Augmenting Datum Flow Chain Method to Support the Top-Down
Design Process for Mechanical Assemblies
by
Gaurav Shukla
B. Tech., Mechanical Engineering
Indian Institute of Technology at Kanpur, 1999
Submitted to the Department of Mechanical Engineering
In Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2001
( 2001 Massachusetts Institute of Technology
All rights reserved
Signature of Author.
Department df Mechanical Engineering
May 11, 2001
Certified by
Dr. Da 'el E. Whitney
Senior Research Scientist
Center for Technology, Policy and Industrial Development
Lecturer, Department of Mechanical Engineering
Thesis Supervisor
Accepted by_
Prof. Ain A. Sonin
Chairman, Departmental Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 16 2001
LIBRARIES
BARKER
2
Augmenting Datum Flow Chain Method to Support the Top-Down
Design Process for Mechanical Assemblies
by
Gaurav Shukla
Submitted to the Department of Mechanical Engineering
In Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
Abstract
The aim of this thesis is present tools which support the top-down design process for assemblies
by analyzing the locating scheme or constraint structure of assemblies in absence of detailed
level part geometry. The top-down design process has received attention both in academia and
industry. However, there have been few analytical tools to support it. The bottom-up approach
supported by CAD systems is good for detailed level design of a single part. The representation
and manipulation of assemblies involves structural and spatial relationships between individual
parts at a higher level of abstraction than the representation of single parts. This thesis uses the
Datum Flow Chain (DFC) for symbolic representation of mechanical assemblies and screw
theory for representation of constraints between two parts. DFC captures the design intent by
recording location scheme of assemblies. Screw theory can represent constraints in three
dimensions.
This thesis presents the design steps and corresponding analytical tools for a top-down design
process in a logical progressive way. The approach of bottom-up process supported by CAD
systems is compared all along the presentation. A method to generate the screw theory
representation of relative constraints between two arbitrary contacting surfaces is presented first.
A procedure has been outlined to generate the screw representation of an assembly feature
constructed by several contacting surface pairs. These tools can be used to construct screw
theory representation of an arbitrarily complex assembly feature. A method of finding the
constraint properties of assemblies, which uses screw theory, is presented next. The method of
motion analysis can find under-constraints for all assemblies. This can be used for analysis of
instantaneous kinematics of a general mechanism as well. Finding over-constraints in an
assembly is a separate problem and it requires different procedure of analysis than motion
analysis. This thesis presents a method of finding over-constraints of assemblies. Quantitative
information about over-constraint of all assemblies may not be found in cross-coupled
assemblies. Motion and constraint analyses can help assembly designers in evaluating the
nominal design.
A method to calculate the sensitivity of the location of a part due to variation in the location of
an assembly feature is presented next. This method uses the screw theory representation of
constraints and information about location of assembly features. Clearance is introduced on bi-
3
directional assembly features to reduce the probability of interference but it introduces
uncertainty in the location of parts. A method is proposed to analyze uncertainty in the location
of parts due to clearance on the size dimensions of assembly features. These analysis tools can be
used to check robustness of the nominal design. A classification of assemblies based upon
constraint properties is presented next. This classification relates properties of constraint
structure of assemblies to design context. Finally, this thesis lays out a coherent scheme of
design steps forming a procedure for designing mechanical assemblies in a top-down fashion.
Thesis Supervisor: Daniel E. Whitney
Title: Senior Research Scientist
4
Acknowledgements:
First I would like to thank my advisor Dr. Daniel Whitney for giving me the opportunity to
conduct this research. I acknowledge the freedom that he gave me in finding the research topics
and pursuing them. I must acknowledge him for his mentorship as well. I cannot imagine, at this
moment, that my research could have evolved the way it has in absence of his guidance. I think I
inherited from him various aspects of his personality during our discussions in last two years.
I must thank Dr. Whitney for providing me various opportunities to visit industry and to interact
with engineers there. I would like to thank Dr. Nancy Wang in Knowledge-Based Engineering at
Ford Motor Co. who helped me in understanding CAD systems better. Craig Moccio, Mike
Trygar and Chuck Voelker in Total Vehicle Geometry group at Ford gave me practical problems
to validate our theory. I must thank Dr. Chris Magee for helping me in finding the right audience
within Ford. Jack Chung and Jeff Wang at Structural Dynamics Research Corporation helped me
in focusing on the loose aspects of our research by their constructive criticism. I should also
acknowledge the support Dr. Allan Jones at Boeing provided me by answering my questions.
It is impossible not to mention friends that I made at MIT. It is because of them I would
remember this place the most. I thank Alberto, Gennadiy, Fredrik and Pung for providing a
stimulating work environment. This list will remain incomplete if I don't thank Shivanshu for
putting up with me in the same apartment for two years.
I would also like to acknowledge the support that my parents provided throughout the period of
my studies. It is impossible for me to think that I could have thought anything worthwhile
without their support.
This material is based upon work supported by National Science Foundation (NSF) under Grant
no. DMI-9610163 and Ford Motor Company. Any opinions, findings and conclusions or
recommendations expressed in this material are those of author and do not necessarily reflect the
views of the NSF or Ford Motor Company.
5
6
Contents
ABSTRACT
3
ACKNOWLEDGEMENTS
5
CONTENTS
7
11
LIST OF FIGURES AND TABLES
1
17
INTRODUCTION
1.1
Motivation
17
1.2
Goal of Research
18
1.3
Thesis Overview
19
1.3.1
Top-Down Design Process for Mechanical Assemblies
19
1.3.2
Bottom-Up Design Process for Mechanical Assemblies
20
1.3.3
Comparison between the Top-Down and the Bottom-Up Methods
21
1.3.4
The Organization of the Thesis
22
DATUM FLOW CHAIN (DFC)
23
2
24
Datum Flow Chain
2.1
2.1.1
Background and Prior Work
25
2.1.2
Properties of DFC
26
2.1.3
Mates and Contacts
28
2.1.4
DFC and Assembly Architecture
28
2.2
CAD Systems and Assembly Analysis during Conceptual Stage Design
30
2.3
Conceptual Stage Design of Mechanical Assemblies using DFC Approach and the
32
same using CAD Systems
2.4
Subsequent Design Steps in Top-Down and Bottom-Up Approaches
33
2.5
Summary
34
3
35
BUILDING AN ASSEMBLY FEATURE
Construction of an Assembly Feature
35
3.1.1
Twist and Wrench Representation
37
3.1.2
Basic Surfaces and Types of Contacts
38
3.1.3
Basic Surface Contacts and their Twist-Matrices
42
3.1.4
Method to Calculate the Constraint Representation of the Assembly Feature
47
3.1
7
3.1.5
3.2
Examples
48
Identification of Chain of Mates in CAD
3.2.1
51
TTRS
53
3.2.1.1
Definitions
53
3.2.1.2
Analogy between TTRS and Screw Representation of Contacting Surface Pairs
54
3.2.1.3
Identification of Chain of Mates in TTRS
56
3.2.1.4
Inadequacy of the Process of Identifying and Analyzing the Independent Loops
59
in TTRS
3.3
Comparison between the Feature-Based Approach of Top-Down Method and
60
Feature Recognition Approach of Bottom-Up Method
3.4
4
Summary
61
MOTION AND CONSTRAINT ANALYSIS
4.1
Graphical Technique for Evaluation of Constraint Properties
63
64
4.1.1
Previous Work
64
4.1.2
Graphical Representation of DFC
66
4.1.3
Motion Analysis for A Part (Evaluation of Under-Constraints)
67
4.1.3.1
Constructing the Paths for Motion Analysis
67
4.1.3.2
Constructing the Effective Twist-Matrix of the Paths
72
4.1.3.3
Intersecting the Effective Twist-Matrices of the Paths
74
4.1.3.4
Cross Coupling (Dependent Degrees of Freedom)
74
4.1.4
Comparison of the Method of Motion Analysis
82
4.1.5
Constraint Analysis for A Part (Evaluation of Over-Constraints)
91
4.1.6
Examples
94
4.1.7
Limitations of Motion and Constraint Analysis in the Context of Assembly
103
Problems
4.2
Constraint Analysis in CAD System
104
4.3
Comparison of the Constraint Analysis in Top-Down and Bottom-Up Approaches
105
4.4
Summary
106
5 VARIATION AND CONTRIBUTION ANALYSIS
107
5.1
108
5.1.1
Connective Model of Assemblies
Variation Analysis using Connective Assembly Models (e.g. DFC)
8
110
111
CAD Model of Assemblies
5.2
5.2.1
World Model
111
5.2.2
Surface-Constrained Model
111
5.2.3
Variation Analysis using CAD Assembly Models
112
5.2.4
Tolerance Allocation
113
5.3
Comparison between the Top-Down and Bottom-Up Assembly Models
115
5.4
Contribution Analysis for Location of Parts in an Assembly
116
5.4.1
Approach of Modeling Variation in Assembly Feature Location
117
5.4.2
Sensitivity in the Part Location to the Variation in Assembly Feature Location
117
5.4.3
Examples
123
5.4.4
Facts of Contribution Analysis
128
5.5
6
129
Summary
UNCERTAINTY DUE TO DESIGN-IN-CLEARANCE
Design-in-clearance and Size Variations in Top-down Design Process
6.1
131
132
6.1.1
Design-in-clearance and Uncertainty in the Location of Assembly Features
132
6.1.2
Design-in-Clearance and Multiple Tolerance Chains
134
6.1.3
Modeling Uncertainty in Assembly Feature Location due to Design-in-Clearance
136
6.1.4
Analyzing Uncertainty in the Location of Parts due to Design-in-Clearance
137
Analysis of Design-in-clearance in Properly Constrained Assemblies
6.1.4.1
6.1.4.1.1
Statistical Simulation of Uncertainty in Properly Constrained Assemblies
Analysis of Design-in-clearance in Over-Constrained Assemblies
6.1.4.2
6.1.4.2.1
Statistical Simulation of Uncertainty in Over-Constrained Assemblies
138
140
144
144
6.2
Size Tolerance in Bottom-Up Design Process
150
6.3
Comparison between the Size Variation Analysis Approach of Top-Down Method
152
and that of Bottom-Up Method
7
152
Summary
6.4
155
CLASSIFICATION OF ASSEMBLIES
7.1
Previous Work
156
7.2
Classification of Mechanical Assemblies
156
7.2.1
Under-Constrained Assemblies
157
7.2.2
Properly Constrained Assemblies
157
9
7.2.3
Over-Constrained Assemblies
158
7.2.3.1
Over-Constraint Needed for Function
158
7.2.3.2
Over-Constraint Needed for Assembly
160
7.2.3.3
Over-Constraint as Mistake
161
7.3
8
Summary
161
DESIGN PROCESS & DETECTION OF MISTAKES
8.1
Design Procedure
8.1.1
163
164
Nominal Design Phase
164
8.1.1.1
Identification of Key Characteristics
164
8.1.1.2
Selection of a Conceptual Framework of the Design (Making a DFC)
164
8.1.1.3
Selection or Construction of the Assembly Features (Realizing the DFC)
165
8.1.2
Constraint Analysis Phase
166
8.1.2.1
Motion & Constraint Analysis (Checking DFC)
166
8.1.2.2
Making Corrections in DFC
167
8.1.2.3
Identification and Selection of Assembly Sequences
168
8.1.2.4
Detection of KC Conflict
168
8.1.3
Variation Design Phase
169
8.1.3.1
Checking Robustness of the DFC
169
8.1.3.2
Allocating tolerances to the KCs and to the Mates
169
8.1.3.3
Variation and Contribution Analysis
170
8.2
Meeting Assembly Tolerances
172
8.2.1
Deterministic Coordination
172
8.2.2
Statistical Coordination
172
8.2.3
No Coordination
174
8.3
9
Summary
174
CONCLUSION AND FUTURE WORK
175
9.1
Review and Contribution
175
9.2
Scope for Future Research
179
REFERENCES
181
APPENDIX A
189
APPENDIX B
201
10
List of Figures:
Fig. 1-1: Top-Down Design Process
20
Fig. 1-2: Bottom-Up Design Process
20
Fig. 2-1: DFC
27
Fig. 2-2: Facilities Offered by Turnkey CAD Systems
31
Fig. 3-1: Square-Peg in Square-Hole
36
Fig. 3-2: Different type of "Line" Contacts
39
Fig. 3-3: Cylinder on Plane
43
Fig. 3-4: Wrench-Matrix of a Point Contact
45
Fig. 3-5: Two-Dimensional "Line" Contact
45
Fig. 3-6: Three-Dimensional "Line" Contact
47
Fig. 3-7: Square Peg in a Square-Hole Assembly Feature
49
Fig. 3-8: Pin-Slot Assembly Feature
50
Fig. 3-9: Prismatic Pair
50
Fig. 3-10: Variation in Prismatic Pair
54
Fig. 3-11: Motions for Cylinder on Plane Contacting Pair
55
Fig. 3-12: TTRS and Assembly Graph
58
Fig. 4-1: Two Paths of the Four-Bar
67
Fig. 4-2: Serial Path
68
Fig. 4-3: Path with a Parallel Branch
69
Fig. 4-4: Branches of a Path
69
Fig. 4-5: Path as a Parallel Branch
70
Fig. 4-6: Paths that can be Intersected
70
Fig. 4-7: Path with Cross Coupling
71
Fig. 4-8: Paths with Shared Nodes
71
Fig. 4-9: Process of Analyzing Cross Coupling
75
Fig. 4-10: Velocity Components at the Origin of Assembly Feature
77
Fig. 4-11: Path as a Parallel Branch
83
Fig. 4-12: Two DOF Manipulator
84
Fig. 4-13: Five-Bar Structure
86
11
Fig. 4-14: Method of Finding Over-Constraints: A Set-Theory Analogy
91
Fig. 4-15: Two Plates Joined by Four Features
94
Fig. 4-16: Over-Constraint
96
Fig. 4-17: Parallelogram Mechanism
97
Fig. 4-18: Paths for "L2" and "L4"
98
Fig. 4-19: Paths for "L4" when "L2" is locked
99
Fig. 4-20: Parallel Manipulator
102
Fig. 5-1: Three Parts Joined by a Connective Assembly Model
109
Fig. 5-2: An Assembly of Three Parts in a World Coordinate Frame
109
Fig. 5-3: A Surface-Constrained Assembly Model of Two Parts
112
Fig. 5-4: A Connective Assembly Model of Two Parts
115
Fig. 5-5: Pin in a Slot Assembly Feature
117
Fig. 5-6: Multiple Chains on Part-Feature Diagram
120
Fig. 5-7: Velocity Components at the Origin of Assembly Feature
120
Fig. 5-8: A Five-Bar Linkage
123
Fig. 5-9: Two Plates
123
Fig. 5-10: Variation in the Five-Bar Linkage
126
Fig. 6-1: Unidirectional Constraint
133
Fig. 6-2: Bi-directional Constraint
133
Fig. 6-3: Properly Constrained Assembly
134
Fig. 6-4: Over-Constrained Assembly
134
Fig. 6-5: Over-Constrained Assembly
135
Fig. 6-6: Properly Constrained Assembly
135
Fig. 6-7: Square Peg in Square Hole
136
Fig. 6-8: Design-in-Clearance in Over- and Properly Constrained Assemblies
139
Fig. 6-9: Properly Constrained Assembly
141
Fig. 6-10: Uncertainty in X-location (Properly Constrained Assembly)
143
Fig. 6-11: Uncertainty in 0 -location (Properly Constrained Assembly)
143
Fig. 6-12: Ambiguous Tolerance Chains for Over-Constrained Assemblies
145
Fig. 6-13: Ambiguous Tolerance Chains for Over-Constrained
146
Fig. 6-14: Over-Constrained Assembly
147
12
Fig. 6-15: Multiple Tolerance Chains
148
Fig. 6-16: Uncertainty in X-location (Over-Constrained Assembly)
150
Fig. 7-1: Simple Assembly Classification
157
Fig. 7-2: Classification of Assemblies
160
Fig. 8-1: Design Process Chart
171
Fig. 8-2: Classification of Techniques of Achieving Tolerance Specifications
173
Fig. 9-1: Top-Down Design Process
178
Fig. 9-2: Bottom-Up Design Process
178
Fig. A-1: Any Surface with Any Surface
190
Fig. A-2: Any Surface with Helical Surface
190
Fig. A-3: Any Surface with Surface of Revolution
191
Fig. A-4: Any Surface with Cylindrical Surface
191
Fig. A-5: Any Surface with Planar Surface
191
Fig. A-6: Any Surface with Spherical Surface
192
Fig. A-7: Helical Surface with Helical Surface
192
Fig. A-8: Helical Surface with Surface of Revolution
193
Fig. A-9: Helical Surface with Cylindrical Surface
193
Fig. A-10: Helical Surface with Planar Surface
193
Fig. A-11: Helical Surface with Spherical Surface
194
Fig. A-12: Surface of Revolution with Surface of Revolution
194
Fig. A-13: Surface of Revolution with Cylindrical Surface
194
Fig. A-14: Surface of Revolution with Planar Surface
195
Fig. A-15: Surface of Revolution with Spherical Surface
195
Fig. A-16: Cylindrical Surface with Cylindrical Surface (Unidirectional Line Contact)
195
Fig. A-17: Cylindrical Surface with Cylindrical Surface (Unidirectional Point Contact)
196
Fig. A-18: Cylindrical Surface with Cylindrical Surface (Bi-directional Contact)
196
Fig. A-19: Cylindrical Surface with Planar Surface
196
Fig. A-20: Cylindrical Surface with Spherical Surface (Unidirectional Contact)
196
Fig. A-21: Cylindrical Surface with Spherical Surface (Bi-directional Contact)
197
Fig. A-22: Planar Surface with Planar Surface
197
Fig. A-23: Planar Surface with Spherical Surface
197
13
Fig. A-24: Spherical Surface with Spherical Surface (Unidirectional Contact)
198
Fig. A-25: Spherical Surface with Spherical Surface (Bi-directional Contact)
198
Fig. A-26: Wrench for a Point Contact between An Edge and A Surface
198
Fig. B-1: Properly Constrained Assembly
202
Fig. B-2: Over-Constrained Assembly
203
14
List of Tables:
Table 3-1: Surface-to-Surface Contacts
40
Table 3-2: Changing and Unchanging Vectors for "Cylinder on Plane" Assembly Feature
56
Table 3-3: Twist and Wrench Directions for "Cylinder on Plane" Assembly Feature
56
Table A-1: Surface-to-Surface Contacts
189
Table A-2: Edge-to-Surface Contacts
199
Table B-1: Unidirectional and Bi-directional Degrees of Freedom of Assembly Features
201
15
16
Chapter 1: Introduction
1.1 Motivation:
Now, the customers are being given more importance during the design activities. Cost used to
be the most important factor of consideration during design of mechanical assemblies but now
quality assumes greater significance and reduction in cost is given second priority. Delivering
quality requires more attention to what customer wants and translating the customer needs in
terms of design requirements. Several researchers [Ulrich and Eppinger, Pahl and Beitz, Suh]
emphasize a top-down or requirements-driven design process. [Whitney, Mantripragada, Adams
and Rhee, 1999] presented the different phases of a top-down design process for mechanical
assemblies. The top-down design process relates customer requirements to the concept and
details of the design. It starts with the customer requirements and proceeds systematically to
create functional concepts, physical embodiments of these concepts and then decompositions of
the main embodiments into smaller and smaller assemblies, sub-assemblies and finally individual
parts. It is argued that the top-down design process can reduce the design time and it can avoid
potential mistakes during initial conceptual design phase.
Geometric reasoning is one of the most important connections between design and
manufacturing. Since the top-down design process calls for the attention of the design team to
geometric reasoning in the initial phase of design, it does make the design team more focused
towards potential manufacturing problems. Current computer-aided design (CAD) systems are
part centric (i.e. The CAD systems do not provide functionality for making assembly level
design decisions before filling up the details of the parts). Intelligent CAD systems must support
a top-down design process. To support the top-down design process a CAD system need to have
functionality for representation of assembly models without the detailed level part design. It
should be able to reason in the domain of geometry, handle geometric constraints and satisfy
these constraints in an appropriate, complete and unambiguous manner. There exist a need to
extend the geometric modeling technology to represent assemblies of parts, since most
engineering problems are solved by assemblies rather than single parts.
17
The representation and manipulation of assemblies involves structural and spatial relationships
between individual parts at a higher level of abstraction than the representation of single parts.
Such a representation must support association of form features, mating surfaces involved in
kinematic connections and determination of degrees of freedom from the mating conditions.
Additionally, to support manufacturing, design tools must provide support for representation of
tolerances, interference checking and tolerance allocation.
The top-down design can lead to the greater level of customer satisfaction. So far, there have
been very few analytical tools that can support the top-down design process which requires
analysis tools to evaluate certain design decisions in absence of details of geometry. Mechanical
assemblies, where geometrical locations of different parts are important, are main focus of this
research. This piece of work focuses on laying out a way to design mechanical assemblies using
a top-down design process right from the conceptual stage till the stage of variation analysis. All
along the description the analytical tools, which can support a top-down design environment for
mechanical assemblies, are discussed. The approach of the bottom-up design process and the
tools available for analysis are also discussed to highlight the contribution of this research.
1.2 Goal of Research:
The ultimate goal of this research is to develop a CAD system that supports a top-down
designing environment for mechanical assemblies. It should start from a sketcher where the
designers can play with several initial concepts. There should be a user-friendly interface to
convert the data in the concepts in terms of physical relationships between parts (assembly
features) to a schematic form (DFC). There need to be analysis tools to check the various
concepts at this stage itself. Analysis tools to carry out the robustness check of assembly level
dimensions against part level variations are also required. After this, the sub-assemblies could be
sourced out to different design teams for similar exercises. Since all the interfaces among subassemblies are coming from the top, there cannot be any problem of co-ordination as long as the
databases are shared among different design teams. This thesis is a step towards this goal. It
provides foundation to some of the basic analysis techniques that need to be supported.
18
1.3 Thesis Overview:
This section presents a brief overview of the top-down and bottom-up design processes. A
comparison between the two processes is also presented. The organization of this thesis is
presented in the final sub-section.
