy ζ y + λβ ζ β

[ζ y , ζ β ]

X, Y

Z

δx w , δyw , δα w , δβ w

λz ζ z + λγ ζ γ

[ζ z , ζ γ ]

Y, Z

X

δyw , δzw , δβ w , δγ w

λx ζ x + λα ζ α

[ζ x , ζ α ]

X, Y, Z

X

δx w , δyw , δzw , δβ w , δγ w

λα ζ α

ζα

Directions of Datum machining requirements

“plane X to plane” Y or “line to plane” Z

“line to line”

Y

Z

(continued)

38

2 Analysis of Locating Determination

Table 2.1 (continued) Type

“plane to line”

Directions of Datum machining requirements

Theoretical Theoretical DOF δq ∗w constrains

Base vector ζ*

X, Y, Z

Y

δx w , δyw , δzw , δα w , δγ w

λβ ζ β

ζβ

X, Y, Z

Z

δx w , δyw , δzw , δα w , δβ w

λγ ζ γ

ζγ

X

Y

δx w , δγ w

λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ y , ζ z , ζ α, ζ β]

Z

δx w , δβ w

λy ζ y + λz ζ z + λα ζ α + λγ ζ γ

[ζ y , ζ z , ζ α, γ ζγ ]

X

δyw , δγ w

λz ζ z + λx ζ x + λα ζ α + λβ ζ β

[ζ z , ζ x , ζ α, ζ β]

Z

δyw , δα w

λz ζ z + λx ζ x + λβ ζ β + λγ ζ γ

[ζ z , ζ x , ζβ, ζγ ]

X

δzw , δβ w

λx ζ x + λy ζ y + λα ζ α + λγ ζ γ

[ζ x , ζ y , ζ α, ζ γ ]

Y

δzw , δα w

λx ζ x + λy ζ y + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζβ, ζγ ]

X, Z

Y

δx w , δzw , δα w , δγ w

λy ζ y + λβ ζ β

[ζ y , ζ β ]

X, Y

Z

δx w , δyw , δα w , δβ w

λz ζ z + λγ ζ γ

[ζ z , ζ γ ]

Y, Z

X

δyw , δzw , δβ w , δγ w

λx ζ x + λα ζ α

[ζ x , ζ α ]

X, Y, Z

X

δx w , δyw , δzw , δβ w , δγ w

λα ζ α

ζα

X, Y, Z

Y

δx w , δyw , δzw , δα w , δγ w

λβ ζ β

ζβ

X, Y, Z

Z

δx w , δyw , δzw , δα w , δβ w

λγ ζ γ

ζγ

Y

Z

Fig. 2.13 Parallelism of plane to plane in the Z direction

2.3 DOF Level Model of Orientations

39

As shown in Fig. 2.13, the bottom plane of the block workpiece is the datum whereas its top plane is the plane to be machined. The parallelism tolerance 0.05 between them is required to satisfy during the machining process. The coordinate system XYZ is established based on the bottom plane, as shown in Fig. 2.13. Thus, the parallelism between the top plane to be machined and the XOY plane is easily known to be in the Z direction. In order to guarantee the parallelism between them, the position variation of the plane to be machined along the Z direction must be limited. Therefore, the following equation can be obtained according to Eq. (2.11). δαw y P − δβw x P = 0

(2.58)

Because point P is any point on the process datum, that is, point P is on the XOY plane, x P and yP are arbitrary values except that zP = 0. In order to ensure that the Eq. (2.58) is always true for any point P, δα w = δβ w = 0 must be satisfied, that is, DOFs δα w and δβ w should be limited. Similarly, it can be known that when the datum plane is the YOZ plane, the DOFs should be restricted to δβ w and δγ w . When the datum plane is the ZOX plane, the DOFs should be restricted to δα w and δγ w .

2.3.1.2

Parallelism of Line Relative to Plane

If the required parallelism tolerance is that the line to be machined is relative to the datum plane, such a parallelism tolerance is called “line to plane” parallelism tolerance requirement. As shown in Fig. 2.14, a through hole is machined to obtain the parallelism of the hole axis to the bottom plane. Obviously, the bottom plane is the process datum of the through hole with the parallelism requirement. According to the requirement of the parallelism tolerance, the hole axis must be in the area between two planes which are 0.05 mm apart and parallel to the datum plane. Because the parallelism tolerance is in the Z direction and its datum is the XOY plane, thus the equation can easily obtained as δαw y P − δβw x P = 0 Fig. 2.14 Parallelism of line to plane in the Z direction

(2.59)

40

2 Analysis of Locating Determination

Because point P is on the XOY plane, thus zP = 0, x P and yP are arbitrary values. In order to ensure that the above equation is always true for any point P, there must be δα w = δβ w = 0. In other words, if and only if DOFs δα w and δβ w must be constrained can the parallelism tolerance in the Z direction be guaranteed. By analogy, it can be known that when the datum plane is the YOZ plane, the DOFs should be restricted to δβ w and δγ w to ensure the parallelism tolerance in the X direction. When the datum plane is the ZOX plane, the DOFs should be restricted to δα w and δγ w to ensure the parallelism tolerance in the Y direction. It can be seen from Sects. 2.3.1.1 and 2.3.1.2, the parallelism tolerance with plane as datum feature requires the same DOFs to be limited. Therefore, the parallelism tolerance of “plane to plane” and “line to plane” can be summarized together when building the level model.

2.3.1.3

Parallelism of Plane to Line

The parallelism requirement of the surface to be machined is based on the straight line where the surface to be machined is the plane. If this so called “plane to line” parallelism requirement is based on the different datum in the different direction, the DOFs to be restricted are different. Now the axis of the hole is taken as the datum of the plane to be machined for specifying a parallelism tolerance of 0.05 mm. Therefore, the X axis of the coordinate system is established based on the hole axis, as shown in Fig. 2.15. Thus, the plane is parallel to the XOY plane and the specified parallelism tolerance is in the Z direction. In order to ensure the parallelism of the plane to be machined to the datum, it is necessary to limit the variation of the plane to be machined to the datum in the Z direction, namely, δαw y P − δβw x P = 0

(2.60)

Because point P is on the X axis, thus yP = zP = 0 and x P is an arbitrary value. By substituting them into Eq. (2.60), if and only if δβ w = 0 can Eq. (2.60) be always hold. This means that the DOF δβ w must be limited. The X axis is still used as the datum whereas the plane to be machined is required to parallel to the XOZ plane, as shown in Fig. 2.16. Fig. 2.15 Parallelism of plane to line in the Z direction

2.3 DOF Level Model of Orientations

41

Fig. 2.16 Parallelism of plane to line in the Y direction

Obviously, the machining requirement is in the Y direction. In order to ensure the parallelism tolerance of the plane to be machined to the datum in the Y direction, it must meet the requirement of δγw x P − δαw z P = 0

(2.61)

Because point P is on the X axis, thus yP = zP = 0 and x P is an arbitrary value. By substituting them into Eq. (2.61), it can easily be conclude δγ w = 0 which means the DOF δγ w to be limited. It can be seen from the above that, if the rotation axis of the workpiece is on the X axis, the DOFs to be limited, corresponding to the different position of the plane to be machined, are different. If the plane to be machined is parallel to the XOY plane, δβ w should be limited. If the plane to be machined is parallel to the XOZ plane, δγ w should be limited. Likewise, if the Y axis is taken as the datum, δα w must be limited when the plane to be machined is required to be parallel to the XOY plane whereas δγ w must be limited when the plane to be machined is required to be parallel to the YOZ plane. However, if the Z axis is taken as the datum, δα w must be limited when the plane to be machined is required to be parallel to the XOZ plane whereas δβ w must be limited when the plane to be machined is required to be parallel to the YOZ plane.

2.3.1.4

Parallelism of Line to Line

The parallelism of “line to line” refers to the case where the existed straight line is taken as a datum of a straight line to be machined. As shown in Fig. 2.17, on the connecting rod workpiece, the axis of the lower through hole is taken as the process datum to drill the upper through hole. Here, the axis of the upper through hole has the parallelism requirement of 0.05 mm to the process datum. Obviously, the parallelism requirement is in the Z direction, so it must satisfy δαw y P − δβw x P = 0

(2.62)

42

2 Analysis of Locating Determination

Fig. 2.17 Parallelism requirement in the single direction

Again, point P is on the X axis, as shown in Fig. 2.17. Therefore, yP = zP = 0 and x P is an arbitrary value. They are substituted into Eq. (2.62) to obtain δβ w = 0. That is, δβ w should be limited. The line to be machined is still based on the X axis. If the parallelism requirement of the line to be machined to the X axis is in the Y direction, there exists δγw x P − δαw z P = 0

(2.63)

Because point P is on the X axis, yP = zP = 0 and x P is an arbitrary value. In order to make Eq. (2.63) always hold, it is necessary to make δγ w = 0, that is to say, δγ w should be limited. Under the condition that the process datum is in coincidence with the Y axis, if the parallelism requirement is in the X direction, δγ w should be limited. If the parallelism requirement is in the Z direction, δα w should be limited. If the Z axis is taken as the datum, δβ w must be limited when the parallelism requirement is in the X direction whereas δα w must be limited when the parallelism requirement is in the Y direction. If the parallelism of the axis of the through hole to be machined to the datum is required in two directions. As shown in Fig. 2.18, the axis of the hole to be machined is required to locate in a square prism parallel to the Y axis and with a cross section of 0.05 mm × 0.05 mm. Because the parallelism requirements are in the X and Z directions, the following equation can be obtained. 

δβw z P − δγw y P = 0 δαw y P − δβw x P = 0

(2.64)

Because point P is on the Y axis, x P = zP = 0 and yP is an arbitrary value. Thus, they are substituted into Eq. (2.64) to solve δα w = δγ w = 0. Accordingly, DOFs δα w and δγ w must be limited to ensure the parallelism tolerances in the in the X and Z directions. Equally, if the X axis is established on the process datum, DOFs δβ w and δγ w should be constrained when the parallelism requirements are in the Y and Z directions.

2.3 DOF Level Model of Orientations

43

Fig. 2.18 Parallelism requirements in two directions

But if the Z axis is established on the process datum, DOFs δα w and δβ w should be constrained when the parallelism requirements are in the X and Y directions. If the axis of the hole to be machined is specified to have the parallelism tolerance with respect to the axis of the known hole in any direction, the axis of the known hole can be taken as arbitrary coordinate axis to establish a coordinate system. As shown in Fig. 2.19, the axis of the hole to be machined must be located in a cylinder with a diameter of 0.05 and parallel to the datum. Here, the datum is the axis of the hole with a diameter of ΦD. The coordinate system XYZ is established with the datum as the Y axis. In fact, a line in any direction in a plane can be represented by two directed segments perpendicular to each other. If there is a deviation between the practical axis and the theoretical requirement, the practical axis can be projected into the XOZ plane perpendicular to the direction of the datum (i.e., Y axis). According to the above theory, the parallelism requirement in any direction of “line to line” can be changed to the parallelism requirement in two directions perpendicular to each Fig. 2.19 Parallelism tolerance in any direction

44

2 Analysis of Locating Determination

other. Therefore, the parallelism requirement in any direction with the Y axis as the datum is same as the case that parallelism requirement in the X and Z directions, the following equation can be obtained according to Eq. (2.11). 

δβw z P − δγw y P = 0 δαw y P − δβw x P = 0

(2.65)

Because point P is on the Y axis, x P = zP = 0 and yP is an arbitrary value. Thus, they are substituted into Eq. (2.65) to obtain δα w = δγ w = 0. Accordingly, DOFs δα w and δγ w must be limited. By analogy, if the X axis is established on the process datum, DOFs δβ w and δγ w must be constrained. If the Z axis is established on the process datum, DOFs δα w and δβ w must be constrained. In order to understand clearly the relationship between the parallelism tolerance and the theoretical DOFs, the level model can be established, as listed in Table 2.2.

2.3.2 Relation Between Perpendicularity/Inclination Tolerance and DOFs The relationship between the parallelism tolerance and the theoretical DOFs is analyzed, and then the other two kinds of orientation tolerances are analyzed. Because the perpendicularity can be regarded as a special case of inclination (the included angle between the measured feature and the datum feature is 90°), we can only analyze the relationship between the more general inclination and the theoretical DOFs by studying them together.

2.3.2.1

Perpendicularity/Inclination of Plane to Plane

In the inclination tolerance with the plane as the datum feature and the plane to be machined as the measured feature (i.e., the inclination of plane to plane), its corresponding machining requirement is measured in the direction with perpendicular the datum plane. As shown in Fig. 2.20, an inclined plane with the angle of 60° to the bottom plane is milled on the workpiece. The plane to be machined has inclination requirement of 0.1 mm to the bottom plane. Therefore, the bottom plane is taken as the datum to establish the coordinate system, as shown in Fig. 2.20. Thus, the inclination requirement can be measured in the Z direction. The XOY plane is datum feature. Therefore, in order to guarantee the specified inclination tolerance, the following equation can be obtained according to Eq. (2.11). δαw y P − δβw x P = 0

(2.66)

2.3 DOF Level Model of Orientations

45

Table 2.2 Level model between parallelism tolerance and theoretical DOFs Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ*

Plane to plane

X

YOZ

δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α

[ζ x , ζ y , ζ z, ζ α]

Y

ZOX

δα w , δγ w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β

[ζ x , ζ y , ζ z, ζ β ]

Z

XOY

δα w , δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ

[ζ x , ζ y , ζ z, ζ γ ]

X

Y

δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ x , ζ y , ζ z, ζ α, ζβ]

Z

δβ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λγ ζ γ

[ζ x , ζ y , ζ z, ζ α, ζγ ]

X

δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ x , ζ y , ζ z, ζ α, ζβ]

Z

δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζ z, ζ β , ζγ ]

X

δβ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λγ ζ γ

[ζ x , ζ y , ζ z, ζ α, ζγ ]

Y

δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζ z, ζ β , ζγ ]

Y

δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ x , ζ y , ζ z, ζ α, ζβ]

Z

δβ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λγ ζ γ

[ζ x , ζ y , ζ z, ζ α, ζγ ]

X

δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ x , ζ y , ζ z, ζ α, ζβ]

Z

δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζ z, ζ β , ζγ ]

X

δβ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λγ ζ γ

[ζ x , ζ y , ζ z, ζ α, ζγ ]

Y

δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζ z, ζ β , ζγ ]

Plane to line

Y

Z

Line to line

X

Y

Z

(continued)

46

2 Analysis of Locating Determination

Table 2.2 (continued) Type

Line to plane

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ*

X, Z

Y

δα w , δγ w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β

[ζ x , ζ y , ζ z, ζ β ]

X, Y

Z

δα w , δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ

[ζ x , ζ y , ζ z, ζ γ ]

Y, Z

X

δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α

[ζ x , ζ y , ζ z, ζ α]

X

YOZ

δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α

[ζ x , ζ y , ζ z, ζ α]

Y

ZOX

δα w , δγ w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β

[ζ x , ζ y , ζ z, ζ β ]

Z

XOY

δα w , δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ

[ζ x , ζ y , ζ z, ζ γ ]

Fig. 2.20 Machining of the inclined plane

Because point P is on the process datum which is in coincident with the XOY plane, x P and yP are arbitrary values besides zP = 0. In order to ensure that Eq. (2.66) is always true for any point P, it must satisfy δα w = δβ w = 0. In other words, it is necessary for the parallelism tolerance of 0.1 mm to constrain DOFs δα w and δβ w . Similarly, if the process datum is the XOZ plane, the DOFs to be limited are δα w and δγ w . If the process datum is taken as the YOZ plane to establish the coordinate system XYZ, the DOFs to be limited are δβ w and δγ w .

2.3.2.2

Perpendicularity/Inclination of Plane to Line

The perpendicularity/inclination of plane to line is a case where the surface to be machined is a plane whereas its process datum is a line.

2.3 DOF Level Model of Orientations

47

Fig. 2.21 Parallelism of inclined plane to axis line

As shown in Fig. 2.21, an inclined plane with the angle of 60° to the axis line is milled on the stepped shaft workpiece. Moreover, an inclination tolerance of 0.05 mm is specified for the plane to be milled. According to the definition of inclination, the inclined plane must be located between two parallel planes with a distance of 0.05 mm and an angle of 60° from the axis line. If the datum line of the stepped shaft is taken as the X axis to established the coordinate system XYZ, the specified inclination tolerance is in the Y and Z directions. In order to ensure the inclination of the plane to be machined, the machining requirement model of Eq. (2.11) can be used to obtain 

δγw x P − δαw z P = 0 δαw y P − δβw x P = 0

(2.67)

Because point P is on the X axis, yP = zP = 0 except that x P is arbitrary value. By substituting them into Eq. (2.67), we can conclude δβ w = δγ w = 0. In other words, only by restricting DOFs δβ w and δγ w can the inclination tolerance of 0.05 mm be guaranteed. Likewise, if the process datum is the Y axis, the DOFs to be limited are δα w and δγ w . If the process datum is taken as the Z axis to establish the coordinate system XYZ, the DOFs to be limited are δα w and δβ w .

2.3.2.3

Perpendicularity/Inclination of Line to Line

If the corresponding line of the surface to be machined is required to locate between two parallel planes with a distance and an angle from the axis of the known hole, the distance is called as the perpendicularity/inclination of line to line. As shown in Fig. 2.22, a hole whose axis is at an angle of 60° from the hole with a diameter of D is bored on the cylinder workpiece. Again, the axis of the hole to be machined is required to have an inclination of 0.05 mm to the axis of the hole with a diameter of D. Therefore, the axis of the hole with a diameter of D is taken as the X axis to establish the coordinate system XYZ, as shown in Fig. 2.22. Thus, the inclination requirement

48

2 Analysis of Locating Determination

Fig. 2.22 A inclined hole bored on the cylinder workpiece

of 0.05 mm is in the Z direction. In order to obtain the specified inclination of the plane to be machined, the following equation can be concluded in light of the machining requirement model of Eq. (2.11). δαw y P − δβw x P = 0

(2.68)

Because point P is on the X axis, yP = zP = 0 except that x P is an arbitrary value. In order for any point P Eq. (2.68) to be true, δβ w must be zero. In other words, only by restricting the DOF δβ w can the inclination tolerance of 0.05 mm be guaranteed. Next, the process datum is still taken as the X axis. But the inclination of the axis of the surface to be machined is in the Y direction. To ensure the machining requirements in the Z direction, it is necessary to δγw x P − δαw z P = 0

(2.69)

Because point P is on the X axis, yP = zP = 0 except that x P is arbitrary value. In order for any point P Eq. (2.69) to be true, δγ w must be zero. Therefore, the limit of the DOF δγ w is the prerequisite to guarantee the inclination tolerance of 0.05 mm relative to the X axis in the Y direction. As can be seen from the above example that, even though the datum is the same, but if the direction of machining requirements is different, the DOFs to be limited are also different. Thus, if the Y axis is taken as the datum, δγ w must be limited when the machining requirement is in the X direction whereas δα w must be limited when the machining requirement is in the Z direction. Of course, if the Z axis is taken as the datum, δβ w must be limited when the machining requirement is in the X direction whereas δα w must be limited when the machining requirement is in the Y direction.

2.3 DOF Level Model of Orientations

2.3.2.4

49

Perpendicularity/Inclination of Line to Plane

The variation of the measured feature of the inclined hole to be machined with respect to the datum plane can be in one direction, two directions and any direction. Generally, the measured feature is the axis of the hole to be machined. Therefore, this kind of the variation is referred to as the perpendicularity/inclination of line to plane. First of all, we will discuss the theoretical DOFS for the perpendicularity/inclination of line to plane in one direction. As shown in Fig. 2.23, the inclined hole will be drilled on the cuboid workpiece, which requires the inclination of the hole to be machined to the bottom plane. Thus, the axis of hole with a diameter of d is located between two parallel planes with a distance of 0.05 mm and an angle of 60° from the datum plane. Except that the datum plane is chosen as the XOY coordinate plane, the projected line of the axis of hole to be machined on the XOY plane is taken as the X axis to establish the coordinate system. The machining requirement in the Z direction can be used as a basis to derive the following equation δαw y P − δβw x P = 0

(2.70)

Because the point P is on the XOY plane, x P and yP are arbitrary value. In order for any point P Eq. (2.70) to be true, δα w and δβ w must be zero. Therefore, the limit of the DOFs δα w and δβ w is the prerequisite to guarantee the inclination tolerance of 0.05 mm relative to the XOY plane in the Z direction. Therefore, if the datum plane is the XOZ plane, δα w and δγ w must be limited. But if the datum plane is the YOZ plane, δβ w and δγ w must be limited. When the measured line of the surface to be machined has inclination requirements relative to the datum plane in two directions, for example, the center line of the hole to be processed has the inclination requirement with the A plane, as shown in Fig. 2.24. Fig. 2.23 An inclined hole drilled on the cuboid workpiece

50

2 Analysis of Locating Determination

Fig. 2.24 An inclined hole drilled on the cuboid workpiece

Thus, the A plane is taken as the XOY plane to establish the coordinate system. Therefore, the inclination requirements are in the Z direction and the following condition must be satisfied.  δβw z P − δγw yP = 0 (2.71) δαw yP − δβw xP = 0 Because the inclination in the X direction is based on the YOZ plane, yP and zP in δβ w zP -δγ w yP = 0 are arbitrary numbers. Again, the inclination in the Z direction is based on the XOY plane, x P and yP in δα w yP -δβ w x P = 0 are arbitrary numbers. Therefore, there is δαw = δβw = δγw = 0

(2.72)

Likewise, if the inclination requirements are respectively relative to the YOZ and ZOX planes, it should be in the X and Y directions. Thus the the DOFs to be limited should be δα w , δβ w and δγ w . If the inclination requirement are based on the ZOX and YOZ planes, it should be in the Y and X directions. Thus the the DOFs to be limited should be δα w , δβ w and δγ w . Therefore, the level model between the perpendicularity/inclination and the theoretical DOFs can be concluded to list in Table 2.3.

2.3 DOF Level Model of Orientations

51

Table 2.3 Level model between the perpendicularity/inclination and the theoretical DOFs Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ*

Plane to plane

X

YOZ

δβ w, δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α

[ζ x , ζ y , ζ z, ζ α]

Y

ZOX

δα w, δγ w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β

[ζ x , ζ y , ζ z, ζ β ]

Z

XOY

δα w, δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ

[ζ x , ζ y , ζ z, ζ γ ]

Plane X to line

X

δβ w, δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α

[ζ x , ζ y , ζ z, ζ α]

Y

Y

δα w, δγ w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β

[ζ x , ζ y , ζ z, ζ β ]

Z

Z

δα w, δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ

[ζ x , ζ y , ζ z, ζ γ ]

Line X to line

Y

δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ x , ζ y, ζ z, ζ α, ζ β]

Z

δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ + λα ζ α

[ζ x , ζ y , ζ z, ζ γ , ζ α]

X

δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α + λβ ζ β

[ζ x , ζ y, ζ z, ζ α, ζ β]

Z

δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζ z, ζ β , ζγ ]

X

δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ + λα ζ α

[ζ x , ζ y , ζ z, ζ γ , ζ α]

Y

δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β + λγ ζ γ

[ζ x , ζ y , ζ z, ζ β , ζγ ]

X

YOZ

δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z + λα ζ α

[ζ x , ζ y , ζ z, ζ α]

Y

ZOX

δγ w , δα w

λx ζ x + λy ζ y + λz ζ z + λβ ζ β

[ζ x , ζ y , ζ z, ζ β ]

Z

XOY

δα w , δβ w

λx ζ x + λy ζ y + λz ζ z + λγ ζ γ

[ζ x , ζ y , ζ z, ζ γ ]

Z、X

XOY, YOZ

δα w , δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z

[ζ x , ζ y, ζ z]

X、Y

YOZ, ZOX

δα w , δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z

[ζ x , ζ y, ζ z]

Y

Z

Line to plane

(continued)

52

2 Analysis of Locating Determination

Table 2.3 (continued) Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ*

Y, Z

ZOX, XOY

δα w , δβ w , δγ w

λx ζ x + λy ζ y + λz ζ z

[ζ x , ζ y, ζ z]

2.4 DOF Level Model of Locations The location tolerance refers to the total amount of allowable variation in the position of the associated feature relative to the datum. The position of the location tolerance zone is fixed with respect to the datum. The location tolerance zone not only controls the position error of the measured feature, but also controls the direction error and form error of the measured feature. The orientation tolerance zone not only controls the direction error of the measured feature, but also controls its form error. The form tolerance zone can only control the form error of the measured feature. The location tolerance includes the coaxiality, the symmetry and the position.

2.4.1 Relation Between Coaxiality Tolerance and DOFs The coaxiality is used to control the coaxiality error between the measured axis and the datum axis of shaft workpieces. The coaxiality tolerance zone is the area within a cylinder whose diameter is the tolerance value t and is coaxial to the datum axis. Figure 2.25 is the coaxiality based on the X axis. The coaxiality error based on the X axis means that there is an error in the Y and Z directions, so 

Fig. 2.25 Stepped shaft with the requirement of coaxiality

δyw + δγw x P − δαw z P = 0 δz w + δαw y P − δβw x P = 0

(2.73)

2.4 DOF Level Model of Locations

53

Table 2.4 Level model of coaxiality and theoretical DOFs Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ *

Line to line

Y 、Z

X

δyw , δzw , δβ w , δγ w

λx ζ x +λα ζ α

[ζ x , ζ α ]

Z、X

Y

δx w , δzw , δα w , δγ w

λy ζ y + λβ ζ β

[ζ y , ζ β ]

X、Y

Z

δx w , δyw , δα w , δβ w

λz ζ z + λγ ζ γ

[ζ z , ζ γ ]

Because point P is on the X axis, yP = zP = 0 and x P is arbitrary value. By substituting them into Eq. (2.73), we can obtain δβw = δyw = δz w = δγw = 0

(2.74)

In other words, in order to ensure the requirement of the coaxiality error based on the X axis, the DOFs δzw , δyw , δβ w and δγ w must be restricted. By analogy, in order to ensure the requirement of the coaxiality error based on the Y axis, the DOFs δzw , δyw , δβ w and δγ w must be restricted. In order to ensure the requirement of the coaxiality error based on the Z axis, the DOFs δx w , δyw , δα w and δβ w must be restricted, as listed in Table 2.4.

2.4.2 Relation Between Symmetry Tolerance and DOFs The symmetry is used to control the coplanar (or collinear) error of the center plane (or axis) of the measured feature. The symmetry tolerance zone is the area between two parallel planes (or straight lines) that are symmetrically arranged relative to the datum center plane (or center line, or axis) with a distance of tolerance value t.

2.4.2.1

Symmetry of Plane to Line and Line to Line

As shown in Fig. 2.26, line b is machined with the symmetry of 0.05 mm to the center line of line a and line c. So the center line is taken as the Y axis to establish the coordinate system XYZ. Thus, the symmetry requirement is in the X direction and there is δxw + δβw z P − δγw yP = 0

(2.75)

Because point P is on the Y axis, thus x P = zP = 0, yP is an arbitrary value. In order to make Eq. (2.75) to be true, if and only if

54

2 Analysis of Locating Determination

Fig. 2.26 Three scale lines with the symmetry requirement

δxw =δγw =0

(2.76)

In other words, two DOFs, δx w and δγ w , must be limited for the symmetry requirement of the measured feature relative to the Y axis line in the X direction. Obviously, two DOFs, δx w and δβ w , must be limited for the symmetry requirement of the measured feature relative to the Z axis line in the X direction. Likewise, two DOFs, δyw and δγ w , must be limited for the symmetry requirement of the measured feature relative to the X axis line in the Y direction. two DOFs, δyw and δβ w , must be limited for the symmetry requirement of the measured feature relative to the Z axis line in the Y direction. Moreover, two DOFs, δzw and δγ w , must be limited for the symmetry requirement of the measured feature relative to the X axis line in the Z direction. two DOFs, δzw and δβ w , must be limited for the symmetry requirement of the measured feature relative to the Z axis line in the Z direction.