1.3.1 Top-Down Design Process for Mechanical Assemblies:
Fig. 1-1 summarizes the steps of the top-down design process (the boxes in thick borders shall
be discussed later in the chapters of the thesis). The top-down design process begins with
customer requirements. Customer requirements are translated into key engineering requirements
(Key Characteristics or KCs) and some concepts are chosen to fulfill the KCs.
The next step is to layout the concepts in terms of the geometric reasoning among sub-systems.
A methodology to capture the design intent called Datum Flow Chain (DFC) was introduced by
[Mantripragada and Whitney, 1998]. It provides a method, together with a vocabulary and a set
of symbols, for documenting a location strategy for the parts and relating that strategy explicitly
to the achievements of customer requirements.
The next step becomes identifying the assembly features which will realize the connections
between parts. The assembly features can be picked off the shelf (from a library) or they can be
built from basic surfaces. The assembly features constrain relative degrees of freedom between
parts. It is important to understand how one can build new assembly features and how the new
assembly features would constrain the degrees of freedom.
The next step becomes doing constraint analysis of the DFC to ensure proper constraint structure
of assembly and necessary fixtures. Depending upon the results of the constraint analysis, one
may want to make changes in the assembly features or the DFC itself. After achieving the
desired constraint structure of the assembly, one would like to check the robustness of the
location of parts (assembly level dimensions) and that of constraint strategy itself. Variations in
location, size or shape of assembly features may propagate to assembly level dimensions and
these variations may change the constraint structure of assembly as well.
19
The detailed level design of parts should be done after checking the robustness of those assembly
level dimensions which are related to the achievement of customer requirements.
Bottom-Up Design Process
Top-Down Design Process
Customer
Requirements
Customer
Requirements
Concepts
Concepts
Datum Flow
Chain (DFC)
Detailed
Level Part
1
The Assembly
Features
How do they
constrain?
I
Mating
Surfaces
How are they
built?
i
Changing &
Unchanging
Directions
Kinematic
Loop (TTRS)
Constraint
Analysis
I
Due to Changes
in Shape & Size
Propagation
of Variation
F Vin
Due to Changes
in Location
Variation
Analysis
Due to Changes
Location
Due to Changes
in Shape & Size
Customer
Requirements
Detailed Level
Part Design
Fig. 1-2
Fig. 1-1
1.3.2 Bottom-Up Design Process for Mechanical Assemblies:
Fig. 1-2 summarizes the steps of the bottom-up design process (the boxes in thick borders shall
be discussed later in the chapters of the thesis). The bottom-up approach is supported by the
20
existing CAD systems because CAD systems are much better equipped to support detailed and
precise design than a rough sketch of a concept identified early in the top-down design process.
The bottom-up design process also starts with a set of concepts which are aimed at satisfying the
ultimate customer requirements. However, after selection of concept, the design team jumps to
detailed level part design. Usually, the concepts are in form of legacy designs.
CAD systems do support the assembly of parts after detailed level design. However, this
assembly process can at best be described as putting perfect pictures next to each other. After
detailed design, the main concern of the design team becomes tolerance allocation and tolerance
analysis. One requires a tolerance model of the assembly in order to perform the tolerance
analysis. Prof. Clement introduced the idea of "Technologically and Topologically Related
Surfaces" (TTRS) to create tolerance models of three-dimensional solid models [Clement, 1991].
TTRS is a technique which finds the mating surfaces in an assembly which pass the constraints
from one part to other. After finding the mating surfaces, the tolerance chains are formed to
analyze an assembly level dimension. There are various methods for creating tolerance models
from CAD parts. TTRS is only one such method.
Finally, the variation in assembly level dimensions is checked against customer requirements. If
some of the requirements are not satisfied often the tolerances on part geometry are modified to
achieve the functionality. Of course, the detailed part geometry can also be changed at much
higher cost because this design iteration will require starting from the very beginning.
1.3.3 Comparison between the Top-Down and the Bottom-Up Methods:
The main difference between the top-down and the bottom-up methods is that whereas the
former calls for the design intent in form of a structure of the assembly, the latter tries to find one
from the collage of parts. DFC is a declaration of the spatial locations of the key assembly
features so the design team knows what the delivery path is for an assembly level dimension.
Whereas a methodology like TTRS tries to find the tolerance chain and associated mating
surfaces for an assembly level dimension by inspecting neighboring parts.
21
However, the bottom-up design process may save time and money by reusing existing designs of
parts and sub-assemblies. Designs, tools, equipments, process and test plans can all be reused.
The top-down design process can be very challenging intellectually. It requires seeing ahead at
each stage of the process, imagining sub-assemblies and parts before they are known in the
detail.
So, it may be advantageous to have some elements of bottom-up design process like legacy parts,
legacy systems and legacy DFCs. This will imply that the top-down design process may have to
meet the existing parts to result a consistent design. This design usually will be a compromise
between novelty, optimal performance, lower cost and faster time.
1.3.4 The Organization of the Thesis:
This thesis presents how mechanical assemblies can be designed in a top-down design way. The
top-down design process will be compared against the bottom-up design process all along the
presentation. Chapter 2 presents the methodology of DFC. This chapter also describes how
design teams may select one concept out of various possible legacy concepts in case of bottomup design process. Chapter 3 presents how assembly features can be built and used. It also
presents how mating surfaces are identified and grouped together in TTRS. Chapter 4 presents
the constraint analysis of DFC using screw theory. It also presents how CAD systems do
constraint analysis of assemblies constituted by fully designed parts. This chapter is based on the
article [Shukla and Whitney, 2001]. Chapter 5 and 6 presents how variation propagation can be
analyzed in case of both top-down and bottom-up design processes. Chapter 5 presents the effect
of location variation. Chapter 6 presents the effect of size and shape variations. Chapter 7 and 8
summarize the whole thesis and presents the way to use the content of this thesis for practical
design purposes. Chapter 7 presents the classification of assemblies based upon their constraint
properties. Chapter 8 presents the way to systematically start the design process using DFC
methodology and several techniques to achieve the customer requirements. Chapter 7 and 8 are
based on the article [Whitney, Shukla and Von-Praun, 2001]. Chapter 9 concludes the thesis by
summarizing the main findings of this research and it also presents the topics for future research.
22
Chapter 2: Datum Flow Chain (DFC)l
A generic product development process for mechanical assemblies should have the system level
design stage merging with the initial concept selection process. The Datum Flow Chain (DFC)
provides a set of tools and techniques for defining, documenting and evaluating the system level
design decisions. In case of mechanical assemblies, the assembly itself and the manufacturing
set-up (tools, dies, assembly sequence, production facility layout etc.) constitute the system.
However, the kinematic structure of the assembly itself is the most important of all. A substantial
amount of all quality problems that arise during assembly can be referred to the geometrical
design and especially the geometrical concept of the product, i.e. the way parts are designed and
located to each other. Special emphasis thus must be put on geometry design, especially during
the early design phases, to try to find robust concepts and avoid solutions that may cause
downstream production problems.
Current CAD systems provide rudimentary assembly modeling capabilities once part geometry
exists, but these capabilities basically simulate an assembly drawing. Most often, the
dimensional relations that are explicitly defined to build an assembly model in CAD are those
most convenient to construct the CAD model and are not necessarily the ones that need to be
controlled for proper functioning of the assembly. What is missing is a way to represent and
display the designer's strategy for locating the parts with respect to each other, which amounts to
the underlying structure of dimensional references. The DFC is intended to capture this logic.
This chapter is organized in the following way. Section 1 presents DFC and its associated
terminology. Section 2 describes that there is a vacuum of tools that can support the
documentation and analysis of the assembly in early stage of design. CAD systems force the
design team to detail part-level design after concept selection process. Section 3 compares the
process of conceptual stage design of mechanical assemblies using DFC (top-down) and the
same using CAD systems (bottom-up). Section 4 presents what the next steps are in both the
approaches (top-down and bottom-up). Section 5 presents a summary of the chapter.
1The first section of this chapter is based on article [Mantripragada and Whitney, 1998].
23
2.1 Datum Flow Chain:
Our aim is to be able to present a unified way to layout, analyze, outsource, assemble, and debug
complex assemblies. To accomplish this, one needs to capture their fundamental structure in a
top-down design process that shows how the assembly is supposed to go together and deliver its
Key Characteristics (KCs) 2 . This process should be able to
*
Represent the customer level requirements (top-level goals) for the assembly.
*
Link these goals to engineering requirements on the assembly and its parts in the form of
KCs.
*
Show how the parts will be constrained, and what features will be used to establish
constraint, so that the parts will acquire their desired spatial relationships that achieve the
KCs.
* Show where the parts will be in space relative to each other both under nominal conditions
and under variation.
* Show how each part should be designed, dimensioned, and toleranced to support the plan.
*
Assure that the plan is robust.
*
A clear statement of these elements for a given assembly is called the design intent for that
assembly.
A "Datum Flow Chain" (DFC) captures assembly design intent. It provides a method, together
with a vocabulary and a set of symbols, for documenting a location strategy for the parts and
relating that strategy explicitly to achievement of the product's key characteristics. It helps the
designer choose mating features on the parts and provides the information needed for assembly
sequence and tolerance analyses.
This section is organized in the following way. The first sub-section presents the background and
prior work which is used for the representation of DFC. The second sub-section presents DFC
and it will also be shown how it represents common assembly situations. The third sub-section
presents a classification of assembly features. Assembly features are divided into two classes,
called mates and contacts: mates pass dimensional constraint from part to part, while contacts
2 Key Characteristics are the customer requirements translated in terms of engineering requirements.
24
merely provide support, reinforcement, or partial constraint along axes that do not involve
delivery of a KC. The fourth sub-section presents the classification of assemblies based upon the
DFC. The assemblies are divided into two types: Type-i assemblies are fully constrained. The
assembly process for Type-1s puts their parts together at their pre-fabricated mating features.
Type-2 assemblies are under-constrained. The assembly process for Type-2s involves fixtures
and can incorporate in-process adjustments to redistribute variation. DFC for a Type-1 assembly
directly defines the assembly itself. However, the DFC for a Type-2 assembly directly defines
the process for creating it and thus only indirectly defines the assembly.
2.1.1 Background and Prior Work:
Assemblies have been modeled systematically by [Lee and Gossard, 1985], [Sodhi and Turner,
1992], [Srikanth and Turner, 1990], and [Roy, Bannerjee and Liu, 1989] among others. Such
methods are intended to capture relative part location and function, and enable linkage of design
to functional analysis methods like kinematics, dynamics, and, in some cases, tolerances. Almost
all of them need detailed descriptions of parts to start with, in order to apply their techniques.
[Gui and Mantyla, 1994] have attempted to apply a function-oriented structure modeling to
visualize assemblies and represent them in varying levels of detail. DFC doesn't attempt to
model assemblies functionally. DFC begins at the point where the functional requirements have
been established and there is at least a concept sketch.
Top-down design of assemblies emphasizes the shift in focus from managing design of
individual parts to managing the design of the entire assembly in terms of mechanical
"interfaces" between parts. [Hart-Smith, 1997] proposes eliminating or at least minimizing
critical interfaces in the structural assembly rather than part-count reduction as a means of
reducing costs. He emphasizes that, at every location in the assembly structure, there should only
be one controlling element that defines location, and everything else should be designed to
"drape to fit." In our terms, the controlling element is a mate and the joints that drape to fit are
contacts. [Muske, 1997] describes the application of dimensional management techniques on 747
fuselage sections. He describes a top-down design methodology to systematically translate key
characteristics to critical features on parts and then to choose consistent assembly and fabrication
methods. These and other papers by practitioners indicate that several of the ideas to be
25
presented here are already in use in some form but that there is a need for a theoretical
foundation for top-down design of assemblies.
Academic researchers have generated portions of this foundation. [Shah and Zhang, 1992]
proposed an attributed graph model to interactively allocate tolerances, perform tolerance
analysis, and validate dimensioning and tolerancing schemes at the part level. This model defines
chains of dimensional relationships between different features on a part and can be used to detect
over and under dimensioning (analogous to over- and under-constraint) of parts. [Wang and
Ozsoy, 1990] provide a method for automatically generating tolerance chains based on assembly
features in one-dimensional assemblies. [Shalon et. al., 1992] show how to analyze complex
assemblies, including detecting inconsistent tolerancing datums, by adding coordinate frames to
assembly features and propagating the tolerances by means of 4x4 matrices. [Zhang and Porchet,
1993] present the Oriented Functional Relationship Graph, which is similar to the DFC,
including the idea of a root node, propagation of location, checking of constraints, and
propagation of tolerances. A similar approach is reported by [Tsai and Cutkosky, 1997] and by
[Johannesson and Soderberg, 2000]. The DFC is an extension of these ideas, emphasizing the
concept of designing assemblies by designing the DFC first, then defining the interfaces between
parts at an abstract level, and finally providing detailed part geometry.
CAD today bountifully supports design of individual parts. It thus tends to encourage premature
definition of part geometry, allowing designers to skip systematic consideration of part-part
relationships. Most textbooks on engineering design also concentrate on design of machine
elements (i.e., parts) rather than assemblies.
2.1.2 Properties of DFC:
A datum flow chain is a directed acyclic (a graph with no cycles) graphical representation of an
assembly with nodes representing the parts and arcs representing mates between them. Every
node represents a part or a fixture and every arc transfers dimensional constraint along one or
more DOFs from the node at the tail to that at the head (Fig. 2-1). Loops or cycles in a DFC
would mean that a part locates itself once the entire cycle is traversed and hence are not
permitted. Every arc constrains certain degrees of freedom depending upon the type of mating
26
conditions it represents. Each arc has an associated 4*4 transform matrix that represents
mathematically how the part at the head of the arc is located with respect to the part at the tail of
the arc. A typical DFC has only one root node that has no arcs directed towards it, which
represents the part from which the assembly process begins. This could be a base part or a
fixture.
Root
Fig. 2-1: DFC
Every arc is labeled to show which degrees of freedom it constrains, which depends on the type
of mating conditions it represents. The sum of the unique degrees of freedom constrained by all
the incoming arcs to a node in a DFC should be equal to six (less if there are some kinematic
properties in the assembly or designed mating conditions such as bearings or slip joints which
can accommodate some amount of pre-determined motion; more if locked-in stress is necessary
such as in preloaded bearings). This is equivalent to saying that each part should be properly
constrained, except for cases where over- or under-constraint is necessary for a desired function.
The following assumptions are made to model the assembly process using a DFC:
1. All parts in the assembly are assumed rigid. Hence, each part is completely located once its
position and orientation in the three dimensional space are determined.
2. Each assembly operation completely locates the part being assembled with respect to existing
parts in the assembly or an assembly fixture. Only after the part is completely located is it
fastened to the remaining parts in the assembly.
Assumption 1 states that each part is considered to be fully constrained once three translations
and three rotations are established. If an assembly, such as a preloaded pair of ball bearings, must
contain locked-in stress in order to deliver its KCs, the parts should still be sensibly constrained
and located kinematically first, and then a plan should be included for imposing the over-
27
constraint in the desired way, starting from the unstressed state. If flexible parts are included in
an assembly, they should be assumed rigid first, and a sensible locating plan should be designed
for them on that basis. Modifications to this plan may be necessary to support them against
sagging under gravity or other effects of flexibility that might cause some of their features to
deviate from their desired locations in the assembly.
Assumption 2 is included in order to rationalize the assembly process and to make incomplete
DFCs make sense. An incomplete DFC represents a partially completed assembly. If the parts in
a partially completed assembly are not completely constrained, by each other or by fixtures, it is
not reasonable to expect that they will be in a proper condition for receipt of subsequent parts, inprocess measurements, transport, or other actions that may require an incomplete assembly to be
dimensionally coherent and robust.
2.1.3 Mates and Contacts:
A typical part in an assembly has multiple joints with other parts in the assembly. Not all of these
joints transfer locational and dimensional constraint, and it is essential to distinguish the ones
that do from the ones that are redundant location-wise and merely provide support or strength.
We define the joints that establish constraint and dimensional relationships between parts as
mates, while joints that merely support and fasten the part once it is located are called contacts.
Hence mates are directly associated with the KCs for the assembly because they define the
resulting spatial assembly relationships and dimensions. The DFC therefore defines a chain of
mates between the parts. If we recall that the liaison diagram includes all the joints between the
parts, then it is clear that the DFC is a subset of the liaison diagram. The process of assembly is
not just of fastening parts together but should be thought of as a process that first defines the
location of parts using the mates and then reinforces their location, if necessary, using contacts.
2.1.4 DFC and Assembly Architecture:
Most models of assemblies represent the assembly as complete, i.e. with all its parts in place and
all mates and contacts fastened. Therefore, these models are not capable of addressing issues that
occur during the act of assembling. Assembly planning considers a series of successively more
complete assemblies. Incomplete assemblies may have unconstrained degrees of freedom that
28
will be constrained when the assembly is complete. They may be subject to shape and size
variations that the final assembly will not be subject to. Yet these uncontrolled degrees of
freedom or variations may cause the next assembly step to fail or may result in a misshapen final
assembly and thus have to be considered during design. In order to manage these issues
systematically, assemblies are distinguished in the following two types:
Type-1 Assemblies:
Type-1 comprises typical machined or molded parts that have mating features fully defined by
their respective fabrication processes prior to final assembly. These are called part-defined
assemblies because the variation in the final assembly is determined completely by the variation
contributed by each part in the assembly, assuming all the 'rules' of the assembly (correct bolt
torque, cleanliness, etc.) are followed. The assembly process merely puts the parts together by
joining their pre-defined mating features. The mating features are almost always defined by the
desired function of the assembly and the designer of assembly process has little or no freedom in
selecting mating features. Defined in terms of the DFC, a type-1 assembly is one where every
part has at least one mate with at least one other part in the assembly. Fixtures, if present, merely
immobilize the base sub-assembly and present it to the part being assembled in the desired
position and orientation.
Type-2 Assemblies:
The second type of assembly includes aircraft and automotive body parts that are usually given
some or all of their assembly features or relative locations during the assembly process.
Assembling these parts requires placing them in proximity and then drilling holes or bending
regions of parts as well as riveting or welding. The locating scheme for these parts must include
careful consideration of the assembly process itself since function by no means is a sufficient
guide. Final assembly quality depends crucially on achieving desired final relative locations of
the parts, something that is by no means assured because at least some of the parts lack definite
mating features that tie them together unambiguously. A different datum flow logic, assembly
sequence, etc. will result in quite different assembly configurations, errors and quality. It is
possible to build a perfect assembly out of imperfect parts and vice versa by choosing an
appropriate or inappropriate datum flow chain logic.
29
Defined in terms of the DFC, a type-2 assembly is one where it is possible to have only contacts
between all parts in the assembly. In such cases, the parts will have mates with fixtures used to
locate them. Typically, a type-2 assembly will have a mixture of mates and contacts, making inprocess adjustments or absorption possible only at certain locations and not at others.
2.2 CAD Systems and Assembly Analysis during Conceptual Stage Design:
In a bottom-up design process also, the design team starts with a set of customer requirements
and then they move to the concept selection process. One of the main electronic aids available to
the design teams is in form of CAD systems. Current CAD systems are inherently part centric.
Accordingly, design teams show the tendency of jumping to the detail part-level design after
selection of a concept. Not much time is spent on establishing the structure of the concept. The
detailed level part design precedes the assembly or layout design. The tendency to do the detailed
level part design before assembly design has become deep rooted in most organizations. [Pugh,
Total Design, Page 189, 1991] confirms this:
"However, progressively over the last 20 years, we seem to have lost our way
by concentratingmainly on CAD, almost regardlessof the tasks that confront us
and certainly almost regardless of the efficiency and utilization of such systems.
In fact, many companies have purchased CAD systems to their cost, have had to
use them to justify these costs and are now removing them in certain
circumstances, to be returnedto later."
In case of product development process for a "new" product (mechanical assembly), it is
expected that the design team will spend required time and efforts in establishing the validity of
the concept. The concept can be in the form of a layout or scheme drawing and checking its
validity shall require some analysis tools that can analyze the kinematic structure of the concept
with respect to the customer requirements. CAD systems offer the analysis tools which take
input from the detail part-level design.
On the other hand, CAD may be very attractive where there are significant benefits in increasing
the carryover content of the design. Automotive and aircraft industries are the two front-runners
30
as far maximizing the carryover content is concerned. [Pugh, Total Design, Page 190, 1991]
confirms this:
"CAD grew from the needs of the automotive and aerospace industries in the
fifties."
"About 80% of a typical design is a modification of various parts of earlier
designs."
In a large organization developing a new design may be trivialized to selecting one legacy design
and improving upon it. It may be driven by the lock-in of the organization due to investment in
the inflexible manufacturing system or supplier lock-in or due to other business drivers. CAD
seems to be favoring the designs with a fixed concept where the process of design becomes
based on convention or based on product line. Most of the CAD systems offer capabilities for
handling detailed part design or manufacturing related activities. [Pugh, Total Design, Page 190,
1991] says: (see Fig. 2-2)
".. in a detailed study of the design activity in 1984, relating to CAD systems,
where some 85 turnkey systems were examined in great detail and correlated to
the design core .. The conclusions were that: the 2D drafting mainly aimed at
detail drawing and the remainingfacilities all stemming from this base (of detail
drawing), with a strong bias towards manufacturing are the main facilities that
CAD systems offer."
2D Drafting
3D Modeling
Geometric Analysis
Interference Analysis
Part List
2 D Visualization
Fig. 2-2: Facilities Offered by Turnkey CAD Systems
The current CAD systems support the bottom-up design process and they are inefficient at
handling the design and analysis of assembly structure in the conceptual stage. It may lead to
31
design of assemblies that create problems in manufacturing due to their under- or overconstrained structure. [Pugh, Total Design, Page 192, 1991] echo similarly:
"Too much emphasis on utilization of CAD systems during early stages of
design may seriously curtail conceptual options and therefore designs may lead
to increasedprocess losses."
2.3 Conceptual Stage Design of Mechanical Assemblies using DFC
Approach and the same using CAD Systems:
Assembly is the point in a product's life cycle where parts from different sources come together
and the product first comes to life. The assembly process should be viewed as a proxy for a wide
range of decisions, events and relationship between different stages of the product development
process. Assembly is really the chaining together of dimensional relationship and constraints.
The success of these chains determines the success of the product's quality from an assembly
point of view. The goal of top-down assembly modeling is to permit these chains to be
determined first and followed by design of individual parts. Datum Flow Chain (DFC)
implements this approach to assembly modeling and design of assemblies.
However, it is common to view assembly as a process that merely fastens parts together. The
bottom-up design methodology supports this view. The design team following a bottom-up
design process jumps from the concept development stage to the detailed level part design. It
puts additional demands on the testing and refinement stage later on which essentially refers to
the tolerance allocation and tolerance analysis. The design team usually resorts to such routines
as "tolerance chain identification" instead of designing the tolerance chain. Such a part-centric
product design approach that ignores assembly and system issues may create many fit-up
problems. Finding the source of these fit-up problems is a very difficult and time-consuming task
and most of the time the exact causes cannot be identified. The time and cost involved to make
engineering changes, in-process adjustments, etc., to fix these problems increase rapidly as the
product development process evolves. Early anticipation and avoidance of these problems can
have a huge impact in reducing the product development time, cost and production fit-up
problems and can improve final product quality.
32
Moreover during the design of a component, the context of that component to its design is the
most important. The context of the component is defined and documented during the system
level design. If the system level design has been skipped or not taken care of very thoroughly,
the designer putting the details in the component may completely lose the context of the
component in the system which might create the problems later on. Customer requirements do
not drive the design process directly after concept selection in a bottom-up design process.
Whereas the very advent of DFC methodology is to provide tools and techniques for evaluating
design decisions against customer requirements at each step of design.
However, the bottom-up design process may save time and money by reusing existing designs of
parts and sub-assemblies. Designs, tools, equipments, process and test plans can all be reused.
The top-down design process can be very challenging intellectually. It requires seeing ahead at
each stage of the process, imagining sub-assemblies and parts before they are known in the
detail. So, it may be advantageous to have some elements of bottom-up design process like
legacy parts, legacy systems and legacy DFCs. This will imply that the top-down design process
may have to meet the existing parts to result a consistent design. This design usually will be a
compromise between novelty, optimal performance, lower cost and faster time.
2.4 Subsequent Design Steps in Top-Down and Bottom-Up Approaches:
Top-down:
The DFC comprises design intent for the purpose of locating the parts but it does not say how the
parts will be located. Providing location means providing constraint. Assembly features are the
vehicles which apply constraint between parts. Thus the next step after defining the DFC is to
choose features to provide the constraint. Once features have been declared, one can calculate the
nominal locations of all the parts by chaining their 4x4 transforms together and one can check for
over- or under-constraint, using methods that will be described in later chapters. In order to be
precise about locating scheme, however, one needs to keep distinguishing between mates and
contacts. The constraint representation and the information in the DFC will be used for
calculating the sensitivity of the part locations to manufacturing variations.
33
Bottom-up:
In a bottom-up design process, tolerance allocation and analysis is an important stage of design
after detailed level part design. Tolerance allocation is often achieved by trial and error
(Allocate-Analyze-Modify). Tolerance analysis is a well-researched area and there are several
techniques which attempt to find the tolerance chains from the CAD solid models. After
identification of tolerance chains, the analysis predicts sensitivity of part locations to
manufacturing variations.
2.5 Summary:
"What attracts and delights customers in a product and what is compelling in a
process, is system performance."3
The fundamental challenge of the product development process is to combine engineering detailspecific dimensions, assembly dimensions, part dimensions, materials etc.-into a coherent whole.
This chapter presented the DFC method which ensures that customer requirements drive the
assembly architecture. The detailed part design comes after that, once the context of the part in
the system is known. DFCs express the designer's logical intent concerning how parts are to be
related to each other geometrically to deliver the KCs repeatedly.
The bottom-up design process suits CAD systems. The design teams tend to jump to detailed
level part design without evaluating the concepts thoroughly. The main drivers for the bottom-up
design process are rudimentary properties of CAD systems as far as conceptual and system level
design is concerned and some business reasons (carryover designs, inflexible manufacturing
systems).
Next chapter presents how the assembly features constrain the relative degrees of freedom and
how the assembly features can be built. The constraint representation shall be used for
calculating the variation sensitivities as well. The presentation shall compare this with the
approach of identification of tolerance chains in case of a bottom-up design process.
3 [Wheelwright S. C. and Clark K. B., Revolutionizing Product Development, The Free Press, New York, 1992]
34
Chapter 3: Building An Assembly Feature
Assembly features carry constraints (by locking the DOFs of one part with respect to the other
part). In a top-down approach, the designer tries to find assembly features that can realize the
connections (for mates and contacts) represented on the DFC. In a bottom-up approach, the parts
are designed individually and then the parts are brought together for assembly. CAD systems try
to identify a chain of mates from the collage of parts to solve for the configuration of the
assembly and later on the same or different chain of mates is used to perform the variation
analysis. However, CAD systems do not know how to differentiate between mates and contacts
and any constraint can be selected as mate.
This chapter presents a method of constructing assembly features from basic surfaces and
calculating the relative degrees of freedom allowed by the same. This chapter also presents the
way the chain of mates is found out in the bottom-up approach. Finally, the process of designing
and building the chain of mates (in top-down) shall be compared with the process of identifying
the chain of mates (in bottom-up). This chapter is organized in the following fashion. First
section presents the method of construction of assembly features. Assembly features are created
by a set of contacting surface pairs. This section presents the different basic surfaces, the types of
contact among basic surfaces, the method to construct constraint representation of the contacting
surface pairs and the method to calculate the relative degrees of freedom allowed by the
assembly feature. Second section presents the methodology of finding the chain of mates.
Appropriate references to some methods that try to find a chain of mates from the CAD solid
models shall be given. The theory of TTRS is one such technique that finds the chain of mates
from CAD solid models. This section describes the theory of TTRS. Third section compares the
process of building the chain of mates (in top-down) with the process of identifying the chain of
mates (in bottom-up). Fourth section presents the summary of the chapter.
3.1 Construction of An Assembly Feature
Every assembly feature involves two sets of surfaces. One set of surfaces belongs to one part and
the other set of surfaces belongs to the other part. An assembly feature can be as simple as just
35
one pair of surfaces (plane on plane) or it can involve multiple sets of surfaces (e.g. a square-peg
in a square-hole). At nominal dimensions, the surfaces in one set of surfaces remain in contact
with the corresponding surfaces in the other set. Normally, some clearance is allowed in the
assembly features, the clearance on assembly features and uncertainty introduced by it shall be
discussed in chapter 6. This chapter assumes no clearance on assembly features. Fig. 3-1(a)
shows the square-peg in square-hole assembly feature with no clearance. Fig. 3-1(b), (c) show
the same assembly feature when some clearance is allowed. This chapter shall consider the
configurations of assembly features where contact on all the potentially contacting surface pairs
is maintained.
No Clearance
Clearance
Clearance
B
D
A
C
(a)
(b)
(c)
Fig. 3-1: Square-Peg in Square-Hole
This section presents the basic types of surfaces first. The types of contacts among the surfaces
shall be presented next. The assembly features are made by a set of contacting surface pairs. The
method to construct the assembly features from contacting surface pairs is presented in this
section. The assembly features are characterized by the degrees of freedom allowed by them.
This section is organized in the following fashion. First sub-section presents the concept of twists
and wrenches. Twist-matrix representation is used in this chapter to represent the degrees of
freedom allowed by the assembly features. Second sub-section presents the basic surfaces and
the types of contact among them. Third sub-section presents the method to calculate the twistmatrix of the contacts between any two basic surfaces. Fourth sub-section presents the method to
construct the twist-matrix representation of assembly features constructed from multiple
contacting surface pairs. Fifth sub-section presents solved examples.
36
3.1.1 Twist and Wrench Representation:
[Ball, 1900] or [Waldron, 1966] can be used for detailed reference to screw theory. The
following definitions to twists and wrenches should suffice for the purpose of analysis in this
chapter.
Twists:
A twist is a screw which describes to first order the instantaneous motion of a rigid body: T =
[ox, oy, Oz, vX, vy, vz]. The first triplet represents the angular velocity of the body with respect to
a global reference frame. The second triplet represents the velocity, in the global reference
frame, of that point on the body or its extension that is instantaneously located at the origin of the
global frame. The line vectorI represents the rotation vector, if any, of the body, and is called the
instantaneous spin axis (ISA). The free vector2 represents the body's translation, whose
magnitude may depend on the location of the unique point associated with it. If a body can
undergo more than one independent motion, there is a separate twist for each one, and the set of
all these independent motions is represented by combining all the twists as a stack of rows called
a twist-matrix. If a twist represents only linear motion, the first triplet entries are zero. If the axis
of the rotation passes through the global reference frame, the second triplet entries are zero.
Wrenches:
A wrench is a screw which describes the resultant force and moment of a force system acting on
a rigid body. The first triplet describes the resultant force in a global reference frame. The second
triplet represents the resultant moment of the force system about the origin of the global frame. A
wrench is also written as a row vector W =[fx, fy, fz, m., my, mz]. The first triplet represents
independent force that can be resisted by the wrench, while the second triplet represents moment.
If a body is acted on or can resist several independent forces or moments, there is a separate
wrench for each one, and the set of all these independent forces and moments is represented by
combining all the wrenches as a stack of rows called a wrench-matrix.
1Line vector is a vector with a fixed location in the space. Rotation about an axis is a line vector.
2 Free vector can float in the space. It has no fixed location. Translation in a direction is a free vector.
37
Relation between Twists and Wrenches:
When two rigid bodies interact by contacting without friction, they restrict each other's motions
and exert forces and torques on each other. Twists express the motions and the wrenches express
forces and torques. Under these conditions, the wrench and twist are such that the wrench cannot
do any work along the direction of the twist. Thus, the reciprocal of a twist is a wrench and vice
versa (Mathematically speaking, wrench-matrix is the (orthogonal) complementary space of the
twist-matrix). If the rank of a twist is n, then the rank of its reciprocal wrench is 6-n. The
wrench-twist pair that are reciprocal of each other, form complementary spaces: if the twist
describes directions along which motion is allowed, then the wrench describes directions that can
resist forces or moments. The function "reciprocal" is a combination of two operations: 1)
computing the null space of the screw matrix S, and 2) "flipping" the first three elements of the
result with the last three. "Flipping" exchanges the columns of the matrix according to the
following pattern: i .i+3 mod(6) 3.
3.1.2 Basic Surfaces and Types of Contacts:
It is becoming increasingly feasible to use complex surfaces with the help of advanced
manufacturing techniques in order to satisfy functional requirements. However, most of the
surfaces involved in the assembly features are planar, cylindrical or spherical. [Clement, 1991]
presented a classification of basic surfaces. A similar classification of surfaces has been used in
this chapter. The surfaces have been divided in the following categories:
Any Surface, Helical Surface, Surface of Revolution, Cylindrical Surface,
Planar Surface, Spherical Surface.
Any surface includes all the surfaces that are not included in other five categories. The contact
between two surfaces is the building block of the assembly features. The contact area between
two surfaces can be of different types: a surface patch, a "line" 4 segment or a point.
1. Contact Area:: Surface Patch:
Surface contact occurs when the two contacting surfaces are identical in a finite region
around a point.
3 Strictly speaking, the flip operation is not fundamental to the concept of reciprocal. It is necessary in order for the
elements of the resulting wrench to come out in the order [f M].
4 The word line is written within quotes ("") and it refers to a topologically one-dimensional entity (e.g. curve,
straight line).
38
2. Contact Area:: "Line" Segment:
"Line" contact occurs when the two contacting surfaces are touching along a curve (or
straight line). The curved (or straight-line) contact is referred as "line" contact.
3. Contact Area:: Point:
Point contact occurs when the contacting surfaces have different curvatures locally.
A surface contact may constrain anywhere from three to six degrees of freedom. Similarly, a
"line" contact may constrain anywhere from two to six degrees of freedom (Fig. 3-2). Fig. 3-2(a)
shows a "line" contact on a plane. This contact provides four relative degrees of freedom
between the contacting surfaces. Fig. 3-2(b) shows a "line" contact that is in one plane. The
contact here is a curve in a plane. This type of contact may provide zero, one, two, three or four
relative degrees of freedom between the contacting surfaces depending upon the curvature of the
contacting curve and the gradient of the contacting surfaces along the contacting curve. Fig. 32(c) shows a "line" contact that is three-dimensional. Such a "line contact" may constrain all six
relative degrees of freedom between the contacting surfaces. However, a point contact constrains
one and only one degree of freedom.
Line Contact
with four
DOFs
(a)
Line Contact
with three
DOFs
Line Contact
with two DOFs
Line Contact
with one DOF
(b)
Line Contact
with zero
DOFs
(c)
Fig. 3-2: Different type of "Line" Contacts
The following table presents the types of contacts possible between the basic surfaces 5 .
Appendix A presents more explanation regarding all the combinations among the basic surfaces.
5 This table only considers the contact between interiors of two surfaces. The contact between the edge of one
surface with the interior of other surface or the contact between edges of two surfaces shall be discussed in
Appendix A.
39
Appendix A shall also cover the contacts between basic surfaces and their edges and contacts
between edges of two basic surfaces.
Table 3-1: Surface-to-Surface Contacts
Any
Surface of
Helical
Cylinder
Planar
Spherical
Revolution
"
Any
Helical
Surface of
Revolution
Point
0
Point
*
Point
*
Point
0
Point
0
Point
6
*
Line
*
Line
*
Line
*
Line
*
Line
*
Line
*
Surface
0
Surface
e
Surface
e
Surface
9
Surface
*
Surface
*
Line
*
Point
*
Point
*
Surface
Point
*
0
Point
*
Point
*
*
Point
L
Line
Point
0
Point
0
Line
*
Line
*
*
Surface
Line
*
Point
*
Point
*
Line
0
Line
*
Surface
0
Line
*
Point
*
Point
*
Surface
Cylinder
Planar
0
Spherical
6 Some
Surface
of the entries in the table are in italics because these entries are possible only if any surface is locally
matching to the contacting surface (e.g. If a helical surface comes in contact with an any surface the contact area can
be a surface only if the any surface is locally a matching helical surface.
40
Contact between two surfaces can also be classified based upon its load bearing capacity, as
follows:
Unidirectional Contacts:
All the degrees of freedom constrained by unidirectional contacts will be such that it will not
support 'force" bi-directionally in any direction. Unidirectional contacts result into force
closure. The examples of unidirectional contacts are as follows:
1. Surface Contact:
A unidirectional surface contact may constrain from three to six degrees of freedom.
Examples:
a. Any Surface with Any Surface: Relative DOFs = None, one or two
b. Plane with Plane: Relative DOFs = Two translational, one rotational
2. Line Contact:
A unidirectional line contact may constrain from two to six degrees of freedom.
Examples:
a. Any Surface with Any Surface: Relative DOFs = None, one, two or three
b. Surface of Revolution with Cylinder: Relative DOFs = One translational, one
rotational
c. Surface of Revolution with Plane: Relative DOFs = Two translational, two rotational
d. Cylinder with Plane: Relative DOFs = Two translational, two rotational
3. Point Contact:
All point contacts are unidirectional in nature. A point contact constrains one and only
one degree of freedom
Examples:
a. Any Surface with Any Surface
b. Any Surface with Helix, Surface of Revolution, Cylinder, Plane or Sphere
c. Helix with Surface of Revolution, Cylinder, Plane or Sphere
d. Surface of Revolution with Cylinder, Plane or Sphere
e. Cylinder with Cylinder or Sphere
f. Plane with Sphere
g. Sphere with Sphere
41
Bi-directional Contacts:
The bi-directional contacts will have at least one degree of freedom constrained in such a way
that any instantaneous motion will not break the contact. (It will have the ability to support the
force bi-directionally at least in one direction). Bi-directionally constrained directions result into
form closure. All the degrees of freedom constrained by an assembly feature may not be bidirectionally constrained because of assemblability.
1. Surface Contact:
Example:
a. Helix with Helix: Relative DOFs = One rotational (coupled with translation)
b. Surface of the revolution with the same: Relative DOFs = None, one or two
c. Cylinder with Cylinder: Relative DOFs = One translational, one rotational
d. Sphere with Sphere: Relative DOFs = Three rotational
2. Line Contact:
Example:
a. Surface of Revolution with Sphere: Relative DOFs = One translational, three
rotational
b.
Cylinder with Sphere: Relative DOFs = One translational, three rotational
3.1.3 Basic Surface Contacts and their Twist-Matrices:
When two surfaces touch and one is considered fixed in space, the other loses some of its
degrees of freedom. Many surface contacting pairs are created by several combinations of two
basic surfaces. The contacting area of the two contacting surfaces may be a surface patch, a
"line" segment or a point depending upon the two contacting surfaces. These possibilities have
been shown in Table 3-1 (the details can be found in Appendix A).
To determine what degrees of freedom remain once two surfaces contact, one can make use of
Screw Theory. For example, the cylinder-plane contact illustrated in Fig. 3-3. If the plane is
assumed stationary, then the cylinder can move in four degrees of freedom (The cylinder can
translate along Y and Z axes and it can rotate about X and Z axes). The same result is achieved if
the cylinder is assumed stationary and the plane moves, of course.
42
z
R=
Fig. 3-3: Cylinder on Plane
Each of these motions can be described by a twist matrix. In this case, the matrix will have four
rows, one for each of the possible relative motions. The contact area for this contacting surface
pair is a straight line parallel to Z-axis. When the cylinder rocks on the plane about the
contacting line, an imaginary point on the cylinder that coincides with the origin of its frame
moves in the Y-direction. Thus the twist contains a non-zero entry in the third place representing
unit rotation about Z and a non-zero entry in the fifth place representing the resulting translation
along X. Similarly, one can find the entries for other rows of the twist-matrix. The twist-matrix
will be:
T=[0 0 10 10; 10 0 0 0 0; 0 0 0 0 1 0; 0 0 0 0 0 1]
There may be situations where the geometry of contact area is not so simple. The visualization
would not help in such situations. This section presents a method to determine the twist-matrix of
the contacting surface pair in such situations. This method does not depend on visualization and
it can compute the twist-matrix of a contacting surface pair of arbitrarily complex geometry.
Exact number of DOFs constrained by a surface contact or a "line" contact depends upon the
geometry of the contact. It can be calculated by dividing the contacting surface or contacting
curve (contact area) into multiple points. Each point contact will constrain only one degree of
freedom. A point contact can support only one force along the direction of the normal vector to
the tangent plane passing through the contact point7 . So, any point contact will allow five
degrees of freedom (five independent motions). If the wrench-matrices of multiple point contacts
7 Tangent plane will be defined as long as the contact point is on the interior of the two surfaces and the two surfaces
have continuous second order derivative. The two surfaces in such situations will have a common tangent plane at
the contact point.
43
are combined it will give the wrench-matrix of the surface or "line" contact. The method of
calculating the wrench-matrix of a surface or "line" contact consists of the following three steps:
1. Dividing the Contact Area (Curve or Surface) into Points
The contact area can be a surface or a curve. There exist a set of minimum points which can
determine the degrees of freedom constrained by the contact area. The points in this set are
such that the union of their wrenches spans the wrench-space of the contact area. The
condition number of the union of the wrenches of the chosen points should be satisfactory.
This set of minimum points cannot have more than six points because a rigid body can only
have at most six degrees of freedom. The number and choice (in terms of location on the
contact area) of the points depend upon the actual geometry of the contact area. Here things
like rate of change of curvature of the contact area at a point on it will become important
because otherwise the set of points, which one may choose, may not be representative of the
contact area (The union of the wrenches of the chosen points may not span the wrench-space
of the contact area). This set of points is not unique. The optimal location of points depends
upon the actual geometry of the contact area. However, there is no harm if one takes more
points than those in the minimum set. Here, an approach of dividing the whole contact area
into uniformly distributed hundred points was taken.
2. Generating Wrench-Matrices of the chosen Points
The point contact constrains only one degree of freedom. The point contact supports the
force along the normal vector to the tangent. The wrench-matrix of the point contact will
have only one row and it will be populated by the force along the normal vector to the
tangent plane and its moment about the origin of the co-ordinate system. Fig. 3-4 shows a
schematic example.
44
Wrench - Matrix = [fi; f * ii]
Fig. 3-4: Wrench-Matrix of a Point Contact
3. Generating the Twist-Matrix for the Contacting Surface Pair
The union of the wrench-matrices of all the chosen points on the contact gives the wrenchmatrix of the contacting surface pair. The union of wrench-matrices is defined as follows:
WU = [W1;W2;W3;....;Wn];
The twist-matrix of the contacting surface pair is the reciprocal of its wrench-matrix. One can
get the wrench-matrix from the twist-matrix by calculating its null space. The following
figures (3-5 and 3-6) give examples of the different types of "line" contacts. Fig. 3-5 shows
a simple two-dimensional example. The contacting surfaces touch along the circular arc
which lies in XY plane. The center of the contacting curve is at (0,0,0). The analysis predicts
the following twist-matrix for the contacting surface pair:
T=[1 0 0 0 0 0; 0 1 0 0 00; 0 0 1 0 00; 0 0 0 0 0 1]
Thus, the contacting surface pair allows rotations about all three axes and translation about
Z-axis which is expected.
Fig. 3-5: Two-Dimensional "Line" Contact
45
Fig. 3-6 shows a three-dimensional "line" contact. The two contacting surfaces are
represented by dotted lines in the figure. The entire curve is constructed from three different
circular arcs. The first segment of the curve from (-1,-1,1) to (0,0,1) is a circular arc in Z=1
plane with center at (0,-1,1). The normal vector in this segment of the curve lies in the Z=1
plane and they are perpendicular to the contacting curve. The second segment of the curve
from (0,0,1) to (1,0,0) is a circular arc in Y=O plane with center at (0,0,0). The normal vector
in this segment of the curve is constant and it is along Y-axis. The third segment of the curve
from (1,0,0) to (1,1,-1) is again a circular arc in plane X=1 with center at (1,1,0). The normal
vector in this segment of the curve lies in the X=1 plane and it is perpendicular to the
contacting curve.