2.4.2.2

Symmetry of Plane to Plane

As shown in 2.27, the key slot to be milled on the workpiece has the symmetry requirement relative the plane A. So the plane A is chosen as the XOZ coordinate plane to establish the coordinate system XYZ. Thus, the symmetry requirement is in the Y direction. The following equation can be obtained according to Eq. (2.10). (Fig. 2.27) δyw +δγw xP − δαw z P = 0

(2.77)

Again, yP = 0, x P and zP are the arbitrary values. To make Eq. (2.77) hold, if and only if δyw = δαw = δγw = 0

(2.78)

2.4 DOF Level Model of Locations

55

Fig. 2.27 Workpiece with symmetry requirement of plane to plane

That is, three DOFs, δyw , δα w and δγ w , must be constrained, the symmetry requirement relative to the XOZ plane can be guaranteed. By analogy, if the XOY and YOZ based symmetry are required to achieve, the DOFs to be limited are respectively δzw , δα w , δβ w and δx w , δβ w , δγ w .

2.4.2.3

Symmetry of Line to Plane

As shown in Fig. 2.28, a hole is bored on the workpiece for the symmetry of 0.1 mm with respect to the datum plane A-B. Therefore, the plane A-B is select as the XOZ coordinate plane to establish the coordinate system XYZ. Because the symmetry requirement is in the Y direction, the following equation can be obtained according to Eq. (2.10). δyw + δγw xP − δαw z P = 0

(2.79)

According to the condition that the point P is on the XOZ plane, it is easily known that x P and zP are arbitrary values except of yP = 0. Thus, Eq. (2.79) can be hold if and only if Fig. 2.28 Workpiece with symmetry of line to plane in one direction

56

2 Analysis of Locating Determination

Fig. 2.29 Workpiece with symmetry of line to plane in two directions

δyw = δαw = δγw = 0

(2.80)

Accordingly, three DOFs δyw , δα w and δγ w must be limited, the symmetry requirement of the axis to the XOZ plane can be guaranteed. Likewise, in order to ensure the XOY plane and YOZ plane based symmetries, the DOFs to be limited are respectively δzw , δα w , δβ w and δx w , δβ w , δγ w . As illustrated in Fig. 2.29, the hole will be drilled on the workpiece. The axis of hole has the symmetry requirements relative to the XOZ plane and the XOY plane, respectively. It is clear that the symmetry requirements are in the Y and Z directions. Therefore, 

δyw + δγw xP − δαw z P = 0 δz w + δαw y P − δβw x P = 0

(2.81)

Because the symmetry in the Y direction is based on the XOZ plane, x P and zP in δyw + δγ w x P -δα w zP = 0 are arbitrary numbers. Again, the symmetry in the Z direction is based on the XOY plane, x P and yP in δzw + δα w yP -δβ w x P = 0 are arbitrary numbers. Therefore, there is δβw = δyw = δαw = δz w = δγw = 0

(2.82)

Consequently, five DOFs δyw , δzw , δα w , δβ w and δγ w must be limited, the symmetry requirements of the axis to the XOZ plane and the XOY plane can be guaranteed. Likewise, in order to ensure the XOY plane and YOZ plane based symmetries, the DOFs to be limited are δx w , δzw , δα w , δβ w and δγ w . But, in order to ensure the symmetry requirements of the axis to the XOZ plane and the YOZ plane, DOFs δx w , δyw , δα w , δβ w and δγ w must be limited. As stated above, the level model of the symmetry and the theoretical DOFs can be conclude as listed in Table 2.5.

2.4 DOF Level Model of Locations

57

Table 2.5 Level model of the symmetry and the theoretical DOFs Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ *

Plane to plane

X

YOZ

Δx, δβ, δγ

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z , ζ α ]

Y

ZOX

δy, δα, δγ

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z , ζβ]

Z

XOZ

δz, δα, δβ

λx ζ x + λy ζ y +λγ ζ γ

[ζ x , ζ y , ζγ ]

Plane to line X or line to Y line

X

δx, δβ, δγ

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z , ζ α ]

Y

δy, δα, δγ

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z , ζβ]

Z

Z

δz, δα, δβ

λx ζ x + λy ζ y +λγ ζ γ

[ζ x , ζ y , ζγ ]

X

X

δx, δβ, δγ

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z , ζ α ]

Y

Y

δy, δα, δγ

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z , ζβ]

Z

Z

δz, δα, δβ

λx ζ x + λy ζ y +λγ ζ γ

[ζ x , ζ y , ζγ ]

X, Y

YOZ, ZOX

δx, δy, δα, δβ, δγ

λz ζ z

ζz

Y, Z

ZOX, XOY

δy, δz, δα, δβ, δγ

λx ζ x

ζx

X, Z

YOZ, XOY

δx, δz, δα, δβ, δγ

λy ζ y

ζy

Line to plane

2.4.3 Relation Between Position Tolerance and DOFs The position is used to control the position error of the measured feature (point, line and plane) to the datum. According to the functional requirements of the workpiece, the position tolerance can be divided into three types: given one direction, given two directions and arbitrary directions.

2.4.3.1

Position of Point

A hole will be drilled on the sheet workpiece. The axis of the hole has a position relative to plane A and plane B. Therefore, the coordinate system XYZ can be established as shown in Fig. 2.30. Obviously, the position requirements are in the X and Z directions. Thus, we can be obtain 

δxw + δβw z P − δγw γP = 0 δz w + δαw y P − δβw x P = 0

(2.83)

58

2 Analysis of Locating Determination

Fig. 2.30 Position of the point on the sheet workpiece

Because the position in the X direction is based on the YOZ plane, yP and zP in δx w + δβ w zP -δγ w yP = 0 are arbitrary numbers. On the other hand, the position in the Z direction is based on the XOY plane, x P and yP in δzw + δα w yP -δβ w x P = 0 are arbitrary numbers. Equation (2.83) can always be hold, under the condition of δxw = δz w = δαw = δβw = δγw = 0

(2.84)

That is, it is necessary for the position requirements of a point to the XOY and YOZ planes to limit DOFs δx w , δα w , δβ w , δzw and δγ w . In the same way, only by limiting DOFs δx w , δα w , δyw , δβ w and δγ w can the the position requirements of a point to the YOZ and ZOX planes be guaranteed. In order to ensure the position requirements of a point to the ZOX and XOY planes, DOFs δα w , δyw , δβ w , δzw and δγ w must be limited. If the point is in space, in order to determine the position of the point, the position error of the point in the X, Y and Z directions must be limited, as shown in Fig. 2.31. Therefore ⎧ ⎨ δxw + δβw z P − δγw yP = 0 (2.85) δy + δγw xP − δαw z P = 0 ⎩ w δz w + δαw y P − δβw x P = 0 It is seen from Fig. 2.31, the position in the Z direction is based on the XOY plane, x P and yP in δzw + δα w yP -δβ w x P = 0 are arbitrary numbers. The position in the X direction is based on the XOY plane, zP and yP in δx w + δβ w zP -δγ w yP = 0 are arbitrary numbers. The position in the Y direction is based on the XOY plane, x P and zP in δyw + δγ w x P -δα w zP = 0 are arbitrary numbers. Equation (2.83) can always be hold, under the condition of δxw = δβw = δγw = δαw = δz w = δγw = 0

(2.86)

2.4 DOF Level Model of Locations

59

Fig. 2.31 Position of point in space

In other words, six DOFs, δx w , δα w , δyw , δβ w , δzw and δγ w , must be limited, the position requirements of point in space can be guaranteed.

2.4.3.2

Position of Line

As shown in Fig. 2.32, four lines will be scaled on the sheet workpiece. Every scale line has the position tolerance in the X direction. The plane A is the most important datum of four sale lines. According to Eq. (2.10), the following equation can be obtained δxw + δβw z P − δγw γP = 0

Fig. 2.32 Position requirement of the line in single direction

(2.87)

60

2 Analysis of Locating Determination

Because point P is on the plane A, yP and zP in Eq. (2.87) are arbitrary values. Thus, if and only if the following condition is satisfied can Eq. (2.87) be always hold. δxw = δβw = δγw = 0

(2.88)

Therefore, if three DOFs δx w , δβ w and δγ w are constrained, the position of the line to be machined in the X direction can be guaranteed. By analogy, if the line to be machined has a position requirement in the Y direction or the Z direction, the DOFs to be constrained are respectively δyw , δα w , δγ w or δzw , δα w , δβ w . According to the union operation of DOF, the following conclusions can be obtained. In order to ensure the positions of the line to the YOZ and ZOX planes in the X and Y directions, it is requisite to constrain five DOFs δx w , δyw , δα w , δβ w and δγ w . In order to ensure the positions of the line to the ZOX and XOY planes in the Y and Z directions, it is requisite to constrain five DOFs δyw , δzw , δα w , δβ w and δγ w . In order to ensure the positions of the line to the XOY and YOZ planes in the Z and X directions, it is requisite to constrain five DOFs δzw , δx w , δα w , δβ w and δγ w . Moreover, In order to ensure the positions of the line to the XOY, YOZ and ZOX planes in the Z, X and Y directions, it is requisite to constrain six DOFs δx w , δyw , δzw , δα w , δβ w and δγ w .

2.4.3.3

Position of Plane

If a plane to be machined has the position requirement with the YOZ plane, the machining requirement is in the X direction. Therefore, the following equation can be achieved according to Eq. (2.10). δxw + δβw z P − δγw γP = 0

(2.89)

Again, x P = 0, yP and zP are arbitrary numbers. In combination with Eq. (2.89), the solution on DOFs can be obtained as δxw = δβw = δγw = 0

(2.90)

As a result, it is essential to limit DOFs δx w , δβ w and δγ w for the position requirement of the plane to be machined with the YOZ plane. Likewise, in order to guarantee the position of the plane to be machined with the XOY plane, DOFs δzw , δα w and δβ w must be limited. If the plane to be machined is required a position with respect to the XOZ plane, DOFs δyw , δα w and δγ w must be limited. In order to clarify the relationship between the position requirement of the plane to be machined surface and the theoretical DODs, the level model can be concluded as listed in Table 2.6.

2.4 DOF Level Model of Locations

61

Table 2.6 Level model of the position and the theoretical DOFs Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ*

Point

Z, X

XOY, YOZ

δx, δz, δα, δβ, δγ

λy ζ y

ζy

X, Y

YOZ, ZOX

δx, δy, δα, δβ, δγ

λz ζ z

ζz

Line

Plane

Y, Z

ZOX, XOY

δy, δz, δα, δβ, δγ

λx ζ x

ζx

X, Y, Z

XOY, YOZ, ZOX

δx, δy, δz, δα, δβ, δγ

0

0

X

YOZ

δx, δβ, δγ

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z, ζ α]

Y

ZOX

δy, δα, δγ

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z, ζ β ]

Z

XOY

δz, δα, δβ

λx ζ x + λy ζ y +λγ ζ γ [ζ x , ζ y, ζγ ]

X、Z

YOZ, XOY

δx, δz, δα, δβ, δγ

λy ζ y

ζy

X、Y

YOZ, ZOX

δx, δy, δα, δβ, δγ

λz ζ z

ζz

Y 、Z

ZOX, XOY

δy, δz, δα, δβ, δγ

λx ζ x

ζx

X、Y 、Z

XOY, YOZ, ZOX δx, δy, δz, δα, δβ, δγ

0

0

X

YOZ

δx, δβ, δγ

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z, ζ α]

Y

ZOX

δy, δα, δγ

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z, ζ β ]

Z

XOY

δz, δα, δβ

λx ζ x + λy ζ y +λγ ζ γ [ζ x , ζ y, ζγ ]

2.4.4 Relation Between Runout Tolerance and DOFs The runout tolerance is a kind of tolerance set on the basis of a specific testing method. The runout tolerance can be categorized into the circular runout and the total runout. The circular runout is divided into the radial circular runout, the end face circular runout and the oblique circular runout. The total runout is divided into the radial total runout and the end face total runout. The radial circle runout tolerance zone is the area between two concentric circles in any plane perpendicular to the datum axis, where the difference in radius of two concentric circles is the tolerance value t. The radial circle runout tolerance of the cylinder to be machined is shown in Fig. 2.33. Obviously, the requirement of radial circle runout tolerance based on X axis indicates that the workpiece cannot move in Y and Z direction. Therefore  δyw + δγw xP − δαw z P = 0 (2.91) δz w + δαw y P − δβw x P = 0

62

2 Analysis of Locating Determination

Fig. 2.33 Stepped shaft with the radial circular runout tolerance

Because point P is on the X axis, yP = zP = 0, and x P is any number. In order to make Eq. (2.91) to be hold, if and only if δβw = δyw = δz w = δγw = 0

(2.92)

Consequently, if four DOFs including δzw , δyw , δβ w and δγ w are limited, the radial circle runout tolerance based on the X axis can be guaranteed. By analogy, in order to ensure the radial circle runout tolerance based on the Y axis, DOFs δx w , δα w , δzw and δγ w must be limited. If the radial circle runout tolerance is specified for the surface machined based on the Z axis, it is necessary to limit DOFs δx w , δα w , δyw and δβ w . The end face circular runout tolerance zone is a cylindrical area with a width of tolerance value t along the direction of the generatrix on the measuring cylindrical surface which is coaxial with the datum axis and at any diameter. As shown in Fig. 2.34, an end face in the YOZ plane will be machined on the stepped shaft with the the end face circular runout tolerance 0.05 mm relative to the X axis. Thus, the machining requirement of the end face is in the X, Y and Z directions and in turn, there is ⎧ ⎨ δxw + δβw z P − δγw yP = 0 (2.93) δy + δγw xP − δαw z P = 0 ⎩ w δz w + δαw y P − δβw x P = 0 Fig. 2.34 Stepped shaft with the end face circular runout tolerance

2.4 DOF Level Model of Locations

63

Table 2.7 Level model of the total/circular runout and the theoretical DOFs Type

Directions of machining requirements

Datum

Theoretical constrains

Theoretical DOF δq ∗w

Base vector ζ *

Radial circular runout or total runout

Y, Z

X

δx, δβ, δγ

λy ζ y + λz ζ z + λα ζ α

[ζ y , ζ z , ζ α]

Z, X

Y

δy, δα, δγ

λx ζ x + λz ζ z + λβ ζ β

[ζ x , ζ z , ζβ]

X, Y

Z

δz, δα, δβ

λx ζ x + λy ζ y +λγ ζ γ

[ζ x , ζ y , ζγ ]

X

X

δx, δy, δz, δβ, δγ

λα ζ α

ζα

Y

Y

δx, δy, δz, δα, δγ

λβ ζ β

ζβ

Z

Z

δx, δy, δz, δα, δβ

λγ ζ γ

ζγ

End face circular runout, oblique circular runout or total runout

Because the measuring feature is the process datum, that is the X axis, thus x P = 0, yP and zP are arbitrary values. Equation (2.93) can always be hold, if and only if δxw = δβw = δyw = δz w = δγw = 0

(2.94)

Accordingly, in order to ensure the end face circular runout based on the X axis, the DOFs δx w , δyw , δβ w , δzw and δγ w must be limited. Likewise, in order to ensure the end face circular runout based on the Y axis, the DOFs δx w , δα w , δyw , δzw and δγ w must be limited. in order to ensure the end face circular runout based on the Z axis, the DOFs δx w , δα w , δyw , δβ w and δzw must be limited. Nevertheless, it can be concluded that the theoretical DOFs required for total runout is consistent with that of circular runout, so it will not be repeated. It can be seen from the above derivation that the level model of total/circular runout and DOFs is shown in Table 2.7.

2.5 Model of Locating Point Layout In the rectangular coordinate system of space, if the workpiece can translate along or rotate around the X axis (or Y axis, Z axis), the position variation δx w and δα w (or δyw and δβ w , δzw and δγ w ) caused by the translation or rotation of the workpiece is called the theoretical DOF (Wu et al. 2007). Thus δx w = λx (or δyw = λy , δzw = λz ) and δα w = λα (or δβ w = λβ , δγ w = λγ ). Otherwise, it is called the theoretical constraint where δx w = 0 (or δyw = 0, δzw = 0) and δα w == 0 (or δβ w = 0, δγ w = 0). The obtainment of the theoretical constraint depends on the determination of the values of 6 position parameters x w , yw , zw , α w , β w , γ w or some position parameters of the

64

2 Analysis of Locating Determination

workpiece by reasonably laying out the locating points. In this way, the workpiece has a correct position relative to the fixture. This process is the so-called locating.

2.5.1 Establishment of Locating Point Layout Model In the process of workpiece locating, the DOF can be divided into theoretical DOF and practical DOF. Accordingly, constraints are naturally divided into theoretical constraints and practical constraints. As shown in Fig. 2.35a, in order to satisfy the of the stepped surface, the workpiece can only machining requirement l0+δl and h +δh 0 move in the Z direction during the process of locating. In other words, in order to machine a qualified step surface, the theoretical DOF of workpiece is δzw whereas δx w , δyw , δα w , δβ w and δγ w are theoretical constraints. Therefore, the DOFs to be theoretically limited is δq ∗w = λz ζ z . the corresponding form of set can be represented as {δq ∗w } = {δx w , δyw , δα w , δβ w , δγ w } which is called the theoretical constraint (Qin et al. 2006, 2008). The theoretical constraints of the workpiece are limited by the locating layout scheme of the fixture. Figure 2.33b is a layout scheme of three locating points, all of which are located on the bottom surface of the workpiece. Thus, the workpiece can neither move in the Y direction nor rotate around the X and Z axes, that is to say, the DOFs of the workpiece are δx w , δzw , δβ w , and the constraints are δy, δα w , δγ w . This is called the practical DOF and the practical constraint. If δq hw is denoted as the practically limited DOF, its set form should be {δq hw } = {δy, δα w , δγ w } will be named the practical constraint. By comparing Fig. 2.35b with Fig. 2.35a, δy, δα w and δγ w in the theoretical DOF δq ∗w are limited by the locating layout scheme of Fig. 2.35b. As shown in Fig. 2.36, the locating layout scheme consists of k locating points. Suppose that the workpiece is a rigid body with a surface represented by a piecewise differentiable function in WCS. f (r w ) = f (x w , y w , z w ) = 0

Fig. 2.35 DOFs and constraints

(2.95)

2.5 Model of Locating Point Layout

65

Fig. 2.36 Workpiece locating

where r w = [x w , y w , z w ]T is the coordinate of any point on the workpiece with respect to WCS. rw = [x w , yw , zw ]T denotes the position of the origin of WCS in GCS. Θ w = [α w , β w , γ w ]T is the orientation of WCS with respect to GCS. If the orientation and position of the workpiece are known, the workpiece point rw can be mapped from WCS to GCS following r = T (Θ w )r w + r w

(2.96)

⎤ cβw cγw −cαw sγw + sαw sβw cγw sαw sγw + cαw sβw cγw where T (Θ w ) = ⎣ cβw sγw cαw cγw + sαw sβw sγw −sαw cγw + cαw sβw sγw ⎦ is −sβw sαw cβw cαw cβw an orthogonal rotation matrix with c = cos and s = sin. By substituting Eq. (2.96) into Eq. (2.95), the surface equation of the workpiece in GCS can be obtained as ⎡

f (T (Θ w )T (r − r w ))= 0

(2.97)

where T (Θ w )T is the transpose matrix of T (Θ w ).

T

T = xw , yw , z w , αw , βw , γw represents six position If q w = r Tw , Θ Tw parameters of the workpiece, Eq. (2.97) can be further rewritten as     f q w , r = f T (Θ w )T (r − r w ) = 0

(2.98)

It is well known, the coordinate ri = [x i , yi , zi ]T for the i-th locating point in GCS

T has the following relationship with the corresponding coordinate r iw = xiw , yiw , z iw defined in WCS

66

2 Analysis of Locating Determination

r i = T (Θ w )r iw +r w

(2.99)

Since the i-th locating point is located on the workpiece surface, according to Eq. (2.98), the relationship between every locating point and the workpiece is       f i q w = f q w , r i = f T(Θ w )T (r i − r w ) = 0, 1 ≤ i ≤ k

(2.100)

Obviously, this is a simultaneous equation, i.e.,  ⎧  f q w , r1 = 0 ⎪ ⎪ ⎨ f (qw , r2 ) = 0 ⎪ ··· ⎪ ⎩ f (qw , rk ) = 0

(2.101)

In the actual locating process, if the locating is not reasonable, the workpiece must not be in the theoretical position q ∗w . Assume that the position of the workpiece changes in the neighborhood δq hw of q ∗w . However, no matter how the position of the workpiece changes, the locating point should theoretically always keep contact with the surface of the workpiece. Otherwise, it will lose the practical significance of locating. By substituting any position qw of the workpiece into Eq. (2.101), it can further be rewritten as     f q w , r i = f q ∗w + δq hw , r i , 1 ≤ i ≤ k

(2.102)

If the higher order terms are ignored, Taylor expansion of Eq. (2.102) at the theoretical position q ∗w of the workpiece can be described as     f q ∗w + δq hw , r i = f q ∗w , r i + J i δq hw , 1 ≤ i ≤ k where J i =  T n iwx , n iwy , n iwz =





∂ fi , ∂ fi , ∂ fi , ∂ fi , ∂ fi , ∂ fi ∂ xw ∂ yw ∂z w ∂αw ∂βw ∂γw T  ∂ fi ∂ fi ∂ fi ∂ fi = is w = w, w, w ∂ ri ∂ xi ∂ yi ∂z i

(2.103)

is the gradient vector, niw

=

the unit normal vector of workpiece

T surface f (r ) = 0 at the i-th locating point = xiw , yiw , z iw . If all locating points are always in contact with the workpiece surface, and r iw

w

J i δq hw = 0, 1 ≤ i ≤ k

(2.104)

then the workpiece is in the theoretical position according to Eq. (2.101) and Eq. (2.103). The compact expression of Eq. (2.104) can be rewritten in matrix form as Jδq hw = 0 where J is the locating Jacobian matrix and

(2.105)

2.5 Model of Locating Point Layout

⎡

67 ∂ f1 ∂ xw ∂ f2 ∂ xw

∂ f1 ∂ yw ∂ f2 ∂ yw

∂ f1 ∂ f1 ∂z w ∂αw ∂ f2 ∂ f2 ∂z w ∂αw

∂ f1 ∂βw ∂ f2 ∂βw

∂ f1 ∂γw ∂ f2 ∂γw

⎤

⎢ ⎥ ⎢ ⎥ J =⎢ ⎥ ⎣ ··· ··· ··· ··· ··· ··· ⎦

(2.106)

∂ fk ∂ fk ∂ fk ∂ fk ∂ fk ∂ fk ∂ xw ∂ yw ∂z w ∂αw ∂βw ∂γw

In Eq. (2.105), the determination of Jacobian matrix J is the key to solve δq hw .

2.5.2 Solution of Locating Point Layout Model Assumed that the 3D workpiece rotates α w , β w and γ w around the X axis, Y axis and Z axis successively, then its coordinate transformation matrix should be respectively ⎤ 10 0 T (αw ) = ⎣ 0 cos αw − sin αw ⎦ 0 sin αw cos αw ⎤ ⎡ cos βw 0 sin βw ⎦ T (βw ) = ⎣ 0 10 − sin βw 0 cos βw ⎡ ⎤ cos γw − sin γw 0 T (γw ) = ⎣ sin γw cos γw 0 ⎦ 0 0 1 ⎡

(2.107)

(2.108)

(2.109)

In this way, the direction cosine matrix (i.e. transformation matrix T(Θ w )) of a rigid body in any position and direction can be obtained by multiplying the direction cosine matrix of three rotations T (Θ w ) = T (γw )T (βw )T (αw ) ⎡ ⎤⎡ ⎤ ⎤⎡ cγw −sγw 0 10 0 cβw 0 sβw = ⎣ sγw cγw 0 ⎦⎣ 0 1 0 ⎦⎣ 0 cαw −sαw ⎦ 0 sαw cαw 0 0 1 −sβw 0 cβw ⎡ ⎤ cβw cγw −cαw sγw + sαw sβw cγw sαw sγw + cαw sβw cγw = ⎣ cβw sγw cαw cγw + sαw sβw sγw −sαw cγw + cαw sβw sγw ⎦ (2.110) −sβw sαw cβw cαw cβw where c and s represent function cos and function sin, respectively. Because Eq. (2.110) is a function of α w , β w and γ w , it is differentiated with respect to α w , β w and γ w such that

68

2 Analysis of Locating Determination

⎤  ⎡0  0 0 ∂ T (Θ w )T = ⎣ sαw sγw + cαw sβw cγw −sαw cγw + cαw sβw sγw cαw cβw ⎦ ∂αw cαw sγw − sαw sβw cγw −cαw cγw − sαw sβw sγw −sαw cβw (2.111) ⎡ ⎤   −sβw cγw −sβw sγw −cβw ∂ T (Θ w )T = ⎣ sαw cβw cγw sαw cβw sγw −sαw sβw ⎦ (2.112) ∂βw cαw cβw cγw cαw cβw sγw −cαw sβw ⎤   ⎡ −cβw sγw cβw cγw 0 ∂ T (Θ w )T = ⎣ −cαw cγw − sαw sβw sγw −cαw sγw + sαw sβw cγw 0 ⎦ (2.113) ∂γw sαw cγw − cαw sβw cγw −cαw cγw − sαw sβw sγw 0 For the 2D workpiece, its orthogonal coordinate transformation matrix is

cos γw sin γw T (Θ w ) = T (γw ) = − sin γw cos γw

 (2.114)

and there is  

 ∂ T (Θ w )T − sin γw − cos γw = cos γw − sin γw ∂γw

(2.115)

By combining Eq. (2.110) with Eq. (2.108), it can be obtained as     ∗ f i q w = f q ∗w , r i = 0 r i = T (Θ w )r iw + r w

(2.116)

By differentiating Eq. (2.116) with respect to x w , yw , zw , α w , β w and γ w , it is put in a compact form as ⎧  T ⎪ ∂ fi ∂ fi ⎪ = − T (Θ w )T ∂∂ xr ww ⎪ ∂ xw ∂riw ⎪ ⎪   ⎪ T ⎪ ∂ fi ∂ fi ⎪ ⎪ = − ∂r T (Θ w )T ∂∂ ryww w ⎪ ∂ y w ⎪ i ⎪   T ⎪ ⎪ ⎨ ∂ fi = − ∂ fwi T (Θ w )T ∂ r w ∂z w ∂ri ∂z w  T ∂ T (Θ w )T ) w ( ∂ f ∂ f ⎪ i i ⎪ ri ⎪ ∂αw = ∂riw ∂αw ⎪ ⎪  T ⎪ T ⎪ (∂ T (Θ w ) ) r w ∂ fi ∂ fi ⎪ ⎪ = ∂r w ⎪ i ∂βw ∂βw i ⎪ ⎪   T T ⎪ ⎪ ⎩ ∂ fi = ∂ fwi (∂ T (Θ w ) ) r w i ∂γw ∂r ∂γw i

(2.117)

2.5 Model of Locating Point Layout ∂ fi ∂ r iw

=



∂ fi , ∂ fi , ∂ fi ∂ xiw ∂ yiw ∂z iw

69

T

is the normal vector of workpiece surface f (r w ) = 0

T at i-th locating point r iw = xiw , yiw , z iw . If niw = [n iwx , n iwy , n iwz ]T is further denoted as the normal vector of the workpiece at the i-th locating point r iw , then

where

⎧ w ⎪ ⎨ ni x = n iwy = ⎪ ⎩ nw = iz

∂ fi ∂ xiw ∂ fi ∂ yiw ∂ fi ∂z iw

(2.118)

Without loss of generality, WCS and GCS can be always assumed to have the identical orientation (Song and Rong 2005; Rong et al. 2002), i.e., αw = βw = γw = 0. By substituting Eq. (2.111) and Eq. (2.112) into Eq. (2.117), there is ⎧ ∂ fi  w T  w T ∂ fi ⎪ ⎨ ∂ xw = − ni  ex , ∂ yw = − ni e y T T ∂ fi ∂ fi = − niw ez , ∂α = niw I x r iw ∂z w w ⎪  T  T ⎩ ∂ fi ∂ fi = niw I y r iw , ∂γ = niw I z r iw ∂βw w

(2.119)

where I x , I y and I z are the skew symmetric matrixes. ex , ey , ez are the unit vectors and ⎧ ⎡ ⎤ 0 0 0 ⎪ ⎪ ⎪ ⎪ Ix = ⎣0 0 1 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ 0 −1 0 ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ 0 0 −1 ⎨ Iy = ⎣0 0 0 ⎦ (2.120) ⎪ ⎪ ⎪ 1 0 0 ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ 0 10 ⎪ ⎪ ⎪ ⎪ I = ⎣ −1 0 0 ⎦ ⎪ ⎪ ⎩ z 0 00 ⎧ T ⎪ ⎨ ex = [1, 0, 0] (2.121) e y = [0, 1, 0]T ⎪ ⎩ T ez = [0, 0, 1] Thus, the locating Jacobian matrix can be obtained from Eq. (2.106) as     T  T J i = − niw , − niw × r iw

= −n iwx , −n iwy , −n iwz , n iwz yiw − n iwy z iw , n iwx z iw − n iwz xiw , n iwy xiw − n iwx yiw (2.122) For a 2D workpiece, Eq. (2.122) can be simplified as

70

2 Analysis of Locating Determination

J i = −n iwx , −n iwy , n iwy xiw − n iwx yiw

(2.123)

According to Eq. (2.105), the position variation of the workpiece in the locating layout scheme can be expressed as δq hw = ker( J)λ

(2.124)

where ker( J) is a matrix composed of the standard orthogonal basis for the null space of J, rank(J) is the rank of matrix J, and λ is a vector of arbitrary constant. In fact, according to the definition of δq hw , δq hw is the so-called pracitcal DOF. It is known from Eq. (2.124), the value of δq hw can be solved by using the function null in MATLAB.