All three circular arcs have radius of one unit. The contacting surfaces have been assumed in
such a way that the common tangent plane exists at every point of the contacting curve. The
result of the analysis predicts the following twist-matrices for each segment of the curve:
For first segment of the curve:
[0 1-100 0; 10001 0;00 1-1 00;00000 1]
For second segment of the curve:
[0 10000;000 100;00000 1]
For third segment of the curve:
[1 1 00 00; 0 10 0 0 1; 0 0 10 -1 0; 0 0 0 10 0]
The following twist-matrix is predicted for the contacting surface pair (i.e. for the entire
curve taking into account all three segments):
T=[0 10 -10 1]
46
Fig. 3-6: Three-Dimensional "Line" Contact
*
The twist-matrix for the contacting surface pair can be constructed by considering the entire
curve as one entity and it is not required at all to consider how the curve has been
constructed. The results for the twist-matrices of different segments have been given to help
readers in visualization. However, one can get the twist-matrix of the contacting surface pair
by intersecting the twist-matrices of three curve segments8 . Exactly, same result is obtained
in both cases.
The contacting surface pair allows rotations about an axis parallel to Y-axis passing through
the point (1,0,1). The last three entries represent the velocity of the point on the moving body
(or its imaginary extension) which is located at the origin due to the unit rotation about the
axis of rotation.
3.1.4 Method to Calculate the Constraint Representation of the Assembly Feature:
An assembly feature may allow certain motions and it shall restrict motion in the complementary
directions. Twist-matrix represents the space of the instantaneous motion allowed by the
assembly feature. Wrench-matrix represents the instantaneous constraints imposed by one part
on the other.
47
The following method is used to calculate the twist-matrix of an assembly-feature:
1. Identify twist-matrices of all the contacting surface pairs of the assembly feature
An assembly feature may or may not have different contacting surface pairs. The twistmatrix of the contacting surface pair may be constructed by visual inspection in simple
cases. The method described in the previous section can be used if the geometry of
contact area is complex.
Example: A lap joint between two planes has just one contacting surface pair of a plane
touching another plane. On the other hand, a square peg in a square-hole has five
contacting surface pairs.
2. Intersecting Twist-Matrices of all contacting surface pairs
[Konkar, 1993] defined intersection of twist-matrices. Intersection of "n" twist-matrices
is defined as follows:
Sintersection = Reciprocal
(U
Reciprocal Si)
i=1
Reciprocal (Si)
=Reciprocal
Reciprocal (S2)
-Reciprocal (Sn)
Si represents a twist-matrix. Reciprocal of a twist-matrix is wrench-matrix. One can get
the wrench-matrix from the twist-matrix by calculating its null space. Rows of the twistmatrix and that of the wrench-matrix span complementary orthogonal subspaces of a sixdimensional space.
3.1.5 Examples:
1. Square Peg in Square-Hole:
Fig. 3-7 shows a square hole. A square peg in a square-hole assembly feature has five contacting
surface pairs (fifth contacting surface pair "T5" is not shown in the figure). All the contacting
surface pairs are planar lap joints. Intersecting the twist-matrices of all the contacting surface
8 Intersection of twist-matrices is defined in next sub-section
(3.1.4).
48
pairs give the twist-matrix of the assembly feature. This assembly feature constrains all six
relative degrees of freedom.
z
T
4
F
Fig. 3-7: Square Peg in a Square-Hole Assembly Feature
The twist-matrix of first contacting surface pair:
T1=[0 0 0 1 0 0; 0 0 0 0 0 1; 0 1 0 0 0 01;
The twist-matrix of second contacting surface pair:
T2=[0 0 0 0 1 0; 0 0 0 0 0 1; 1 0 0 0 0 0];
The twist-matrix of third contacting surface pair:
T3=[0 0 0 1 0 0; 0 0 0 0 0 1; 0 1 0 0 0 0];
The twist-matrix of fourth contacting surface pair:
T4=[0 0 0 1 0 0; 0 0 0 0 1 0; 0 0 1 0 0 0];
The twist-matrix of fifth contacting surface pair (at bottom of the hole, not shown in the figure):
T5=[0 0 0 1 0 0; 0 0 0 0 0 1; 0 1 0 0 0 0];
The intersection of the contacting surface pa
T= An Empty Matrix
2. Pin-Slot Assembly Feature:
The pin-slot assembly feature has two contacting surface pairs. The slot will have two parallel
walls made from the two planes. These walls will be spaced apart exactly the diameter of the pin.
The pin forms the contacting pair with either of the planar wall on the slot (see Fig. 3-8).
The twist-matrix of the first contacting surface pair:
T1=[0 0 0 1 0 0; 0 0 0 0 0 1; 0 0 1 1 0 0; 0 1 0 0 0 0];
The twist-matrix of the second contacting surface pair:
49
T2=[000 100;00000 1;00 1-100;0 10000];
The intersection of the two contacting surface pairs:
T=[O 0 0 10 0; 0 0 0 0 0 1; 0 0 10 0 0; 0 10 0 0 0];
9*x
T
y
T2
Fig. 3-8: Pin-Slot Assembly Feature
The intersection of these twist-matrices produces the twist-matrix of the pin-slot
assembly feature. The intersection shows that the assembly feature allows four
relative degrees of freedom (the pin can translate in two directions and rotate about
two others).
3. Prismatic Pair
The prismatic assembly feature has three contacting surface pairs (see Fig. 3-9). All the
contacting surface pairs are planar lap joints. Intersecting the twist-matrices of all the contacting
surface pairs give the twist-matrix of the assembly feature.
x
T2
T3
Fig. 3-9: Prismatic Pair
50
The twist-matrix of first contacting surface pair:
T1=[O 0 0 1 0 0; 0 0 0 0 0 1; 0 1 0 0 0 0];
The twist-matrix of second contacting surface pair:
T2=[O 0 0 1 0 0; 0 0 0 0 0 1; 0 1 0 0 0 0];
The twist-matrix of third contacting surface pair:
T3=[0 0 0 1 0 0; 0 0 0 0 1 0; 0 0 1 0 0 0];
The intersection of the contacting surface pairs:
T=[ 0 0 10 0];
3.2 Identification of Chain of Mates in CAD
CAD systems also use the features such as slots, ribs and holes etc. However, these features
should not be confused with assembly features. The features provided in CAD systems help
users in creating the solid model of one part. These features do not carry any information about
the constraint they may carry due to assembly of one part to another. [Hoffman and John-Arinyo,
1998] extended the functionality of CAD features by proposing a mechanism by which the user
can build its own feature for solid modeling.
In CAD systems, an assembly is created by constraining detailed solid models of parts with one
another. Most often, the dimensional relations that are explicitly defined to build an assembly
model in CAD are those most convenient to construct the CAD model and are not necessarily the
ones that need to be controlled for proper functioning of the assembly. Now, CAD systems are
trying to automate the process of assembly. This means that the designer provides fewer input
and most of the positioning of the parts is decided automatically. [Chang and Perng, 1997]
presented a method for automatically positioning parts in an assembly with some input from
user.
51
To understand the degrees of freedom of different parts in final assembly, the kinematic
information need to be associated with the mating relationships of an assembly (e.g. a peg and
hole assembly feature allows only one motion: rotation about the axis of the pin). [Mullins and
Anderson, 1998] presented a technique to automatically identify the geometric constraints in
mechanical assemblies. They developed a method of identifying the constraints from the
algebraic representation of mating surfaces. For example, the constraint passed from one planar
surface to another planar surface may be represented by an equation. The technique developed in
this paper differentiated between mating conditions and kinematic joints. Geometric relationship
of a mating condition is static (e.g. gap between two static surfaces or two static surfaces in
contact). Kinematic joints allow motion (e.g. revolute joints etc.).
The variation analysis and allocation of tolerances is also an important task to ensure
functionality and to limit the cost of the product. Variation analysis of an assembly level
dimension requires a chain of mates. Variation analysis is often performed by separate computer
aided tolerancing (CAT) tools. CAT tools take the input from CAD solid models. The tolerance
models may be created with the help of input from a user or it can be created automatically from
the CAD solid models. Prof. Clement introduced the idea of "Technologically and Topologically
Related Surfaces" (TTRS) to create tolerance models of three-dimensional solid models
[Clement, 1991]. TTRS is a methodology that identifies the chain of mates from solid models of
assemblies. TTRS methodology looks for basic surfaces and it identifies the contacting surface
pairs to find a chain of mates for an assembly level dimension. TTRS was used in CATIA
software (CATIA v. 4.17). It is important to describe this approach here because of two reasons.
First, this approach also uses the surface pairs to represent the kinematic structure of the
assembly. The information about degrees of freedom is associated with surface pairs. It will be
shown that TTRS is inadequate regarding handling some of the relative degrees of freedom that
are very normal in assemblies. The second and more important reason to discuss TTRS is that it
has no formal rules for identifying the chain of mates. This approach shall be described in brief
in the following sub-section.
52
3.2.1 TTRS:
This sub-section briefly introduces the methodology of TTRS. This sub-section is organized in
the following fashion. First sub-section presents the definitions about TTRS. Second sub-section
presents the analogy between TTRS and screw representation of contacting surface pairs. This
sub-section shows that TTRS, in its current form, is inadequate to represent the degrees of
freedom between two parts in an assembly. Third sub-section describes the process used by
TTRS to identify the chain of mates in an assembly through an example. Fourth sub-section
describes the inadequacies of this process of identifying the chains of mates.
3.2.1.1 Definitions:
Clement divided the surfaces in the following seven categories:
Any Surface, Prismatic Surface, Surface of Revolution, Helical Surface,
Cylindrical Surface, Planar Surface, Spherical Surface
The definition of a TTRS is as follows:
"A TTRS is defined as an assembly formed by two surfaces (or surface and TITRS or between two
TTRS) belonging to the same solid (topological aspect) and located in the same kinematic loop
in a given mechanism (technologicalaspect)."
The TTRS were created to do tolerance analysis of assemblies. Clement proposes to draw an
"assembly graph" corresponding to the assembly. Twenty-eight unique combinations of these
seven surfaces were identified by Clement. All the surfaces were considered unbounded. All the
different combinations were assigned an "unchanging vector" and a complementary "changing
vector". Changing vectors represent the directions along which the variation is not allowed or in
other words the variation along these directions shall affect the geometry of the part and the
assembly. Similarly, unchanging vectors represent the directions along which the variation does
not affect the part geometry or assembly configuration. The definitions of these terms are as
follows:
Unchanging Vector:
The independent directions, along which the movement of a particle on the TTRS, does not take
it off the TTRS, are defined as unchanging directions. Example: The rotation about the axis and
the translation along the axis are two independent unchanging vectors for a cylindrical surface.
53
Changing Vector:
The independent directions, along which the movement of a particle on the TTRS, does take it
off the TTRS, are defined as changing directions. Example: Except the rotation about the axis
and the translation along the axis, the other four independent degrees of freedom are changing
vectors for a cylindrical surface.
3.2.1.2 Analogy between TTRS and Screw Representation of Contacting Surface Pairs:
Unchanging vectors are analogous to twist-matrix of the contacting surface pair and changing
vectors are analogous to wrench-matrix of the same. The variation along the directions of twistmatrix has no meaning because the motion is allowed along these directions.
Example: The variation in the x-direction for a prismatic pair has no meaning because the
prismatic pair allows translation along x-axis (see Fig. 3-10).
The variation along the directions of wrench-matrices affects the assembly configuration.
Example: If the slot in the prismatic pair is shifted in y-direction the part on which the pin is
located will also get shifted by same amount in y-direction. Y-direction is constrained by this
assembly feature (see Fig. 3-10).
Fig. 3-10: Variation in Prismatic Pair
However, the analogy is neither complete nor perfect. The unchanging and changing vectors
have been defined in such a manner that TTRS misses certain relative motions between two
surfaces on two different parts. This makes TTRS inadequate for analyzing assembly problems.
The theory of TTRS does not handle all the relative degrees of freedom between two surfaces on
two different parts. The "changing" and "unchanging" directions are assigned in such a fashion
that they do not take into account all the possible relative motions between two surfaces on two
different parts. Consider the case of a cylinder on a plane (see Fig. 3-11). Fig. 3-11(b), (c), (d)
and (e) show the possible relative motions between the two surfaces. However, theory of TTRS
54
forms the following changing and un-changing vectors for the same contacting surface pair
(cylinder on plane):
The changing vector for this configuration:
C=[1 0 0 0 0 0; 0 1 0 0 00; 0 0 10 0 0; 0 0 0 10 0; 0 0 0 0 10];
The unchanging vector for this configuration:
UC=[O 0 0 0 0 1];
Translation along Z-axis is the only motion that will not take a point off from both of the
surfaces. Consider, any point on cylinder. Translation along Z-axis and rotation about the axis of
the cylinder is allowed for a point on the cylinder. However, rotation about Z-axis shall take a
point on the plane off it. Hence, only translation along Z-axis survives. Table 3-2 shows the
changing and unchanging directions of TTRS for "cylinder on plane" contacting surface pair.
Table 3-3 shows the twist and wrench directions for the same contacting surface pair.
Unchanging vector corresponds to twist direction and changing vector corresponds to wrench
directions. Tolerancing problem of an assembly, which has an assembly feature having
cylindrical surface on one part and planar surface on another part, cannot be solved properly by
TTRS because relative motion is possible along some of the changing vectors of this contacting
surface pair (cylinder on plane). Relative motion is not same as variation.
zz
z
R= R=y
x
z
y
y
I
Cylinder on
Plane
(a)
z
X-Rotation
(b)
Il
Y-Translation,
Pure Sliding
Z-Translation,
Pure Sliding
(c)
(d)
Xx
Y-Translation
+ Z-Rotation,
Pure Rolling
(e)
Fig. 3-11: Motions for Cylinder on Plane Contacting Pair
TTRS methodology is fine for tolerancing of one part because the question of relative motion
simply does not arise in case of multiple surfaces on one part. However, an assembly has at least
two parts and TTRS is not capable of handling all the relative motions between parts.
55
Table 3-2: Changing and Unchanging Vectors for "Cylinder on Plane" Assembly Feature
TTRS Changing
Directions
All the Others
TTRS Unchanging
Directions
Z-Translation
Table 3-3: Twist and Wrench Directions for "Cylinder on Plane" Assembly Feature
Twist Directions
Z-Translation
Y-Translation
X-Rotation
Wrench Directions
X-Translation
Y-Rotation
Z-Rotation +Y-Translation
3.2.1.3 Identification of Chain of Mates in TTRS9 :
In Clement's method, the first step is to identify the different TTRS and their related datum coordinate frames belonging to the different parts of the assembly. Clement calls the datum
associated with a TTRS as Minimum Geometric Datum Element (MGDE). The formal definition
of MGDE is given in this sub-section. Clement also proposes to draw an "assembly graph"
corresponding to the contacting surface pairs of assembly to identify the chain of mates. Clement
calls the chain of mates as loops of TTRS.
The definition of MGDE shall be given first. The process of making the assembly graph,
identifying the independent loops and then arriving at the sequence of analysis is described with
the help of an example later.
MGDE:
To represent and to localize any surface in a Euclidian space, it is necessary to associate the
surface to a datum system. The Minimum Geometric Datum Element (MGDE) plays this role for
9 Section 3.2.1.3 gives details about TTRS methodology and this section is based upon an internal report submitted
to Center of Technology Policy and Industrial Development at MIT MIT by Benoit Marguet [Marguet, 1998].
56
the different type of TTRS. This concept was proposed by [Clement, 1993] through the
following definition:
"The Minimum Geometric Datum Element, or MGDE, of a TTRS is the minimum set of points,
lines or planes necessary and sufficient to define the reference frame corresponding to the
invariant sub-group of that TTRS".
According to Clement, MGDE are useful for:
" To give mathematical representation of TTRS. MGDE allows representation of position and
orientation of TTRS. Moreover, MGDE could be used to represent degrees of freedom or
invariant displacements.
*
To put tolerance specification for TTRS.
*
To ensure assembly feasibility. Verification of assembly for a mechanical product could be
performed by fitting each MGDE representing TTRS associated at different assembly parts.
The information about degrees of freedom is derived from changing and unchanging vectors as
described in previous sub-section (3.2.1.2). Tolerance specification can be associated to the
changing directions of MGDE because variations propagate only along the changing directions.
Process of Identifying and Analyzing Independent Loops in the Assembly Graph:
Fig. 3-12(a) shows an assembly of three cubes. Fig. 3-12(b) shows the surfaces on the parts. Fig.
3-12(c) shows the graph of this assembly. The arc between surface S12 and S32 is shown in
dotted line because it corresponds to non-functional surfaces. The goal of TTRS methodology is
to find the tolerance specifications for functional surfaces. Functional surfaces correspond to
"contact surface" (i.e. surfaces which pass constraint). The tolerance specifications are chosen so
as to keep the variation on non-functional surfaces with in requirements. S12 and S32 are nonfunctional surfaces in case of example assembly shown in Fig. 3-12. TTRS methodology finds
changing vectors for non-functional surfaces. Changing vectors represent directions along which
variation is possible. Variation is analyzed using the following process:
" Identify the TTRS on parts
" Construct the MGDE on parts
" Find the kinematic loops in assembly
" Identify the sequence of analysis for different kinematic loops
57
* Use TTRS methodology to predict directions where variation shall propagate or not
*
Analyze kinematic loops in a pre-identified sequence using directions of variations
Kinematic loop refers to a closed chain of mates in an assembly. It is identified from the solid
model. In order to identify each TTRS (that means each functional surface belonging to the same
part and the same kinematic loop) all independent loops of the graph are determined. For
example, three independent loops are found for the assembly shown in Fig. 3-12. The
independent loops are as follows:
Loop: S11-S12-S32-S31-S22-S21
Loop2: S12-S32-S33-S13
Loop3: S11-S21-S23-S13
S21
S22
S32
A
B
B
S23
S31 S33
Base
Base
Sil
(a)
S12
(b)
Base
S12
S13
2
A
S23
S13
S22
BS3
(c)
Fig. 3-12: TTRS and Assembly Graph
After identification of loops, the following tasks need to be resolved:
1. Determine the starting loop (to begin the analysis).
2. Determine the sequence in which the loops need to be analyzed.
[Clement et. al., 1991] proposed the following criteria for choosing the starting loop and the
sequence:
58
"
Choose one-dimensional loops passing through contacts having their normal vectors pointing
in opposite direction.
" Choose loops generating the TTRS that were designed using dimensional assistance or
through technical functions.
Jonge Poerink [Poerink, 1994] added two additional criteria for loop selection:
" Avoid loops that contain only one surface on a part.
" Choose loops containing the simplest basic geometry.
The theory of TTRS does not give any proof or reasoning for these rules. TTRS acknowledges
that if different sequence of graph exploration is selected the tolerance specification will be
different. However, it claims that the different specifications corresponding to different sequence
of exploration of graphs will be correct.
3.2.1.4 Inadequacy of the Process of Identifying and Analyzing the Independent Loops in
TTRS:
The process of identifying kinematic loops (closed chains of mates in an assembly) cannot
differentiate between mates and contacts. Moreover, the process of identifying the start loop
from the independent loops and then finding the sequence in which the loops need to be analyzed
seems to have arisen by trying different approaches on simple examples. No theoretical
reasoning has been provided for these rules. Moreover, theory of TTRS is unclear regarding
which loop should be analyzed in the context of an assembly level dimension. There might be
several independent loops in an assembly and several assembly level dimensions to be analyzed.
Even worse, if the proper care has not been taken in designing the parts, there may be multiple
sets of loops which can be used for analyzing one assembly level dimension. The analyses of
different sets of loops may give conflicting results. This phenomenon corresponds to multiple
chains of mates for one assembly level dimension. TTRS does not have rules for finding the
relationships between kinematic loops (closed chains of mates) and assembly level dimensions.
The rules for finding the sequence of analysis are vague.
59
3.3 Comparison between the Feature-Based Approach of Top-Down Method
and Feature Recognition Approach of Bottom-Up Method:
The bottom-up approach relies on automatic detection of design intent instead of explicit
declaration of the same. Methodologies like TTRS have inadequate processes of identifying and
analyzing the chains of mates for assembly level dimensions. TTRS, in its current form, is also
incapable of representing all the relative motions between two parts in an assembly due to the
way the unchanging and changing vectors are defined. If attempts are made to rectify theory of
TTRS to handle relative degrees of freedom between two parts it shall amount to duplicating the
screw theory.
DFC supports the top-down design process. Designers can configure the chain of mates to
achieve desired functionality and assembly features can be selected from a library or constructed
to realize the chain of mates. Constraint and variation analyses can be performed using the
constraint representation of assembly features.
The context of features on a part may not be obvious to a designer in case of the bottom-up
approach because it is not known to him/her what assembly level dimensions may be affected by
a feature on his/her part. Upfront deliberations regarding designing the chain of mates shall also
enable explicit relationship between key characteristics (customer requirements) and assembly
architecture. Moreover, the analysis of assembly can be done without detailed level part design.
Making changes at this stage may cost much less than it would cost when the detailed level part
design has already been done and the identified chain of mates reflect that assembly level
dimension cannot be held within desired specifications in the scope of allowed variations on
part-level dimensions.
Another aspect of the comparison between the process of designing the chain of mates and that
of identifying one from solid models is identification of mistakes. Automatic constraint detection
techniques may find the incorrect chain of mates or it may find one out of the multiple chains of
mates. Over-constrained assemblies may have multiple chains of mates for an assembly level
dimension and assembly may be over-constrained in the first place because the features on the
parts were added without understanding their need or context in the assembly. The mistakes may
60
not be identified at all during the analysis or it may become increasingly hard to locate the main
source of variation accurately.
3.4 Summary:
This chapter presented a method to construct assembly features using the basic surfaces. This
method requires information about the configuration of contacting surface pairs and it can handle
arbitrarily complex surfaces. No visualization is required in this method. The method produces
the constraint representation of an assembly feature in terms of screw theory. Assembly features
realize the constraint structure represented by the design team through DFC as their intent of
design.