2.6 Judgment Criteria of Locating Determination Only the position parameters that affect the machining accuracy are generally determined, so it is not necessary to limit 6 DOFs in the process of workpiece locating. According to the locating methods of complete locating, partial locating, over locating and under locating, the following theorems can be derived as the quantitative basis for judging the locating determination.

2.6.1 Theoretical Condition On the basis of the machining requirements of the workpiece, the theoretical DOF δq ∗w can be in advance calculated according to Eqs. (2.10, ⎤ and the corresponding ⎡ 2.11) δxw ⎢ δy ⎥ ⎢ w⎥ ⎥ ⎢ ⎢ δz ⎥ ∗ level model. For example, it is known from δq w = ⎢ w ⎥ = λx ζ x + λ y ζ y + λγ ζ γ ⎢ δαw ⎥ ⎥ ⎢ ⎣ δβw ⎦ δγw in Eq. (2.14) that, the solution space of δq ∗w is composed of basic solution system ζ x , ζ y and ζ γ . Therefore, the rank of the solution space is rank(δq ∗w ) = rank(ζ * ) = rank([ζ x , ζ y , ζ γ ]) = 3. Or, the theoretical constraint {δq ∗w } = {δzw , δα w , δβ w }. Obviously there is rank(δq ∗w ) = rank(ζ x , ζ y , ζ γ ) = 3. Comparatively, δq hw can be calculated by Eq. (2.31) corresponding to the real displacement of the workpiece after the locating setup. Suppose {I } = {δx w , δyw , δzw , δα w , δβ w , δγ w } is the full set. Obviously, the theoretical constaint set {δq ∗w } and the practical constaint set {δq hw } are the subset of {I }. Thus, the logical relationship between {δq ∗w } and {δq hw } can be obtained by

2.6 Judgment Criteria of Locating Determination

71

Venn diagram and categorized into inclusion, intersection and difference, as shown in Fig. 2.37. Again, the relation of {δq hw } ⊇ {δq ∗w } signifies that the theoretical constraint is a subset of the practical constraint, whereas {δq hw } ⊂ {δq ∗w } denotes that the practical constraint is a proper subset of the theoretical constraint. Thus, if and only if {δq ∗w } is included in {δq hw }, i.e., {δq ∗w } ⊆ {δq hw } ⊆ {I}

(2.125)

can the locating point layout scheme be correct (in other words, the locating point layout scheme has the locating correctness). Otherwise, the locating point layout scheme be incorrect. In the logical relationship where δq ∗w is included in δq hw , there must be rank(δq hw ) = rank(δq ∗w )—rank ( Jδq ∗w ). According to Eq. (2.105), the rank of coefficient matrix J in homogeneous linear equations is assumed to be rank(J). Then, the rank of the solution vector of δq hw is rank of δq hw is equal to rank(δq hw ) = 6—rank (J) (Qin et al. 2008). Therefore, the theoretical condition of locating correctness can be described as rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) = 6

(2.126)

Simultaneously, it is easy to know the sufficient and necessary conditions for incorrect locating rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) < 6

Fig. 2.37 Logical relationship of {δq ∗w } and {δq hw }

(2.127)

72

2 Analysis of Locating Determination

2.6.2 Process Condition It can be seen from Eq. (2.105) that for a locating scheme with k locating points, the rank of the locating Jacobian matrix always has the following relationship with the number of locating points rank( J) ≤ k

(2.128)

It is worthy to notice that the locating point layout scheme with rank( J) < k is unreasonable. Though the application of the great number of locators can increase the workpiece stiffness, the addition of a locator into a fixture may increase the workpiece set-up time, fixture manufacturing cost and weight, the complexity of cutting tool access to the workpiece. Therefore, from the point of view of manufacturing process, if and only if rank( J) = k

(2.129)

is hold, can the locating point layout scheme be correct.

2.6.3 Corollary and Flowchart According to the theoretical condition of Eq. (2.126) and the process condition of Eq. (2.129), some corollaries can be used to verify the correctness of the locating point layout scheme mathematically dominated by Eq. (2.105). Definition 1 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) = 6 and rank( J) = k, then the locating point layout scheme is called deterministic locating (Wu et al. 2002; Chou et al. 1994; Cai et al. 1997). Definition 2 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) < 6, then the locating point layout scheme is called under locating. Definition 3 If rank( J) < k, then the locating point layout scheme is called over locating. Corollary 1 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) = 6 and rank( J) = k = 6, then this deterministic locating is called complete locating. Corollary 2 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) = 6 and rank( J) = k < 6, then this deterministic locating is called partial locating. Corollary 3 If rank( J)+rank(δq ∗w )−rank( Jδq ∗w ) = 6, rank( J) < k and rank( J) < 6, then this over locating is called partial over locating.

2.6 Judgment Criteria of Locating Determination

73

Corollary 4 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) = 6 and rank( J) = 6 < k, then this over locating is called complete over locating. Corollary 5 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) < 6 and rank( J) < k, then this under locating is called under over locating. Corollary 6 If rank( J) + rank(δq ∗w ) − rank( Jδq ∗w ) < 6 and rank( J) = k, then this under locating is called partial under locating. According to the above definitions and corollaries, the logical relationship among the locating point layout schemes can be shown in Fig. 2.38. It is worthy to notice that both complete locating and partial locating belong to correct locating scheme whereas neither under over locating nor partial under locating are viable. This two infeasible locating should be avoided in the design of locating scheme. From the viewpoint of engineering application, the locating scheme being complete over locating or partial over locating is thought to be incorrect. The flowchart is given in Fig. 2.39 to summarize judgement criteria for different locating scheme.

Fig. 2.38 Relationship between the locating point layout schemes

74

2 Analysis of Locating Determination

Fig. 2.39 Flowchart of verifying the locating determination

2.7 Application and Analysis Generally speaking, a workable locating scheme is able to constrain the undesired DOFs of the workpiece. The fixture designer must determine the layout of a given number of locators so as to locate the workpiece in its desired position. In fact, a variety of locating schemes will be obtained by changing the locator number and locating datum, as shown in Fig. 2.40. However, not all the locating point layout schemes are correct and reasonable (Qin et al. 2006). To have a good understanding, Fig. 2.40a and b show a 2D workpiece constrained by two locators represented by triangles. However, the difference is that two locating points in Fig. 2.40a are distributed on the uniform locating datum, while two locating points in Fig. 2.40b are respectively placed on different locating datum. Thus, the workpiece is obviously free to either translate along the X direction or rotate about point O. This means that the DOF to be limited of the workpiece in case (a) is different with in case (b). On the contrary, the workpiece is determinately constrained in case (c) where the locating datum is the same as case (b). In this section, two typical examples are given to illustrate in detail the whole process of analyzing and verifying the rationality of the number of locating points and their layout by using the locating determination judgment method.

2.7 Application and Analysis

75

Fig. 2.40 Locating point layout scheme

2.7.1 Verification of Locating Point’s Number Figure 2.41 shows the locating point layout schemes for machining the stepped surface, which consists of 4 and 5 locating points respectively. The positions and unit normal vectors of every locating point are shown in Table 2.8. The machining requirements of workpiece are size b in the X direction and size h in the Y direction. The first step is to determine the theoretical DOF. For the first locating point layout scheme shown in Fig. 2.41a, it is assumed that {GCS} coincides with {WCS}. Because there are size requirements in the X and Y directions, five DOFs δx w , δyw ,

Fig. 2.41 The locating schemes with different number of locating points

76

2 Analysis of Locating Determination

Table 2.8 Coordinates and unit normal vetors of locating points The locating scheme with 4 locating points

The locating scheme with 5 locating points

Locating point

Coordinate ri

Unit normal vector ni

Locating point

Coordinate ri

Unit normal vector ni

1

[x 1 , 0, z1 ]T

[0, 0, 1]T

1

[x 1 , 0, z1 ]T

[0, 0, 1]T

2

[x 2 , 0, z2 ]T

[0, 0, 1]T

2

[x 2 , 0, z2 ]T

[0, 0, 1]T

3

[x 3 , 0, z3

]T

[0, 0,

1]T

3

[x 3 , 0, z3

]T

[0, 0, 1]T

4

[0, y4 , z4

]T

[0, 1,

0]T

4

[0, y4 , z4

]T

[0, 1, 0]T

–

–

5

[0, y5 , z5 ]T

[0, 1, 0]T

–

δα w , δβ w and δγ w must be limited according to Table 2.1. Therefore, the theoretical DOF δq ∗w = [δx w , δyw , δzw , δα w , δβ w , δγ w ]T = λζ with ζ = ζ z . In other words, the DOF that should be limited is {δq ∗w } = {δx w , δyw , δα w , δβ w , δγ w }. The second step is to calculate the practical DOF. According to Eq. (2.122), the ⎤ ⎡ 0 −1 0 −z 1 0 x1 ⎢ 0 −1 0 −z 2 0 x2 ⎥ ⎥ locating Jacobian matrix can be obtained as J = ⎢ ⎣ 0 −1 0 −z 3 0 x3 ⎦. The

−1 0 0 0 z 4 −y4 function rank in MATLAB can be used to calculate rank(J) = 4. The practical DOF calculated by function null in MATLAB is ⎤ ⎡ 0 δxw ⎢ δy ⎥ ⎢ 0 ⎢ w⎥ ⎢ ⎥ ⎢ ⎢ ⎢ δz ⎥ ⎢ 1 δq hw = ⎢ w ⎥ = ⎢ ⎢ δαw ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎣ δβw ⎦ ⎣ 0 δλw 0 ⎡

⎤ z4 0⎥ ⎥  ⎥ 0 ⎥ λ1 ⎥ 0 ⎥ λ2 ⎥ 1⎦ 0

(2.130)

where w56 and w66 are arbitrary constants. It is known from Eq. (2.130), I16 . Therefore, the practical DOF can be further written as δq hw = λx ζ x + λz ζ z + λβ ζ β . In other words, the DOF limited practically by the locating point layout scheme is {δq hw } = {δyw , δα w , δγ w }. ∗ The third step is to judge the locating determination. Because rank(δq  w∗ ) = ∗ h 1, rank(  Jδq w ) = 0 and rank(δq w ) = 3, then there are rank( J) + rank δq w − rank Jδq ∗w = 4 + 1 − 0 = 5 and rank( J) = k = 4. According to the judgement flowchart in Fig. 3-39, the locating scheme belongs to the partial under locating. In the simular way, it can be obtained that the DOF limited by the locating point layout scheme shown in Fig. 2.41b is δq hw = λz ζ z whose set form is {δq hw } = {δx w , δyw , δα w , δβ w , δγ w }. Because rank( J) = 5, rank(δq ∗w ) = 1, rank( Jδq ∗w ) = 0 and rank(δq hw ) = 1, then rank(J) + rank(δq ∗w )–rank ( Jδq ∗w ) = 5+1–0 = 6 and rank(J) = k = 5. In light of the judgement flowchart in Fig. 3.39, such a locating scheme is identified to be a partial locating.

2.7 Application and Analysis

77

2.7.2 Verification of the Layout of Locating Points Figure 2.42 shows a locating point layout scheme for milling a keyway on the top of the workpiece. This locating scheme consists of six locating points including three locating points laid out on the bottom surface, two locating points on the left side surface and one locating point on the behind surface. Assumed that the global coordinate system GCS coincides with the workpiece coordinate system WCS, the position of each location point and the unit normal vector are listed in Table 2.9. Because the non-through has the machining requirements in all three directions, the theoretical DOF is then δq ∗w = 0. Only when six DOFs are constrained, the machining operation is correct. In other words, the theoretical constraint is {δq ∗w } = {δx w , δyw , δzw , δα w , δβ w , δγ w }. According to the information in Table 2.9, the Jacobian matrix J can be obtained Fig. 2.42 Locating point layout scheme of milling a non-through slot

Table 2.9 Positions and directions of locators

Locator

Coordinate ri

Unit normal vector ni

[x 1 , 0, z1

]T

[0, 0, 1]T

2

[x 2 , 0, z2

]T

[0, 0, 1]T

3

[x 3 , 0, z3 ]T

[0, 0, 1]T

4

[0, y4 , z4

]T

[0, 1, 0]T

5

[0, y5 , z5

]T

[0, 1, 0]T

6

[x 6 , y6 , 0]T

[0, 0, 1]T

1

78

2 Analysis of Locating Determination

⎡

0 ⎢ 0 ⎢ ⎢ ⎢ 0 J =⎢ ⎢ −1 ⎢ ⎣ −1 0

−1 −1 −1 0 0 0

0 0 0 0 0 −1

−z 1 −z 2 −z 3 0 0 y6

0 0 0 −z 4 −z 5 −x6

⎤ x1 x2 ⎥ ⎥ ⎥ x3 ⎥ ⎥ −y4 ⎥ ⎥ −y5 ⎦ 0

(2.131)

Therefore, the determinant of Jacobian matrix J can be described as | J| = (z 4 − z 5 )(−x1 z 2 + z 1 x2 − z 1 x3 + x1 z 3 + z 2 x3 − z 3 x2 )

(2.132)

Obviously, if z 4 − z 5 = 0

(2.133)

−x1 z 2 + z 1 x2 − z 1 x3 + x1 z 3 + z 2 x3 − z 3 x2 = 0

(2.134)

| J| = 0

(2.135)

or

there exists

Nevertheless, the rank of Jacobian matrix J is rank( J) = 6

(2.136)

Again, the conditions described in Eqs. (2.133, 2.134), there is rank(J) + rank(δq ∗w )—rank ( Jδq ∗w ) = 6 + 0-0 = 6. Therefore, the locating scheme belongs to an applicable complete locating being to determine correctly the workpiece position. However, if z 4 − z 5 = 0, locating point 4 and locating point 5 are arranged in parallel with the Y axis, as shown in Fig. 2.43a. Thus, there is the following relationship ⎡

0 ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ −1 ⎢ ⎣ −1 0

−1 −1 −1 0 0 0

0 0 0 0 0 −1

−z 1 −z 2 −z 3 0 0 y6

0 0 0 −z 4 −z 5 −x6

⎤ ⎤⎡ δxw x1 ⎢ ⎥ x2 ⎥ ⎥⎢ δyw ⎥ ⎥ ⎥⎢ x3 ⎥⎢ δz w ⎥ ⎥=0 ⎥⎢ −y4 ⎥⎢ δαw ⎥ ⎥ ⎥⎢ −y5 ⎦⎣ δβw ⎦ 0 δλw

(2.137)

When the function null in MATLAB is used to solve Eq. (2.137), the practical DOF can be achieved as

2.7 Application and Analysis

79

Fig. 2.43 The locating schemes with the same number of locating points

⎤ ⎡ ⎤ z4 δxw ⎢ δy ⎥ ⎢ 0 ⎥ ⎢ w⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ δz −x ⎢ ⎥ ⎢ ⎥ w 6 δq hw = ⎢ ⎥=⎢ ⎥λ ⎢ δαw ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ ⎥ ⎣ δβw ⎦ ⎣ 1 ⎦ δγw 0 ⎡

(2.138)

where λ is an arbitrary value. It is known from Eq. (2.138), δyw = δαw = δγw = 0. Equation (2.138) can further be described as δq hw = λx ζ x + λz ζ z + λβ ζ β . That is, the practical comstraint is {δq hw } = {δyw , δα w , δγ w } which is also called the DOF limited practically by the locating point layout scheme.

80

2 Analysis of Locating Determination

In reality, the rank of the Jacobian matrix J can be calculated by directly using the function rank in MATLAB as rank( J) = 5

(2.139)

    Moreover, rank( J) + rank δq ∗w − rank Jδq ∗w = 5 + 0 − 0 = 5. Obviously, the locating point layout scheme with locating point 4 and 5 parallel to the Y axis belongs to under over locating. Therefore, the locating scheme is not correct and can not guarantee the machining requirements of the workpiece. If the locating points 1, 2 and 3 are collinear, as shown in Fig. 2.43b, it is known that the triangle area formed by the locating points 1, 2 and 3 is

S 123

  x z 1 1  1 1  =  x2 z 2 1  2 x3 z 3 1  1 = (x1 z 2 − z 1 x2 + z 1 x3 − x1 z 3 − z 2 x3 + z 3 x2 ) 2

(2.140)

Therefore, if −x1 z 2 + z 1 x2 − z 1 x3 + x1 z 3 + z 2 x3 − z 3 x2 = 0, S 123 = 0 which indicates that the location points 1, 2 and 3 are collinear. Thus, the practical DOF is ⎤ ⎤ ⎡ 5 z4 − z5 yz45 −y δxw −z 4 z 3 x2 −z 2 x3 ⎥ ⎢ δy ⎥ ⎢ ⎥ ⎢ w⎥ ⎢ z −z ⎥ ⎢ x3 −x2 y 3− 2y5 −y4 x ⎥ ⎢ δz ⎢ ⎥ ⎥ ⎢ 6 6 w δq hw = ⎢ ⎥ = ⎢ z3 −z2 x3 −x2z5 −z4 ⎥ λ ⎥ ⎢ δαw ⎥ ⎢ z 3 −z 2 ⎥ ⎥ ⎢ ⎢ y5 −y4 ⎦ ⎣ δβw ⎦ ⎣ − z5 −z4 δγw 1 ⎡

(2.141)

with the arbitrary number λ. In conclusion, δx w = 0, δyw = 0, δzw = 0, δα w = 0, δβ w = 0 and δγ w = 0. Therefore, the practical DOF is an empty set, i.e., {δq hw } = {}. In the case shown  in Fig.2.41b, the computing gives rise to rank( J) = 5 and  rank( J) + rank δq ∗w − rank Jδq ∗w = 5 + 0 − 0 = 5. This means that the locating scheme is also an under over locating based on Corollary 5.

References Cai W, Hu SJ, Yuan JX. A variational method of robust fixture configuration design for 3-D workpieces [J]. Trans J Manuf Sci Eng. 1997;119:593–602. Chou YC, Srinivas RA, Saraf S. Automatic design of machining fixtures: conceptual design [J]. Int J Adv Manuf Technol. 1994;9(1):3–12.

References

81

He L. Independence comprehensiveness and relativity of freedom in location of work parts [J]. J Sichuan Inst Technol. 1997;5:50–4 (in Chinese). Kang YZ. Computer aided fixture design verification [D]. MA: Worcester Polytechnic Institute; 2001. Liao NZ, Gu Y, Mo YS, Li SG, Yang XJ. Interchangeability and technical measurement [M]. China Metrology Press, 2000. (in Chinese). Li HG, Liu M. The fundamental concepts and terminology of shape and position tolerance [J]. Shangdong Agricult Mechan. 2005;4:20 (in Chinese). Ma JJ, Zhang FR. Researching on the structure and properties of the solutions of system of linear equations by geometric method [J]. J Sci Teach Coll Univ. 2013;33(6):25–7 (in Chinese). Qin GH, Zhang WH, Wan M. A mathematical approach to analysis and optimal design of fixture locating scheme [J]. Int J Adv Manuf Technol. 2006;29(3–4):349–59. Qin GH, Wu ZX, Zhang WH. Modeling and optimal design of fixture locating scheme [J]. China Mechan Eng. 2006;17(23):2425–2429. (in Chinese). Qin GH, Zhang WH, Wan M. A machining-dimension-based approach to locating scheme design [J]. Trans ASME J Manuf Sci Eng. 2008;130(5):05101–01–08. Qin GH, Hong LH, Wu TJ. Analysis technology of degrees of freedom of workpiece based on homogenous linear equations [J]. Comput Integ Manuf Syst. 2008;14(3):466–9 (in Chinese). Qin GH, Zhang WH, Wan M, Sun SP, Wu TJ. Locating correctness analysis and modification for fixture design [C]. Proceedings of the 13th International Conference on Advanced Design And Manufacture, Sanya, China, January 14–16, 2008, pp. 551–562. Qin GH, Zhang WH, Wan M, Sun SP. A novel approach to fixture design based on locating correctness [J]. Int J Manuf Res. 2010;5(4):429–48. Rong YM, Zhu YX, Luo ZB. Computer-aided fixture design [M].China Machine Press, 2002. (in Chinese). Song H, Rong Y. Locating completeness evaluation and revision in fixture plan [J]. Robot Comput Integ Manuf. 2005;21:368–78. Sun ZL, Zhu FY. Mechanical engineering drawing and tolerance [M]. Chinese Forestry Press, 2006. (in Chinese). Wang H, Qian WW, Liu P. Right dimensions mark on symmetry structure of common aperture and key slot in spares [J]. J Henan Mechan Elect Eng Coll. 2001;9(1):28–32 (in Chinese). Wu YX. The research and designing of hierarchical module for working precision in connection with restricting degrees of freedom [J]. Jiangsu Machine Build Autom. 2003;8(4):5–8 (in Chinese). Wu YG, Gao SM, Chen ZC. Geometric theory on automated modular fixture planning [J]. Chinese J Mechan Eng. 2002;38(1):117–22 (in Chinese). Wu ZX, Wu TJ, Xiao J, Peng CM. Analysis method of workpiece degree-of-freedom based on rigid kinematics [J]. Mod Machine Tool Autom Manuf Techn. 2007;12:7–11 (in Chinese). Wu ZX, Xiao J, Wu TJ, Peng CM. Analysis model of constraint of degrees of freedom of a workpiece based on fixture [J]. Machine Tool Hyd. 2007;35(12):19–22 + 63. (in Chinese).

Chapter 3

Analysis of Workpiece Stability

The fixturing of workpiece in a fixture is a unified process in which locating and clamping are closely related to each other. After being located on a fixture or machine bed, a workpiece will be subject to gravity and cutting forces during the machining operation. In order to keep the locating precision as well as the production safety, it is necessary to maintain the workpiece stability. Workpiece stability is an indispensable part of fixture design. It is the main basis for determining or optimizing the locating/clamping scheme of workpiece and judging its reliability.

3.1 Modeling of Workpiece Stability The so-called workpiece stability refers to the ability to ensure that the position of workpiece in the fixture, which is determined by the locating point layout, can still stop vibration or movement from the action of cutting force/torque and other external forces in the process of machining (Jiang et al. 2010; Liao and Roy 2002). Based on the qualitative analysis of the stability of the above workpiece, it can be seen that if and only if the workpiece satisfies the static equilibrium equation and the friction constraint conditions in every clamping step, the workpiece is stable in the clamping sequence scheme. Figure 3.1 shows the multiple clamping scheme of a workpiece. Fmach is the machining force, F1 and F2 are the clamping forces supplied by clamps C 1 and C 2 , respectively. This clamping scheme has two clamping sequence schemes in which they can be decomposed into three clamping steps, as listed Table 3.1. In order to explain the workpiece stability simply and clearly, gravity and friction are ignored. Obviously, in steps 1 and 2 of clamping sequence scheme A, the workpiece can always be in equilibrium to maintain its stability. In step 3, as long as the clamping force provided by clamp C 1 and clamp C 2 is large enough, the workpiece can still keep equilibrium.

© Shanghai Jiao Tong University Press 2021 G. Qin, Advanced Fixture Design Method and Its Application, https://doi.org/10.1007/978-981-33-4493-8_3

83

84

3 Analysis of Workpiece Stability

Fig. 3.1 Multiple clamping

Table 3.1 Decomposition of clamping process No

Clamping sequence A

Clamping step 1

Clamping sequence B

F1

L3 F2 L3

L1

L1

Clamping step 2

L2

L2 C1

F1

L3 F2

L3 C2

L1

Clamping step 3

L2

L1

Fmach

L2

Fmach

C1

C1

L3

L3

C2 L1

L2

C2 L1

L2

3.1 Modeling of Workpiece Stability

85

Fig. 3.2 Unstable state of workpiece

Fig. 3.3 Mapping of the workpiece stability in a certain clamping sequence

However, in steps 1 of clamping sequence B, no matter how much clamping force F 2 provided by clamp C 2 is, the workpiece will rotate δqwh clockwise which can upset the equilibrium state of the workpiece. Thus, the workpiece is unstable so that it cannot occupy an accurate position, as shown in Fig. 3.2. Therefore, the purpose of stable clamping is to ensure the static equilibrium of the workpiece in every clamping step by exerting the sufficient clamping forces at the reasonably arranged clamps (i.e. clamping points). From a purely mathematical point of view, workpiece stability is a kind of mapping relationship between clamping layout scheme and workpiece equilibrium state, as shown in Fig. 3.3.

3.1.1 Static Equilibrium Conditions Suppose that the fixturing layout scheme of a workpiece consists of k locators and n clamps, as shown in Fig. 3.4. The workpiece is subjected to external loads such as machining force wrench W mach , gravity wrench W grav , etc. (Chou et al. 1989). Contacts between the workpiece and fixels (i.e., fixture elements) including locators and clamps are considered as frictional contacts.

86

3 Analysis of Workpiece Stability

Fig. 3.4 Fixturing scheme of the workpiece

Denote ni , t i and bi to be the unit inner normal vector and two orthogonal unit tangential vectors of the workpiece at contact point ri = [x i , yi , zi ]T of the i-th fixture element. When the machining forces and moments vary with respect to machining time t, the contact forces at all fixture elements will correspondingly change with respect to the machining time. As shown in Fig. 3.4, for the i-th fixture element, Fi = Fin + Fit + Fib denotes the i-th contact force resultant expressed in the global coordinate system GCS with Fin , Fit and Fib being the three components of Fi along ni = [n i x , n i y , n i z ]T , t i = [ti x , ti y , ti z ]T and bi = [bi x , bi y , bi z ]T , respectively. Hence ⎧ ⎪ ⎨ Fin = Fin ni Fit = Fit ti (3.1) ⎪ ⎩ Fib = Fib bi Thus, the contact force wrench produced by the contact force Fi on the workpiece can be expressed as  Wi =

Fi r i × Fi

 = Gi Fi

(3.2)

3.1 Modeling of Workpiece Stability

87



ni , r i × ni

where G i = [G in , G it , G ib ] is a layout matrix of the i-th fixel with G i n =   ti bi and G i b = . Gi t = ri × ti r i × bi However, every fixel has different effect on the workpiece in the process of locating, clamping and machining. If a fixel can exert a force on the workpiece, it is called an active element. If the fixel provides a motion constraint for the workpiece to support clamping and external forces, it belongs to a passive element. At the contact point with the workpiece, both the normal pressure and tangential friction force exist. An active element, however, cannot generate frictional forces by itself. So the contact force at the i-th active element characterized by layout matrices Gi can be expressed as Fi = [F in , 0, 0]T , whereas the contact force at the i-th passive element characterized by layout matrices Gi can be expressed as Fi = [F in , F ib , F it ]T . Moreover, the contact force at the active element is defined as the active contact force and the contact force at the passive element is called the passive contact force. According to the practical operation process, the arbitrary clamping sequence can be decomposed into n + 2 clamping steps and the workpiece must keep static equilibrium in each clamping step (Qin and Zhang 2007; Qin et al. 2005). In clamping step 1, only gravity force W grav is applied as the external load onto the workpiece, as shown in Fig. 3.5. Locators 1, 2, …, k are passive elements (Qin et al. 2008). Then, the static equilibrium system of equations of the workpiece is dominated by G 1,pas F 1,pas + W 1,ext = 0 Fig. 3.5 Clamping step 1

(3.3)

88

3 Analysis of Workpiece Stability

Fig. 3.6 Clamping step 2

where G1,pas = [G1 , G2 , …, Gk ] is the layout matrix of passive elements in step 1. F1,pas = [F T1 , F T2 , · · · , F Tk ]T = [F 1n , F 1t , F 1b , F 2n , F 2t , F 2b , …, F kn , F kt , F kb ]T is the passive contact force vector in step 1. W 1,ext = W grav is the external wrench in step 1. In clamping step 2 shown in Fig. 3.6, the clamp k + 1 is now set up as an active element with a prescribed clamping force F k+1 applied to the workpiece whereas locators 1, 2, …, k are passive elements. Therefore, the current static equilibrium equation system in step 2 can be obtained as G 2,pas F 2,pas + G 2,act F 2,act + W 2,ext = 0

(3.4)

where the layout matrix of passive elements, G2,pas , in step 2 is identical to G1,pas . F2,pas = [F T1 , F T2 , · · · , F Tk ]T = [F 1n , F 1t , F 1b , F 2n , F 2t , F 2b , …, F kn , F kt , F kb ]T is the passive contact force vector in step 2. W 2,ext = W grav is the external wrench in step 2. G2,act = G(k+1)n and F2,act = F (k+1)n are the layout matrix of the active elements and the corresponding clamping force vector in step 2, respectively. In step 3, suppose the second active clamp numbered k + 2 is put in use to the workpiece with a known clamping force of F k+2 . Note that clamp k + 1 used in Step 2 now becomes a passive element. Therefore, the current static equilibrium equation system of the workpiece can be written as G 3,pas F 3,pas + G 3,act F 3,act + W 3,ext = 0

(3.5)

3.1 Modeling of Workpiece Stability

89

where the layout matrix of passive elements, G3,pas , is an extension of G2,pas with G3,pas = [G1 , G2 , …, Gk , Gk+1 ]. F3,pas = [F T1 , F T2 , …, F Tk , F Tk+1 ]T = [F 1n , F 1t , F 1b , F 2n , F 2t , F 2b , …, F (k+1)n , F (k+1)t , F (k+1)b ]T is the resultant contact force vector of total passive elements in Step 3. W 3,ext = W grav is the external wrench in step 3. G3,act = G(k+2)n and F3,act = F (k+2)n are the layout matrix of the active elements and the corresponding clamping force vector in step 3, respectively. Fig. 3.7 It is worth noting that if the clamp k + 1 in step 2 is a hydraulic or rigid element, the clamping force applied by the clamp k + 1 can be converted to preload in step 3. Thus, the layout matrix of the passive elements in step 3 is G3,pas = [G1 , G2 , …, Gk , G(k+1)b , G(k+1)t ], the passive contact force vector in step 3 is F3,pas = [F T1 , F T2 , …, F Tk , F (k+1)t , F (k+1)b ]T = [F 1n , F 1t , F 1b , …, F kn , F kt , F kb , F (k+1)t , F (k+1)b ]T , the external wrench in step 3 is W 3,ext = W grav , the layout matrix of the active elements in step 3 is G3,act = [G(k+1)n , G(k+2)n ] whereas the corresponding clamping force vector is F3,act = [F (k+1)n , F (k+2)n ]T . By analogy, in the clamping step j shown in Fig. 3.8, the clamp k + j-1 provides a clamping force of F k+j-1 for the workpiece. The clamp k + j-1 is an active element whereas the locators 1, 2, …, k and clamps k + 1, k + 1, …, k + j-2 are passive elements. Therefore, the static equilibrium equation in Step j is generally stated as G j,pas F j,pas + G j,act F j,act + W j,ext = 0

(3.6)

where Gj,pas = [G1 , G2 , …, Gk , …, Gk+j-2 ] is the layout matrix of the passive elements in step j. Fj,pas = [F T1 , F T2 , …, F Tk , …, F Tk+ j−2 ]T = [F 1n , F 1t , F 1b , F 2n , F 2t , F 2b , …, F (k+j-2)n , F (k+ j-2)t , F (k+ j-2)b ]T is the passive contact force vector at the passive element

Fig. 3.7 Clamping step 3

90

3 Analysis of Workpiece Stability

Fig. 3.8 Clamping step j



W grav , 1 ≤ j ≤ n + 1 is the external wrench in step j. W grav + W mach , j = n + 2 Gj,act = G(k+j-1)n and Fj,act = F (k+j-1)n are respectively the layout matrix of the active element exerted in step j and the corresponding clamping force vector. Here, if the clamps k + 1, k + 2, …, k + j-1 are the constant force elements in step j, for example, the hydraulic elements, the layout matrix of the passive elements in step j is Gj,pas = [G1 , G2 , …, Gk , G(k+1)b , G(k+1)t , …, G(k+j-2)b , G(k+j-2)t ], the passive contact force vector in step j is Fj,pas = [F T1 , F T2 , …, F Tk , F (k+1)t , F (k+1)b , …, F (k+j-2)b , F (k+j-2)t ]T = [F 1n , F 1t , F 1b , F 2n , F 2t , F 2b , …, F (k+1)t , F (k+1)b , …, F (k+j-2)b , F (k+j-2)t ]T ,

W grav , 1 ≤ j ≤ n + 1 the external wrench in step 3 is W j,ext = , the layout W grav + W mach , j = n + 2 matrix of the active elements in step j is Gj,act = [G(k+1)n , G(k+2)n , …, G(k+j-1)n ], the clamping force vector is F3,act = [F (k+1)n , F (k+2)n , …, F (k+j-1)n ]T , respectively.

in step j. W j,ext =

3.1.2 Friction Cone Constraints As shown in Fig. 3.9, in order to prevent the workpiece detachment from the fixels in the fixturing process, the normal forces at any contact point between the workpiece and fixel must be in compression such that

3.1 Modeling of Workpiece Stability

91

Fig. 3.9 Friction cone

⎧ ⎪ ⎨ k, j = 1 Fin ≥ 0, 1 ≤ i ≤ k + j − 1, 2 ≤ j ≤ n + 1 ⎪ ⎩ k + n, j = n + 2

(3.7)

Furthermore, the resultant of normal and frictional forces at any contact point must also lie within the friction cone to prevent the workpiece from slipping (Kang 2001; Kang et al. 2003). In view of Coulomb’s Friction Law shown in Fig. 3.9, it follows ⎧ ⎪ ⎨ k, j = 1 2 2 2 (3.8) (Fit ) + (Fib ) ≤ (μi Fin ) , 1 ≤ i ≤ k + j − 1, 2 ≤ j ≤ n + 1 ⎪ ⎩ k + n, j = n + 2 where μi is the friction coefficient between the workpiece and the i-th fixel.

3.1.3 Analysis Model In the fixture design, stability analysis is concerned with the evaluation of the workpiece static equilibrium and friction constraints under given fixturing conditions and machining forces (Wang and Pelinescu 2003; Roy and Liao 2002). Thus, in clamping sequence j + 1, the necessary and sufficient conditions for a workpiece to be stable are obtained as

92

3 Analysis of Workpiece Stability

G j,pas F j,pas + G j,act F j,act + W j,ext = 0 s.t. 

Fi n ≥ 0 (Fi t )2 + (Fi b )2 ≤ (μi Fi n )2

(3.9)

The workpiece stability without friction is a special case of the workpiece stability with friction. In other words, if  

ti = 0 bi = 0

(3.10)

Fi t = 0 Fi b = 0

(3.11)

By substituting Eqs. (3.10, 3.11) into Eq. (3.9), the workpiece stability model can be simplified as G j,pa F j,pa + G j,ac F j,ac + W j,ext = 0 s.t. F j,pa ≥ 0, F j,ac ≥ 0

(3.12)

where G j,pa = [G1n , G2n , …, Gkn , …, G(k+j-2)n ], F j,pa = [F 1n , F 2n , …, F kn , …,  Wgrav , 1 ≤ j ≤ n + 1 T , and G j,ac = G(k+j-1)n , F j,ac = F (k+j-2)n ] , W j,ext = Wgrav + Wmach , j = n + 2 F (k+j-1)n . Specially, if the clamp k + 1 in step j is a constant force element such as a hydraulic element, then G j,pa = [G1n , G2n , …, Gkn ], F j,pa = [F 1n , F 2n , …, F kn ]T ,  Wgrav , 1 ≤ j ≤ n + 1 , G j,ac = [G(k+1)n , G(k+2)n , …, G(k+j-1)n ] and W j,ext = Wgrav + Wmach , j = n + 2 F j,ac = [F (k+1)n , F (k+2)n , …, F (k+j-1)n ]T . The stability analysis in fixture design is to evaluate and analyze the static equilibrium of workpiece under certain fixturing and machining mode (Trappey and Liu 1992; Liu and Strong 2003). Therefore, if Eqs. (3.9) or (3.12) is satisfied, the workpiece is said to be stable. In other words, this implies that at least a solution of Eqs. (3.9) or (3.12) exists. From the mechanistic viewpoint, the workpiece stability can be classified as locating, clamping and machining stability in detail. Such a classification is as follows: When j = 1, the workpiece is only affected by its own gravity. If Eqs. (3.9) or (3.12) has solutions for j = 1, the workpiece is of locating stability. When 2 ≤ j ≤ n + 1, the workpiece is affected by clamping force as well as gravity. If Eqs. (3.9) or (3.12) has solutions for 2 ≤ j ≤ n + 1, the workpiece is of clamping stability.

3.1 Modeling of Workpiece Stability

93

When j = n + 2, the workpiece is affected by both clamping force and machining force as well as gravity. If Eqs. (3.9) or (3.12) has solutions for j = n + 2, the workpiece is of machining stability.

3.2 Solution Techniques In principle, as long as it is judged that there is a solution to the passive contact force in the stability model, it can be shown that the workpiece is in a equilibrium state without further solving the size of the passive contact force. In other words, the analysis of workpiece stability can be transformed into the judgment of the solution existence of the passive contact force (Kang et al. 2003; Hurtado and Melkote 2002; Liang et al. 2019).

3.2.1 Linear Programming Techniques A set of linear simultaneous equations with bj ≥ 0 (1 ≤ j ≤ s) is considered as follows ⎧ ⎪ a x + a12 x2 + · · · + a1 j x j + · · · + a1r xr = b1 ⎪ ⎪ 11 1 ⎪ ⎪ ⎪ ⎪ a21 x1 + a22 x2 + · · · + a2 j x j + · · · + a2r xr = b2 ⎨ ······ ⎪ x a i1 1 + ai2 x 2 + · · · + ai j x j + · · · + air xr = b j ⎪ ⎪ ⎪ ⎪ · ····· ⎪ ⎪ ⎩ as1 x1 + as2 x2 + · · · + as j x j + · · · + asr xr = bs s.t. x 1 , x 2 , · · · , xr ≥ 0

(3.13)

Mathematically, the solution existence of Eq. (3.13) can be examined equivalently by solving the following linear programming problem max w = c1 x1 + c2 x2 + · · · + c j x j + cr xr s.t. ⎧ a x + a12 x2 + · · · + a1 j x j + · · · + a1r xr ≤ b1 ⎪ ⎪ ⎪ 11 1 ⎪ ⎪ a x + a22 x2 + · · · + a2 j x j + · · · + a2r xr ≤ b2 ⎪ ⎪ 21 1 ⎪ ⎪ ······ ⎨ ai1 x1 + ai2 x2 + · · · + ai j x j + · · · + air xr ≤ bi ⎪ ⎪ ⎪ ······ ⎪ ⎪ ⎪ ⎪ x a ⎪ s1 1 + as2 x 2 + · · · + as j x j + · · · + asr xr ≤ bs ⎪ ⎩ x 1 , x 2 , · · · , xr ≥ 0

(3.14)

94

3 Analysis of Workpiece Stability

where c j =

s

ai j (1 ≤ j ≤ r) is the sum of all elements in column j of coefficient i=1 ⎡ ⎤ a11 a12 · · · a1 j · · · a1r ⎢a a ··· a ··· a ⎥ ⎢ 21 22 2j 2r ⎥ ⎢ . . . . . . ⎥ ⎢ . . . . . . ⎥ ⎢ . . . . . . ⎥ matrix A = ⎢ ⎥. ⎢ ai1 ai2 · · · ai j · · · air ⎥ ⎢ ⎥ ⎢ .. .. .. .. .. .. ⎥ ⎣ . . . . . . ⎦ as1 as2 · · · as j · · · asr It is important to note that equalities in Eq. (3.13) are all relaxed into inequalities in Eq. (3.14). Theoretically, it proves that a solution satisfying Eq. (3.13) exists if and only if max(w) =

s 

bi

(3.15)

i=1

where bi (1 ≤ i ≤ s) is the i-th element in the vector b = [b1 , b2 , …, bs ]T .

3.2.2 Solution to the Model of Form-Closure If the positions and orientations of contact points on the workpiece are given in addition to the external loads, the existence of the solution is determined for Eqs. (3.9) or (3.12). This kind of workpiece stability belongs to form closure (Wu et al. 1997; Asada and By 1985; Asada and Kitagawa 1989) which is called the force existence.

3.2.2.1

Form Closure Analysis Without Friction

Under the condition that the external loads such as gravity and cutting force are known, the analysis model of force existence without friction can be obtained according to Eq. (3.12), i.e., G j,pa F j,pa + G j,ac F j,ac = −W j,ext s.t. F j,pa ≥ 0, F j,ac ≥ 0

(3.16)

Because any element –W u (1 ≤ u ≤ 6) in –W j, ext can be positive or negative, it is not in accordance with the constraint condition, –W u ≥ 0, of Eq. (3.13). If the sign function of –W u is

3.2 Solution Techniques

95

 sgn( − Wu ) =

1, −Wu ≥ 0 ,1 ≤ u ≤ 6 −1, −Wu < 0

(3.17)

Then the matrix form of Eq. (3.17) can be described as sgn( − W j,ext ) = [sgn( − W1 ), sgn( − W2 ), · · · , sgn( − W6 )]T

(3.18)

Denote now a diagonal function of sgn (–W j,ext ) is ⎡ ⎢ ⎢ ⎢ ⎢ diag(sgn( − W j,ext )) = ⎢ ⎢ ⎢ ⎣

⎤ sgn( − W1 ) 0 0 0 0 0 ⎥ 0 0 0 0 0 sgn( − W2 ) ⎥ ⎥ 0 0 0 0 0 sgn( − W3 ) ⎥ ⎥ ⎥ 0 0 0 0 0 sgn( − W4 ) ⎥ ⎦ 0 0 0 0 0 sgn( − W5 ) 0 0 0 0 0 sgn( − W6 )

(3.19) If two members of Eq. (3.16) are simultaneously multiplied by diag(sgn( − W j,ext )), then Eq. (3.16) can be transformed as AX = Y s.t. X ≥0

(3.20)

where A = diag (sgn (–W j,ext )[Gj,pa , Gj,ac ], X = [F Tj,pa , F Tj,ac ]T and Y = [diag (sgn (–W j,ext )] (–W j,ext ). Now, a comparison between Eqs. (3.20) and (3.13) shows that both systems have the same form. Therefore the following linear programming can be solved max Q Nonf = C1 X 1 + C2 X 2 + · · · + Ck+ j−2 X k+ j−2 s.t. ⎧ ⎪ A11 X 1 + A12 X 2 + · · · + A1(k+ j−2) X k+ j−2 ≤ Y1 ⎪ ⎪ ⎪ ⎪ ⎨ A21 X 1 + A22 X 2 + · · · + A2(k+ j−2) X k+ j−2 ≤ Y2 ······ ⎪ ⎪ ⎪ A 61 X 1 + A62 X 2 + · · · + A6(k+ j−2) X k+ j−2 ≤ Y6 ⎪ ⎪ ⎩ X 1 , X 2 , · · · , X k+ j−2 ≥ 0

(3.21)

where Y u (1 ≤ u ≤ 6) is the u-th element in vector Y, Auv (1 ≤ u ≤ 6, 1 ≤ v ≤ k + j-2) 6 Auv . is the element of the u-th row and the v-th column in matrix A, and Cv = u=1

It turns out that Eq. (3.16) has solutions provided that the following criterion holds

96

3 Analysis of Workpiece Stability

max(Q Nonf ) =

6 

Yu

(3.22)

u=1

Therefore, the contact forces supplied by clamps (i.e., clamping forces) have solutions as well as the contact forces at the locators (i.e., supporting reaction forces). It shows that the clamping forces can be applied on the workpiece at the given clamping placements. 6 Here, max(QNonf ) and Yu are defined as the internal force measurement and u=1

the external force measurement. Thus, if and only if the internal force measurement is equal to the external force measurement, namely the existence index is I exist = 6 max(Q Nonf ) − Yu = 0, Eq. (3.16) has solutions. u=1

3.2.2.2

Form Closure Analysis with Friction

By analogy, it is known from Eq. (3.9) that, if there exists friction between the workpiece and the fixels, the analysis model of force existence can be described as G j,pas F j,pas + G j,act F j,act = −W j,ext s.t. Fi n ≥ 0, (Fi b )2 + (Fi t )2 ≤ (μi Fi n )2

(3.23)

Obviously, Eq. (3.23) is quadratic in terms of contact forces, and both F ib and F it may take positive or negative values, Eq. (3.23) cannot be solved directly by using Eq. (3.14). To circumvent this difficulty, the quadratic inequality (F it )2 + (F ib )2 ≤ (μi F in )2 in Eq. (3.23) will be approximately linearized and tangential forces F ib and F it will be substituted with non-negative variables. As shown in Figs. (3.10 and 3.11), the friction cone is approximated by a polyhedron whose approximation accuracy can be improved with the increase of the plane number. Figure 3.11 is a projection of 4q (q is a natural number) sided polyhedral cone, with α s being an inclination angle of the line perpendicular to side s. Thus αs =

π π + (s − 1), 1 ≤ s ≤ 4q 4 2q

(3.24)

with 0 ≤ αs ≤ 2π . So, any plane of a 4q-sided polygon can be described by the equation μi Fin − Fit cos αs − Fib sin αs = 0

(3.25)

3.2 Solution Techniques

97

Fig. 3.10 Friction cone linearized by inscribed polyhedron

Fig. 3.11 Friction cone linearized by circumscribed polyhedron

Consequently, the friction cone (F it )2 + (F ib )2 ≤ (μi F in )2 can be approximated as μi Fin − Fit cos αs − Fib sin αs ≥ 0

(3.26)

with 1 ≤ s ≤ 4q. Equation (3.26) can be further described by matrix form as H i Fi ≥ 0 with the i-th circumscribed polyhedron matrix being Fig. 3.10

(3.27)

98

3 Analysis of Workpiece Stability

⎡

μi ⎢ μi Hi = ⎢ ⎣··· μi

− cos α1 − cos α2 ··· − cos α4q

⎤ − sin α1 − sin α2 ⎥ ⎥ ⎦ ··· − sin α4q

(3.28)

Obviously, the more the tangent planes of the circumscribed polyhedron are, the higher the approximation degree is. Thus, the relative error of the friction cone linearized by a circumscribed polyhedron is   π 4q tan −1 Er = π 4q

(3.29)

If an inscribed regular polyhedron with 4q planes is used to approximate the friction cone, the linear equation of the friction cone can be expressed as 

π μi Fin cos 4q

 − Fit cos αs − Fib sin αs ≥ 0

(3.30)

where 1 ≤ s ≤ 4q, and the i-th inscribed polyhedron matrix being ⎡

 

π  4q  ⎢ ⎢ π ⎢ μi cos 4q ⎢

μi cos

− cos α1 − sin α1

⎤

⎥ ⎥ − cos α2 − sin α2 ⎥ Hi = ⎥ ⎥ ⎢··· ··· ⎦ ⎣   ··· π μi cos 4q − cos α4q − sin α4q

(3.31)

As done conventionally in linear programming, tangential force components of F ib and F it will be represented as a difference between two non-negative variables with  Fit = u1i − v1i (3.32) Fib = u2i − v2i where u1i , u2i , v1i , v2i ≥ 0. By substituting Eqs. (3.26, 3.30, 3.32) into Eq. (3.23), the workpiece stability model with friction may be converted into the following equation        diag sgn −W j,ext G F = diag sgn −W j,ext −W j,ext s.t. F ≥ 0, h F ≥ 0 where

(3.33)

3.2 Solution Techniques

99

  G = G1 , G2 , · · · , Gk+ j−1

(3.34)

is the extended layout matrix of passive elements, G i = [G i n , G i t , −G i t , G i b , −G i b ], 1 ≤ i ≤ k + j − 1

(3.35)

is the layout matrix of the i-th fixel, T  T T T F = F 1 , F 2 , · · · , F k+ j−1

(3.36)

is the extended contact force vector in step j, F i = [Fi n , u1i , v1i , u2i , v2i ]T

(3.37)

is the extended contact force vector of the i-th fixel,   h = diag H 1 , H 2 , · · · , H k+ j−1

(3.38)

is the extended polyhedron matrix, and ⎤ ⎧⎡ μi − cos α1 cos α1 − sin α1 sin α1 ⎪ ⎪ ⎪ ⎢ μi − cos α2 cos α2 − sin α2 sin α2 ⎥ ⎪ ⎪ ⎢ ⎥, circumscribed polyhedron ⎪ ⎪ ⎣··· ··· ⎦ ⎪ · · · · · · · · · ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎡ μi − cos  α4q cos α4q − sin α4q sin α4q ⎤ π Hi = μ − cos α cos cos α − sin α sin α i 1 1 1 1 ⎪  4q  ⎢ ⎥ ⎪ ⎪ ⎢ μ cos π − cos α cos α − sin α sin α ⎥ ⎪ ⎪ ⎢ i 2 2 2 2 ⎥ ⎪ 4q ⎪ ⎢ ⎥, inscribed polyhedron ⎪ ⎪ ⎢ ⎥ ⎪ ···   ··· ··· ··· ··· ⎪ ⎣ ⎦ ⎪ ⎪ ⎩ π μi cos 4q − cos α4q cos α4q − sin α4q sin α4q (3.39) is the extended polyhedron matrix of the i-th polyhedron. Let a = diag(sgn(−W j,ext ))G, y = diag(sgn(−W j,ext ))(−W j,ext ) and x = F, the standard form of Eq. (3.33) can be obtained as ax = y s.t. x ≥ 0, hx ≥ 0

(3.40)

100

3 Analysis of Workpiece Stability

Therefore, the force existence with friction can be verified by solving the following linear programming problem with the same criterion. max Q Frict = C TFrict x s.t. ⎧ ⎪ ⎨ ax ≤ y x≥0 ⎪ ⎩ hx ≥ 0 where C Fric = [C1 , C2 , · · · , C5(k+ j−1) ]T with C m =

(3.41)

6

alm , alm is the l-th row and

l=1

the m-column element in matrix a. Thus, if and only if the the internal force measurement max(QFrict ) is equal to the 6 external force measurement yu , namely the existence index is I exist = 0, Eq. (3.23) u=1

has solutions.

3.2.3 Solution to the Model of Force-Closure If the positions and orientations of contact points on the workpiece are given in addition to the external loads and bounds of active clamping forces, the existence of the solution is determined for Eqs. (3.9) or (3.12). This kind of workpiece stability belongs to force closure which is called the force feasibility.

3.2.3.1

Force Closure Analysis Without Friction

If the friction between the workpiece and the fixels is ignored, the force closure model can be deduced by Eqs. (3.12, 3.20), i.e., BX = Z s.t. X ≥0

(3.42)

where B = diag(sgn(-W j,ext -Gj,ac Fj,ac )Gj,pa , Z = diag(sgn(-W j,ext -Gj,ac Fj,ac )(-W j,ext Gj,ac Fj,ac ) and X = Fj,pa . If Fj,pa in Eq. (3.42) has solutions, the workpiece can resist the external load W j,ext to be in equilibrium under the action of clamping force Fj,ac . Thus, Eqs. (3.43, 3.44) can be used to analyze the feasibility of clamping forces in Eq. (3.42)

3.2 Solution Techniques

101

max qNonf = DTNonf X s.t. 

where the j-th element, D j =

6

BX ≤ Z X ≥0

(3.43)

Bi j , in vector DNonf is the sum of all elements in

i=1

column j of matrix B.

6

Here, define Ifeas = max(qNonf ) −

Z i as the fesibility index. And then, if

i=1

and only if the internal force measurement max(qNonf ) is same as the external force 6 measurement Z i , namely, the feasibility index is i=1

Ifeas = 0

(3.44)

Equation (3.42) has solutions.

3.2.3.2

Force Closure Analysis with Friction

In connection with Eqs. (3.33 ~ 3.40, 3.9) can be used to obtain the force closure model with friction as bx = z s.t. x ≥ 0, hx ≥ 0

(3.45)

where b = diag(sgn(-W j,ext -Gj,act Fj,act )Gj,pas , z = diag(sgn(-W j,ext -Gj,act Fj,act )(-W j,ext Gj,act Fj,act ) and x = Fj,pas . Likewise, the internal force measurement can be calculated by the following equation max qFric = d TFric x s.t.  bx ≤ z z≥0

(3.46)

102

where the j-th element, d j =

3 Analysis of Workpiece Stability 6

bi j , in vector d Fric is the sum of all elements in

i=1

column j of matrix b. Therefore, if and only if max(qFric ) is equal to the external force measurement 6 z i , Eq. (3.45) has solutions. That is, the feasibility index is I feas = 0, Eq. (3.45) has i=1

solutions. At this time, the workpiece is in a stable state which shows the clamping force Fj,act is feasible.

3.3 Numerical Examples In this section, several examples are utilized to illustrate the proposed method. Different loading conditions are considered. Example 1 is used to judge the existence of clamping forces. Example 2 and 3 aim at investigating the effects of the clamping force and clamping sequence upon the stability over the entire machining time, respectively.

3.3.1 Analysis of Force Existence In order to explain the application of the stability analysis model of workpiece simply and clearly, it is assumed that the friction between fixture and workpiece is ignored. As shown in Fig. 3.12, a fixture configuration is designed for a 2D workpiece of 80 mm × 50 mm. Four locators L 1 , L 2 , L 3 , L 4 and one clamp C 5 are set up to restrain the workpiece. The weight of the workpiece is 50 N. The clamping force supplied for

Fig. 3.12 Fixturing layout with four locators and one clamp

3.3 Numerical Examples

103

the workpiece by clamp C 5 is 60 N to 6000 N. Coordinates of five fixture elements and their normal unit vectors are given in Table 3.2. Above all, Based on Eq. (3.12), the problem about workpiece stability is stated in a standard form ⎧ ⎨ F4n − F5n = 0 F + F2n + F3n = 50 ⎩ 1n 70F1n + 30F2n + 10F3n − 40F4n + 10F5n = 2000 s.t. 