In case of bottom-up approach, parts are designed individually and the chain of mates is often
found through automatic constraint detection techniques. TTRS is one such technique which
finds the chain of mates from 3D solid models. It has been shown that it is inadequate, in its
current form, regarding identifying the sequence of chains and regarding representing some of
the relative motions among parts.
The next chapter presents motion and constraint analysis. Motion analysis finds underconstraints and constraint analysis finds over-constraints. The approach of CAD systems for
motion and constraint analysis shall also be presented in the next chapter.
61
62
Chapter 4: Motion and Constraint Analysis'
This chapter presents a new method for evaluation of constraint properties of assemblies. This
comprehensive method is applicable to all DFCs. Certain types of DFC require detailed
kinematic analysis. This shall be discussed while presenting the method. This method finds all
under-constraints in the assembly. All over-constraints can be found when detailed kinematic
analysis is not required. When detailed kinematic analysis become necessary for finding underconstraints, exact information about only some of the over-constraints may be found. Qualitative
information about other over-constraints can be found in these cases too. The terms "constraint
analysis" and "mobility analysis" have been used by several researchers for evaluation of
degrees of freedom of a mechanism. However, in this chapter the term "motion analysis" shall be
used to refer to the method which finds the degrees of freedom of a rigid body in a mechanism or
structure. The term "constraint analysis" shall be reserved to refer to the method which finds the
degrees of freedom that are over-constrained. These methods use the screw theory based
constraint representation of assembly features. One can represent all physical mating conditions
in form of assembly features in the DFC using screw theory. DFC is a symbolic model of the
assembly.
The chapter is organized in the following fashion. First section presents the graphical technique
and associated algorithm for evaluation of constraint properties of the assembly. This section
also presents relevant references to previous work, detailed explanation, comparison among the
methods of motion analysis and solved examples. Second section presents the underlying process
of evaluating constraint properties of assemblies in CAD systems. Third section compares the
constraint analysis of the two approaches (top-down and bottom-up). Fourth section presents the
summary of this chapter.
1 This chapter is based on article [Shukla and Whitney, 2001]. Significant improvements have been made in this
chapter as far as the method for motion analysis of assemblies is concerned.
63
4.1 Graphical Technique for Evaluation of Constraint Properties:
This section presents a logical way of evaluating the constraint properties of the DFC. This
section presents two types of analyses: Motion analysis and Constraint Analysis. Motion analysis
finds under-constraints. Constraint analysis finds over-constraints. If no under-constraints and
over-constraints are found then the assembly is called properly constrained.
This section also presents the relevant references to the previous-work, explanation of the
method, limitations of the method, comparison of the method to algorithms proposed by earlier
researchers and solved examples. This section has been organized in the following fashion. First
sub-section provides references to the previous work. Second sub-section introduces the
graphical method. Third sub-section presents the method of motion analysis. Fourth sub-section
presents the comparison of the proposed method of motion analysis with the other methods
suggested by previous researchers. Fifth sub-section presents the method of constraint analysis.
Sixth sub-section presents solved-examples. Seventh sub-section discusses the limitations of the
method of motion and constraint analysis.
4.1.1 Previous Work:
Motion analysis of rigid body mechanisms is a more than hundred years old research topic
(Here, "motion analysis" refers to evaluation of degrees of freedom of a mechanism.
Researchers may have used the term "constraintanalysis" for the same.). Mobility equations
and other degree of freedom equations have been studied in the past by a number of investigators
such as Chebyshev, Sylvester, Grubler, Somov, Hochmon, Kutzbach etc. [Chebyshev, 1945] and
[Grubler, 1917] proposed formulae for mobility of planar mechanisms. Several other researchers
extended the capabilities of these formulae later on. [Voinea and Atanasiu, 1962] used theory of
the instantaneous screw axis ("Screw Theory") to overcome the problems of these equations.
[Ball, 1900] presented Screw Theory in a more concrete fashion. It took another sixty-five years
before Screw Theory was proposed as a tool to analyze mechanisms by [Waldron, 1966].
Waldron presented the concept of twist- and wrench-matrix. Twist-matrix is collection of screws
that represents relative motions between two rigid bodies. Wrench-matrix is also a collection of
screws that represents constraints exerted by one body on other. Waldron introduced the series
and parallel laws of instantaneous kinematics. Series and parallel law are important for serial
64
chains and purely parallel chains respectively. However, Waldron had no algorithm to evaluate
either under-constraints or over-constraints in an assembly. Several researchers augmented the
capabilities of Screw Theory in due course of time. [Davies and Primrose, 1971] pointed out for
the first time that series and parallel laws of Waldron are insufficient for determining the relative
freedom between any two bodies divided by cross coupling (see Fig. 4-8). This article proposed
a solution for planar linkages with cross coupling. [Baker, 1980] extended this method and he
proposed an algorithm to solve for the degrees of freedom of link with respect to other when the
two links are separated by cross coupling. He used of screw theory to represent threedimensional assembly joints. This algorithm was limited to closed loop problems. [Davies, 1981]
used kirchoff's circulation law to develop loop equations in the mechanisms in terms of relative
velocities of assembly joints (relative screws). This method was also limited to closed loop
problems. [Mohamed and Duffy, 1985] proposed an algorithm for solving the degrees of
freedom of a fully parallel manipulator using screw theory. They also used the approach of
forming loop equations. The problem of fully parallel manipulator can in fact be solved by using
series and parallel laws of Waldron and forming loop is not required. [Konkar, 1993] developed
an algorithm to intersect the twist-matrices. This algorithm can find the degrees of freedom of a
body under multiple constraints. [Konkar, 1993; Konkar 1995] also proposed an algorithm to
find the degrees of freedom of any link in a general mechanism. He claimed that his algorithm
would work for any general mechanism. However, this chapter shows that it does not even
implement Waldron's parallel law satisfactorily and it certainly fails in case of cross couplings.
The shortcomings of Konkar's algorithm are highlighted in this chapter. This chapter proposes a
method for motion analysis which can be used both for open and closed chains. It implements
Waldron's series and parallel law correctly and it proposes a procedure for solving the cross
coupled situations. However, screw theory alone is not sufficient to solve all assemblies. In some
cases, detailed kinematic analysis is necessary.
Analysis of over-constraints in mechanisms and structural assemblies did not get as much
attention as analysis of mobility (under-constraints). Kinematicians often referred to mechanisms
with less than desired degrees of freedom as over-closed or over-constrained. This chapter uses
the term "over-constraint" in a very strict sense to refer to degrees of freedom which are being
multiply constrained. [Davies, 1983] talked about redundancy formally. He defined redundancy
65
as number of constraints which are not required for intended purpose. He also gave a formula for
degree of redundancy for a mechanism. Other researchers like [Kriegel, 1994] have emphasized
the importance of properly constrained assemblies and evaluation of over-constraints but they
had no systematic procedure to evaluate the constraint situation. A systematic method to find all
over-constraints associated with every part in an assembly has not been presented so far.
4.1.2 Graphical Representation of DFC:
The method requires conversion of DFC into Part-Feature diagram. Part-Feature diagram is
another representation of DFC. Nodes on right hand side represent parts and nodes on left hand
side list all assembly features. An assembly feature is typically between two parts. However,
there may be assembly features that relate more than two parts. Each assembly feature node is
connected to the corresponding part nodes. An example Part-Feature diagram is presented in Fig.
4-1. There must be one and only one fixed part in the Part-Feature diagram. This part will
correspond to the root of the DFC. If multiple parts are grounded in the physical assembly, all of
them should be grouped together. Each assembly feature carries some constraints. The
constraints carried by an assembly feature can be represented by its twist-matrix. Every feature
node has a twist-matrix associated with it that represents the relative degrees of freedom between
two parts. The reciprocal of twist-matrix is wrench-matrix that represents the constraints
imposed by one body on the other. A twist-matrix will always have six columns. The number of
rows in a twist-matrix corresponds to number of degrees of freedom allowed by the
corresponding assembly feature. Each row in the twist-matrix represents motion allowed by any
one independent degree of freedom. The numbers in the twist-matrix become co-ordinate frame
dependent because of this reason. Details about twist-matrix representation and twist-matrices
for different assembly features can be found in the section 3.1.1 of chapter 3.
66
R2
L3
R3
O Ll
RI
L2
L2
L4
R2
L3
R3
Li
RI
L4
R4
R4
Fig. 4-1: Two Paths of the Four-Bar
4.1.3 Motion Analysis for A Part (Evaluation of Under-Constraints):
Motion analysis checks whether a part is under-constrained or not. Motion analysis needs to be
performed for every part in the assembly, if the under-constraints for every part need to be found
out. For evaluating whether a part is under-constrained, the procedure is as follows:
4.1.3.1 Constructing the Paths for Motion Analysis:
*
Identify all paths from the part in question to the fixed part. A path is defined as a sequence
of successive part and feature nodes starting from the part being analyzed and ending at the
fixed part. A path may have branches. Parts other than binary links give rise to branches in
the paths. For example, a ternary link will give rise to one branch. The paths and their
branches are identified in a depth-first manner starting from the part being analyzed. This
branch may be connected to the fixed link, it may be connected to the part being analyzed
itself, or it may be connected to some other branch of the same path or it may be connected to
another path (or branch thereof). At least one branch of the path must terminate at the fixed
part node. A part other than binary link in a branch shall give rise to sub-branches and so on.
Some of the branches may be intersected 2 . Intersection of branches and paths shall be
described later in this sub-section.
2 Intersection of branches refers to intersection of their effective twist-matrices. The process of constructing an
effective twist-matrix of a path is explained in section 4.1.3.2.
67
There are some rules that a path needs to obey otherwise it shall become an invalid path.
These rules are as follows:
1. A valid path (or branch) should not revisit any feature or part node. It creates a redundant
loop. One branch may visit nodes of another branch.
2. If a path visits a feature node that is linked to fixed part node, path should immediately
terminate to fixed part node after such a feature node. It should not go to any other node
from such a feature node except the fixed part node. This rule is applicable when an
assembly feature connects more than one part.
Example: If three links are connected to a grounded pivot, the path should terminate after
reaching to feature node corresponding to grounded pivot instead of moving on to any of
the other links connected to this pivot.
* The part being analyzed for under-constraints may be connected to multiple parts and it shall
have exactly same number of paths. A path may have no branches and it shall look like path
shown in Fig. 4-2. Part nodes are shown by black dots and feature nodes are shown by empty
circles.
Start Node
End Node
Fig. 4-2: Serial Path
If two or more branches emanate from a part node in the middle of a path and if all of these
branches come together to a part node (with all their sub-branches intersected already) these
branches need to be intersected. Fig. 4-3 shows these types of paths. Fig. 4-4 shows a
physical mechanism which has such a path. The path originates at L6 and it gives rise to two
branches at L7. The process of intersecting such branches shall be described later. In such
68
cases the shared nodes of the branches are same and the shared nodes appear in exactly same
sequence (if one traverses the path from the part being analyzed to the fixed part).
Start Node
Start Node
Start Node
Sub-Branch
Branch
End Node
End Node
End Node
Fig. 4-3: Path with a Parallel Branch
Li
RI
R2
L3
L2
R3R
R5
L5
R2
L3
L4
L2
L6
R4
RI
R6
9,1
I
R3
L4
R7R4
Li
R5
This pat hs gives rise to
two branches here. The
two bran ches need to be
in Lersected.
R6
L5
L6
R7
Fig. 4-4: Branches of a Path
If two or more paths come together at a part node (with all their branches and sub-branches
intersected already) such paths shall be intersected for the purpose of analysis. The process of
intersecting such paths is exactly same as the process of intersecting branches. It shall be
described later in this section. Fig. 4-5 shows two situations where paths need to be
intersected. Fig. 4-6 shows two paths which originate at "L7" and they come together to
69
"L4". One path is shown by thick gray line and the other is shown by normal-width gray line.
A third path is also shown in this figure with a normal-width black line from "L7" to "Li".).
Start Node
Start Node
Branch
End Node
End Node
Fig. 4-5: Path as a Parallel Branch
Li
RI
L3
R2
L2
*
R3R
R5
L5
R7
R2
R6
L6
L7
R3
L2
L3
L4
R4
L4
R
Liv
R9
5
L5
L6
R6
These tw o paths need to
be inter sected because
they sha re exactly same
nodes] here onwards.
L7
R7 40
R8
R9
0
Fig. 4-6: Paths that can be Intersected
If a branch of one path shares some nodes with another branch of the same path or two paths
branch out and their branches share some nodes such mechanisms or structures may require a
more detailed procedure of analysis. Such paths will have cross coupling in the middle of
them. Fig. 4-7 shows such a path. Fig. 4-8(a) shows a mechanism where "L6" is the part
70
being analyzed. Two paths are also shown. Fig. 4-8(b) shows the first path. The path
originates at "L6" and it gives rise to two branches at "L4". Fig. 4-8(c) shows the second
path. The path originates at "L6" and it gives rise to two branches at "Li".
Fig. 4-7: Path with Cross Coupling
L
R6
L6
R7
RI
L2
L3
L5
R3
L4
Rj
U
I
L4
R4
iLl
R4
R5
L2
R6*
R,2
2
R1
.
L2
L3
3R3
--
L4
R4
L5
R5
L6
R6
L5
L6
R7
R7
(a) Paths with
Shared Nodes
-----
R2
R2
R5
Li
R1--
(b) First Path
(c) Second Path
Fig. 4-8: Paths with Shared Nodes
*
For each path an effective twist-matrix need to be constructed. The effective twist-matrix for
a path will represent the motion space of the part being analyzed when the connections of the
part being analyzed are broken with the first links of all other paths. The second part node in
71
a path is referred as first link and this part-node corresponds to a part directly connected to
the part being analyzed. The effective twist-matrix cannot be constructed for paths having
cross coupling in the middle of them. These cases require detailed kinematic analysis.
Effective twist-matrix can be constructed for the type of paths shown in Fig. 4-2 and 4-3.
Some paths may also be intersected and in this case they shall be represented by one effective
twist-matrix. Fig. 4-5 shows such paths.
Next sub-section (4.1.3.2) describes the process of constructing the effective twist-matrix of
a path. It covers the process of intersecting the branches and paths. Sub-section 4.1.3.3
describes the Konkar's algorithm for intersecting the twist-matrices. This algorithm is used to
calculate the degrees of freedom of a part after calculating the effective twist-matrices of the
paths. Sub-section 4.1.3.4 describes how the situations resulting in cross coupling (sharing of
nodes among branches of two different paths or sharing of nodes between two different
branches of one path) can be analyzed.
4.1.3.2 Constructing the Effective Twist-Matrix of the Paths:
For each path, construct the twist-matrix for each feature on the path (including branches and
sub-branches), using the same reference coordinate frame (such as one attached to the fixed
part). If a path has no branches it shall emanate from a feature node on the part being
analyzed (say "G") and it shall terminate to fixed part node (see Fig. 4-2). This type of path
has features in series. In such cases, all the twist-matrices associated with the feature nodes
of the path need to be combined into one union twist-matrix (twist-union). The twist-union
(TU) of multiple twist-matrices (T1, T2 and so on) is defined as follows:
TU = [TJ;T2;T3;....;Tn]
This twist-union will be the effective twist-matrix of the path. This process implements
Waldron's series and parallel law. As explained earlier, bodies other than binary links give
rise to branches in the paths. The branches need to be intersected using the parallel law of
Waldron. This procedure is described in the following points.
72
Intersecting the Branches:
o
Suppose a path starts from part node G and it gives rise to multiple branches at a part
node (say "I1"). If all of these branches come together at another part node (say "12")
with all their sub-branches intersected already, these branches should be intersected. The
procedure is as follows:
o
Construct the twist-unions of the branches emanating from part node I (where the
multiple branches are emanating from) and coming together at part node 12. These
twist-unions will be the union of twist-matrices of feature nodes that lie between I
and I2.
o
Intersect these twist-unions (intersection of twist-unions is described in the section
4.1.3.3).
o
Construct the twist-union of G with respect to I and the twist-union of 12 w.r.t. fixed
part node. Twist-union of G w.r.t. I1 (say Ul) will be given by union of the twistmatrices of all features nodes between these two part nodes (G and I1). Similarly, the
twist-union of 12 w.r.t. fixed part node (say U2) will be given by union of the twistmatrices of all features nodes between these two part nodes (12 and fixed part node).
Union of U1, U2 and the resultant intersection obtained in previous step will
represent the effective twist-matrix of the path.
o
This process of intersecting the branches of a path should be done until all such
branches have been taken into account. The same process applies in case of branches
giving rise to such sub-branches (which originate at a part node and come together at
another part node with all their sub-branches intersected already). If part node I
corresponds to the part being analyzed (G) then this procedure will essentially
combine all paths starting at G and coming together at 12. The effective twist-matrix
in this case shall represent all the paths coming together at 12.
This step generates effective twist-matrices for each valid path. At this time, it should be
noted that effective twist-matrix cannot be constructed for paths with cross coupling. Cross
73
coupling requires detailed kinematic analysis which shall be presented in section 4.1.3.4. The
effective twist-matrix of a path (or combination thereof) should have a rank less than six,
otherwise the path (or combination thereof) is useless for the purposes of constraint
evaluation of the part being analyzed. Such a path will not constrain the part being analyzed.
Example: Consider the terminal body of a serial manipulator. The terminal body may have
six degrees of freedom. If another body is connected to this terminal body by any assembly
joint the path from the new body to the fixed part is not useful because the terminal body is
already free in six degrees of freedom. It cannot pass the constraints to any body through any
assembly feature.
4.1.3.3 Intersecting the Effective Twist-Matrices of the Paths:
* Form the intersection of all the effective twist-matrices representing different paths. [Konkar,
1993] defined intersection of twist-matrices. Reciprocal of a twist-matrix is called wrenchmatrix. One can get the wrench-matrix from the twist-matrix by calculating its null space.
For an assembly feature, the rows of the twist-matrix and that of the wrench-matrix span
complementary orthogonal subspaces of a six-dimensional DOF space. Intersection of "n"
twist-matrices is defined as follows:
Sintersection = Reciprocal
(U Reciprocal Si)
i=1
Reciprocal (Si)
=Reciprocal
Reciprocal (S2)
[Reciprocal (Sn)
If the intersection of effective twist-matrices results into a non-empty matrix it will represent
under-constraints. The part will have as many degrees of freedom as the number of
independent rows in the resultant intersection. An empty matrix shall mean that the part has
no allowed motions. An empty matrix has no rows and no columns.
4.1.3.4 Cross Coupling (Dependent Degrees of Freedom):
*
If the two or more paths branch out and their branches share some nodes or if two branches
of a path give rise to sub-branches and some of the sub-branches share some nodes, detailed
74
-
kinematic analysis shall be required for motion analysis. The process of analyzing situations
resulting into cross coupling is as follows: One needs to take the part being analyzed ("G")
off the assembly. "G" may be attached to multiple parts. Degrees of freedom of all these
parts need to be found out. For finding out the degrees of freedom of these parts same rules
(regarding type of paths) apply as in the case of "G". Algorithm for motion analysis is
recursive in nature. Degree of freedom of "G" depends upon all the degrees of freedom of all
the parts it connects to. However, the degrees of freedom of all the parts "G" connects to may
not be independent. For example, if "G" is connected to "a" and "b" it is possible that both
"a" and "b" have one degree of freedom each but the degree of freedom of "b" may be
dependent on that of "a". So, after finding degrees of freedom of all the parts "G" connects
to, the dependence among the degrees of freedom of these parts needs to be found out. The
procedure to find the dependence in degrees of freedom of the parts is explained in next
bullet point. Fig. 4-9 explains the process of finding degree of freedom of "G". The path
from "G" to fixed part has cross coupling. So, "G" needs to be removed from the system.
Degrees of freedom of "G" depends on that of "a". However, "a" also needs to be taken off
from the system because the paths for this part also have the cross coupling. Degrees of
freedom of "a" depends on that of "b" and "c". Degrees of freedom of "b" and "c" can be
found out because the paths for these parts are such that there is no cross coupling 3
G
Dependence
a
c
b
Fig. 4-9: Process of Analyzing Cross Coupling
3Dangling portions are discarded while considering the rest of the system.
75
*
Suppose degree of freedom of "G" depends on n parts. Each of these parts will have a screw
a screw (Si) representing its twist-space. Rank of each of these screws must be less than six
otherwise the part will be completely free in space and it will not provide any constraints to
"G". For every degree of freedom of a part, it needs to be checked whether it is dependent on
degrees of freedom of some other parts.
* To check the dependence of the degrees of freedom of parts, one can start with any part (say
part-1). It needs to be checked what happens to the degrees of freedom of other parts if each
of the degree of freedom part-1 is locked one by one. Locking one degree of freedom of a
part can be modeled by creating a fictitious assembly feature between the part and the fixed
part. This fictitious assembly feature should constrain the degree of freedom to be locked.
The wrench matrix of this assembly feature should be reciprocal of the degree of freedom to
be locked. The degrees of freedom of rest of the parts need to be evaluated again when the
fictitious assembly feature has been inserted in the system. The degrees of freedom that
disappear are dependent degrees of freedom. Some or all degrees of freedom of a part may be
dependent. The results of the process of finding dependence can be represented in terms of
following upper triangular matrix:
S1
S2
S3
Sn
S1
X
V21
V31
Vnl
S2
S3
X
X
V32
Vn2
x
x
X
Vn3
Sn
x
x
x
x
vii = Dependence Vector representingdependence of Si on Sj;
No. of rows in vii = Rank(Si);
No. of non-zero entries in vii = Dependence of Si on j5 min (rank(Si),rank(Sj))
* Dependence of a degree of freedom of a part (say part-2) on a degree of freedom of another
part (say part-1) means that motion in this particular degree of freedom of part-i shall cause
some motion in the corresponding degree of freedom of part-2. Dependence among degrees
of freedom of different parts gives rise to new wrenches (i.e. due to dependence in degrees of
freedom additional force carrying capacity may be attained). In the previous bullet point, a
76
method was described which finds the pairs of degrees of freedom which are dependent on
each other (each degree of freedom on a different part). This method does not find the exact
relationship between degrees of freedom (i.e. if unit motion is effected along a degree of
freedom what the magnitude of the motion will be along the dependent degree of freedom).
The exact dependence between degrees of freedom needs to be found out in order to find the
degrees of freedom of the part that connects to the set of bodies with dependent degrees of
freedom. The process of finding exact dependence among degrees of freedom is described in
next bullet point.
Suppose, the dependence between a degree of freedom of part-1 and a degree of freedom of
part-2 needs to be found. It will require forming a chain of parts which are physically
connected to each other by assembly features. The chain must start from part-1 and it must
end at part-2. The nodes corresponding to the fixed part (the part which is grounded in the
physical mechanism) and the nodes corresponding to all assembly features connected to the
fixed part should be avoided in the middle of the path. The grounded part does not move due
to variation in the location of any assembly feature. Parts on this chain share the points on the
successive origins of the co-ordinate frames of the assembly features. If an assembly feature
connects two parts there exist two points on the origin of the co-ordinate frame of the
assembly feature belonging to either of the parts (or their imaginary extensions). Fig. 4-10
shows an assembly feature. There exist two points Op and Oq belonging to part-p and part-q
respectively both lying on the origin of the co-ordinate frame of the assembly feature (0).
These points will have same velocity components along the constrained direction of the
assembly feature.
Op,
x
Assembly Feature
Y Co-ordinate Frame
Oq
x
Slot
0
Pin
Part-p
Part-p
Part-q
Part-q
Fig. 4-10: Velocity Components at the Origin of Assembly Feature
77
The magnitude of the motion in the dependent degree of freedom of part-2 due to unit
magnitude of motion in the corresponding degree of freedom of part-1 can be found out by
analyzing the chain of parts from part-1 to part-2. If multiple chains are found it is possible
that more than one chain can be used for finding the magnitude of the motion of part-2 in its
dependent degree of freedom due to unit magnitude of motion in the corresponding degree of
freedom of part-1. It is also possible that some paths cannot be used for this purpose. This
can happen if such an assembly feature (Apq) is picked to move from one part (part-p) to the
other (part-q) that the velocity of a point on part-p (at the origin of the co-ordinate frame of
the assembly feature (Apq) which connects part-p to part-q) lies entirely in the twist-space of
the assembly feature (Apq). In this case, it would not be possible to move to next part in the
chain of parts. However, since the dependence between the two degrees of freedom of part-i
and part-2 has already been established, there must exist at least one chain of parts which can
be used for the purpose of establishing the relationship between the magnitudes of the
corresponding degrees of freedom. It can be understood in the following way. It is certain
that the degree of freedom of part-2 is dependent upon part-1 so there must be a chain of
parts connected by assembly features responsible for transferring the motion from part-i to
part-2.
The actual process of finding the relationship between magnitudes of dependent degrees of
freedom shall be explained in detail in the solved example.
* The process of finding degrees of freedom of part (say "G") connected to a set of bodies
which may have dependent degrees of freedom is as follows: one needs to make the twistunions for all the connections of part G with multiple bodies and then these twist-unions need
to be intersected. If there is no dependence in the degrees of freedom of multiple bodies
intersection of these twist-unions shall give the degrees of freedom of part G. The twistunion of the connection refers to the union of twist-matrix of the part in question and the
twist-matrix of assembly feature realizing the connection. Twist-matrix intersection was
described in section 4.1.3.3. If however dependence in some of the degrees of freedom is
found, it is possible that the motion due to these dependent degrees of freedom may not be
78
possible. One needs to check whether motion due to dependent degrees of freedom is
possible or not.
*
If part G connects to part-1 and part-2 and there one degree of freedom (say d2) of part-2 is
dependent on some degree of freedom (say dl) of part-1 the exact relationship between the
magnitudes of the dependent degrees of freedom is ascertained by finding an appropriate
chain of parts. There may be one or more assembly features between part G and part-1. The
twist-matrices of all assembly features between part-1 and part G should be intersected. This
intersection (Tgl) shall represent twist-space of part G relative to part-1 when it is only
connected to part-1. Similarly all assembly features between part-2 and part G should be
intersected to get the twist-space (Tg2) of part G relative to part-2 when it is only connected
to part-2. The following two statements list the conditions under which the motion of part-1
and part-2 along degree of freedom "d1" and "d2" respectively can be passed on to part G.
if rank(Union(Tgi,d1))> rank(Tgi) then motion along "d" will be passed to part-G.
if rank(Union(Tg2,d2)) > rank(Tg2) then motion along "d2" will be passed to part-G.
If it is possible that both part-1 and part-2 can pass the motion to part G along dl and d2
respectively, it needs to be checked whether the motion along these dependent degrees of
freedom violate rigid body law or not.
* Due to unit motion along dependent degree of freedom (dl) of part-1 one needs to find the
velocity at the origin of co-ordinate frames all assembly features between part-1 and part G
(say v1', v2', .., vk' are the velocities at the origins of the co-ordinate frames of the assembly
features). The components of these velocities along wrench directions of respective assembly
features will be passed to points of part G which are coincident on the origins of the coordinate frames of assembly features. For example, if v1' is the velocity of a point on part-1
(say point 01) which lies on the origin of the co-ordinate frame of an assembly feature
between part-1 and part-G, the components of v1' along the wrench direction of the same
assembly feature will be passed to a point on part G which is coincident with point 01 of
part-1 So, the wrench components of the velocities may be denoted by (vi, v2, ..., vk). Unit
motion along dl in part-1 will cause some motion in dependent degree of freedom (d2) of
79
part-2. This relationship is already known. One needs to find the velocity at the origin of coordinate frames of all assembly features between part-2 and part G (say ul', u2', .., um' are
the velocities at the origins of the co-ordinate frames of the assembly feature). Similarly the
wrench components of these velocities may be denoted by (ul, u2, ..., um). The mobility of
part G due to motion along dependent degrees of freedom can be checked by using the
equation (4-1). Va and Vb represent velocities of two points on a rigid body. R is the position
vector from point "b" to point "a". "Q" is the angular velocity of the body. This equation is
valid for any co-ordinate frame as long as all the vectors are in the same frame. Velocities of
multiple points of part G has been found due to motion along dependent degree of freedom.
Suppose velocities on "k" points of part G due to part-1 and velocities on "m" points of part
G due to part-2 have been found. One needs to form k*m pairs of velocities (one velocity due
to part-1 and other due to part-2) and one need to check whether there exist a valid solution
for the equation (4-1) for all these velocity pairs. All the pairs should produce same result for
the angular velocity otherwise motion along dependent degree of freedom will not be
possible. If all the pairs give exactly same solution to the value of angular velocity motion
along dependent degree of freedom will be possible. However, this mobility may come at the
cost of over-constraint because the mobility may become critical. In other words, if the
mobility becomes dependent on the certain part level dimensions (i.e. slight variation in
certain part dimensions makes the part immobile in the dependent degrees of freedom), it
will amount to over-constraint. This shall be discussed in detail in sub-section 4.1.5 that
presents constraint analysis.
(41)
Va =Vb+ i*xR
One needs to check all dependent degrees of freedom regarding whether motion is possible
along the different set of dependent degrees of freedom 4 . After doing this analysis, one
should discard all the sets of dependent degrees of freedom. Suppose part-2 has three degrees
of freedom and one of them is dependent on part-1. Part-1 may have just one degree of
freedom. Twist-matrices of part-1 (Ti) and part-2 (T2) may be as follows:
T 1= [( aop;rpx wp)]
4 A set of dependent degree of freedom refers to degrees of freedom of different parts which move together (i.e.
motion along a degree of freedom in one part causes motion along the dependent degrees of freedom of other parts.
80
-
T2
= [(Oa ;rax oa);(o)b ;rbx(ob);( wc,;rcx a,)]
The dependent degrees of freedom of either part are shown in italics (these degrees of
freedom are dependent on each other). It may be found that motion along the dependent
degree of freedom pair is not possible. Then, one needs to discard the dependent degree of
freedom from the twist-matrices of part-1 and part-2. The modified twist-matrix for part-2
will have two independent degrees of freedom and modified twist-matrix of part-1 will be an
empty matrix:
TI'=[
]
T2'= [(Oa
;ra x oa);(ob ;rbx
0b)]
After modifying the twist-matrices of all the parts which connect to the part (part G) one
needs to recalculate the twist-unions of twist-matrices of the parts with that of respective
assembly features. Suppose part G is connected to part-i through an assembly feature Fl.
Lets say that the modified twist-matrix of part-1 is T1' and the twist-matrix of assembly
feature Fi is Tfi. The twist-union corresponding to part-1 shall be given by union of T1' and
Tfl. One needs to construct twist-unions for all the parts connected to part G. The
intersection of these modified twist-unions will give the correct information about degrees of
freedom of part G.
*
This point presents the summary of this rather complicated process of finding degrees of
freedom of a part:
1. Find the paths from part being analyzed (G) to the fixed part.
2. If paths are such that their effective twist-matrix can be formed (i.e. there is no cross
coupling and Waldron's series and parallel law are sufficient for analyzing them) then
form the effective twist-matrices and intersect the matrices to get the degrees of freedom
of G with respect to fixed part.
3. If paths have cross coupling detach the body from the system and try to find the degrees
of freedom of the parts G connects to. This process is recursive in nature. However, it is
81
certain that at some stage one can find the paths for a body when there are no cross
couplings in the paths for it 5 .
4. After finding the degrees of freedom of all the parts G connects to, one needs to find the
dependence in the degrees of freedom these parts. Motion along dependent degrees of
freedom may or may not be possible. The procedure to find dependence and the method
to check the mobility along the set of dependent degrees of freedom has been described
in detail in this sub-section. Solved example will further explain this process. One needs
to discard the dependent degrees of freedom from the twist-matrices of the bodies G
connects to. After this, one needs to form the twist-unions of modified twist-matrices of
the set of bodies G connects to with the twist-matrices of respective assembly features.
The intersection of these twist-unions will give the correct answer about the degrees of
freedom of part G with respect to the fixed part. Mobility due to dependent degrees of
freedom may come at the cost of over-constraints. This shall be explained further in the
sub-section 4.1.5 that presents constraint analysis.
The solved examples can be found in sub-section 4.1.6.
4.1.4 Comparison of the Method of Motion Analysis:
There are alternate methods of finding out instantaneous degrees of freedom using screw theory.
These methods also lack in generality. The two most relevant of the methods ([Davies, 1981] and
[Konkar, 1995]) are discussed here to highlight the contribution of this research.
Konkar's Algorithm:
[Konkar, 1995] presented an algorithm for motion analysis and claimed that his method can
work for all classes of mechanisms and structure but his algorithm doesn't work for mechanisms
with cross-couplings. In fact his algorithm doesn't work even for some mechanism which do not
have any cross coupling and can be analyzed just by series and parallel law of Waldron.
Suppose the degrees of freedom of a body (G) need to be found and it is connected to a set of
bodies (A1, .., An). Konkar's algorithm is as follows:
5 There cannot be any cross coupling for an assembly of three parts. In the worst case, one will have to go to this
level.
82
1. One needs to find the degrees of freedom of A1, A2 and so on when G has been detached
from the system.
2. Form the twist-unions of twist-matrices of bodies Al, A2 and so on with twist-matrices
of respective assembly features. For example, the twist-matrix of the assembly feature
between body Al and body G may be represented by Tgl and the degrees of freedom of
body Al when G has been detached from the system may be represented by Tal. The
twist-union for body Al will be the union of twist-matrix Tgl and Tal.
3. Intersect all the twist-unions in the previous step to get the degrees of freedom of body G.
Konkar' algorithm fails to understand the phenomenon of dependent degrees of freedom among
parts. Degrees of freedom of part Al may not be independent from those of part A2. It has been
shown earlier that degrees of freedom become dependent due to cross coupling in the paths for a
body (see section 4.1.3.4). Degrees of freedom may also become dependent due to parallel paths
or parallel branches of a path. If two paths emanate from the part being analyzed and both of
them come together at a part node other than fixed part, the paths shall have some dependent
degrees of freedom (see Fig. 4-11). However, Waldron's parallel law is sufficient to handle these
situations. The method of motion analysis presented in this chapter implements Waldron's
parallel law properly. Such parallel paths (as shown in Fig. 4-11) shall have one effective twistmatrix. This process is described in section 4.1.3.2. However, Konkar's algorithm cannot analyze
such paths properly. Konkar's algorithm shall work only for purely parallel paths where all
parallel paths originate at the part being analyzed and terminate at the fixed part node.
Start Node
Start Node
Branch
End Node
End Node
Fig. 4-11: Path as a Parallel Branch
83
The following two examples demonstrate that Konkar's algorithm does not work properly for
cross coupling and it doesn't work even for some cases where there is no cross coupling (parallel
paths terminating at parts other than fixed part). The method of motion analysis presented in this
chapter overcomes the shortcomings of Konkar's algorithm. Waldron's series and parallel law
are implemented correctly and the procedure of detailed kinematic analysis is proposed for
problems with cross couplings.
Example-1:
Li
R4
R4'
R4
y
Ri
L3
L2
L3
L2
R3
R2
R3
R2
L1'
X
Li
RI
(a)
(b)
Fig. 4-12: Two DOF Manipulator
Fig. 4-12(a) presents a mechanism. All the joints are planar revolute joints in this mechanism.
The twist-matrices of the joints are as follows:
R1=[0 0 10 0 0];
R2=[0 0 12 10];
R3=[ 01 2 -10];
R4=[O 0 1 10 0];
If Konkar's algorithm is used to find the DOFs of "L4".It will have the following steps:
1. Detach "L4" from the system and find the degrees of freedom of "L2" and "L3".
2. "L2" is connected to "Li" and "Li" is connected to the ground. The degrees of freedom
for "L2" when "L4" is not in the system will be the union of degrees of freedom of "Li"
and the degrees of freedom allowed by the assembly feature between "L2" and "Li".
Essentially, the degree of freedom of "L2" (when "LA" is not in the system) will be given
84
by the series law of Waldron. It will be represented by union of twist-matrices of
assembly features RI and R4.
R14=[00 1 000; 00 1 100]
3. Similarly, the degrees of freedom of "L3" when "L4" is not in the system will be given
by the union of twist-matrices of assembly features RI and R4.
4. Konkar's algorithm does not recognize that the degrees of freedom of "L2" and "L3"
have some dependence. It will form the twist-union of degrees of freedom of "L2" with
the degrees of freedom allowed by the assembly feature between "L2" and "L4".
a. The twist-union for "L2" will be given by union of the twist-matrices of assembly
features R2, R4 and R1.
R142=[0 0 1 0 00; 0 0 11 00; 0 0 12 10]
b. Similarly, the twist-union for "L3" will be given by the union of the twistmatrices of assembly features R3, R4 and R1.
R143=[0 0 1 0 00; 0 0 1 1 00; 0 0 12 -10]
c. Intersecting these two twist-unions obtained in step-b and c will answer that "L4"
has three degrees of freedom which is incorrect. In fact, Konkar's algorithm
analyzes the mechanism shown in Fig. 4-12(b). "L4" in the mechanism shown in
4-12(b) has three degrees of freedom.
T 14 =[00 1 000;000 1 00;0000 10]
The method presented in this chapter shall recognize that two constraint paths emanate from L4
and both come together to another part Li hence the two paths must be intersected. The Degree
of freedom of L4 shall be given by the following twist-matrix:
T14= R1U((R2UR4)f(R3U R4))
T14 =[00 1 000;000 100]
This example shows that Konkar's algorithm cannot apply Waldron's parallel law properly.
85
Example-2:
L.2
R2
L3
L2 R
L5
R3
14
6jR6
x,
RI
0.3
++
0.5
0.6
R4
L7
1.7
Fig. 4-13: Five-Bar Structure
Fig. 4-13 presents a mechanism. All the joints are planar revolute joints in this mechanism. The
twist-matrices of assembly features are as follows:
R1=[O 0 10 0 0];
R2=[0 0 1 2 -0.5 0];
R3=[0 0 1 2 -1.5 0];
R4=[0 0 10 -20];
R5=[0 0 1 1.2 -0.3 0];
R6=[0 0 1 0.6 -1.7 0];
If Konkar's algorithm is used to find the degrees of freedom of "L3". It will have the following
steps:
1. Detach "L3" from the system and find degrees of freedom of "L2" and "L4".
2. For degrees of freedom of "L2" when "L3" is not in the system, one needs to find degrees
of freedom of "L5".
a. Detach "L2" also from the system and find the degrees of freedom of "L5". Now
the degrees of freedom of "L5" will be given by the union of twist-matrices of
assembly features R6 and R4.
R64=[0 0 10.6 -1.7 0; 0 0 10 -2 0]
86
b. Form the twist-union of twist-matrix of "L5" (from previous point) with twistmatrix of the assembly feature between "L5" and "L2" (i.e. twist-matrix of
assembly feature R5).
R645=[O 0 10.6 -1.7 0; 0 0 10 -20; 0 0 1 1.2 -0.3 0]
c. Intersect the twist-matrix of assembly feature RI with twist-union obtained in
previous step. The resultant will be the twist-matrix of "L2" when "L3" is not in
the system (say TI2).
T12 =[0 0 10 0 0]
3. For degrees of freedom of "L4" when "L3" is not in the system, one needs to find degrees
of freedom of "L5".
a. Detach "L4" also from the system and find the degrees of freedom of "L5". Now
the degrees of freedom of "L5" will be given by the union of twist-matrices of
assembly features R5 and Ri.
R51=[0 0 1 1.2 -0.3 0; 0 0 10 0 0]
b. Form the twist-union of twist-matrix of "L5" (from previous point) with twistmatrix of the assembly feature between "L5" and "L4" (i.e. twist-matrix of
assembly feature R6).
R516=[O 0 1 1.2 -0.3 0; 0 0 10 0 0; 0 0 10.6 -1.7 0]
c. Intersect the twist-matrix of assembly feature R4 with twist-union obtained in
previous step. The resultant will be the twist-matrix of "L4" when "L3" is not in
the system (say T14 ).
T 14 =[0 0 10-20]
4. Form the twist union of T12 with twist-matrix of assembly feature R2.
U1=[0 0 1 0 00; 0 0 12 -0.5 0]
5. Form the twist union of T 14 with twist-matrix of assembly feature R3.
87
U2=[ 0 1 0 -2 0; 0 0 12 -1.5 0]
6. Intersect the twist-unions obtained from step-5 and 6. The resultant will give the degrees
of freedom of "L3". This result is as follows:
TI=[00 14-10]
This result is incorrect. "L3" has no degrees of freedom in this configuration.
This example shows that Konkar's algorithm doesn't work on problems which have cross
coupling. The method of motion analysis presented in this chapter recognizes that cross coupling
may make degrees of freedom dependent. Degrees of freedom of L2 and L4 are not independent
in the presence of L5. Detailed analysis of a similar example shall be given in the sub-section
that presents solved example. For making the comparison between this method of motion
analysis and Konkar' method complete, the main steps of motion analysis are given as follows:
1. Find the degrees of freedom of L2 and L4 after removing L3 from the mechanism
because there is cross coupling in the paths for L3.
TA=[0 0 10 0 0]
T14=[O 0 10 -2 0]
2. Establish the dependence between the degrees of freedom of L2 and IA. Establishing the
dependence has a detailed procedure and it is described on a similar example in solved
examples. For the purpose of being brief, it is being omitted here. It is obvious that
degrees of freedom of L2 and L4 are dependent due to L5.
3. Establish the relationship between the magnitudes of motion of L2 and L4. For this
purpose, a chain of parts from L2 to L4 shall be formed. Chain of parts goes to L4 from
L2 via L5. Unit angular velocity is assumed along the rotational degree of freedom of L2.
This angular motion shall induce a velocity at the origin of the co-ordinate frame of
assembly feature R5. This velocity is given by (-1.2, 0.3, 0). The degree of freedom of L5
is also known when L3 is disconnected. It is:
T15=[0 0 1 2.67 -0.67 0]
88
The instantaneous center of rotation can be found using the information in the results of
twist-matrix itself. This process will be described in detail when presenting the solved
example. The instantaneous center of rotation for L5 will be at point (0.67, 2.67, 0). Since
the instantaneous center of rotation of L5 and complete velocity (both magnitude and
direction) of a point on it are known, one can find the magnitude of the angular velocity
of L5 due to unit motion of L2. It will be -1.22 units (i.e. L5 shall rotate about its
instantaneous center of rotation with 1.22 magnitude in the clockwise direction). The
angular motion of L5 shall induce a velocity at the origin of co-ordinate frame of R6.
This velocity is given by (-2.53, -1.22, 0). This velocity can be used to find the magnitude
of angular velocity of link L4. It shall be 4.21 units. Hence, unit angular velocity of L2 in
counter clockwise direction induces 4.21 unit angular velocity in L4 in counter clockwise
direction.
4. Use the relationship between the dependent degrees of freedom to check whether motion
will be possible along dependent degrees of freedom after L3 is attached. For this find the
velocity of a point (that coincides with origin of co-ordinate frame of R2) on L2 due to
unit angular motion (about RI). This velocity (say vi) will be (-2, 0.5, 0). Due to motion
of L2, an angular velocity will be induced in L4. Motion of L4 shall induce a velocity on
the point that coincides with the origin of co-ordinate frame of R3. This velocity (say ul)
will be (-8.42, -2.11, 0). "ul" and "vi" represent velocities of two different points on L3.
One needs to check whether "v1" and "ul" are possible when L3 is attached. One can
check this by trying to solve equation (4-1) which relates velocities of two points on a
rigid body. The solution to this equation is not possible in this case because these
velocities violate rigid body conditions 6. The x-component of "vi" is -2 and xcomponent of "ul" is -8.42. However, for this particular configuration x-components of
"ul" and "v1" need to be same to satisfy the rigid body conditions. Hence, the motion
along dependent degrees of freedom will not be possible.
6 Rigid body condition implies that components of the velocities of two points should be same along the line joining
the two points.