F1n , F2n , F3n , F4n , F5n ≥ 0 60 ≤ F5n ≤ 6000

(3.47)

And then, the stability model of workpiece can be converted the following linear programming problem in light of Eq. (3.20), i.e., min w1 = −w = −(71F1n + 31F2n + 11F3n − 39F4n + 9F5n ) s.t. ⎧ ⎪ F4n − F5n ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎨ F1n + F2n + F3n ≤ 50 70F1n + 30F2n + 10F3n − 40F4n + 10F5n ≤ 2000 ⎪ ⎪ ⎪ −F 1n , −F2n , −F3n , −F4n , −F5n ≤ 0 ⎪ ⎪ ⎩ 60 ≤ F ≤ 6000 5n

(3.48)

The function linprog in MATLAB toolbox can be used to solve Eq. (3.48). Analysis results of the workpiece stability are shown in Table 3.3. Because the existence index I exist is not 0 but -7.5, the workpiece is unstable. In other words, the clamping force F 5n has no solution in the range from 60 to 6000. Table 3.2 Coordinates and unit normal vectors of fixels Fixel

L1

L2

L3

L4

C5

Coordinate

[70,0]T

[30,0]T

[10,0]T

[0,40]T

[80,10]T

Unit normal vector

[0,1]T

[0,1]T

[0,1]T

[1,0]T

[–1,0]T

Table 3.3 Stability analysis Contact force/N L1

L2

L3

L4

C5

50

0

0

52.5

60

Internal force measurement

External force measurement

Existence index

2042.5

2050

– 7.5

104

3 Analysis of Workpiece Stability

3.3.2 Analysis of Force Feasibility Without Friction In order to hold the workpiece, the designed fixture is shown in Fig. 3.13, in which L 1 , L 2 , L 3 , L 4 are four locators whereas C 1 , C 2 are two clamps supplied clamping forces F 5n , F 6n for the workpiece. The workpiece has a weight of 50 N. The workpiece is subject to an external machining force Fmach of [–150 N,– 100 N]T in the position rmach = [0.5 mm, 1 mm]T . The gravity center of the workpiece is located at rgrav = [–.5 mm, –0.5 mm]T . The positions and the corresponding unit normal vectors of each fixels are listed as Table 3.4. In this case, two different clamping sequences shown in Table 3.5 are investigated here. The function linprog of MATLAB toolbox is used to solve the linear programming problem (3.43). Stability analysis results are shown in Tables 3.6 and 3.7. Effects of the clamping sequence on workpiece stability are shown in this example. As shown in Figs. 3.14 and 3.15, in clamping sequence A, the workpiece is said to be strongly stable in step 2 and step 3 as equality I feas = 0 is valid all the time for arbitrary positive values of F 6n and F 5n . However, the workpiece is said to be weakly stable in step 2 of clamping sequence B provided that F 5n varies within the closed interval [0,75]. Otherwise, the workpiece

Fig. 3.13 Fixturing layout with four locators and two clamps

Table 3.4 Position and unit normal vector of each fixel Fixel

L1

L2

L3

L4

C5

C6

Coordinate

[–3.5,0]T

[–3,–2]T

[–1,–2]T

[0,–2]T

[0.5,–1]T

[–1.5,1]T

Unit normal vector

[1,0]T

[0,1]T

[0,1]T

[0,1]T

[–1,0]T

[0,–1]T

3.3 Numerical Examples

105

Table 3.5 Clamping sequence schemes Clamping step

Clamping sequence A

Clamping sequence B

L1

L1

1 Fgrav

Fgrav

L2

2

L3

L2

L4

L3

L4

F6n L1

Fgrav

L2

Fgrav

L2

3

L2

L2

L1

L1

C6

F6n

L1 Fgrav

C5

F5n

C6

L1

Fgrav

L2

F5n

C5

C6

L1

L2

C5

L1 L2

4

Fmach

C6

L2

L1

Fmach

C6

L1

L1 Fgrav

L2

Fgrav

C5

L2

L1

L2

C5

L2

L1

becomes unstable when F 5n varies within the open interval (75, + ∞). Similarly, the workpiece is found to be strongly stable in step 3 of clamping sequence B no matter how F 6n varies within the interval [0, + ∞).

106

3 Analysis of Workpiece Stability

Table 3.6 Analysis of workpiece stability in clamping sequence A Workpiece stability

Locating stability

Clamping stability

Clamping step

1

2 75]T

Machining stability 3

[0, –50,

75]T

4

[0, –50,

75]T

External force wrench

[0, –50,

[–150, –150, –100]T

External force measurement

125

125

125

400

Internal force measurement

125

125

125

400

Result analysis

Stable

Strongly stable

Strongly stable

Stable

Table 3.7 Analysis of workpiece stability in clamping sequence B Workpiece stability

Locating stability

Clamping stability

Clamping step

1

2 75]T

Machining stability 3

[0, –50,

75]T

4

External force wrench

[0, –50,

External force measurement

125

125

125

400

Internal force measurement

125

125

125

400

Result analysis

Stable

Weakly stable

Strongly stable

Stable

[–150, –150, –100]T

200

400 Internal force measurement External force measurement

350

Objective function/N

Objective function/N

[0, –50,

75]T

300 250

200 150 100 0

10 20

30

40

50 60

70

80

90 100

Clamping force/N

(a) Clamping step 2

180

Internal force measurement External force measurement

160 140 120 100 80 60 0

10

20

30

40

50

60

70

80

90 100

Clamping force/N

(b) Clamping step 3

Fig. 3.14 Stability of clamping sequence A

3.3.3 Analysis of Force Feasibility with Friction A workpiece is shown in Fig. 3.16, which consists of a base block of 120 mm × 80 mm × 50 mm and a semi-ellipsoid top. In accordance with the “3–2-1” principle, six spherical locating pins are placed together with one clamp. Their coordinates,

3.3 Numerical Examples

107 400

180

Internal force measurement External force measurement

160

Objective function/N

Objective function/N

200

140 120 100 80 60 0

10

20

30

40

50

60

70

80

90 100

Clamping force/N

(a) Clamping step 2

Internal force measurement External force measurement

350 300 250 200 150 100

0

10

20

30

40

50

60

70

80

90 100

Clamping force/N

(b) Clamping step 3

Fig. 3.15 Stability of clamping sequence B

Fig. 3.16 The workpiece in a milling process

normal and tangential unit vectors are listed in Table 3.8. The workpiece has a weight of 50 N and considered as the external force in stability analysis. The static friction coefficient between the workpiece and the fixture elements is assumed to be 0.3. The normal clamping force F 7n is prescribed to be 200 N. A keyway milling operation with a feed-rate of 5 mm/s will be performed on the workpiece to produce a through keyway. The instantaneous milling forces (f tx , f ty , f tz ) and a couple mtz defined in the local tool frame x t yt zt are imposed on the workpiece with f tx = 25sin(π t/4) N, f ty = 30 N, f tz = –20 N, and mtz = 800 Nmm, in which the cutting force component f tx is supposed to be time-varying. Suppose the friction cone is approximated by a 4-sided circumscribed polyhedron with q = 1. Now, locating stability, clamping stability and machining stability will be all studied in detail. We utilize linprog of the MATLAB toolbox to solve the linear programming problem (3.46). Results of the first two ones are shown in Table 3.9.

108

3 Analysis of Workpiece Stability

Table 3.8 Coordinates and orientations of fixels Fixel

Cooedinate

Unit normal vector ni

Unit tangential vector t i

Unit tangential vector bi

1

(–30, 50, –50)

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

2

(30, 50, –50)

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

3

(–30, –50, –50)

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

4

(40, 50, –20)

(–1, 0, 0)

(0, 1, 0)

(0, 0, –1)

5

(40, –50, –20)

(–1, 0, 0)

(0, 1, 0)

(0, 0, –1)

6

(0, –60, –20)

(0, 1, 0)

(1, 0, 0)

(0, 0, –1)

( √4 , −

(− √2 , −

7

√ (−20, 30, 15 2)

Table 3.9 Analysis of locating and clamping stability

(

√

5 , √5 4 10 15 )

−

√ 2 5 15 ,

−

70

√6 , 70

√ 3 2 √ ) 70

√ 7 √2 ) 3 7

5 √ , 3 7

−

Workpiece stability

Locating stability

Clamping stability

External force wrench

[0, 0, –50, 0, 0, 0]T

[0, 0, –50, 0, 0, 0]T

External force measurement

50

50

Internal force measurement

50

50

Result analysis

Stable

Stable

Finally, machining stability is discussed. Because the machining force is a timevarying load, a set of linear programming problems is solved at different time increments. Corresponding results are listed in Table 3.10. It turns out that the workpiece is of stability during the entire machining process. Therefore, the placement and magnitude of clamping force F 7n are reasonably chosen.

3.3 Numerical Examples

109

Table 3.10 Analysis of machining stability Machining time/s

External force wrench

External force measurement

Internal force measurement

Result analysis

0

[0, 30, –70, 1200, 0, 800]T

2100

2100

Stable

2

[25, 30, –70, 502.5, 414.6, 2050]T

3092.1

3092.1

Stable

4

[0, 30, –70, 129.2, 0, 800]T

1029.2

1029.2

Stable

6

[–25, 30, –70, –179.4, –649.5, 50]T

1003.9

1003.9

Stable

8

[0, 30, –70, –448.5, 0, 800]T

1348.5

1348.5

Stable

10

[25, 30, –70, –687.4, 739.5, 1050]T

2601.9

2601.9

Stable

12

[0, 30, –70, –900, 0, 800]T

1800

1800

Stable

14

[–25, 30, –70, –1087.4, –739.5, 1050]T

3001.9

3001.9

Stable

16

[0, 30, –70, –1248.5, 0, 800]T

2148.5

2148.5

Stable

18

[25, 30, –70, –1379.4, 649.5, 50]T

2203.9

2203.9

Stable

20

[0, 30, –70, –1470.8, 0, 800]T

2370.8

2370.8

Stable

22

[–25, 30, –70, –1497.5, –414.6, 2050]T

4087.1

4087.1

Stable

24

[0, 30, –70, –1200, 2100 0, 800]T

2100

Stable

References Asada H, By AB. Kinematic analysis of workpiece fixturing for flexible assembly with automatically reconfigurable fixtures [J]. IEEE J Robot Automat. 1985; RA-1(2):86–94. Asada H, Kitagawa M. Kinematic analysis and planning for form closure grasps by robotic hands [J]. Robot Comp Integ Manuf. 1989;5(4):293–9. Chou YC, Chandru V, Barash MM. A mathematical approach to automatic configuration of machining fixtures: analysis and synthesis [J]. Trans ASME J Eng Ind. 1989;111(11):299–306. Hurtado JF, Melkote SN. Modeling and analysis of the effect of fixture-workpiece conformability on static stability. ASME J Manuf Sci Eng. 2002;124:234–41.

110

3 Analysis of Workpiece Stability

Jiang A, Fan QL, Zheng C, Yu KG, Jin S. Stability evaluation of fixture locating layout and research in locator-searching algorithm [J]. J Shanghai Jiaotong Univ. 2010;44(4):484–8 (in Chinese). Kang YZ. Computer aided fixture design verification [D]. Ph.D. Thesis, Worcester Polytechnic Institute, 2001. Kang YZ, Rong YM, Yang JC. Computer-aided fixture design verification. Part 3. Stability analysis [J]. Int J Adv Manuf Technol. 2003;21:842–9. Liang ZP, Zhao CJ, Zhou HW, Zhou YH. Investigation on fixture design and precision stability of new-type double collect for machining of long ladder shaft gear [J]. J Mech Sci Technol. 2019;33(1):323–32. Liao JM, Roy U. Fixturing analysis for stability consideration in an automated fixture design [J]. ASME J Manuf Sci Eng. 2002;124(1):98–104. Liu JJX, Strong DR. Machining fixture verification for nonlinear fixture systems [J]. Int J Adv Manuf Technol. 2003;21:426–37. Qin GH, Zhang WH. Modeling and analysis of workpiece stability based on the linear programming method [J]. Int J Adv Manuf Technol. 2007;32(1–2):78–91. Qin GH, Zhang WH, Wan M. Modeling and application of workpiece stability based on the linear programming method [J]. J Mechan Eng. 2005;41(9):33–37+41. (in Chinese). Qin GH, Wang XY, Wu TJ, Ao ZQ, Xiao J. Prediction of workpiece-fixture contact forces based on multiple clamping [J]. Comput Integr Manuf Syst. 2008;14(7):1421–6 (in Chinese). Roy U, Liao J. Fixturing analysis for stability consideration in an automated fixture design system [J]. Trans ASME J Manuf Sci Eng. 2002;124:98–104. Trappey AJC, Liu CR. An automatic workholding verification system [J]. Robot Comput-Integ Manuf . 1992;9(4–5):321–6. Wang MY, Pelinescu DM. Contact force prediction and force closure analysis of a fixtured rigid workpiece with friction [J]. Trans ASME J Manuf Sci Eng. 2003;125:325–32. Wu NH, Chan KC, Leong SS. Static interactions of surface contacts in a fixture-workpiece system [J]. Int J Comput Appl Technol. 1997;10:133–51. Yan HC, Ahmad S. Kinematic analysis of fixturing systems for robot aided assembly [C]. In: IEEE International Conference on Systems Engineering, 1990, 9–11, pp 499–502.

Chapter 4

Analysis of Clamping Reasonability

Because the workpiece is affected by machining force, inertia force, gravity and other external forces in the machining process, some clamps are used to exert clamping forces to press the workpiece against locators. However, insufficient clamping forces cannot prevent the workpiece from slipping or detaching from locators whereas excessive clamping forces may cause strongly the workpiece deformations and overall workpiece motions. Therefore, besides gravity and machining forces, clamping forces have a significant impact on the machining quality. From a mathematical point of view, the essence of clamping layout scheme is a mapping from clamping source error to clamping deformation, as shown in Fig. 4.1. In order to solve the contradiction between the clamping reliability and the clamping deformation, a clamping reasonability model must be established to describe the relationship among the magnitude of clamping force, the placement of clamping force and the clamping error.

4.1 Local Deformation Mathematically, gravity, machining forces, clamping forces and corresponding moments may be described by a resultant wrench vector W e , as shown in Fig. 4.2a. Under their solicitations, three kinds of deformations will occur, i.e. The first is the fixel (including the locator and the clamp) deformation, as shown in Fig. 4.2b. The second is the contact deformation at the local region between the workpiece and the fixels, as shown in Fig. 4.2c. And the final is the workpiece deformation as illustrated in Fig. 4.2d. However, only the contact deformations at the locators and the locator deformations can lead to the workpiece position error, whereas the contact deformations at the clamps and the clamp deformations have no direct impact on the workpiece position error. © Shanghai Jiao Tong University Press 2021 G. Qin, Advanced Fixture Design Method and Its Application, https://doi.org/10.1007/978-981-33-4493-8_4

111

112

4 Analysis of Clamping Reasonability

Fig. 4.1 Mapping of clamping. source to objective error

We

Yw

Xw

Yw Xw Z

w

Y Z

ni fin

Zw

ηi

Y

fiη

X

fiτ τ i

X

Z

(a) Before deformation

(b) Fixel deforamtion

Yw

Yw

Xw

Xw

Zw

Z

X (c) Contact deformation

Zw

Y

Y

X

Z

(d) Workpiece deformation

Fig. 4.2 Source errors due to workpiece-fixture compliance

As can be seen from Fig. 4.2, the position variation of the workpiece coordinate system X w Y w Z w relative to the global coordinate system XYZ, which is caused by the local deformation consisting of the locator deformations and contact deformations, is a factor of clamping error (Qin et al. 2007). Moreover, the workpiece deformation is another factor of clamping error.

4.1 Local Deformation

113

4.1.1 Contact Deformation When two elastic bodies contact with each other, a point on the boundary of the elastic half plane will generate additional displacement, in addition to the general elastic displacement caused by the internal deformation of the body. This kind of displacement is caused by the local deformation of the surface determined by the surface structure of the elastic body, which is called contact deformation (Gui et al. 1996; Li and Melkote 2001). In dealing with the contact problem of general smooth elastic body, Hertz applies the Bushenisk theory of infinite elastic half space to the finite elastic body according to the fact that the contact area is very small compared with the macroscopic size of the elastic body (Johnson 1985). Thus, the contact problem is successfully solved. Figure 4.3 shows the point contact between the sphere and the sphere, in which the workpiece is spherical at the i-th contact point as well as the fixel. Denote E wi and E fi as the Young’s moduli of the workpiece and fixture, respectively, at the i-th contact point. ν wi and ν fi are Possion’s ratios. Gwi and Gfi are the shear moduli of two contact bodies. Rwi and Rfi are the radii of workpiece and the i-th fixel at the i-th contact point, respectively. When the spherical surface is in contact with the spherical surface, the contact surface of the two objects is a circular region with a radius of r i .  ri =

3Ri∗ 4E i∗

 13

1

( f in ) 3

(4.1)

where f in is the normal contact force at the i-th contact point, 1 1 1 = + Ri∗ Rwi Rfi

(4.2)

ni

Fig. 4.3 Point contact between sphere and sphere

Workpiece ηi

Rwi τi

Rfi

Fixel i

114

4 Analysis of Clamping Reasonability

is the equivalent radius, and 2 1 − vwi 1 − vfi2 1 = + E i∗ E wi E fi

(4.3)

is the equivalent contact Young’s modulus. According to Hertz theory and its extension in the field of contact, the contact stiffness at the i-th contact point in the normal direction ni , tangent directions τ i and ηi are respectively ⎧  1 ⎪ ∗ ∗2 3 ⎪ 16R E 1 ⎪ i i ⎪ kci n = ( fi n ) 3 ⎪ ⎪ 2 ⎪ 1 − v ⎪ wi ⎪ ⎪ ⎨  ∗  13 3Ri 1 kciτ = 8G i∗ ( fi n ) 3 ⎪ ∗ ⎪ 4E i ⎪ ⎪ ⎪ ⎪  ∗  13 ⎪ ⎪ 3Ri 1 ⎪ ⎪ ( fi n ) 3 ⎩ kciη = 8G i∗ ∗ 4E i

(4.4)

1 2 − νwi 2 − νfi = + G i∗ G wi G fi

(4.5)

where

is the equivalent contact shear modulus. In Eq. (4.4), if Rwi or Rfi tends tends to infinity, the contact stiffness of point contact model between plane and sphere can be obtained. Under the condition that the fixel and the workpiece are plane at the i-th contact point, if the workpiece surface is much larger than the surface of the fixel, the contact between them is approximately regarded as a point contact. Thus, the contact area between the workpiece and the fixel is a circle with a radius of afi , as shown in Fig. 4.4. Therefore, the contact stiffness between the fixel i and the workpiece is Fig. 4.4 Point contact between plane and plane

Workpiece ni

ηi τi afi

Fixel i

4.1 Local Deformation

115

⎧ 2E wi ⎪ ⎪ kci n = afi ⎪ ⎪ 1 − vwi2 ⎪ ⎪ ⎪ ⎨ 8G wi afi k = ⎪ ciτ π(2 − vwi ) ⎪ ⎪ ⎪ ⎪ 8G wi ⎪ ⎪ ⎩ kciη = afi π(2 − vwi )

(4.6)

where afi is the radius of the planar-tipped fixel (Li and Melkote 2001). Figure 4.5 is that the contact between the workpiece and the i-th fixel is the contact   , Rwi between the curved surface and the curved surface at the i-th contact point. Rwi and R f i , R f i are the principal radii of the workpiece and the i-th fixel at the i-th contact point, respectively. Then the equivalent radius reads Ri∗ =



Rai Rbi

(4.7)

1 (Ai + Bi ) − (Bi − Ai )

(4.8)

where Rai =

1 (Ai + Bi ) + (Bi − Ai )  1 1 1 1 1 Ai + Bi = +   +  + 2 Rwi Rwi R f i Rfi Rbi =

Fig. 4.5 Point contact between curved surface and curved surface

(4.9)

(4.10)

116

4 Analysis of Clamping Reasonability

Bi − Ai ⎤ 21 ⎡ 2 2     1 1 1 1 1 1 1 1 1 = ⎣ + −  +2 −  cos 2θi ⎦  −   −  2 Rwi Rwi R f i Rfi Rwi Rwi R f i Rfi     θi is the angle between main surfaces containing Rwi and R f i or Rwi and R f i (4.11) An elliptical contact area forms when the workpiece is in contact with the i-th fixel, as shown in Fig. 4.5. So major and minor radii of contact ellipse can be written as ⎧  1   13 ⎪ Rai 3 3Ri∗ ⎪ ⎪ fi n ⎪ ⎨ ai = αi R 4E i∗ bi (4.12)  1   13 ⎪ ⎪ Rbi 3 3Ri∗ ⎪ ⎪ fi n ⎩ bi = αi Rai 4E i∗ Thus, the contact stiffness in the contact coorinate system τ i ηi ni at the i-th contact point can be achieved as ⎧  13  ⎪ 16E i∗2 Ri∗ 1 ⎪ ⎪ kci n = fi n ⎪ ⎪ ⎪ 9 βi ⎪ ⎨ 1 kciτ = 8ai G i∗ ⎪ ⎪ γi ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ kciη = 8ai G i∗ 1 λi

(4.13)

where αi , βi , γi , λi are the correction factors whose expressions are as follows  αi ≈ 1 −  βi ≈ 1 −

Rai Rbi Rai Rbi

1.456

0.0602 −1

(4.14)

1.531

0.0684 −1

    ai γi ≈ 1+ 1.4 − 0.8vi∗ log bi     ai λi ≈ 1+ 1.4 + 0.8vi∗ log bi

(4.15)

(4.16) (4.17)

4.1 Local Deformation

117

with the equivalent Poisson’s ratio νi∗ which can be formulized as 1 1 1 = + vi∗ 2vwi 2vfi

(4.18)

In workpiece fixturing, the contact type between curved surface and curved surface is mainly the point contact between the workpiece with curved surface and the fixel with spherical or planar surface. Their contact stiffness can be easily derived according to the equations from Eq. (4.7) to Eq. (4.18).

4.1.2 Fixel Deformation As shown in Fig. 4.6, K fin , K fiτ and K fiη are assumed to be the shiftiness of the fixel in three directions. According to unit force method, the finite element method can be used to calculate the fixel stiffness. In the global coordinate system XYZ, it is assumed that the fixel is affected by unit force Fi = [F i X , F iY , F i Z ]T in X, Y and Z directions respectively at the contact point, then ⎧ ⎪ ⎨ Fi X = 1 F iY = 1 ⎪ ⎩ Fi Z = 1

(4.19)

The fixel deformations at all notepoints, which are in the global coordinate system XYZ, can directly be obtained by the finite element software, and denote Ufi as the maximum notepoint deformation. Fig. 4.6 Model of fixel

ni

FiY ηi

FiZ

τi

Y Z

X

FiX

118

4 Analysis of Clamping Reasonability

By transforming the fixel deformation Ufi in XYZ to the fixel deformation ufi in τ i ηi ni , the following equation can be obtained U fi = T (i )ufi

(4.20)

with 

T (i ) = ni , τ i , ηi



⎡

⎤ n i x τi x ηi x = ⎣ n i y τi y ηi y ⎦ n i z τi z ηi z

(4.21)

Because Fi = [F i X , F iY , F i Z ]T is the unit force in XYZ, it must be transformed to f i = [ f in , f iτ , f iη ]T in ni τ i ηi , i.e., F i = T (i ) f i

(4.22)

In the global coordinate system XYZ, the relationship between the contact force and deformation of the i-th fixel is F i = K fi U fi ⎡ where Kfi = ⎣

(4.23)

⎤

K fin

⎦. K fiη By substituting Eqs. (4.20, 4.22) into Eq. (4.23), the following relationship can be achieved K fiτ

T (i ) f i = K fi T (i )ufi

(4.24)

By putting Eq. (4.24) in order, the stiffness of the i-th fixel in the contact coordinate system ni τ i ηi can be obtained as kfi = T (i )T K fi T (i ) ⎡ with kfi = ⎣

(4.25)

⎤

kfin kfiτ

⎦.

kfiη Specially, the planar-tipped fixels may be modeled as cantilevered beam elements with a cylindrical cross-section of radius r fi and length L fi (Gou 2000). Thus, the stiffness of the i-th fixel in ni τ i ηi is written as

4.1 Local Deformation

119

⎧ π E fi rfi2 ⎪ ⎪ k = ⎪ fi n ⎪ L fi ⎪ ⎪ ⎪ ⎨ 3π G fi rfi2 kfiτ = ⎪ 4L fi ⎪ ⎪ ⎪ ⎪ ⎪ 3π G fi rfi2 ⎪ ⎩ kfiη = 4L fi

(4.26)

4.1.3 Local Deformation At the i-th contact point, the contact stiffness in the contact area of the workpiece and the locator can modeled as a linear spring in the ni , τ i and ηi direction, in addition to the locator stiffness, as shown in Fig. 4.7. According to Hooke’s law, the relationship between the contact force and the locator deformation at the i-th contact point is ⎧ fi n ⎪ δei n = ⎪ ⎪ ⎪ kfi n ⎪ ⎪ ⎨ f iτ δeiτ = kfiτ ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ ⎩ δeiη = iη kfiη

(4.27)

Again, the relationship between contact force and contact deformation at i-th contact point is as follows ni

ni workpiece

kcin

fin

kfin fiη

kfiτ

fiτ

ηi Locator i

τi

(a) Contact between workpiece and locator

Fig. 4.7 Spring modeling of local stiffness

ηi

kciη

kciτ

kfiη

(b) Equivalent spring model

τi

120

4 Analysis of Clamping Reasonability

⎧ fi n ⎪ δci n = ⎪ ⎪ ⎪ kci n ⎪ ⎪ ⎨ f iτ δciτ = k ⎪ ciτ ⎪ ⎪ ⎪ ⎪ f iη ⎪ ⎩ δciη = kciη

(4.28)

Considering the contact deformation and locator deformation presented above, we can write the overall local deformation terms in each direction at the i-th contact point as follows δei + δci = ki−1 f i , i = 1, 2, . . . , m

(4.29)

where f i = [f in , f iτ , f iη ]T is the contact force at the i-th contact point, δei = [δein , δeiτ , δeiη ]T , δci = [δcin , δciτ , δciη ]T are respectively the locator deformation and the contact deformation. ⎡ ⎤ ki n 0 0 (4.30) ki = ⎣ 0 kiτ 0 ⎦, i = 1, 2, . . . , m 0 0 kiη is the local stiffness at the i-th contact point with ⎧ 1 1 1 ⎪ = + ⎪ ⎪ ⎪ ki n kci n kfi n ⎪ ⎪ ⎨ 1 1 1 = + k k k ⎪ iτ ciτ fiτ ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎩ = + kiη kciη kfiη

(4.31)

4.2 Workpiece Position Error When the workpiece is clamped, the contact area between the workpiece and the locator will produce local deformation including the contact deformation and the locator deformation under the action of clamping force. Thus, the workpiece position will vary with the local deformation. This is so-called the workpiece position error.

4.2 Workpiece Position Error

121

4.2.1 Static Equilibrium Equation Suppose that there are m locators and n clamps in the workpiece-fixture system. The workpiece is subjected the machining force wrench W mach , gravity wrench W grav and the clamping forces. Contacts between the workpiece and fixels are considered as frictional contacts. Assumed that ni is the unit normal vector at the position ri = [x i , yi , zi ]T in which the workpiece is contact with the i-th fixel. Moreover, denote τ i and ηi to be two orthogonal unit tangential vectors of the workpiece at ri . As shown in Fig. 4.8, f i = f in + f iτ + f iη is the contact force at the i-th contact point in the global coordinate system XYZ. f in , f iτ and f iη are respectively the three components of f i along ni , τ i and ηi . Therefore, f in , f iτ and f iη can be expressed as ⎧ ⎪ ⎨ f i n = f i n ni f iτ = f iτ τ i ⎪ ⎩ f iη = f iη ηi

(4.32)

According to the forced status of the workpiece, its static equilibrium equation can easily be obtained as m   i=1

fi ri × f i

Fig. 4.8 Static equilibrium of the workpiece



m+n 

+

j=m+1



 f jn + Wm + Wg = 0 r j × f jn

(4.33)

We

Workpiece ni kiτ

fin fiη

ηi

kiη fiτ

τi Y {GCS} Z

kin ri

X

Fixel i

122

4 Analysis of Clamping Reasonability

In order to express Eq. (4.33) concisely and clearly, it can be rewritten using the matrix form as G 1 f 1 = − G cn f cn − W e

(4.34)

where  G1=

τ1 η 1 · · · nm τm ηm n1 r 1 × n1 r 1 × τ 1 r 1 × η 1 · · · r m × nm r m × τ m r m × η m   ··· nm+n nm+1 G cn = r m+1 × nm+1 · · · r m+1 × nm+n

 (4.35)

(4.36)

T  f cn = f (m+1)n , f (m+2)n , · · · , f (m+n)n

(4.37)

W e =W g + W m

(4.38)

4.2.2 Friction Cone In order to prevent the workpiece from detaching from the i-th fixel in the fixturing process, the contact force between the workpiece and the i-th fixel must be in compression such that f in ≥ 0

(4.39)

On the other hand, based on Coulomb’s Friction Law, the contact force at the i-th fixel must be in the friction cone, i.e., 

f iτ2 + f i2n fi n

≤ μi

(4.40)

or ⎡

⎤T ⎡ 2 ⎤ ⎤⎡ fi n μi fi n ⎣ f iτ ⎦ ⎣ ⎦⎣ f iτ ⎦ ≥ 0 −1 f iη f iη −1 where μi is the friction coefficient between the workpiece and the i-th fixel.

(4.41)

4.2 Workpiece Position Error

123

4.2.3 Relationship Between Local Deformation and Contact Force It is known from Eq. (4.31) that f i = [f in , f iτ , f iη ]T is the i-th contact force. Again, according to Eq. (4.28), the relationship between the local deformations and the contact forces can be obtained as f l = kl (δel + δcl )

(4.42)

T  f l = f 1T , f 2T , · · · , f mT

(4.43)

where

is the contact force vector at locators, kl = diag(k1 , k2 , · · · , km )

(4.44)

is the stiffness matrix of locators, and T  δel + δcl = (δe1 + δc1 )T , (δe2 + δc2 )T , · · · , (δem + δcm )T

(4.45)

is the local deformation vector of locators.