89
5. Now, one needs to discard dependent degrees of freedom from the twist-matrices of L2
and L4. The modified twist-matrices of L2 and L4 will be:
T12'=[
T14'=[
L2 and L4 are connected to L3 by assembly features R2 and R3 respectively. The twistunions of modified twist-matrices of L2 and L3 with the assembly features that connect
them to L3 will be as follows:
U12 =[0 0 12 -0.5 0]
U14 =[0 0 1 2 -1.5 0]
One can check the degrees of freedom of L3 by intersecting these two twist-unions. This
shall give the expected result that L3 is locked and it cannot move.
Davies' Algorithm:
[Davies, 1981] used Kirchoffs nodal law for forming the vector loop equations in mechanisms.
[Davies, 1983] improved his method presented earlier by analyzing the loop equations in more
detail. He used partitioning of matrix to improve the method. The fundamental algorithm of
generating the loop equations remained the same. In a linkage composed of rigid bodies, a path
from one body to another can be constructed noting which joints connect the intervening bodies.
Any path that starts and ends at the same body is a loop around which the sum of the joint
velocities must be zero. Thus velocity in a mechanical network is analogous to voltage in an
electrical network and this analogy is the basis of Davies' formulation of Kirchoff's law for
mechanical networks.
Davies' algorithm cannot handle dangling bodies. In other words Davies' algorithm is only for
closed loop mechanisms or structures. Davies proposes to use the loop equations in terms of
screw velocities to solve for the degrees of freedom. However, it may not be required to form
loop equations if the configuration is such that Waldron's series and parallel law are sufficient.
90
4.1.5 Constraint Analysis for A Part (Evaluation of Over-Constraints):
Constraint analysis checks whether a part is over-constrained or not. Constraint analysis needs to
be performed for every part in the assembly, if the over-constraints for every part need to be
found out.
A wrench-matrix represents a set of directions along which the body can support independent
forces. If multiple wrenches are acting on a body, there might be situations when a direction is
being constrained by multiple wrenches. The intersection of multiple wrenches is same as
intersection of multiple twist-matrices. Intersection of multiple twist-matrices was described in
section 4.1.3.3. However, intersection of multiple wrenches may not give all the overconstrained directions. The intersection of all the wrenches may be null but still there might be
over-constrained directions. This section presents a method of finding over-constraints for a
body when multiple wrenches are acting on it. The method intersects two wrenches at a time and
then the intersected wrenches are combined by forming the union. This combined wrench is
intersected with some other wrench and the process of intersecting two wrenches and combining
them together (by forming a union) for next step continues until all the wrenches have been
combined. Fig. 4-14 shows the set theory analogy of this process. The set theory analogy of this
process is finding the intersection between two sets then taking the union of these two sets and
then finding the intersection between this union and another set. Since we know union of
wrenches and intersection between two wrenches, this process works.
Wrench
Wrench 3
1
Wrench 2
Intersection of Wrench
and Wrench 2
Wrench 4
1
Union of Wrench 1
and Wrench 2
Intersection of (Wrenchl union
Wrench2) with Wrench 4
Fig. 4-14: Method of Finding Over-Constraints: A Set-Theory Analogy
91
Systematic constraint analysis begins the same way that motion analysis does, by drawing the
paths and enumerating the valid paths.
Over-Constraints when Paths have no Cross Coupling:
If paths for the part being analyzed are such that their effective twist-matrices can be constructed
(i.e. there is no cross coupling) the following procedure can be used to find all the overconstraints associated with the part being analyzed.
* Find the effective twist-matrices of all paths7 . The method to obtain effective twist-matrix for
a path has been described in the sub-section (4.1.3.2).
* Find the wrench-matrix associated with each path. Wrench-matrix is the reciprocal of the
effective twist-matrix.
* Choose a path and intersect its wrench-matrix with another path's wrench-matrix to check if
this combination over-constrains the parts. Intersection of wrench-matrices is exactly same as
that of twist-matrices. After identifying the over-constraints due to a pair of paths, one needs
to group this pair of paths in to one. i.e. resultant twist-matrix for this pair of paths needs to
be found by intersecting their twists. After this, over-constraints need to be found for this pair
of paths and some other path. One needs to keep combining the paths and keep checking the
over-constraints caused by combined paths and any other path until all the paths have been
combined.
* If some paths have been intersected they shall be represented by one single effective twistmatrix for the purpose of motion analysis. The process of intersecting the paths may
contribute towards over-constraints associated with the part being analyzed. Fig. 4-5 shows
the type of paths which need be intersected. Suppose the paths start from the part node (G)
and they come together to part node (12) with all their branches and sub-branches intersected
already. The effective twist-matrices of sections between G and 12 are intersected in the
process of creating the effective twist-matrix for all the paths being intersected at 12. If the
wrench-matrices of the sections between G and 12 are intersected one by one using the same
process as described in the previous bullet point, one shall get the contribution towards overconstraints associated with G due to the process of intersecting the paths.
7 If some paths have been intersected they shall be represented by one single effective twist-matrix for the purpose
of motion analysis. The process of intersecting the paths may contribute towards over-constraints associated with the
part being analyzed. This is covered in the subsequent bullet points of this section.
92
_
Over-Constraints in Situations Resulting into Cross Coupling:
If the paths for the part being analyzed (G) have cross coupling the method of motion analysis
detaches G from the assembly and the degrees of freedom are found for the parts which connect
to G. Lets say the parts which connect to G are denoted by Al, A2, .., and An. Algorithm for
motion analysis is recursive and it keeps detaching the bodies until such parts are found which
have paths with no cross coupling. It is necessary to find the dependence in the degrees of
freedom of A1, A2 and so on in order to find the degree of freedom of G. Dependence in degrees
of freedom of Al, A2 and so on essentially exerts more wrenches on G. Over-constraints may be
caused by dependence in the degrees of freedom of Al, A2 and so on. Over-constraints
associated with G can be found by analyzing the degrees of freedom of Al, A2 and so on and the
dependence in their degree of freedom. The following procedure should be used to find the overconstraints.
" One needs to make the twist-unions8 for all the connections of G with multiple bodies (Al,
A2, .., An) and then corresponding wrench-matrices need to be intersected one by one as
described in the previous point. If there is no dependence in the degrees of freedom of the set
of bodies this process will give all the over-constraints associated with G.
*
Over-constraints are also caused due to dependent degrees of freedom as mentioned before.
The motion along dependent degrees of freedom may or may not be possible. The possibility
of motion along a set of dependent degrees of freedom is checked by detailed kinematic
analysis as described in sub-section 4.1.3.4. By-product of this analysis is qualitative
information about over-constraints. If the motion along a set of dependent degree of freedom 9
is possible then this motion may accompany over-constraints. If the mobility along the set of
dependent degrees of freedom becomes dependent on part level dimensions, the overconstraints shall be accompanied with mobility definitively. If the mobility along the set of
dependent degrees of freedom does not become dependent on part level dimensions, the
over-constraints shall not be accompanied with mobility. Whether mobility along a set of
dependent degrees of freedom becomes dependent on part level dimensions or not, can be
8 The twist-union of the connection refers to the union of twist-matrix of the part in question and the twist-matrix of
assembly feature realizing the connection. The twist-matrix of the part includes both dependent and independent
degrees of freedom.
9 A set of dependent degree of freedom refers to degrees of freedom of different parts which move together (i.e.
motion along a degree of freedom in one part causes motion along the dependent degrees of freedom of other parts.
93
found out during the process of checking the possibility of mobility itself. If motion along the
set of dependent degree of freedom is not possible there may or may not be over-constraints
associated with part G due to dependent degrees of freedom.
Dependent degrees of freedom create new wrenches and more research is required to find the
over-constraints quantitatively in case of dependent degrees of freedom.
4.1.6 Examples:
First Example:
Fig. 4-15 presents an assembly that has two plates joined by four features. Feature "Fl" allows
translation along X-axis and rotation about Z-axis. Rest of the features are designed in such a
way that they allow five degrees of freedom. Feature "F2" and "F3" allow all motions except
translation along Y-axis. Feature "F4" allows all motions except translation along X-axis. All
these statements are valid for the instance of the assembly shown in Fig. 4-15. The twist-matrices
for these features are presented after Fig. 4-15.
Part-Feature Diagram for Assembly
F3
F4
F1
F2
F3
A
A
F2
' F1
F
Pate-A
EiPlate-B
SF4
Plate B
Plate A
View A-A
Fig. 4-15: Two Plates Joined by Four Features
For"F1":T1=
0
0 1
2 -2
0 0 0 1
0
0
0
94
0
0
6
0
0
0
-2
0
1
0 0 1 4 -4
0 1 0 0 0
0
0 0
1 2 -6
For "F2": T2=0 1 0 0
1 0 0
0 0 0
0
1
0 0 0 0
For "F3": T3
1 0 0 0
0 0 0 1
4
0
0
0
-4
0
4
-2
0
For "F4": T4=0 1 0 0
0
0
1
0
2
-4
0
1
0 0 0 0
0
1
0
0
0
1
0 0 0
0 0 0
0 0 0
1
One can check under-constraints in this assembly using this information. There are four paths
from Part-B to Part-A. Each path has one feature node hence one twist-matrix in it. One needs to
intersect all the four twist-matrices to calculate under-constraints in the assembly. Intersection of
these four twist-matrices will be an empty matrix. Hence, this assembly has no under-constraints.
Now, this example will be used for finding over-constraints in this assembly. This method uses
wrench-matrices associated with assembly features. Wrench-matrix is reciprocal of twist-matrix.
While twist-matrix represents motion allowed by the feature, wrench-matrix represents motion
forbidden by the feature. There are four paths from Part-B to Part-A. One can start with "Fl" and
"F2". One can find over-constraint caused by these two features by intersecting the wrench
matrices associated with these two features. Intersection of wrench-matrices is same as that of
twist-matrices.
This intersection is an empty matrix. So, there is no over-constraint due to
combination of "Fl" and "F2". Now these two features need to be combined into one feature
(say CF12). This combination allows only one degree of freedom (translation along X-axis). The
twist-matrix for this combined feature is the intersection of T1 and T2:
T12= o 0
0 1 0 0
Now, one needs to find over-constraints due to combined feature CF12 and F3. The intersection
of wrench-matrices corresponding to these two features is as follows:
95
0 1
0
0
0
4
This is a screw of unit magnitude force along Y-axis that creates a moment of four units along Zaxis. Corresponding over-constraint is shown in Fig. 4-16. Essentially, assembly is overconstrained along Y-axis.
F3
F2
FlEDIZI
v}
____
Plate-A
____
____
____
___
late-B
Fig. 4-16: Over-Constraint
Now, F3 also need to be combined with this. Let's call combined feature at this stage CF123.
This combined feature also allows translation along X-axis. Finally, one needs to check overconstraints due to combined feature CF123 and F4. The intersection of the wrench matrices of
these two features will yield an empty matrix. So, there is no over-constraint due to this
combination. So, one can conclude that there is only one over-constraint in this assembly. One
can easily miss this over-constraint, if one tries to find this by intersecting wrench-matrices of all
four features together. The intersection of all four wrench-matrices is an empty matrix. This
method of finding over-constraints is extremely rigorous and it cannot miss any over-constraint
in this assembly.
Over-constraint is caused because more than one feature may attempt to constrain same degree
of freedom. This method of finding over-constraints can be used to find over-constrained
directions and the features that create them. Note that choosing the paths in a different sequence
will always result in the same number of degrees of freedom, if any, being detected as overconstrained, but the matrix reporting the over-constraint may appear different. The reason for
this is that as features are added to the combination, one such set may properly constrain the
parts. Any feature added thereafter will necessarily add over-constraint along the direction(s) it is
capable of constraining, and these directions will appear in the results. A different sequence of
96
analysis will eventually arrive at proper constraint with a different subset of the features, and the
next one added will be different this time than last time. Results will then report this feature's
directions rather than another one's. The engineer can use this information to explore the
consequences of establishing joints between parts in different sequences, including deciding
which features, if any, to redesign in order to remove the over-constraint.
Second Example:
2
Li
L3
R2
RI
R3
L2
R2
2
R6
R5
L2
Y
L4
L3
L5
R3
L4
R4
L1
L5
R5
R6
(b)
(a)
Fig. 4-17: Parallelogram Mechanism
R1=[0 0
0 0 0]
R2=[0 0
2 0 0]
R3=[0 0
2 -20]
R4=[0 0
0 -20]
R5=[0 0
1 00]
R6=[0 0
1 -20]
Fig. 4-17(a) shows a planar parallelogram mechanism. Fig. 4-17(b) shows the part-feature
diagram of the mechanism. The problem may be to find the degrees of freedom of L3 when LI is
the fixed link. The paths for L3 will have cross coupling. Hence, one needs to remove L3 from
the mechanism and the degrees of freedom of L2 and L4 need to be found out. After, removing
97
L3 from the mechanism, the degrees of freedom of L2 and L4 can be found out. There will be
two paths for L2. Fig. 4-18(a) shows these two paths. One path is shown by dashed line and the
other path is shown by solid line. Similarly, there will be two paths for L4. Fig. 4-18(b) shows
these two paths.
The twist-unions for the two paths of L2:
R1=[0 0 10 0 0]
R564=[0 0 1 1 00; 0 0 1 1 -20; 0 0 1 0 -20]
Intersection of these two twist-unions gives the twist-matrix for L2 when L3 is not in the
mechanism. The intersection is as follows:
T12=[0 0 10 0 0]
The twist-unions for the two paths of L4:
R4=[0 0 10 -2 0];
R651=[001 1-20;001 100;001000];
Intersection of these two twist-unions gives the twist-matrix for L4 when L3 is not in the
mechanism. The intersection is as follows:
T14=[0 0 10 -20];
-
R1
R5
JI
-Li
j/
R1
-2
L
L2
~R5
RR6
RR6
L5
L5
g*
R4
R4
(a)
(b)
Fig. 4-18: Paths for "L2" and "L4"
Now, the dependence in the degrees of freedom of L2 and 14 need to be found out. A fictitious
assembly feature (say Dl) whose wrench matrix is exactly same as the degree of freedom of L2
98
shall be attached between L2 and fixed part (LI). Now, the degrees of freedom of L4 need to be
re-evaluated. Fig. 4-19 shows the paths for L4 when L2 has been locked by the fictitious
assembly feature. One of the paths of L4 has a branch at L2. The effective twist-matrixes for
both of the paths can be constructed for both of the paths for L4.
Effective Twist Matrix for the path shown by dotted line=>
R4= [0 0 1 0 -2 0]
Effective Twist Matrix for the path shown by solid line=>
(R6 U R5 U (R1lD1))= [0 0 1 1 -2 0; 0 0 1 10 0]
Intersection of these two matrices will give an empty matrix. Hence, L4 also becomes locked by
locking L2. This is an obvious result. However, the process of finding dependence is illustrated
with the help of this simple example. The process remains exactly same for three-dimensional
problems. There may be multiple parts to be checked for dependent degrees of freedom and
some or all of the parts may have more than one degree of freedom.
D1
L1
R1
L
Ri
RIA
R6
R4
Fig. 4-19: Paths for "L4" when "L2" is locked
Now, the dependence has been established and the exact relationship between the magnitude of
the motion of L2 and that of the motion of L4 need to be found out. For this task, one needs to
find a chain of parts starting from L2 and terminating at IA. In this case, there is only one chain
that starts at L2 and terminate at IA. This goes via L5. The other chain of parts between L2 and
L4 (which goes via LI) cannot be selected because Li is the fixed part. The process of finding
the relationship between the magnitudes of the motion of L2 and LA is as follows:
99
Dependence in Degrees of Freedom:
The chain from L2 to 14 is L2-L5-L4. One needs to know the degrees of freedom of L2, L5 and
L4. Motion analysis shall reveal that all these parts have one degree of freedom. L2 and L4 have
rotational degree of freedom and L5 has a translational degree of freedom. The next step should
be finding the respective center of rotation for each of these parts (if applicable). The center of
rotation can be found out using the information from the result of motion analysis itself using
equation no. (4-2).
(4-2)
V=R x
Where;
x = Vector cross product
V = Translational component of the Motion Analysis Results
n = Rotational component of the Motion Analysis Results
R = Location of the center of the rotation with respect to the part co-ordinate frame
A unit motion should be assumed along the degree of freedom of L2. One can find the complete
velocity (both direction and magnitude) of a point on L2 that coincides with the origin of coordinate frame of assembly feature R5 (assuming unit magnitude of angular motion). L5 has a
translational degree of freedom in x-direction. Complete velocity (both direction and magnitude)
of a point on L5 (that coincides with origin of the co-ordinate frame of R5) is known. So, one
can find the magnitude of the motion of link L5. L5 has a translational degree of freedom so the
complete velocity of all point on this link is known. This information in turn may be used to
derive the magnitude of the motion for link L4 (i.e. the magnitude of its angular velocity)
because complete velocity (both direction and magnitude) at a point on this link (coinciding with
the origin of the co-ordinate frame of assembly feature R6) along with the possible directions of
motion for this link (i.e. axis of rotation and its location) are known. So, the relationship between
the magnitude of the degree of freedom of L2 and that of the degree of freedom of L4 is known.
Note that this procedure has nothing specific to this problem. This is a general procedure which
can be used for three-dimensional problems. In this particular case, the relationship between the
magnitudes will indicate a one-to-one ratio (i.e. a unit magnitude of motion of L2 about the Zaxis will cause unit magnitude of motion in L4 about an axis parallel to Z-axis located at (2,0,0)).
100
Degrees of Freedom of L3:
One needs to check whether motion along the dependent degrees of freedom of L2 and L4 is
possible or not. A unit motion in L2 will induce some velocity at the origin of co-ordinate frame
of assembly feature R2. This velocity (say vl) will be given by (-2,0,0). Since the degree of
freedom of L4 is dependent on degree of freedom of L2, the motion of L2 will induce a motion
in IA. This motion of L4 will induce some velocity at the origin of co-ordinate frame of
assembly feature R3. This velocity (say ul) will also be given by (-2,0,0). Now, one needs to
solve equation no. (4-1) for this velocity pair. There will be a valid solution to equation (4-1) in
this case (Q=O). So, the motion due to dependent degree of freedom is possible. Hence, the
dependent degrees of freedom of L2 and L4 shall not be discarded form their twist-matrices. The
twist-matrices of L2 and L4 shall not change. Twist-matrices of L2 and IA are:
T12=[O 0 10 0 0]
T14=[O 0 10 -2 0]
In order to find the degrees of freedom of L3, one needs to form two twist-unions. First twistunion (say "U1", for L2) shall be formed by degrees of freedom of L2 and twist-matrix of
assembly feature R2. Second twist-union (say "U2", for L4) shall be formed by the degrees of
freedom of L2 and twist-matrix of assembly feature R3.
U1=[0 0 1 0 00; 0 0 120 0]
U2=[0 0 1 0 -20; 0 0 12 -20]
The intersection of these two twist-unions will give the degrees of freedom of L3 that is
translation along X-direction.
T13=[O 0 0 10 0]
For over-constraints associated with L3, one can use the two twist-unions (Ul and U2) formed
for finding its degrees of freedom. On needs to find the wrench-matrices of these twist-unions.
Intersection of wrench-matrices shall give the over-constraints. The degrees of freedom of L3 are
found using that of L2 and L4.
Since there exist dependence in the degrees of freedom of L2 and L4, additional over-constraint
might get generated due to it. In the motion analysis, it is found that motion along the dependent
degree of freedom is possible. Mobility along dependent degrees of freedom becomes dependent
101
on part level dimensions in this case. If any of the link lengths have slight variation the entire
mechanism shall become immobile. Slight changes in part level dimensions lock the motion
along dependent degrees of freedom. So qualitatively, it can be said that L3 has one more overconstraint in addition to those given by the intersection of the wrenches corresponding to twistunions Ul and U2. This can be confirmed by making L2 as fixed link and checking for the overconstraints for L3. In this case, the paths for L3 can be analyzed just by using Waldron's series
and parallel law and the additional over-constraint along x-direction will appear in the results of
constraint analysis.
Third Example:
Part-Feature Diagram for Assembly
Plate- 1
L1
late-2
L
L2
4
5
U
L3
Spherica
Joints
3
L6
L2
L5
L5
L3
L6
Cylindrical
Joints
U1
U2
U3
Plate-1
Spherica
Joints
U4
U5
U6
Plate-2
Fig. 4-20: Parallel Manipulator
Fig. 4-20 presents a parallel manipulator, together with its Part-Feature diagram. The problem
may be to find the degrees of freedom of Plate-2. There are six paths from this part to the fixed
part (Plate-1). Each path passes through three feature nodes. First feature node is a spherical
joint, second feature node is a cylindrical joint and third feature node is again a spherical joint in
case of each path. Twist-union for each path will be the union of the three twist-matrices
corresponding to the three features. All six twist-unions will form full rank matrices. i.e. the rank
102
of all twist-unions will be six. It implies that none of the paths are valid. Hence, it can be
concluded that top plate (Plate-2) has six degrees of freedom. On this basis, one need not do
constraint analysis for top plate. Similarly, this method can be used for analyzing degrees of
freedom of any other part as well.
4.1.7 Limitations of Motion and Constraint Analysis in the Context of Assembly Problems:
The method of motion analysis presented in this chapter requires detailed kinematic analysis if
there is cross coupling in the paths of the part being analyzed (G). If part G is connected to a set
of parts cross coupling induces dependence in degrees of freedom of some of the parts connected
to G. Dependence in the degrees of freedom mean that the motion along a degree of freedom of a
part shall induce a motion along a degree of freedom of some other part. Since, the twist-matrix
intersection algorithm does not consider the magnitude of the screws so it considers all degrees
of freedom as independent. A simple use of twist-matrix algorithm cannot solve the problems
with cross coupling. The requirement of detailed kinematic analysis may appear as a limitation
of the method of motion analysis presented in this chapter but in fact it is not. It is imperative
that more detailed kinematic analysis shall be required for solving the problems with cross
coupling.