4.2.4 Relationship Between Workpiece Position Error and Local Deformation As shown in Fig. 4.9, the position of the i-th contact point between the workpiece and the i-th locator can be represented equivalently by the following two equations   ri rw , w , riw = rw + T(w )riw

(4.46)

  ri rfi , fi , rif = rfi + T(fi )rif

(4.47)

and

where rw , w and rfi , fi are respectively the position and orientation of the workpiece and the i-th locator with respect to XYZ. riw = [xiw , yiw , z iw ]T is the position vector of the i-th contact point in X w Y w Z w . rif is the position vector of the i-th contact point in X fi Y fi Z fi . T(w ) is the orthogonal coordinate transformation matrix of X w Y w Z w to XYZ. T(fi ) is the orthogonal coordinate transformation matrix of X fi Y fi Z fi to XYZ.

124

4 Analysis of Clamping Reasonability

Fig. 4.9 The fixturing scheme of the workpiece with m locators

Workpiece Yw Xw Zw

ηi

ni

τi

ri w

rw

Yfi

ri

Y

X

rfi

ri

f

Xfi

Zfi

Z Fixel i

Since the workpiece and the i-th locator are always in contact at the i-th contact point, the following relationship can easily be achieved     ri rw , w , riw = ri rfi , fi , rif

(4.48)

As shown in Fig. 4.10, the local deformation δd i = δei + δci at the i-th contact point can cause the contact point on the locator to deviate from its ideal position for the position error δrif (Xiong et al. 2005). It can also cause the contact point on the workpiece to deviate from its ideal position for the position error δriw . By respectively Yw

Workpiece w

X ni

Zw

ηi

rwi

δei

δrif

rw

τi

δdi Y

δci

ri

rif

Yfi

δriw

X Xfi

Z Zfi Locator i Fig. 4.10 Local deformation of the workpiece-fixture system

4.2 Workpiece Position Error

125

Fig. 4.11 Clamping error

Yw

P Xw

Zw Y

rPw

rP X

Z

differentiating Eq. (4.46) and Eq. (4.47) to obtain   δri rw , w , rwi = δrw + δT(w ) rwi + T(w ) δriw

(4.49)

  δri rfi , fi , rfi = T(fi ) δrfi

(4.50)

where δrw and δw are the deviation of the position rw and orientation w of the workpiece, respectively. During the practical fixturing process, in order to ensure that the workpiece can be keep in touch with the fixel, the following condition must be hold     δri rw , w , rwi = δri rfi , fi , rfi

(4.51)

In combination with Eqs. (4.49, 4.50), the workpiece position error can be deduced from the local deformation as Uiw δqw = δdi , i = 1, · · · , m

(4.52)

where δqw = [δrwT , δTw ]T is the workpiece position error resulting from the workpiece-fixture deformation. Uiw = [I3×3 , −T(w )Riw ] ⎡

(4.53)

⎤ −z iw cαw cβw + yiw sαw yiw cαw + z iw sαw cβw Riw = ⎣ z iw cαw cβw − yiw sαw 0 −xiw + z iw sβw ⎦ w w w w −yi cαw − z i sαw cβw xi − z i sβw 0 (4.54) 0

126

4 Analysis of Clamping Reasonability

δdi = T(fi ) rif − T(w ) δriw = T(i )[δdin , δdiτ , δdiη ] T = T(i )[δcin + δein , δciτ + δeiτ , δciη + δeiη ]T = T(i )(δci + δei )

(4.55)

Without loss of generality, X w Y w Z w and XYZ can be always assumed to have the identical orientation, then αw = βw = γw = 0. Thus, the workpiece position error δqw can be further described as Eiw δqw = δdi , i = 1, · · · , m

(4.56)

⎤ 1 0 0 0 z iw −yiw ⎢ ⎣ 0 1 0 −z w 0 x w ⎦, 3D workpiece ⎢ i i ⎢ w Ei = ⎢ 0 0 1 yiw −xiw 0   ⎢ ⎣ 1 0 yiw , 2D workpiece 0 1 −xiw

(4.57)

where ⎡⎡

In order to clearly understand Eq. (4.56), the workpiece position error can be described by matrix form, i.e., Elw δqw = δdl

(4.58)

w T T ) ] Elw = [(E1w )T , (E2w )T , · · · (Em

(4.59)

where

By substituting Eqs. (4.55, 4.42) into Eq. (4.58), the following equation can finally be concluded  + δq w = E lw T (l )(kl )−1 f l

(4.60)

where (Elw )+ is the Moore–Penrose inverse of matrix Elw , and T(l ) = diag(T(1 ), T(2 ), · · · , T(m ))

(4.61)

4.2 Workpiece Position Error

127

4.2.5 Solution Method It is known from Eq. (4.60) that, the contact force f l is the decisive factor of the workpiece position error δqw . Therefore, contact forces f l must be determined firstly. To this end, we can apply the principle of the total complementary energy stating that for all statically admissible forces satisfying equilibrium, the actual state of forces (the one corresponding to kinematically compatible displacements) leads to an extreme value for the total complementary energy ∗ (Pilkey and Wunderlich 1994). The latter has the form of Π ∗ = U ∗ −W ∗

(4.62)

For a workpiece-fixture model described in Eq. (4.60), as the workpiece is assumed to be rigid and contacts between the workpiece and the fixture elements are considered as frictional point-wise contacts with friction coefficient μi , the complementary strain energy is only concerned with deformable fixels U∗ =

1 T −1 f k fl 2 l l

(4.63)

In our case, as the prescribed displacement vector δl is zero, the related potential W * is also zero with W ∗ = f lT δl = 0

(4.64)

Consequently, the contact force f l can be obtained by finding the extreme value of the following problem Find fl min 21 flT kl−1 fl s.t.  G l fl = −G cn f cn − We flT μl fl ≥ 0

(4.65)

where the first set of constraints is the static equilibrium condition of the workpiece, the second set of constraints is the friction cone condition, and   μl = diag μ1 , μ2 , · · · , μm ⎡ μi = ⎣

μi2

(4.66)

⎤ −1

⎦ −1

(4.67)

128

4 Analysis of Clamping Reasonability

4.3 Workpiece Deformation In fact, the workpiece is an elastic deformable body (Lee and Haynes 1987; Menassa and DeVries 1991; Raghu and Melkote 2005). Generally, the contact force and the workpiece deformation can be calculated by the finite element method. Oppositely to Eq. (4.65), we will solve the mathematical programming problem defined by the complementary strain energy of the workpiece Find fw min 21 fwT kw−1 fw s.t.  Gl fl = −Gcn fcn − We flT μl fl ≥ 0

(4.68)

where kw is the workpiece stiffness matrix. f w is the external load vector including the gravity f g , the machining force f m and the clamping force f cn . Equation (4.68) can be used to solve the contact force fl by commercial finite element software. When contact forces are computed at all locators according to Eq. (4.68), the workpiece deformation can be simultaneously derived by Hooke theorem as Uw = kw−1 fw

(4.69)

where U w = [U w1 , …, U wr ]T = [u1 , v1 , w1 , …, ur , vr , wr ]T is the node deformation of the workpiece.

4.4 Modeling of Clamping Reasonability The clamping reasonability is used to measure the machining error resulting from the clamping deformation of the workpiece. If the clamping error can satisfy the specified machining accuracy of the workpiece, the design of the clamping layout is acceptable. As shown in 4.11, it is assumed that the coordinate of process point P on the workpiece in the workpiece coordinate system X w Y w Z w is rPw = [xPw , yPw , z Pw ]T . Again, X w Y w Z w is assumed to be coincident with XYZ, then the clamping error δrP of process point P is obtained as δrp = E pw δqw + δrpw

(4.70)

4.4 Modeling of Clamping Reasonability

129

where δrPw is the workpiece deformation at process point P whose magnitude is approximately equal to the node deformation UwN = [u N , v N , w N ]T (1 ≤ N ≤ r ) closest to the point P, and ⎡

⎤ 1 0 0 0 z Pw −yPw EPw = ⎣ 0 1 0 −z Pw 0 xPw ⎦ 0 0 1 yPw −xPw 0

(4.71)

4.5 Application and Analysis In this section, two numerical tests are used to illustrate the proposed method in checking step by step the validity and performing the analysis of the machining error. The first example will show the effects of deformations of the workpiece-fixture system upon the workpiece machining error. The second one is used to show the whole procedure of computing the clamping error resulting from local deformations of the high-stiffness workpiece.

4.5.1 Analysis of the Clamping Error Due to the Weak Stiffness Workpiece A workpiece-fixture system model consisting of two clamps and two locators is shown in Fig. 4.12. A through slot will be milled along the centerline of the workpiece. Here, the machining error to be evaluated refers to the asymmetry of the slot with respect to the beam neutral axis C D defining the processing datum line. Suppose that the workpiece and fixture elements are deformable. Locators can be simplified as spring with stiffness K and cantilever beam with the flexural stiffness EI, respectively. The workpiece is a hinged beam with the flexural stiffness EI. Studies are focused on the machining error due to the workpiece-fixture compliance. Firstly, a singular function f (x) is defined as follows  f (x) = x − an =

0, x < a (x − a)n , x ≥ a, n ≥ 0

(4.72)

and this function must obey the integration rule as  x − an d x =

x − an+1 + c, n ≥ 0 (c is an arbitrary number) n+1

(4.73)

130

4 Analysis of Clamping Reasonability

F=qL

Y q

Workpiece

Locator 2 nB

nC C

A

D

B

tC

L

X

tB Locator 1

L

L

Fig. 4.12 A workpiece-fixture system

Suppose that all coordinate systems are identical. As the workpiece is in contact with locators at points C and B whose coordinates respectively are [L, 0]T and [2L, 0]T , contact forces can firstly be determined as T  5 1 f = [(fC ) , (f B ) ] = 0, − q L , 0, q L 2 2 T

T T

(4.74)

In this case, if contact deformations are ignored, locator deformations caused by contact forces are then obtained by using Eq. (4.26) T  5q L q L4 , 0, − δel = [δeC , δe B ]T = 0, 6E I 2K

(4.75)

In addition, the workpiece deformation can be directly evaluated at any arbitrary point, x, on the processing datum line C D. δrpw ≈ Uw = [Uwx , Uwy ]T = [0, Uwy ]T

(4.76)

1 5 q Lx − L3 + 12E q Lx − 2L3 − where Uwy = − 8E1 I q L 4 + 8E1 I q L 3 x − 12E I I 1 1 qx − L4 + 24E qx − 2L4 . 24E I I By observing Fig. 4.12, we can obtain the location matrix of locators

4.5 Application and Analysis

131

⎡

1 ⎢ 0 Elw = ⎢ ⎣1 0

⎤ 0 0 1 −L ⎥ ⎥ 0 0 ⎦ 1 −2L

(4.77)

Therefore, from Eq. (4.60), it concludes that the workpiece position error is  δqw = 0,

1 1 5 5 q L4 + qL, q L3 + q 3E I 2K 6E I 2K

T (4.78)

Furthermore, from Eq. (4.70), the workpiece machining error can be finally solved as δr p = [δx, δy]T = [0, δy]T

(4.79)

where δy =

1 q L 3 (x 4E I

− L) −

5 + 12E q Lx − 2L3 − I



5 2K

+

1 qx 24E I

7L 3 24E I

 q(x − L) −

− L4 +

1 qx 24E I

1 q Lx − L3 12E I − 2L4 + 6E1 I q L 4

(x ≥ L) (4.80)

The magnitude changes from point C to point D and attains the maximum value  2

I at point x = 3L − 23L + 5E . 12 KL From the classical theory of the mechanics of materials (Gou 2000), the deflection function vy (x) of the beam can be obtained as 3

1 5 7L q L x 3 − ( 2K + 24E )qx − L1 + v y (x) = 4E1 I q L 2 x 2 − 12E I I , (x ≥ 0) 3 4 5 1 1 q Lx − 2L − 24E I qx − L + 24E I qx − 2L4 12E I

(4.81)

The comparison of Eq. (4.80) with Eq. (4.81) shows that the predicted value obtained by using Eq. (4.70) is exactly the same as the theoretical calculation value at any point of the workpiece.

4.5.2 Evaluation of the Clamping Error of High Stiffness Workpiece As shown in Fig. 4.13, suppose that the workpiece is a solid block with outer dimensions of 220 mm (±12 µm) × 122 mm (±10 µm) × 112 mm (±9 µm). The fixture consists of 2 clamps and 6 locators that are placed following “3–2–1” locating scheme. The coordinates of these fixels, their normal and tangential unit vectors are defined in Table 4.1.

132

4 Analysis of Clamping Reasonability

Fig. 4.13 Scheme of 3–2–1 fixture setup

Milling tool B C

25

A

0.006

L4

Z

C7

L5

L2 L3

Y

C6

L1

X

L0

Table 4.1 Positions and orientations of fixture elements Fixels

Types

Coordinates X i (mm)

Orientation Y i (mm)

Z i (mm)

ni

τi

ηi

L0

Planar

110

10

0

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

L1

Planar

10

110

0

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

L2

Planar

210

110

0

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

L3

Planar

0

60

60

(1, 0, 0)

(0, 1, 0)

(0, 1, 0)

L4

Planar

210

122

60

(0, −1, 0)

(0, 0, 1)

(−1, 0, 0)

L5

Planar

10

122

60

(0, −1, 0)

(0, 0, 1)

(−1, 0, 0)

C6

Planar

110

0

60

(0, 1, 0)

(0, 0, 1)

(1, 0, 0)

C7

Planar

220

60

60

(−1, 0, 0)

(0, 1, 0)

(0, 0, −1)

The workpiece weighs 59.73 N and is made of aluminum with Young’s modulus E w = 70 GPa and Poisson’s ratio ν w = 0.334. Planar-tipped locators and clamps with a tip radius of 9 mm are made of hardened steel with E f = 207 GPa and ν f = 0.292. Clamping forces of 640 N and 670 N are applied simultaneously by clamp C 6 and clamp C 7 . The static friction coefficient is assumed to be 0.25 between the workpiece and fixture elements. A slot milling operation will be performed on the workpiece to produce a through slot with a cutting speed of 100 mm/min. The milling force and torque are estimated to be (-131 N, 232 N, -55 N) and (0, 0, 2.77 Nm), respectively. In this test, the machining error results from local deformations (i.e., clamping error) while the workpiece deformations are neglected. The machining errors δrP (P = A, B and C) of the workpiece will be measured by the displacements at processing datum points A (220,122,112), B (110,122,112), and C (0,122,112). By solving Eq. (4.65), we can evaluate contact forces numerically as shown in Fig. 4.14. Then the workpiece position error δqw is determined by using Eq. (4.60) and the clamping error δr P is computed at points A, B and C by means of Eq. (4.70). In order to understand the variation, error results are listed in Table 4.2 and plotted

4.5 Application and Analysis

133

Fig. 4.14 Prediction results of contact forces

L0 L3

700

L1 L4

L2 L5

Contact force /N

600 500 400 300 200 100 0 0

22

44

66

88

110

132

Machining time /s

Table 4.2 Clamping error Time (s)

Machining error due to deformation (10−3 mm) δrA

δrB

δrC

0

−0.1427, 0.2960, − 0.0231

−0.1427, 0.2588, −0.0634

−0.1427, 0.2216, −0.1037

22

−0.1417, 0.2819, − 0.0189

−0.1417, 0.2561, −0.0627

−0.1417, 0.2304, −0.1064

44

−0.1407, 0.2678, − 0.0149

−0.1407, 0.2535, −0.0621

−0.1407, 0.2391, −0.1093

66

−0.1397, 0.2538, − 0.0108

−0.1397, 0.2508, −0.0614

−0.1397, 0.2477, −0.1121

88

−0.1388, 0.2398, − 0.0066

−0.1388, 0.2481, −0.0608

−0.1388, 0.2563, −0.1149

110

−0.1381, 0.2256, − 0.0017

−0.1381, 0.2452, −0.0597

−0.1381, 0.2647, −0.1178

132

−0.1373, 0.2114, 0.0032 −0.1373, 0.2423, −0.0587

−0.1373, 0.2732, −0.1207

in Fig. 4.15 versus time. Obviously, the maximum machining error caused by the system compliance occurs at point A. Due to the fact that contact forces are dominant factors influencing the workpiece machining error, it is significant to verify experimentally contact forces between the workpiece and fixels. According to Tao’s literature (Tao et al. 1997), modular fixtures were assembled using 6 locators and 2 camps with piezoelectric sensor, as shown in Fig. 4.16. Piezoelectric force sensors can measure the reaction forces on the fixtures during clamping and machining. Signals from the force sensors were amplified by charge amplifiers and recorded by PC. Figure 4.17 is measured results of contact forces obtained by Tao’s experiment. Based on observation of Fig. 4.17, Tao concluded that curve 0 is very close to zero, curves 1 and 2 are quite low in values, curve 3 is almost constant with very small variation, curve 4 constantly decreases

-0.13 A

-0.135

B

C

-0.14 -0.145 -0.15 0

22

44

66

88

110

132

Machining time /s

Clamping error in direction Y/μm

(a) Machining error in direction X due to deformation 0.32 A

0.3

B

C

0.28 0.26 0.24 0.22 0.2 0

22

44

66

88

110

132

Machining time /s

(b) Machining error in direction Y due to deformation Clamping error in direction Z/μm

Fig. 4.15 Prediction results of the workpiece position error

4 Analysis of Clamping Reasonability Clamping error in direction X/μm

134

0.02

A

B

C

0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 0

22

44 66 88 Machining time /s

110

(c) Machining error in direction Z due to deformation

132

4.5 Application and Analysis

135

Z Clamps Y

Workpiece

Locators

Baseplate

X

Fig. 4.16 Experimental instrument

800

Fig. 4.17 Measured contact forces obtained in Tao’s experiment Contact force /N

700 600 500 400 300 200 100 0

20

0

40

60 80 100 Machining time /s

120

140

whereas curve 5 steadily increases, and they intersect with each other at the vicinity of the gravity center of the workpiece. In addition, according to the friction capacity ration, it is necessary to propose the following contact force model and the corresponding finite element method to predict the contact forces (Tan et al. 2004). Finite element analysis results are showed in Fig. 4.18. Find fl = [f1T , f2T , ... , fmT ]T m  fi ×ni Maximize μ fTn i=1

s.t. 

i i

i

(4.82)

Gl flc = −We − Gcn fcn flT μl fl ≥ 0

Compared with the simulation results and the experimental ones, the maximum relative error of the prediction values can be respectively estimated about 10 and 15% except for contact forces at fixel L 1 when the rigid workpiece is in contact with

136

4 Analysis of Clamping Reasonability

Fig. 4.18 Finite element results for Tan’s model

L0 L3

700

L1 L4

L2 L5

Contact force (N)

600 500 400 300 200 100 0 0

17

50

66

83

116

132

Machining time /s

flexible fixels. The prediction results by the proposed model are quite agreeable with the FEM model prediction, the measured data and Tao’s conclusion (Tao et al. 1997). Therefore the predicted accuracy of the local deformations should be 85%.

References Chetaev NG. Theoretical mechanics [M]. Moscow, 1989. Choudhuri SA, DeMeter EC. Tolerance analysis of machining fixture locators [J]. Trans ASME J Manuf Sci Eng. 1999;121:272–81. Chou YC, Chandru V, Barash MM. A mathematical approach to automatic configuration of machining fixtures: analysis and synthesis [J]. Trans ASME J Eng Ind. 1989;111(11):299–306. DeMeter EC, Xie W, Choudhuri S, Vallapuzha S, Trethewey MW. A model to predict minimum required clamp pre-loads in light of fixture-workpiece compliance [J]. Int J Mach Tools Manuf. 2001;41:1031–54. Gou WX. Mechanics of materials [M]. Northwestern Polytechnical University Press, 2000. (in Chinese). Gui XW, Fuh JYH, Nee AYC. Modeling of frictional elastic fixture-workpiece system for improving location accuracy [J]. IIE Trans. 1996;28:821–7. Hong M, Payandeh S, Gruver WA. Modeling and analysis of flexible fixturing systems for agile manufacturing [J]. In: IEEE International Conference on Systems, Man, and Cybernetics, 1996, Vol. 2, pp. 1231–1236. Johnson KL. Contact mechanics [M]. Cambridge University Press, 1985. Kang YZ. Computer Aided Fixture Design Verification [D]. Ph.D. Thesis, Worcester Polytechnic Institute, 2001. Kashyap S, DeVries WR. Finite element analysis and optimization in fixture design [J]. Struc Optim. 1999;18:193–201. Lee JD, Haynes LS. Finite element analysis of flexible fixturing system [J]. Trans ASME J Manuf Sci Eng. 1987;109(134):9. Li B, Melkote SN. Fixture clamping force optimization and its impact on workpiece location accuracy [J]. Int J Adv Manuf Technol. 2001;17(104):13. Menassa RJ, DeVries WR. Opimization methods applied to selecting support positions in fixture design [J]. Trans ASME J Eng Ind. 1991;113:412–8.

References

137

Pilkey WD, Wunderlich W. Mechanics of structures: variational and computational methods [M]. CRC Press, 1994. Qin GH, Zhang WH, Wu ZX, Wan M. Systematic modeling of workpiece-fixture geometric default and compliance for the prediction of workpiece machining error [J]. Trans ASME J Manuf Sci Eng. 2007;129(4):789–801. Raghu A, Ferreira PM. Analysis of the influence of fixture locator errors on the compliance of work part features to geometric tolerance specifications [J]. Trans ASME J Manuf Sci Eng. 2003;125:609–6161. Raghu A, Melkote SN. Modeling of workpiece location error due to fixture geometric error and fixture-workpiece compliance [J]. Trans ASME J Manuf Sci Eng. 2005;127:75–83. Rong Y, Bai Y. Machining accuracy analysis for computer aided fixture design [J]. J Manuf Sci Eng. 1996;118:289–300. Rong Y, Hu W, Kang Y, Yen DW. Locating error analysis and tolerance assignment for computeraided fixture design [J]. Int J Prod Res. 2001;39(15):3529–45. Tan EYT, Kumar AS, Fuh JYH, Nee AYC. Modeling, analysis, and verification of optimal fixturing design [J]. IEEE Trans Autom Sci Eng. 2004;1:121–32. Tao ZJ, Kumar AS, Nee AYC, Mannan MA. Modeling and experimental investigation of a sensorintegrated workpiece-fixture system [J]. Int J Comput Appl Technol. 1997;10:236–50. Trappey AJC, Liu CR. An automatic workholding verification system [J]. Robot Comput Integ Manuf. 1992;9(4.5):321–6. Wang MY. Tolerance analysis for fixture layout design [J]. Assem Autom. 2002;22:153–62. Xiong CH, Wang MY, Tang Y, Xiong YL. On prediction of passive contact forces of workpiecefixture systems [J]. Proc Ins Mechan Eng Part B-J Eng Manuf. 2005;219(B3):309–24. Xiong CH, Ding H, Xiong YL. Fundamentals of Robotic Grasping and Fixturing [M]. World Scientific, 2007.

Chapter 5

Analysis of Workpiece Attachment/Detachment

The attachment and detachment can reflect the possibility of placing a workpiece in the fixturing layout or displacing a workpiece from the fixturing layout (Asada and By 1985). The objective of attachment and detachment analysis is to help the correct selection of fixturing surfaces and fixturing points on workpiece (Xiong 1991, 1994). According to the practical contact/assembly of workpiece with the fixturing layout, hence, Taylor theorem is used to formulate the attachment and detachment model. After equalizing the possibility of of placing a workpiece in the fixture layout or displacing a workpiece from the fixture layout to the solution existence of the attachment and detachment model, a mathematics trick is created to express an arbitrary number in the attachment and detachment model as a difference of two non-negative numbers. Only by doing so can the analysis of the solution existence of the attachment and detachment model be identified as a linear programming problem and in turn, a judgement method is proposed for the workpiece attachment and detachment. Specially under the condition that the attachment and detachment model has solutions, the direction of attachment/detachment of workpiece to/from the fixturing layout is further considered. By equivalently transforming the directionality of attachment and detachment into the general solution of the attachment and detachment model, the pivot algorithm is suggested for solving a set of homogeneous linear inequalities.

5.1 Attachment and Detachment Model Figure 5.1 is the fixturing layout scheme with m locators and n clamps. Its main purpose is to provide clamping forces by clamps to resist the destructive effect of cutting force/cutting torque on the reasonable position of the workpiece relative to the cutting tool obtained by locators. The global coordinate system {XYZ} in Fig. 5.1 is the fixed coordinate system fastened on the machine tool whereas the workpiece coordinate system {xyz} is the © Shanghai Jiao Tong University Press 2021 G. Qin, Advanced Fixture Design Method and Its Application, https://doi.org/10.1007/978-981-33-4493-8_5

139

140

5 Analysis of Workpiece Attachment/Detachment

Fig. 5.1 The fixturing layout with m locators and n clamps

m+1 y m+j z x

m

ri

m+n i R0

1

Y

Ri X

Z

moving coordinate system fixed on the workpiece. Suppose that the workpiece is a rigid body, and its surface is piecewise differentiable. Thus, the equation of the workpiece surface with respect to {xyz} can be expressed as g() = 0

(5.1)

where Ξ = [x, y, z]T is the coordinate of any point on the workpiece relative to {xyz}. Here, if the point Ξ is outside the workpiece, there is g(Ξ ) > 0; if it is inside the workpiece, there is g(Ξ ) < 0. Again, denote R0 = [X 0 , Y 0 , Z 0 ]T as the position of the origin of {xyz} in {XYZ}. Θ 0 = [α 0 , β 0 , γ 0 ]T as the orientation of {xyz} relative to {XYZ} in which α 0 , β 0 and γ 0 are three independent angles. Thus, the coordinate transformation of Ξ = [x, y, z]T into R = [X, Y, Z]T can be expressed as R = T (Θ 0 )Ξ + R0

(5.2)

where T(Θ 0 ) is the orthogonal coordinate transformation matrix with ⎡

⎤ cβ0 cγ0 −cα0 sγ0 + sα0 sβ0 cγ0 sα0 sγ0 + cα0 sβ0 cγ0 T (Θ 0 ) = ⎣ cβ0 sγ0 cα0 cγ0 + sα0 sβ0 sγ0 −sα0 cγ0 + cα0 sβ0 sγ0 ⎦ −sβ0 sα0 cβ0 cα0 cβ0

(5.3)

c is the cosine function cos, and s is the sine function sin (Hirai and Asada 1993; Qin et al. 2017).

5.1 Attachment and Detachment Model

141

If T(Θ 0 )T is denoted to be the transpose matrix of T(Θ 0 ), the following equation can be achieved according to Eq. (5.2), i.e., Ξ = T (Θ 0 )T (R − R0 )

(5.4)

By substituting the above equation into Eq. (5.1), the surface equation of the workpiece in {XYZ} can be described as   g T (Θ 0 )T (R − R0 ) = 0

(5.5)

If Ri = [X i , Y i , Z i ]T means the position of the i-th locator, then the contact of the i-th locator and the the workpiece can mathematically result in   g(Ri ) = g T (Θ 0 )T (Ri − R0 ) = 0

(5.6)

In principle, a new vector q = [X 0 , Y 0 , Z 0 , α 0 , β 0 , γ 0 ]T of {xyz} relative to {XYZ} can be obtained by combining the position vector R0 of {xyz} relative to {XYZ} with the direction vector Θ 0 . Thus, Eq. (5.6) can further be rewritten as g(q, Ri ) = g(Ri ) = 0

(5.7)

In order to describe Eq. (5.7) clearly, gi (q) is denoted as g(q, Ri ) to obtain   gi (q) = g T (Θ 0 )T (Ri − R0 ) = 0

(5.8)

It is known from Eq. (5.8), three position relationships between the i-th locator and the workpiece (that is, outside the workpiece, on the workpiece, and inside the workpiece) respectively corresponds to ⎧ ⎨ gi (q) > 0 g (q) = 0 ⎩ i gi (q) < 0

(5.9)

Denote q* = [X * , Y * , Z * , α * , β * , γ * ]T to be the theoretical position of the workpiece. When the workpiece is in the theoretical position, m locators can always keep touch with it. So there is gi (q* ) = 0 for 1 ≤ i ≤ m. Again, let q = [δ 1 , δ 2 , δ 3 , δ 4 , δ 5 , δ 6 ]T to be an infinitesimal offset of the practical position q of the workpiece from the theoretical position q* , the Taylor principle is relied on to expand Eq. (5.9) as gi (q) = gi (q ∗ ) + ai q where ai = gi (q) to q.



∂gi ∂gi ∂gi ∂gi ∂gi ∂gi , , , , , ∂ X 0 ∂Y0 ∂ Z 0 ∂α0 ∂β0 ∂γ0

(5.10)

T is the gradient vector by differentiating

142

5 Analysis of Workpiece Attachment/Detachment

Because the external normal vector of the workpiece at the i-th locator is T  ni = ∂g∂ir(q) = ∂g∂ix(q) , ∂g∂iy(q) , ∂g∂zi (q) , there exists the following equation under i 0 0 0 the condition of the identical orientation of {xyz} with {XYZ}, namely

ai =

ni r i × ni

 (5.11)

with 1 ≤ i ≤ m. In order to clearly understand Eq. (5.10), it can be further described in matrix form as follows g(q) = g(q ∗ ) + Jq

(5.12)

where J is the locating Jacobian matrix with   J = aT1 , . . . , aiT , . . . , aTm

(5.13)

By combining Eq. (5.9) with Eq. (5.12), the position relationship between the workpiece and the locators should practically be Jq ≥ 0

(5.14)

Jq = 0

(5.15)

It is noteworthy that if and only if

is hold, the locators are in contact with the workpiece. Otherwise, some locators are separated from the workpiece. In other words, if the Eq. (5.14) has a solution but does not satisfy Eq. (5.15), that is, δ 1 , δ 2 , δ 3 , δ 4 , δ 5 , δ 6 in q are not all 0, the workpiece has the attachment and detachment (Liu and Xiong 2003; Xiong et al. 2007).

5.2 Judgment Method of the Attachment and Detachment As above stated, whether the workpiece has the attachment and detachment, it is necessary to judge whether Eq. (5.14) has a solution which is not a solution of Eq. (5.15). In fact, if ζ i (1 ≤ i ≤ m) is denoted as the infinitesimal non-negative number, Eq. (5.14) is equivalent to the following equation Jq − ζ = 0

(5.16)

5.2 Judgment Method of the Attachment and Detachment

143

with ζ = [ζ 1 , …, ζ i , …, ζ m ]T . Because δ i (1 ≤ i ≤ 6) is an arbitrary number, it can be expressed by a difference between two non-negative numbers, i.e., δi = u i − vi

(5.17)

where ui ≥ 0 and vi ≥ 0. Let a = [J, –J, –I m×m ] and x = [uT , vT , ζ T ] T with u = [u1 , u2 , u3 , u4 , u5 , u6 ]T and v = [v1 , v2 , v3 , v4 , v5 , v6 ]T . Thus, Eq. (5.16) can further be described as ax = 0 s.t. x≥0

(5.18)

where I m×m is the m-by-m identity matrix. Accordingly, the solution existence of Eq. (5.16) can be examined equivalently by solving the following linear programming problem max λ = b1 x1 + b2 x2 + ... + br xr s.t. ⎧ ⎪ a11 x1 + a12 x2 + ... + a1r xr ≤ 0 ⎪ ⎪ ⎪ a x + a x + ... + a x ≤ 0 ⎪ ⎨ 21 1 22 2 2r r ... ... ⎪ ⎪ ⎪ am1 x1 + am2 x2 + ... + amr xr ≤ 0 ⎪ ⎪ ⎩ x1 , x2 , ... , xr ≥ 0

(5.19)

where r = 12 + m, x i (1 ≤ i ≤ r) is the i-th element in vector x, aij (1 ≤ i ≤ m, 1 ≤ j r  ai j . ≤ r) is the element of the i-th row and the j-th column in matrix a, and b j = i=1

Here, max(λ) is defined as the position measurement. Therefore, if and only if the position measurement is the same as the position index, namely max(λ) = 0

(5.20)

Equation (5.18) has a solution. Consequently, Eq. (5.14) has also a solution. However, it is necessary to further exclude the solutions which belong to Eq. (5.15) from solutions of Eq. (5.14). Therefore, in (ui -vi ) or -(ui -vi ) for 1 ≤ i ≤ 6, at least one of them is not 0. In other words, in the following twelve equations, at least one of them has a solution, i.e., ax = 0

144

5 Analysis of Workpiece Attachment/Detachment

s.t. x≥0 (u i − vi ) − σi = ηi

(5.21)

or ax = 0 s.t. x≥0 − (u i − vi ) − σi = ηi

(5.22)

where σ i (1 ≤ i ≤ 6) is an arbitrary non-negative number, ηi (1 ≤ i ≤ 6) is a given infinitesimal positive number. In order to clearly express Eqs. (5.21, 5.22), they can further be described as ak x k = bk s.t. xk ≥ 0

(5.23)



T J − J −I m×m 01×m where ak = , xk = [uT , vT , ζ T , σ k ]T , ok ok −ok 01×m −1 ⎤ ⎧⎡ ⎪ ⎪ ⎪ ⎣0, ... , 0, 1, 0, ... , 0⎦, 1 ≤ k ≤ 6 ⎪ ⎪ ⎪

T  ⎨       0 6−k ⎤ ⎡ k−1 and bk = 1×m . ⎪ ηk ⎪ ⎪ ⎪ ⎣0, ... , 0, −1, 0, ... , 0⎦, 7 ≤ k ≤ 12 ⎪ ⎪    ⎩    k−7

=

12−k

By analogy, the solution existence of Eq. (5.23) can be examined equivalently by solving the linear programming problem of Eq. (5.24) max τk = ek,1 xk,1 + ek,2 xk,2 + ... + ek,w xk,w s.t. ⎧ ⎪ ak,11 xk,1 + ak,12 xk,2 + . . . + ak,1w xk,w ≤ φk,1 ⎪ ⎪ ⎪a x + a x + ... + a ⎪ ⎨ k,21 k,2 k,22 k,2 k,2w x k,w ≤ φk,2 ...... ⎪ ⎪ ⎪ ak,t1 xk,1 + ak,t2 xk,2 + . . . + ak,tw xk,w ≤ φk,t ⎪ ⎪ ⎩ xk,1 , xk,2 , . . . , xk,w ≥ 0

(5.24)

5.2 Judgment Method of the Attachment and Detachment

145

where w = m + 13, t = m + 1, x k,i (1 ≤ i ≤ w) is the i-th element in vector xk , ϕ k,i (1 ≤ i ≤ t) is the i-th element in vector bk , ek,ij (1 ≤ i ≤ t, 1 ≤ j ≤ w) is the element t  of the i-th row and the j-th column in matrix ak , and ek, j = ak,i j . i=1

Thus, if and only if the position measurement max(τ k ) is equal to the position index, that is max(τk ) =

t 

φk,i

(5.25)

i=1

Equation (5.23) has a solution (Qin and Zhang 2007; Qin et al. 2005). If Eq. (5.25) is hold for one arbitrary value k from 1 to 12, Eq. (5.23) has a solution. It shows that the workpiece can be mounted into the fixturing layout scheme or dismounted from fixturing layout scheme.

5.3 Analysis Algorithm of the Attachment and Detachment Direction If the workpiece has the attachment or the detachment, along which direction can the workpiece be mounted into the fixturing layout scheme or dismounted from the fixturing layout scheme? This problem involves the solution of Eq. (5.14). In fact, let A = − J T , Eq. (5.14) can be further rewritten as AT q ≤ 0

(5.26)

where A = [a1 , a2 , …, ai , …, am ] is the gradient vector matrix, and the gradient vector ai is the i-th (1 ≤ i ≤ m) column vector in matrix A. Suppose that the general expression of the solution of Eq. (5.26) is as follows q = Vρ + W π

(5.27)

where ρ is an arbitrary real number vector. π is an arbitrary non-negative real number vector. V and W are the base vector matrix of solution (Castillo et al. 1999, 2002). Obviously, the key to obtain the general solution of q is to solve the basis vector matrix V and W. By combining the gradient vectors, a1 , a2 , …, am , in a non-negative linear way, a polyhedral convex cone Aπ can be generated to be Aπ = {x | x = π 1 a1 + π 2 a2 + … + π m am ; π i ≥ 0; 1 ≤ i ≤ m}. By analogy, the gradient vectors, a1 , a2 , …, am , can be linearly combined to generate a linear space Aρ = {y | y = ρ 1 a1 + ρ 2 a2 + … + ρ k ak ; 1 ≤ i ≤ k}. Here, the column vector ai (1 ≤ i ≤ m) is called the generator of the convex cone Aπ and the linear space Aρ . So the coefficient matrix

146

5 Analysis of Workpiece Attachment/Detachment

A is also called the generator matrix of the base vector matrix (Castillo et al. 1999; Castillo and Jubete 2006). The non-positive dual cone of the convex cone Aπ is APπ = { u | AT u ≤ 0} = { u | aiT u ≤ 0; 1 ≤ i ≤ m}. Moreover, for the convex cone Aπ , the generator matrix A can be classified into B and C, i.e., B m = {bi |bi ∈ Aπ , and − bi ∈ Aπ }

(5.28)

  C = c j |c j ∈ Aπ , but − c j ∈ / Aπ

(5.29)

Thus, the general expression of the polyhedral convex cone Aπ consists of a linear space B and a polyhedral convex cone C, i.e., Aπ = B + C. Therefore, (B + C)p is the non-positive dual cone of the cone (B + C), and APπ = {u|biT u = 0, cTj u ≤ 0}

(5.30)

As stated above, in order to obtain the base vector matrix V and W, it is necessary to find the generator u or the generator matrix U of the non-positive dual cone APπ . Here, for simplicity, V and W are respectively called the first base vector matrix and the second base vector matrix.

5.3.1 Calculation Algorithm of the Generator Matrix of the Non-Positive Dual Matrix Denote the row number and the column number of matrix A to be n and m, respectively. The pivot algorithm of solving the base vector matrix V and W is related as follows. Step 1 is to initiate the generator matrix. Initiate the counter h to be h = 1. Initiate the base vector matrices V h , W h to be V h = I n , W h = Ø to obtain the generator matrix U h = [V h , W h ], where I n is the identity matrix of dimension n, and Ø is the empty set. Initiate the record set of the column in U h to be I jh = Ø, where uhj ∈ U h , vhj ∈ V h , whj ∈ W h , 1 ≤ j ≤ n. Step 2 is to determine a pivot column. Calculate the flag vector t h = aTh Uh according to the dot product of ah and U h . Denote the pivot column to be p = { j |t hj = 0, uhj ∈ V h , 1 ≤ j ≤ n} where n is the column number of U h . If p dose not exist, let p = 0.

5.3 Analysis Algorithm of the Attachment and Detachment Direction

Step 3 is to perform the pivot transformation. Perform the pivot transformation by letting

uhj

=

p = 0. Otherwise, go to step 6 if p = 0.



147

uh

− t hj , j = p p

uhj + t hj uph , j = p,1 ≤ j ≤ n

if

Step 4 is to update the base vector matrix.  h+1 = V h − uph V if a h ∈ B. Otherwise, update Update the generator matrix to be h+1 W = Wh  V h+1 = V h − uph the generator matrix to be if a h ∈ B. W h+1 = W h ∪ uph Update the record set of the column to be I jh+1 = I jh ∪ {h} for j = p and 1 ≤ j ≤ n. Step 5 is to judge the termination condition. Update the counter to be h = h + 1 if h < m, and go to Step 2. Otherwise, stop the iterative calculation process. Step 6 is to construct a new base vector. Add h into I jh corresponding to t hj = 0 for 1 ≤ j ≤ n, i.e., I jh = I jh ∪ {h}. wh

h i Construct a new base vector to be wk(i, j) = |tih | +

wh  j  h t j 

for tih < 0 and t hj > 0.

h h h Calculate the record set of the column to be Ik(i, j) = (Ii ∩ I j ) ∪ {h}. h h h h h h Find the maximal proper subset Z ∗ = {wk(i, j) |Ik(i, j)  ⊂ Ik( p,q) , Ik(i, j)  ⊂ Is , ws ∈ wh

h h i W h , tsh = 0}from Z = {wk(i, j) |wk(i, j) = |tih | +

wh  j , t h  h i t j 

Step 7 is to update the second base vector matrix.  Update the second base vector matrix to be W h =

< 0, t hj > 0}.

W h − whj , t hj > 0 according to W h , t hj ≤ 0

t hj > 0. Update the second base vector matrix  h W − wkh , a h ∈ B, aTh wkh < 0 according to ah ∈ B. W h , others

to

be

Wh

=

Update the second base vector matrix to be W h+1 = {W h , Z ∗ } by adding the vectors in Z * into W h . Update the record set of the column to be I jh+1 = I jh , and go to Step 5.

148

5 Analysis of Workpiece Attachment/Detachment

5.3.2 Classification Method of Coefficient Matrix Whether the vector ai in coefficient matrix An×m is the member of matrix B or matrix C, i.e., ai ∈ B or ai ∈ C, it is only to judge whether there exists –ai ∈ Aπ . / Aπ , then ci = ai . It can be judged If –ai ∈ Aπ , then bi = ai ; Otherwise, if –ai ∈ equivalently by examining whether the equation –ai = π 1 a1 + π 2 a2 + … + π i ai + … + π m am for π 1 , π 2 , …, π i , …, π m has a solution. In other words, provided that the equation has a solution, ai ∈ B. But provided that the equation has no solution, ai ∈ C (Qin et al. 2017). Since ai = [a1,i , a2,i , …, an,i ]T , the equation of –ai = π 1 a1 + π 2 a2 + … + π i ai + … + π m am can in detail be expressed as ⎧ a1,1 π1 + a1,2 π2 + · · · + a1,i πi + · · · + a1,m πm = −a1,i ⎪ ⎪ ⎨ a2,1 π1 + a2,2 π2 + · · · + a2,i πi + · · · + a2,m πm = −a2,i ⎪ ······ ⎪ ⎩ an,1 π1 + an,2 π2 + · · · + an,i πi + · · · + an,m πm = −an,i

(5.31)

s.t. π1 , π2 , · · · , πm ≥ 0 Thus, the solution existence of Eq. (5.31) can be examined equivalently by solving the following linear programming problem max i = λ1 π1 + λ2 π2 + · · · + λm πm s.t. ⎧ ⎪ s1 a1,1 π1 + s1 a1,2 π2 + · · · + s1 a1,m πm = −s1 a1,i ⎪ ⎪ ⎪ ⎪ ⎨ s2 a2,1 π1 + s2 a2,2 π2 + · · · + s2 a2,m πm = −s2 a2,i ······ ⎪ ⎪ ⎪ a s n n,1 π1 + sn an,2 π2 + · · · + sn an,m πm = −sn an,i ⎪ ⎪ ⎩π ,π ,··· ,π ≥ 0 1 2 m where λi =

n 

(5.32)

s j a j,i , and sj is the sign value of –aj,i with

j=1

 sj =

1, −a j,i ≥ 0 −1, −a j,i < 0

(5.33)

Accordingly, if and only if the following condition is hold, max(i ) =

n   j=1

 −s j a j,i = −λi

(5.34)

5.3 Analysis Algorithm of the Attachment and Detachment Direction

149

π 1 , π 2 , …, π m in Eq. (5.31) has solutions. It shows that there exists ai ∈ B. However, the following condition max(i ) =

n    −s j a j,i

(5.35)

j=1

shows that π 1 , π 2 , …, π m in Eq. (5.31) has no solution. Therefore, ai ∈ C.

5.3.3 Application of Pivot Algorithm ⎤ x1 ⎢ x2 ⎥ ⎥ Denote A = (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ) and x = ⎢ ⎣ x3 ⎦, where the column vectors x4 ⎡ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ 0 1 0 0 1 2 ⎢ 0 ⎥ ⎢0⎥ ⎢1⎥ ⎢1⎥ ⎢0⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ are a1 = ⎢ ⎣ 0 ⎦, a2 = ⎣ 1 ⎦, a3 = ⎣ 1 ⎦, a4 = ⎣ 0 ⎦, a5 = ⎣ 0 ⎦, a6 = ⎣ 2 ⎦, −1 2 1 1 1 0 ⎡ ⎡ ⎤ ⎤ 1 1 ⎢ 0 ⎥ ⎢ −1 ⎥ ⎢ ⎥ ⎥ a7 = ⎢ ⎣ −3 ⎦ and a8 = ⎣ −1 ⎦. Therefore, a system of linear inequalities with four ⎡

−2 1 variables can be expressed as ⎧ −x4 ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ x1 + x3 + 2x4 ≤ 0 ⎪ ⎪ ⎪ ⎪ x2 + x3 + x4 ≤ 0 ⎪ ⎪ ⎨ x2 + x4 ≤ 0 ⎪ x1 + x4 ≤ 0 ⎪ ⎪ ⎪ ⎪ 2x1 + x2 + 2x3 ≤ 0 ⎪ ⎪ ⎪ ⎪ x1 − 3x3 − 2x4 ≤ 0 ⎪ ⎪ ⎩ x1 − x2 − x3 + x4 ≤ 0 ⎡

(5.36)

⎤ 0 10012 1 1 ⎢ 0 0 1 1 0 1 0 −1 ⎥ ⎥ Since A = ⎢ ⎣ 0 1 1 0 0 2 −3 −1 ⎦ with m = 8 and n = 4, there are obviously −1 2 1 1 1 0 −2 1 B = Ø and C = A. Consequently, the solving process of x in Eq. (5.36) goes as follows in detail.

150

5 Analysis of Workpiece Attachment/Detachment

Step 1 is the initiation of the base vector matrix. Let h = 1, V 1 = I 4×4 = ⎡ ⎤ v1 v1 v1 v1 ⎡ 1 2 3 4⎤ 1000 1 0 0 0 ⎢0 1 0 0⎥ ⎢ ⎥ and W 1 = Ø. So U 1 = ⎢ 0 1 0 0 ⎥ . ⎣0 0 1 0⎦ ⎥ ⎢ ⎣ 0 0 1 0 ⎦ 0001 0 0 0 1 Again, let the column number record set of each column in the generator matrix of dual cone be an empty set, i.e., I j1 = Ø for 1 ≤ j ≤ 4. Step 2 is the determination of pivot column. Since t 1 = aT1 U 1 = [0, 0, 0, −1], then there is p = 4. That is, the corresponding pivot column is u14 . Step 3 is the performance of pivot transformation. Each column in U 1 is listed in Table 5.1 after it is carried out the pivot transformation. / B, the pivot column Step 4 is the update of base vector matrix. Because a1 ∈ u14 is removed from V 1 and added into W 1 to obtain v21 = [0, 1, 0, 0]T , v22 = [0, 0, 1, 0]T , v23 = [0, 0, 0, 1]T and w21 = [0, 0, 0, 1]T . Therefore, V 2 = [v21 , v22 , v23 ], W 2 = w21 , v2 v2 v2 w2 ⎡ 1 2 3 1⎤ 0 0 0 0 ⎥ and U 2 = ⎢ ⎢ 1 0 0 0 ⎥. ⎣ 0 1 0 0 ⎦ 0 0 1 1 And then, h = 1 is added into the set I j2 (j = 1, 2, 3) to obtain I12 = {1}, I22 = {1}, I32 = {1}, and I42 = Ø. Step 5 is the judgement of termination condition. Because h = 1 is less than m = 8, then h should be updated to be h = 2. Since the first element in t 2 = aT2 U 2 = [1, 0, 1, 2] is non-zero value, the column number of the pivot column is p = 1. Therefore, the pivot column is u21 . The pivot transformation is carried out for U 2 to obtain the data in Table 5.2. / B will remove the pivot column u21 from V 2 and add it into The condition of a2 ∈ 3 2 W such that w1 = [−1, 0, 0, 0]T , v31 = [0, 1, 0, 0]T , v32 = [−1, 0, 1, 0]T , and w32 = [−2, 0, w3 v3 v3 w3 ⎡ 1 1 2 2 ⎤ −1 0 −1 −2 ⎥ 0, 1]T . In consequence, V 3 = [v31 , v32 ], W 3 = [w31 , w32 ], and U 3 = ⎢ ⎢ 0 1 0 0 ⎥. ⎣ 0 0 1 0 ⎦ 0 0 0 1 The next is to add h = 2 into the non-pivot column of I j2 (j = 2, 3, 4). Thus, = {1}, = {1, 2}, I33 = {1, 2}, and I43 = {2}. Because h = 2 is less than m = 8, h = 3. At this time it is easy to calculate t 3 = aT3 3 U = [0, 1, 1, 1]. The column number of pivot column is p = 2. The corresponding pivot column is u32 . Thus, U 3 is carried out the pivot transformation to achieve the result shown in Table 5.3. / B, it is necessary to eliminate the pivot column u32 from V 3 and Because a3 ∈ then merge into W 3 . So w41 = [−1, 0, 0, 0]T , w24 = [0, −1, 0, 0]T , v41 = [−1, −

Before transforming

After transforming

u21

1

0

0

0

I12

1

Before transforming

u11

1

0

0

0

I11

Ø

Ø

I21

0

0

1

0

u12

The second column

The first column

Table 5.1 The first pivot transformation

1

I22

0

0

1

0

u22

After transforming

Ø

I31

0

1

0

0

u13

Before transforming

The third column

1

I32

0

1

0

0

u23

After transforming

Ø

I41

1

0

0

0

u14

Before transforming

The fourth column

Ø

I42

1

0

0

0

u24

After transforming

5.3 Analysis Algorithm of the Attachment and Detachment Direction 151

Before transforming

u22

0

After transforming

u13

−1

0

0

0

I13

1

Before transforming

u12

1

0

0

0

I12

1

1

I22

0

0

1

The second column

The first column

Table 5.2 The second pivot transformation

1, 2

I23

0

0

1

0

u23

After transforming

1

I32

0

1

0

0

u32

Before transforming

The third column

1, 2

I33

0

1

0

−1

u33

After transforming

Ø

I42

1

0

0

0

u42

Before transforming

The fourth column

2

I43

1

0

0

−2

u43

After transforming

152 5 Analysis of Workpiece Attachment/Detachment

Before transforming

After transforming

u41

−1

0

0

0

I14

1, 3

Before transforming

u31

−1

0

0

0

I13

1

1, 2

I23

0

0

1

0

u32

The second column

The first column

Table 5.3 The third pivot transformation

1, 2

I24

0 1, 2

I33

0

1

0

−1 0

−1

u33

Before transforming

0

u42

After transforming

The third column

1, 2, 3

I34

0

1

−1

−1

u43

After transforming

2

I43

1

0

0

−2

u34

Before transforming

The fourth column

2, 3

I44

1

0

−1

−2

u44

After transforming

5.3 Analysis Algorithm of the Attachment and Detachment Direction 153

154

5 Analysis of Workpiece Attachment/Detachment

1, 1, 0]T , and w43 = [−2, −1, 0, 1]T . Obviously, V 4 = v41 , W 4 = [w41 , w24 , w43 ], and w4 w4 v4 w4 ⎡ 1 2 1 3 ⎤ −1 0 −1 −2 ⎥ U4 = ⎢ ⎢ 0 −1 −1 −1 ⎥ . ⎣ 0 0 1 0 ⎦ 0 0 0 1 h = 3 is then added into the non-pivot column of I j2 (j = 1, 3, 4) for obtaining I14 = {1, 3}, I24 = {1, 2}, I34 = {1, 2, 3}, and I44 = {2, 3}. The update of h is continued to obtain h = 4. Thus, t 4 = aT4 U 4 = [0, −1, −1, 0]. Because only the column corresponded to the third element t34 in t 4 belongs to V 4 , the column number of the pivot column is p = 3. By pivoting U 4 , each column can be calculated to be listed in Table 5.4. / B, the pivot column u34 should be canceled from V 4 to add into W 4 such Since a4 ∈ 5 that w1 = [−1, 0, 0, 0]T , w52 = [1, 0, −1, 0]T , w53 = [−1, −1, 1, 0]T and w54 = [−2, −1, w5 w5 w5 w5 ⎡ 1 2 3 4 ⎤ −1 1 −1 −2 ⎥ 0, 1]T . As a result, V 5 = Ø, W 5 = [w51 , w52 , w53 , w54 ], and U 5 = ⎢ ⎢ 0 0 −1 −1 ⎥ . ⎣ 0 −1 1 0 ⎦ 0 0 0 1 4 The addition of h = 4 into the non-pivot column of I j (j = 1, 2, 4) to obtain I15 = {1, 3, 4}, I25 = {1, 2, 4}, I35 = {1, 2, 3} and I45 = {2, 3, 4}. When h = 5, t 5 = aT5 U 5 = [−1, 1, −1, −1]. Even though t 5j = 0 for 1 ≤ j ≤ 4, p = 0 because u5j ∈ / V 5 . Again, because there exist t15 < 0, t25 > 0, t35 < 0 and t45 < 0, three new columns can be obtained as w51(2,1) = [0, 0, −1, 0]T , w52(2,3) = [0, −1, 0, 0]T and w53(2,4) = [−1, −1, −1, 1]T . Therefore, these three new vectors can combine into a new set Z = {w51(2,1) , w52(2,3) , w53(2,4) }. Moreover, the corresponding column number 5 5 5 record sets are I1(2,1) = {1, 4, 5}, I2(2,3) = {1, 2, 5}, I3(2,4) = {2, 4, 5}, respectively. Because there are only non-zero elements in vector t 5 , Ish = Ø. On the other hand, 5 5 5 5 ⊂ I2(2,3) and I1(2,1) ⊂ I3(2,4) , so w51(2,1) ∈ Z * . Likewise, it can easily be since I1(2,1) 5 5 * known that w2(2,3) ∈ Z and w3(2,4) ∈ Z * . In other words, the maximal proper subset of Z is Z * = Z. Since t25 > 0, the corresponding column vector w52 must be removed from W 5 to / B, all vectors in Z * are be W 5 = [w51 , w53 , w54 ]. Finally, because there exists a5 ∈ w6 w6 w6 w6 w6 w6 ⎡ 1 2 3 4 5 6 ⎤ −1 −1 −2 0 0 −1 ⎥ added into W 5 to obtain U 6 = ⎢ ⎢ 0 −1 −1 0 −1 −1 ⎥ , as illustrated in Table ⎣ 0 1 0 −1 0 −1 ⎦ 0 0 1 0 0 1 5.5. Simultaneously, the column number record sets are achieved as I16 = {1, 3, 4}, I26 = {1, 2, 3}, I36 = {2, 3, 4}, I46 = {1, 4, 5}, I56 = {1, 2, 5} and I66 = {2, 4, 5}, respectively.

Before transforming

After transforming

u51

−1

0

0

0

I15

1, 3, 4

Before transforming

u41

−1

0

0

0

I14

1, 3

1, 2

I24

0 1, 2, 4

I25 1, 2, 3

I34

0

1

−1 0

−1

0

0

−1

−1

u43

Before transforming

1

u52

After transforming

The third column

0

u42

The second column

The first column

Table 5.4 The fourth pivot transformation

1, 2, 3

I35

0

1

−1

−1

u53

After transforming

2, 3

I44

1

0

−1

−2

u44

Before transforming

The fourth column

2, 3, 4

I45

1

0

−1

−2

u54

After transforming

5.3 Analysis Algorithm of the Attachment and Detachment Direction 155

156

5 Analysis of Workpiece Attachment/Detachment

Table 5.5 The fifth pivot transformation Before transforming

After transforming

w51

w52

w53

w54

w51(2,1)

w52(2,3)

w53(2,4)

−1

1

−1

−2

0

0

−1

0

0

−1

−1

0

−1

−1

0

−1

1

0

−1

0

−1

0

0

0

1

0

0

1

I15

I25

I35

I45

5 I1(2,1)

5 I2(2,3)

5 I3(2,4)

1, 3, 4

1, 2, 4

1, 2, 3

2, 3, 4

1, 4, 5

1, 2, 3

2, 3, 4

w61

w62

w63

w64

w65

w66

−1

−1

−2

0

0

−1

0

−1

−1

0

−1

−1

0

1

0

−1

0

−1

0

0

1

0

0

1

I16

I26

I36

I46

I56

I66

1, 3, 4

1, 2, 3

2, 3, 4

1, 4, 5

1, 2, 3

2, 3, 4

The calculation process goes on with h = 6. Thus, the flag vector is calculated as t 6 = aT6 U 6 = [ −2, −1, −5, −2, −1, −5]. Because there is only w6j in U 6 but no v6j (1 ≤ j ≤ 6), then p = 0. Since there exists only non-zero element in t 6 such that the column number record sets remain unchanged, i.e., I16 = {1, 3, 4}, I26 = {1, 2, 3}, I36 = {2, 3, 4, 5}, I46 = {1, 4, 5}, I56 = {1, 2, 5}, and I66 = {2, 4, 5}. Since there is only negative value in t 6 , so Z = Ø. Clearly, Z * = Ø. Ultimately, w7 w7 w7 w7 w7 w7 ⎡ 1 2 3 4 5 6 ⎤ −1 −1 −2 0 0 −1 ⎥ the relationship of a6 ∈ / B will result in U 7 = ⎢ ⎢ 0 −1 −1 0 −1 −1 ⎥ . ⎣ 0 1 0 −1 0 −1 ⎦ 0 0 1 0 0 1 Since h = 6 is less than m = 8, the calculation process will still be carried out with h = 7. Though t 7 = aT7 U 7 = [ −1, −4, −4, 3, 0, 0], the pivot column does still not exist in U 7 , i.e., p = 0. This is because an arbitrary element in U 7 is not a member of V 7 . Because t57 = 0 and t67 = 0, h = 7 should be added into I56 and I66 such that I17 = {1, 3, 4}, I27 = {1, 2, 3}, I37 = {2, 3, 4}, I47 = {1, 4, 5}, I57 = {1, 2, 5, 7} and I67 = {2, 4, 5, 7}。 w7 w7 7 = |t 71| + |t 74| = [−1, 0, −1/3, 0]T , Since t17 < 0, t27 < 0, t37 < 0 and t47 > 0, then w1(1,4) 1

w72 |t27 | T

w74 |t47 |

4

w7 w7 = + = [−1/4, −1/4, −1/12, 0] , w73(3,4) = |t 73| + |t 74| = [−1/2, −1/4, 3 4 7 7 7 , w2(2,4) , w3(3,4) }. Accordingly, −1/3, 1/4] which can form a new set Z = {w1(1,4) 7 7 7 I1(1,4) = {1, 4, 7}, I2(2,4) = {1, 7} and I3(3,4) = {4, 7}. 7 7 7 7 7 7 It is easily known that I2(2,4) ⊂I1(1,4) and I3(3,4) ⊂I1(1,4) . So w2(2,4) ∈ / Z * and w3(3,4) 7 7 ∈ / Z * . However, I1(1,4) ⊂ I57 , I1(1,4) ⊂ I67 . Therefore, Z * = {w7k(1,4) }.

w72(2,4)

T

5.3 Analysis Algorithm of the Attachment and Detachment Direction

157

Table 5.6 The seventh pivot transformation Before transforming

After transforming

w71

w72

w73

w74

w75

w76

w71(1,4)

−1

−1

−2

0

0

−1

−1

0

−1

−1

0

−1

−1

0

0

1

0

−1

0

−1

−1/3

0

0

1

0

0

1

0

I17

I27

I37

I47

I57

I67

7 I1(1,4)

1, 4, 5

1, 3, 4

1, 2, 3

2, 3, 4

1, 2, 5, 7

2, 4, 5, 7

1, 4, 7

w81

w82

w83

w84

w85

w86

−1

−1

−2

0

−1

−1

0

−1

−1

−1

−1

0

0

1

0

0

−1

−1/3

0

0

1

0

1

0

I18

I28

I38

I48

I58

I68

1, 3, 4

1, 2, 3

2, 3, 4, 8

1, 2, 5, 7

2, 4, 5, 7

1, 4, 7

The condition of t47 > 0 will make w74 to be removed from W 7 such that W 7 is / B, so all vectors in Z * are added updated as W 7 = [w71 , w72 , w73 , w75 , w76 ]. Again, a7 ∈ 7 8 7 * into W to obtain W = {W , Z }. The seventh iterative results are listed in Table 5.6. The calculation process can not be stopped when h = 7. Consequently, h must be updated to h = 8 at which t 8 = aT8 U 8 = [ −1, −1, 0, 1, 2, −2]. It is thus clear that t38 = 0 such that I18 = {1, 3, 4}, I28 = {1, 2, 3}, I38 = {2, 3, 4, 8}, I48 = {1, 2, 5, 7}, I58 = {2, 4, 5, 7}, and I68 = {1, 4, 7}. Because there exist t18 < 0, t28 < 0, t68 < 0 and t48 > 0, t58 > 0 in t 8 , six new vectors w8

w8

w8

w8

can be constructed as w81(1,4) = |t 81| + |t 84| = [−1, −1, 0, 0]T , w82(1,5) = |t 81| + |t 85| 1 4 1 5 w8 w8 w8 = [−3/2, −1/2, −1/2, 1/2]T , w83(2,4) = |t 82| + |t 84| = [−1, −2, 1, 0]T , w84(2,5) = |t 82| 2 4 2 w8 w8 w8 + |t 85| = [−3/2, −3/2, 1/2, 1/2]T , w85(6,4) = |t 86| + |t 84| = [−1/2, −1, −1/6, 0]T , 6

5

w8

w8

4

w86(6,5) = |t 86| + |t 85| = [−1, −1/2, −2/3, 1/2]T , respectively. In sequence, a new 6 5 w81(1,4) w82(1,5) w83(2,4) w84(2,5) w85(6,4) w86(6,5) ⎧⎡ ⎤⎫ ⎤⎡ 3⎤⎡ ⎤⎡ 3⎤⎡ 1⎤⎡ −1 −1 ⎪ −1 −2 −2 −2 ⎪ ⎪ ⎪ ⎨ ⎥ ⎢ 1 ⎥⎬ ⎥⎢ 1⎥⎢ ⎥⎢ 3⎥⎢ set Z = ⎢ ⎢ −1 ⎥ ⎢ − 2 ⎥ ⎢ −2 ⎥ ⎢ − 2 ⎥ ⎢ −1 ⎥ ⎢ − 2 ⎥ can be merged by ⎣ ⎦ ⎣−1 ⎦ ⎣ 1 ⎦ ⎣−1 ⎦ ⎣−1 ⎦ ⎣−2 ⎦⎪ ⎪ ⎪ ⎪ 0 2 2 6 3 ⎭ ⎩ 1 1 1 0 0 0 2 2 2 8 these six constructed vectors. Furthermore, a little effort is poured out to obtain I1(1,4) 8 8 8 8 = {1, 8}, I2(1,5) = {4, 8}, I3(2,4) = {1, 2, 8}, I4(2,5) = {2, 8}, I5(6,4) = {1, 7, 8}, and 8 I6(6,5) = {4, 7, 8}.

158

5 Analysis of Workpiece Attachment/Detachment

Table 5.7 The eighth pivot transformation Before transforming

w81

w82

w83

w84

w85

w86

w83(2,4)

w85(6,4)

w86(6,5)

−1

−1

−2

0

−1

−1

−1

−1/2

−1

0

−1

−1

−1

−1

0

−2

−1

−1/2

0

1

0

0

−1

−1/3

1

−1/6

−2/3

0

0

1

0

1

0

0

0

1/2

I18

I28

I38

I48

I58

I68

8 I3(2,4)

8 I5(6,4)

8 I6(6,5)

1, 2, 5, 7

2, 4, 5, 7

1, 4, 7

1, 2, 8

1, 7, 8

4, 7, 8

1, 3, 4 1, 2, 3 2, 3, 4, 8 After transforming

w91

w92

w93

w94

w95

w96

w97

−1

−1

−2

−1

−1

−1/2

−1

0

−1

−1

0

−2

−1

−1/2

0

1

0

−1/3

1

−1/6

−2/3

0

0

1

0

0

0

1/2

I19

I29

I39

I69

I79

I89

I99

1, 4, 7

1, 2, 8

1, 7, 8

4, 7, 8

1, 3, 4 1, 2, 3 2, 3, 4, 8

Since the column vector w83 corresponded to t38 = 0 is a member of matrix W 8 , it 8 8 8 8 8 ⊂ I3(2,4) , I2(1,5) ⊂ I6(6,5) , I4(2,5) ⊂ is easily obtained I38 = {2, 3, 4, 8}. Because I1(1,4) 8 8 8 8 8 8 8 8 I3(2,4) and I2(1,5) ⊂ I3 , I4(2,5) ⊂ I3 , the elements, w1(1,4) , w2(1,5) , w4(2,5) , corresponded 8 8 8 to I1(1,4) , I2(1,5) , I4(2,5) should be excluded from Z to obtain Z * = {w83(2,4) , w85(6,4) , 8 w6(6,5) }. Again, because there exist t48 > 0 and t58 > 0, the corresponding column vectors 8 w4 and w85 are also excluded from W 8 . Thus, W 8 = [w81 , w82 , w83 , w86 ]. In addition, / B will make all vectors in Z * to add into W 8 so that W 9 = the relationship of a8 ∈ 8 * {W , Z }. The eighth pivot transformation carried out to obtain the results listed in Table 5.7. Eventually, the value of h is the same as the value of m, i.e., h = m = 8, so the ⎤ ⎡ −1 −1 −2 −1 −1 − 21 −1 ⎢ 0 −1 −1 0 −2 −1 − 1 ⎥ 2⎥ calculation process is over. Therefore, W = ⎢ ⎣ 0 1 0 − 1 1 − 1 − 2 ⎦ and V 3 6 3 0 0 1 0 0 0 21

5.3 Analysis Algorithm of the Attachment and Detachment Direction

159

⎡

⎡

−1 ⎢ 0 = Ø. In other words, x = ⎢ ⎣ 0 0

−1 −1 1 0

−2 −1 0 1

−1 0 − 13 0

−1 −2 1 0

≤ i ≤ 7.

− 21 −1 − 16 0

⎤ π1 ⎤⎢ π2 ⎥ ⎥ −1 ⎢ ⎢ ⎥ 1 ⎥⎢ π3 ⎥ − 2 ⎥⎢ ⎥ ⎢ π ⎥ with π i ≥ 0 for 1 − 23 ⎦⎢ 4 ⎥ ⎢ π5 ⎥ 1 ⎢ ⎥ 2 ⎣ π6 ⎦ π7

5.4 Analysis and Application of Attachment and Detachment This section enumerates several typical examples to illustrate the application of the attachment and detachment analysis method for the workpiece in the fixturing layout.

5.4.1 Three Dimensional Workpiece Figure 5.2 shows the front view and top view of the shift fork held by the drill fixture. The shift fork is located by five locators L 1 , L√2 , L 3 , L 4 , L 5 whose coordinates are √ respectively r1 = [0, 0, 25 mm]T , r2 = [− 252 3 mm, 0, −12.5 mm]T , r3 = [ 252 3 √ √ √ T T mm, 0, −12.5 mm] √ , r4 = [−15 2 mm, 10 mm, −15 2 mm] , r5 = [−15 2 normal vectors are √ n1 = mm, 10 mm, 15 2 mm]T . Obviously, the corresponding √ √ √ 2 2 T 2 T T T [0, 1, 0] , n2 = [0, 1, 0] , n3 = [0, 1, 0] , n4 = [ 2 , 0, 2 ] , n5 = [ 2 , 0, − 22 ]T , respectively. Thus, the analysis process of the attachment and detachment for the shift fork is as follows. First, according to Eq. (5.13), the locating Jacobian matrix can be obtained to be ⎡ ⎤ 0 1 0 −25 0 0√ ⎢ 0 1 0 12.5 0 − 25 3 ⎥ ⎢ √2 ⎥ ⎢ ⎥ J = ⎢ √0 1 √0 12.5 0 252 3 ⎥. It is obvious to be a1 = [0, 1, 0, −25, 0, 0]T , √ √ ⎢ 2 ⎥ ⎣ 0 2√2 5 2 0 −5 2 ⎦ √2 √ √ 2 0 − 22 −5 2 0 −5 2 2 √ √ √ √ √ a2 = [0, 1, 0, 12.5, 0, - 252 3 ]T , a3 = [0, 1, 0, 12.5, 0, 252 3 ]T , a4 = [ 22 , 0, 22 , 5 2, √ √ √ √ √ 0, −5 2]T and a5 = [ 22 , 0, − 22 , −5 2, 0, −5 2]T。Therefore, the classification method of coefficient matrix A = [a1 , a2 , a3 , a4 , a5 ] can be depended on to obtain B = Ø and C = A, as shown in Table 5.8. Because the row number and the column number of A are n = 6 and m = 5, respectively. In light of the pivot algorithm, the base vector matrices V and W can be calculated as follows in detail.

160

5 Analysis of Workpiece Attachment/Detachment

Drill bush

Bush plate

Clamp

Bolt

Base plate

L4

L2

L3

V block

L5

L1

Locating sleeve

Shift fork

Fig. 5.2 The drill fixture for a shift fork

Table 5.8 Classification of coefficient matrix

Vector

Parameter max(Σ i )

Classification matrix -λi

-a1

23.0000

26

C

-a2

32.1506

35.1506

C

-a3

32.1506

35.1506

C

-a4

−2.0491e−11

15.5563

C

-a5

−3.8625e−12

15.5563

C

5.4 Analysis and Application of Attachment and Detachment

161

Table 5.9 The non-pivot column after the first pivot transformation Column vector

u1,i

u2,i

u3,i

u4,i

u5,i

u6,i

1

0

0

0

0

0

u21 after transforming

1

0

0

0

0

0

u13 before transforming

0

1

0

0

0

0

u23 after transforming

0

0

1

0

0

0

u14 before transforming

0

0

0

1

0

0

u24 after transforming

0

25

0

1

0

0

u5

u15 before transforming

0

0

0

0

1

0

after transforming

0

0

0

0

1

0

u6

u25 u16 u26

before transforming

0

0

0

0

0

1

after transforming

0

0

0

0

0

1

u1 u3 u4

u11 before transforming

⎡

⎤ 100000 ⎢0 1 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 0 1 0 0 0⎥ h The first step is to initiate the iteration counter to be h = 1, V = ⎢ ⎥ ⎢0 0 0 1 0 0⎥ ⎢ ⎥ ⎣0 0 0 0 1 0⎦ 000001 and W h = Ø. Again, not pivot transformation makes the initial record set of each column in U h = [V h , W h ] is the empty set, i.e., I j1 = Ø for 1 ≤ j ≤ 6. The second step is to calculate the flag vector t 1 = aT1 U 1 = [0, 1, 0, −25, 0, 0]. Therefore, there exists the pivot column in U 1 which is u12 = [0, 1, 0, 0, 0, 0]T . The third step is to carry out the pivot transformation. Thus, the pivot column is updated to be u22 = [0, −1, 0, 0, 0, 0]T , and other columns are listed in Table 5.9. The fourth step is to obtain the generator matrix. h = 1 is added into the non-pivot column I j1 for j = 2 and 1 ≤ j ≤ 6 to obtain I12 = {1}, I22 = Ø, I32 = {1}, I42 = {1},   I52 = {1} and I62 = {1}. Since a1 ∈ / B, then V 2 = v11 v13 v14 v15 v16 and W 2 = v12 . Accordingly, the result obtained by the first pivot transformation is shown in Table 5.10. Table 5.10 The result of the first iteration Element

v1

w1

v2

v3

1

1

2

0

3

v4

v5

0

0

0

0

0

−1

0

25

0

0

0

0

1

0

0

0

4

0

0

0

1

0

0

5

0

0

0

0

1

0

6

0

0

0

0

0

1

162

5 Analysis of Workpiece Attachment/Detachment

Table 5.11 The non-pivot column after the second pivot transformation Column vector

u1,i

u2,i

u3,i

u4,i

u5,i

u6,i

1

0

0

0

0

0

u31 after transforming

1

0

0

0

0

0

u22 before transforming

0

−1

0

0

0

0

u32 after transforming

0

−0.3333

0

0.0267

0

0

u23 before transforming

0

0

1

0

0

0

after transforming

0

0

1

0

0

0

before transforming

0

0

0

0

1

0

after transforming

0

0

0

0

1

0

before transforming

0

0

0

0

0

1

after transforming

0

14.4338

0

0.5774

0

1

u21 before transforming

u1 u2 u3

u33 u25 u35 u26 u36

u5 u6

The fifth step is to judge the termination condition. Since h = 1 is less then m = 5, h should be increased as h = 2. And then, t 2 = a2T U 2 = [0, −1, 0, 37.5, 0, −21.6506] is calculated to obtain the the column number of the pivot column, p = 4. In other words, there is the pivot column u24 = v23 = [0, 25, 0, 1, 0, 0]T in U 2 , as illustrated in Table 5.10. By performing the pivot transformation on U 2 , the other non-pivot columns are listed in Table 5.11 whereas the pivot column vector is u34 = [0, −0.6667, 0, −0.0267, 0, 0]T . The non-pivot column I j2 (j = 4, 1 ≤ j ≤ 6) is merged h = 2 to I13 = {1, 2}, I23 = {2}, I33 = {1, 2}, I43 = {1}, I53 = {1, 2} and I63 = {1, 2}. Since a2 ∈ / B, then V 3 = [v21 , v22 , v24 , v25 ] and W 2 = [w21 , v23 ], as shown in Table 5.12. The less of h = 2 than m = 5 can result in the continuous increment of h so that h = 3. Since t 3 = aT3 U 3 = [0, 0, 0, −1, 0, 43.3013], the pivot column is the 6-th column whose magnitude is u36 = v34 = [0, 14.4338, 0, 0.5774, 0, 1]T . After the third pivot transformation is carried out, the pivot column is u46 = [0, −0.3333, 0, −0.0133, 0, −0.0231]T , and other columns are listed in Table 5.13. And then, the addition of h = 3 into the record set of the non-pivot column I j3 for j = 6 and 1 ≤ j ≤ 6 can obtain I14 = {1, 2, 3}, I24 = {2, 3}, I34 = {1, 2, 3}, I44 = {1, Table 5.12 The result of the second iteration Element

v1

w1

v2

w2

v3

v4

1

1

0

0

0

0

0

2

0

−0.3333

0

−0.6667

0

14.4338

3

0

0

1

0

0

0

4

0

0.0267

0

−0.0267

0

0.5774

5

0

0

0

0

1

0

6

0

0

0

0

0

1

5.4 Analysis and Application of Attachment and Detachment

163

Table 5.13 The non-pivot column after the third pivot transformation 向量 u1 u2 u3 u4 u5

u1,i

u2,i

u3,i

u4,i

u5,i

u6,i

1

0

0

0

0

0

u41 after transforming

1

0

0

0

0

0

u32 before transforming

0

−0.3333

0

0.0267

0

0

u31 before transforming

u42 after transforming

0

−0.3333

0

0.0267

0

0

u33 before transforming

0

0

1

0

0

0

after transforming

0

0

1

0

0

0

before transforming

0

−0.6667

0

−0.0267

0

0

after transforming

0

−0.3333

0

−0.0133

0

0.0231

before transforming

0

0

0

0

1

0

after transforming

0

0

0

0

1

0

u43 u34 u44 u35 u45

Table 5.14 The result of the third iteration Element

v1

w1

v2

w2

v3

w3

1

1

0

0

0

0

0

2

0

−0.3333

0

−0.3333

0

−0.3333

3

0

0

1

0

0

0

4

0

0.0267

0

−0.0133

0

−0.0133

5

0

0

0

0

1

0

6

0

0

0

0.0231

0

−0.0231

3}, I54 = {1, 2, 3} and I64 = {1, 2}. Again, since a3 ∈ / B, V 4 = [v31 , v32 , v33 ] and W 4 3 3 3 = [w1 , w2 , v4 ], as illustrated in Table 5.14. h = 3 is still less than m = 5. The calculation process must go on with h = 4. So t 4 = aT4 U 4 = [0.7071, 0.1886, 0.7071, −0.2576, 0, 0.0690]. Consequently, the matrix U 4 has also a pivot column u41 = v41 whose the column number is p = 1. The next is to carry out the fourth pivot transformation. Thus, the pivot column is updated to u51 = [−1.4142, 0, 0, 0, 0, 0]T . The transformed non-pivot columns are shown in Table 5.15. By adding h = 4 into the record set of the non-pivot column I j4 for j = 1 and 1 ≤ j ≤ 6, it can easily obtain I15 = {1, 2, 3}, I25 = {2, 3, 4}, I35 = {1, 2, 3, 4}, I45 = {1, 3, 4}, I55 = {1, 2, 3, 4} and I65 = {1, 2, 4}, respectively. Due to a4 ∈ / B, so V 5 4 4 4 4 4 4 5 = [v2 , v3 ] and W = [w1 , w2 , w3 , v1 ], as listed in Table 5.16. The next pivot transformation continues because h = 4 is less than m = 5. Thus, h = 5 so that t 5 = aT5 U 5 = [−1, −0.3771, −1.4142, 0.1886, 0, 0.1886]. There is a pivot column u53 = v51 in U 5 whose column number is p = 3. The pivot column is transformed to obtain u63 = [−0.7071, 0, 0.7071, 0, 0, 0]T . The other columns are transformed as shown in Table 5.17.

164

5 Analysis of Workpiece Attachment/Detachment

Table 5.15 The non-pivot column after the fourth pivot transformation Column vector u2

u3

u4

u5

u6

u1,i

u2,i

u3,i

u4,i

u5,i

u6,i

u42 before transforming

0

−0.3333

0

0.0267

0

0

u52 after transforming

−0.2667

−0.3333

0

0.0267

0

0

u43 before transforming

0

0

1

0

0

0

u53 after transforming

−1

0

1

0

0

0

u44 before transforming

0

−0.333

0

−0.0133

0

0.0231

u54 after transforming

0.3643

−0.3333

0

−0.0133

0

0.0231

u45 before transforming

0

0

0

0

1

0

u55 after transforming

0

0

0

0

1

0

u46 before transforming

0

−0.3333

0

−0.0133

0

−0.0231

u56 after transforming

−0.0976

−0.3333

0

−0.0133

0

−0.0231

Table 5.16 The result of the fourth iteration Element

w1

w2

v1

w3

v2

w4

1

−1.4142

−0.2667

−1

0.3643

0

−0.0976

2

0

−0.3333

0

−0.3333

0

−0.3333

3

0

0

1

0

0

0

4

0

0.0267

0

−0.0133

0

−0.0133

5

0

0

0

0

1

0

6

0

0

0

0.0231

0

−0.0231

The increase of h = 5 into the record set of the non-pivot column I j5 for j = 3 and 1 ≤ j ≤ 6 is done to obtain I16 = {1, 2, 3, 5}, I26 = {2, 3, 4, 5}, I36 = {1, 2, 3, 4}, I46 = {1, 3, 4, 5}, I56 = {1, 2, 3, 4, 5} and I66 = {1, 2, 4, 5}. And then, a5 ∈ / B makes V 6 = v52 and W 6 = [w15 , w52 , w53 , w54 , v51 ], as shown in Table 5.18. Since h = m = 5, the calculation process stops for obtaining the workpiece position variation

5.4 Analysis and Application of Attachment and Detachment

165

Table 5.17 The non-pivot column after the fifth pivot transformation Column vector u1

u2

u4

u5

u6

u1,i

u2,i

u3,i

u4,i

u5,i

u6,i

u51 before transforming

−1.4142

0

0

0

0

0

u61 after transforming

−0.7071

0

−0.7071

0

0

0

u52 before transforming

−0.2667

−0.3333

0

0.0267

0

0

u62 after transforming

0

−0.3333

−0.2667

0.0267

0

0

u54 before transforming

0.3643

−0.3333

0

−0.0133

0

0.0231

u64 after transforming

0.2309

−0.3333

0.1333

−0.0133

0

0.0231

u55 before transforming

0

0

0

0

1

0

u65 after transforming

0

0

0

0

1

0

u56 before transforming

−0.0976

−0.3333

0

−0.0133

0

−0.0231

u66 after transforming

−0.2309

−0.3333

0.1333

−0.0133

0

−0.0231

Table 5.18 The result of the fifth iteration Element

w1

w2

w3

w4

v1

w5

1

−0.7071

0

−0.7071

0.2309

0

−0.2309

2

0

−0.3333

0

−0.3333

0

−0.3333

3

−0.7071

−0.2667

0.7071

0.1333

0

0.1333

4

0

0.0267

0

−0.0133

0

−0.0133

5

0

0

0

0

1

0

6

0

0

0

0.0231

0

−0.0231

⎡

0 ⎢0 ⎢ ⎢ ⎢0 q = ⎢ ⎢0 ⎢ ⎣1 0

⎤ −0.7071 0 −0.7071 0.2309 −0.2309 ⎡ ⎤ π1 ⎢ ⎥ 0 −0.3333 0 −0.3333 −0.3333 ⎥ ⎢ ⎥⎢ π ⎥ ⎥ ⎢ ⎥⎢ 2 ⎥ ⎥ ⎢ −0.7071 −0.0267 0.7071 0.1333 0.1333 ⎥⎢ ⎥ ⎥ ⎥⎢ π3 ⎥ ⎥ρ + ⎢ ⎢ ⎥ 0 0.0267 0 −0.0133 −0.0133 ⎥⎢ ⎥ ⎢ ⎥⎣ π4 ⎦ ⎥ ⎣ ⎦ ⎦ 0 0 0 0 0 π5 0 0 0 0.0231 −0.0231 (5.37) ⎤

⎡

where ρ is the arbitrary value, π i (1 ≤ i ≤ 5) is the non-negative value.

8 mm

m 4

6 mm

m

5 Analysis of Workpiece Attachment/Detachment

8 mm

166

60

7 mm

8 mm

12 mm

Fig. 5.3 Workpieces with different structures

5.4.2 Two Dimensional Workpiece Figure 5.3 shows three workpiece shapes with different fixturing layouts. The diagrams of the layout schemes are listed in the second column of Table 5.19 whose coordinates and normal vectors are in the third column and the fourth column, respectively. Above all, in combination with the data in the third and the fourth columns of Table 5.19, Eq. (5.20) is used to judge whether Eq. (5.14) has solutions. Secondly, if Eq. (5.14) has solutions, Eq. (5.25) are depended on to judge whether Eq. (5.15) has solutions respectively under the condition of six kinds of fixturing layout schemes. Finally, in the case that the workpiece has viable attachment/detachment, the position variation q should be further solved by using the pivot algorithm so that it can be judged the direction of attachment/detachment. Here, it is assumed to take ϕ 1 = ϕ 2 = ϕ 3 = ϕ 4 = ϕ 5 = ϕ 6 = 0.01, the calculated results are listed in the fifth and the sixth columns of Table 5.19.

2

1

L4

L5

O

Y

O

Y

L1

L1

L4

L3

L2

L2

L3

Scheme Fixturing diagram

X

X

r1 r2 r3 r4

= [2, 0]T = [5, 0]T = [3, 6]T = [0, 3]T n1 n2 n3 n4

= [0, 1]T = [0, 1]T = [0, −1]T = [1, 0]T

= [0, 1]T = [0, 1]T = [0, −1]T = [0, −1]T = [1, 0]T

n1 n2 n3 n4 n5

= [2, 0]T = [5, 0]T = [5, 6]T = [2, 6]T = [0, 3]T

r1 r2 r3 r4 r5

Normal vector

Coordinate

Table 5.19 Attachment/detachment analysis of 2D workpiece

i=1

6 

φ1,i

i=1

5 

φ1,i

The workpiece has the attachment and detachment

max(τ1 ) =

max(τ 1 ) = 0.01 b1 = [0,0,0,0,0.01]T

max(λ) = 0

The workpiece has the attachment and detachment

max(τ1 ) =

max(τ 1 ) = 0.01 b1 = [0,0,0,0,0,0.01]T

max(λ) = 0

Attachment/ detachment

(continued)

The workpiece can positively translate only along the X direction

⎡ ⎤ 1 ⎢ ⎥ ⎥ q = π ⎢ ⎣ 0 ⎦(π ≥ 0) 0

The workpiece can positively translate only along the X direction

0

Attachment/detachment direction ⎡ ⎤ 1 ⎢ ⎥ ⎥ q = π ⎢ ⎣ 0 ⎦(π ≥ 0)

5.4 Analysis and Application of Attachment and Detachment 167

3

X

L1

L5

O

L2

L3

L4

Y

Scheme Fixturing diagram

Table 5.19 (continued)

= [−1, 0] T = [−1, 0]T = [0, −1]T = [1, 0]T = [1, 0]T

n1 n2 n3 n4 n5

= [7, 1]T = [7, 5]T = [4, 7]T = [0, 5]T = [0, 1]T

r1 r2 r3 r4 r5

Normal vector

Coordinate

i=1

5 

φ1,i

i=1

5 

φ2,i

i=1

5 

φ3,i

i=1

5 

φ4,i

i=1

5 

φ5,i

The workpiece has the attachment and detachment

max(τ5 ) =

max(τ 5 ) = 0.01 b5 = [0,0,0,0,0.01]T

max(τ4 )