Method for finding over-constraints of a part has a limitation that all over-constraints cannot be
found when dependence in degrees of freedom is found. Though, qualitative information about
over-constraints can be found as a by-product from the detailed motion analysis. However, in the
context of assembly problems, it is not a very severe limitation. The assembly process in car and
aircraft industry goes in such a way that one part is added to the sub-assembly at one time as the
assembly grows. Normally, fixtures are used to locate the new part being added. This process
leaves us with an assembly of three parts at one time. The first part can be considered the subassembly from previous workstation, the second part can be considered the fixture at this
workstation and the third part shall be the part being added at this assembly station itself. The
question of cross coupling cannot arise in case of assembly between three parts.
Though, the problem of finding over-constraints and also under-constraints (mobility or
instantaneous kinematics) in a general mechanism is certainly very important in the field of
103
kinematics. However, a rather simpler solution for finding under- and over-constraints may be
good enough for an assembly designer as long as it can point out the mistakes in small
assemblies. The mistakes refer to undesirable under- or over-constraints in the assemblies.
4.2 Constraint Analysis in CAD System:
The word "constraint" is used in different ways by different researchers. Some researchers mean
consistency of position and orientation equations of a group of parts that are assembled together
with joints that mutually allow certain motions [Thomas, 1991]. Current CAD systems are part
centric. One creates assembly models after completing the part design. Traditionally, assembly
models use algebraic equations to represent geometric constraints in mechanical assemblies.
Assembly constraints are represented in terms of algebraic equations. For example, distance
between two points of two different bodies can be set to a fixed value and this constraint can be
represented by an algebraic equation. The system of algebraic equation which represents the
configuration of the assembly is constructed from the solid model. CAD systems do not
differentiate between mates and contacts. So a constraint, which is actually just stabilizing the
location, can be included in the system of equations and a constraint, which actually passes the
location from one part to another, may not be in the system of equations. The system of
equations can represent only properly constrained assemblies. For over-constrained assemblies,
CAD systems check consistency of constraints. If a designer specifies multiple constraints for a
single degree of freedom of a part CAD systems will check whether all these constraints are
geometrically compatible (i.e. they are all possible without interference). Hence, CAD systems
do not provide any help in determining over-constraints as defined in this chapter (overconstraints refer to degrees of freedom of a part which are multiply constrained). [Serrano and
Gossard, 1988] proposed an algorithm to evaluate the system of equations representing
mechanical assemblies. [Owen, 1991] presented a method to solve for the configuration of the
assembly given constraints of the assembly. These approaches are inherently deficient for
evaluating over-constraints because equations cannot model all the physical constraints in the
assembly. [Mullins and Anderson, 1998] presented a technique to automatically identify the
geometric constraints in mechanical assemblies. They developed a method of identifying the
constraints from the algebraic representation of mating surfaces. For example, the constraint
104
passed from one planar surface to another planar surface may be represented by an equation. The
technique developed in this paper differentiated between mating conditions and kinematic joints.
Geometric relationship of a mating condition is static (e.g. gap between two static surfaces or
two static surfaces in contact). Kinematic joints allow motion (e.g. revolute joints etc.). Their
assembly models also do not differentiate between different types of assembly features (mates
and contacts).
4.3 Comparison of the Constraint Analysis in Top-Down and Bottom-Up
Approaches:
The main deficiency of the approach of CAD systems is that one cannot detect certain overconstraints in the assembly model without having data about tolerances and clearances. Example:
An assembly, having two parts connected to each other with two "peg & hole" mating features,
will not be considered over-constrained by most of the CAD systems*. Over-constraint will be
revealed only if one pursues interference analysis after providing information about clearances
and tolerances. Another weakness of representation of assembly models by algebraic equation is
that one can never include all physical mating conditions if they together create over-constraint.
In order to achieve a properly determined system of equations one needs to remove certain
mating conditions from the assembly model. Example: One will not be able to include the third
"peg & hole" feature in the assembly model for an assembly of two plates connected to each
other by three "peg & hole" features. However, it is perfectly legitimate to include third "peg &
hole" feature in a CAD system as long as the location of third peg and that of corresponding hole
are geometrically
compatible. Alternatively,
sometimes
one can resort to simplified
representation of mating conditions. Example: A peg and hole joint can be represented by two
coincident centerlines. However, it should ideally be represented by two cylindrical surfaces
touching each other. Screw theory on the other hand can handle physically over-constrained
assemblies. Constraint analysis by screw theory may reveal several over-constraints. These
results may appear extraneous at first because clearance on mating conditions may relieve some
of them. However, loose tolerances on some parts may cause some of these over-constraints to
become prominent. i.e. an over-constraint might cause problems of assemblability or it might
* I-DEAS, ProEngineer and Solidworks
were tested.
105
cause deformation or it might create an unnecessary gap. Deformation and unnecessary gap will
lead to problems of variation. Clearance on mating features relieves over-constraint but it creates
uncertainty in location of some parts (to be discussed in chapter 6). Some properly constrained
assemblies may also become over-constrained under variation. DFC will not represent the
tolerance chains in the assembly under these circumstances. Complete information about the
over-constraints in the assembly and classification of all the circumstances when DFC does not
represent the tolerance chains intended by engineer can be two very important pieces of
information for an engineer. It will communicate to engineers the possible scope of
improvements or required points of precautions.
4.4 Summary:
A new graphical technique has been presented to systematically evaluate constraint properties
using DFC as the assembly model. The method of motion analysis enables the use of screw
theory for the problem of instantaneous kinematics of a general mechanism of arbitrary
complexity. This method of motion analysis has been compared with previous methods of
motion analysis to highlight the contribution of this research.
A new method of finding over-constraints has also been presented. This method cannot miss any
over-constraint if the wrenches being applied on a part are independent. If the wrenches become
dependent, qualitative information about over-constraints can be inferred from motion analysis
as a by-product. Some of the over-constraints may appear extraneous. However, if this technique
is combined with information about assembly sequence it can become extremely beneficial for
an engineer. Engineer can find over-constraints affecting key characteristics at each subassembly station and he/she can take decisions accordingly. There cannot be more than six overconstraints when only one part is added to a sub-assembly. So, if one is analyzing overconstraints when one part is being added at a time it will not be tedious to decide which overconstraint may affect assembly level requirements.
DFC is not limited only to motion and constraint analysis. It can represent tolerance chains in the
assembly under certain circumstances. Tolerance chains shall be discussed in chapter 5 and 6.
106
Chapter 5: Variation and Contribution Analysis
Design of mechanisms and assemblies use the perfect model of the part. These models are used
for simulating and verifying the kinematic and dynamic behavior. However, manufacturing
processes are inherently imprecise and the products they yield vary in form, material properties
and performance. Hence, the analysis of the effect of the manufacturing uncertainties is
necessary to control sensitivity and robustness of the design. The design team usually creates a
class of interchangeable and functionally equivalent parts by specifying the tolerances on part
dimensions [Requicha, 1983; Srinivasan and Jayaraman, 1989; Jayaraman and Srinivasan, 1989].
The nominal part with a perfect shape is only a particular member of the class. The tolerance
specifications define the authorized variations of the surfaces of the parts (tolerance zones). Each
class represents the domain of the acceptance of parts.
In chapter 3, it is presented that assembly features are made of surfaces. So, the tolerances on
surfaces can be linked to the tolerances on location, size and form of assembly features. The
process of finding variation in the location of a part due to variations in the locations of assembly
features shall be referred as "variation analysis". Location of a part in an assembly is affected by
location of multiple assembly features. Hence the "contribution analysis", to find the effect of
variation in the location of any particular assembly feature on the location of a part, becomes
important. This chapter has its focus on variation and contribution analysis. The following
chapter shall focus on the effect of variation in the size dimensions of assembly features which
causes uncertainty in the locations of parts. Form tolerance can also be covered under tolerance
on size dimensions.
This chapter is organized in the following fashion. First section of this chapter presents the
connective assembly models required for top-down design method. Datum Flow Chain (DFC)
described in second chapter is a connective assembly model. This section describes the process
of performing variation analysis using the connective assembly models in general and using DFC
in particular. Appropriate references are given to the methods of variation analysis. Second
section of this chapter presents the assembly models supported by CAD and their underlying
process of doing variation analysis. Third section compares the variation analysis performed in
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the top-down and bottom-up approaches. Fourth section of this chapter presents a new method to
perform the contribution analysis to find the sensitivity of the part location to the variation in the
location of an assembly feature. This method of contribution analysis uses the information about
DFC in terms of locations of assembly features and their screw-theory based constraint
representation. Finally, fifth section presents the summary of the chapter.
5.1 Connective Model of Assemblies:
This section shall present the connective assembly models first. Appropriate references to the
methods of variation analysis which use connective assembly models shall be presented later.
Connective models of assembly require matrix transformations to locate the parts with respect to
each other. Each part is assumed to have a base coordinate frame. An assembly will be modeled
as a chain of these frames, and each transformation will allow us to walk from frame to frame
and thus from part to part. Mating features on parts will each have their own frame. A
transformation will allow us to say where each feature is on each part with respect to that part's
base frame. The assembly can be formed by creating relationships that join the feature frames on
mating parts. Later the same mathematical representations can be used to do variation analysis.
The mathematical representation takes the form of a 4x4 matrix. This method of modeling spatial
relations between objects dates at least as far back as [Denavit and Hartenberg, 1955], who used
it to represent kinematic linkages. Researchers in assembly and robotics began using it in the late
1960s and early 70s [Paul, 1981], [Simunovic, 1976], [Popplestone, 1979], [Wesley, Taylor, and
Grossman, 1980].
The connective model of assembly defines a part as having a central coordinate frame plus one
or more assembly features, each feature having its own frame. A transform relates each feature's
location on the part to the part's central coordinate frame. Features can be placed on a part by
defining a transform from part center coordinates to the feature frame. Alternatively, the
transform to the feature frame can be directed from another feature frame. When two parts join,
assembly features on one part are made to coincide with assembly features on the other part. This
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is done by defining an assembly transform that relates the frame on one part's assembly feature
to the frame on the other part's assembly feature. If the axes of these two frames are identical,
then the assembly transform is the identity. If not, then typically an interface assembly transform
must be written to account for the difference between the axes of the two feature frames. The
interface transform can also represent design-in-clearance on assembly feature.
In a connective assembly model, the user joins parts by connecting them at their assembly
features. This can be done by applying the methods of surface constraint to surfaces on the
features, or the frames representing the features can be constrained directly. Fig. 5-1 shows three
parts joined this way. On the left is the nominal situation while on the right a varied situation,
caused by an error in placing an assembly feature on part B, is shown. This error can be detected
even if the parts are modeled only approximately, as long as the assembly features are modeled
and placed on the parts accurately. By contrast, detection of errors in a world coordinate model
like that of Fig. 5-2 requires that the parts be modeled accurately, since no distinction is made
when modeling them between assembly feature surfaces and other surfaces.
(b): The Situation under Variation
(a): The Nominal Situation
Fig. 5-1: Three Parts Joined by a Connective Assembly Model
I
A
I
I
B
CA
I
I
C
(b): The Situation under Variation
(a): The Nominal Situation
Fig. 5-2: An Assembly of Three Parts in a World Coordinate Frame
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Datum Flow Chain (DFC) was discussed in the second chapter is a connective assembly model.
It can represent parts, assembly features, and surfaces individually and can tell the difference
between them. This makes it possible to model different kinds of variation correctly and to
distinguish in the model different sources of error.
5.1.1 Variation Analysis using Connective Assembly Models (e.g. DFC):
Variation analysis can be done in three basic ways: worst-case, statistical and Monte Carlo. Each
has advantages and drawbacks. Worst-case probably incurs excessive manufacturing costs.
Monte Carlo can use any probability distribution for individual errors but takes computer time.
Statistical methods are limited to simple probability distributions (normal and uniform) but yield
answers quickly. [Bjorke, 1989] provides a comprehensive approach to statistical tolerance
analysis (i.e. variation analysis) for one-dimensional stack-ups.
A top-down design process uses the connective assembly models such as DFC. The assembly
features populate the DFC. Feature-based design was extended to assemblies by [DeFazio et. al.,
1993]. The nominal locations of assembly features were stored in 4X4 matrix transforms.
[Veitschegger and Wu, 1986] performed the fundamental calculations for finding the uncertainty
in the relative part locations using the matrix transforms in the domain of robot uncertainty
prediction. Variation in the location of assembly features is modeled as variation in their matrix
transforms with respect to the part co-ordinate frame. [Whitney, Gilbert and Jastrzebski, 1994]
extended the variation analysis based on matrix transform approach to statistical analysis of
GD&T.
Essentially, this approach of calculating accumulated variation in the part location due to
variations in part-level dimensions combines the variations in the location of assembly features
by multiplying the matrix transforms representing the variations along the tolerance delivery
chain. The tolerance delivery chain reflects the intent of the design team which is captured by
DFC. This approach can be used to perform all three types of variation analysis methods (worstcase, statistical and Monte Carlo).
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5.2 CAD Model of Assemblies:
This section shall present the type of assembly models used by CAD systems. Appropriate
references to the methods of doing variation analysis which use the CAD assembly models shall
be presented later. The problem of tolerance allocation is intertwined with that of variation
analysis. The references to the work of tolerance allocation will be given after discussing
variation analysis. CAD model of assemblies can be classified primarily in the following two
groups:
5.2.1
World Model:
In a world model, assemblies are placed in a world coordinate frame by giving each part's
coordinate frame and (x, y, z) coordinate location in the world frame. The origin of the world
frame of a car or airplane, for example, is normally placed in front of the vehicle a bit
beneath the ground plane. This ensures that each part and point in each part has positive
coordinates. Each part may be found by estimating its world coordinates and asking for a
picture on the computer screen of parts near those coordinates.
A model like that in Fig. 5-2 is often made by drawing each part separately and then carefully
placing them in the picture until the desired surfaces touch. A variety of modeling errors
could occur. In Fig. 5-2 (b), one such error is shown, namely that part B is in the wrong
position. The result is that it inter-penetrates part A, an event called interference. CAD
systems can detect interferences. This interference is shown by a thick line in the figure.
However, the same or similar interference could be caused by either part A or part B being
the wrong shape even if they are in the correct location, or by Part A being in the wrong
location. Because this kind of model does not represent the fact that part A should assemble
to part B, these kinds of errors cannot be distinguished.
5.2.2
Surface-Constrained Model:
In a surface-constrained assembly model, the user joins items by establishing relationships
between different surfaces. Two planes can be made coincident, or two cylinders can be
made coaxial, for example. Such operations are often used to build up parts made of
elementary surfaces and simpler objects. In some CAD systems, assemblies are built up the
same way. The result is that the CAD model cannot distinguish parts and their subparts from
assemblies. In Fig. 5-3, a surface-constrained assembly model is shown. Part A in this figure
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is joined to part B by making a surface on one part coincident with a surface on the other.
The figure 5-3(a) shows nominal situation, while the figure 5-3(b) shows varied situation. All
sources of error must be attributed to mis-located surfaces, and all surfaces are treated
identically. In some CAD systems, it will be hard to tell if the error is on part A or on part B.
A
B
A
(a): The Nominal Situation
B
(b): The Situation under Variation
Fig. 5-3: A Surface-Constrained Assembly Model of Two Parts
5.2.3
Variation Analysis using CAD Assembly Models:
Generally, CAD systems provide interference analysis capabilities but provide no or limited
tolerance analysis capabilities. Tolerance analysis is sometimes done by a different group of
people other than who designed the parts, and they use specialized computer aided tolerancing
(CAT) tools which perform the variation analysis on the 3D assembly models. These CAT tools
take input from the CAD assembly models. In addition to variation analysis, traditional CAT
tools such as VSA, 3DCS, CE/Tol and Valisys provide contribution analysis as well. These
analyses are used during detail design to optimize the selection of tolerances.
These CAT tools often require input in form of tolerance chains to perform the various analyses.
Tolerance chain for a given assembly level dimension is identified from the 3D CAD models of
assemblies. The third chapter described the TTRS method which finds tolerance chains from 3D
solid models. Tolerance chain identification refers to forming either an open or a closed loop of
the part level dimensions in order to analyze a given assembly level dimension. In variation
analysis, statistical data such as standard deviation, mean value, tolerance range and acceptance
rate for total population are calculated for the specified critical assembly dimensions. In
contribution analysis, the 3D influence of variation in each geometrical feature, according to
specified tolerance and distribution, is ranked for specified critical dimensions, including effects
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of assembly feature position, direction and variation magnitude. [Salomonsen et. al., 1997]
presented a review of commercial CAT tools.
The tolerance chains are represented using different types of constraint representation in CAD
and CAT systems. The following two ways are the most prominent:
1. Vectorial Representation
In vectorial representation, a system of equations is formed for the assembly level
dimension (or the location of the part) using the part-level dimensions as the known
variables. The part-level tolerances are variations in the corresponding dimensions. Using
the system of the equations, one can find the net variation on the assembly level
dimension due to the part-level variations. Commercial CAT tools use this approach. The
tolerance chains can be analyzed for each of the three types of analysis methods (worstcase, statistical, Monte Carlo).
2. Matrix Representation
The matrix representation is same as the one described in the first section of this chapter.
The relative locations of parts are represented by 4X4 matrix transforms. The tolerance in
the part-level dimensions is represented as variation in the transform that in turn can be
modeled by an error transform matrix. In CAD, researchers like Steven Coons used
matrix transforms to represent locations of objects in a computer in the 1960s [Ahuja and
Coons, 1968]. In the 1980s, CAD researchers made assembly models of mechanical parts
this way [Lee and Gossard, 1985]. The same mathematical model can be used for both
chains of links in a linkage and chains of more general parts in an assembly. [Gao, Chase
and Magleby, 1998] presented the method for 3D tolerance analysis of mechanical
assemblies using matrix transform based approach. In this case as well, the tolerance
chains can be analyzed for each of the three types of analysis methods (worst-case,
statistical, Monte Carlo).
5.2.4
Tolerance Allocation:
After identification of tolerance chains, the assembly level dimension can be analyzed using
different techniques. However, tolerances need to be assigned before the variation analysis can
be performed in this case. The contribution analysis is also possible only after the tolerances are
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assigned to part-level dimensions. Hence, the tolerance allocation becomes an iterative exercise
(The assembly level dimension will have different sensitivities to different part-level dimensions
and the tolerances assigned to different part-level dimensions may be decided to be inversely
proportional to the sensitivity of assembly level dimension to corresponding part-level
dimensions).
Tolerance allocation or tolerance control is an important design issue. Several researchers have
proposed methods for tolerance allocation in this process. [Turner, 1990] presented the linear
programming based approach to solve the problem of variation in relative positions of parts in an
assembly due to part level variations. This method was limited to simple geometry objects
(polygonal, cylindrical etc.). [Sodhi and Turner, 1994] later improved this method using different
contact states formed among parts during assembly process. They tried to utilize fine motion
planning for variation analysis and gross motion analysis for nominal positioning. [Nassef and
ElMaraghy, 1997] proposed a tolerance synthesis method for geometric tolerances. Geometric
tolerances include dimensional features such as perpendicularity, angular dimensions etc. It is
relatively harder to include the geometric dimensions in the tolerance allocation methods as
compared to linear dimensions (e.g. distance between two points). The cost associated with
manufacturing steps and variation introduced by each manufacturing step was main variables in
the proposed optimization procedure. [Inui, Miura and Kimura, 1996] presented a tolerance zone
based approach for calculating the variation in the position of a part in an assembly. This
approach tried to include shape variations but it was limited to 2D polygonal machined parts.
[Ashiagbor et. al., 1998] proposed the tolerance control and propagation method for a product
assembly modeler. His method used assignment of cost functions to the variations. [Bennis and
Fortin, 1999] presented a configuration space based approach for analyzing uncertainty in the
position of a part in the assembly. The main focus of the work is on the robotic grippers etc. for
the assembly of mechanical parts. The matrix-based representation of mating conditions is also
investigated by some researchers. [Gao, Chase and Magleby, 1998] presented the method for
generalized 3D tolerance analysis of mechanical assemblies using matrix-based representation of
relative locations of parts. [Voelcker, 1993] presented a review of tolerancing and metrology in
1993. [Voelcker, 1997] presented a state of current affairs in dimensional tolerancing 1997 again.
The main difference between the two reviews was that the first review was only focused on
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different techniques for tolerancing of parts whereas the second review was focused on
tolerancing for assembly and tolerancing for function. This is a clear sign that researchers are
now focused on the problem of assembly tolerancing and part tolerancing is considered fairly
well understood.
5.3 Comparison between the Top-Down and Bottom-Up Assembly Models:
The CAT tools essentially take assembly with parts designed up to detailed levels. Constraint
decomposition and tolerance chain identification is an integral part of CAT tools. Constraint
decomposition refers to identifying the constraints from the assembly of parts to form a properly
constrained model. The third chapter described the TTRS method which finds tolerance chains
from 3D solid models. Tolerance chain identification refers to forming either a open or closed
loop of the part level dimensions in order to analyze a given assembly level dimension.
Information about tolerance chain is required in order to analyze an assembly level dimension.
[DeMello and Lee, page 82, 1984] summarize this in the following way:
"In the assembly domain, it does not suffice to make the workpiece models
produced by a CAD system available in the programming environment, but a
description of the way the different pieces should be fitted together is also
required. This description can be provided in full detail by either the designer or
the programmer, or rather be automatically inferred, at least in part, from
constraints derived from both the shapes of the workpieces involved in the
assembly, after trying to find matings of complementary subparts between them,
and the mechanics of the assembly operationsthemselves."
A
f
B
A
B
(b): The Situation under Variation
(a): The Nominal Situation
I
f
Fig. 5-4: A Connective Assembly Model of Two Parts
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The bottom-up approach attempts to find the tolerance chain. On the other hand, top-down
approach requires the design team to construct one. The bottom-up approach has a world model
or a surface-constrained model of assembly. The top-down approach requires a connective model
(such as DFC). In Fig. 5-3 a surface-constrained assembly model was shown. Fig. 5-4 shows the
same assembly as shown in Fig. 5-3 but this assembly model is a connective model. In Fig. 5-4
part B mates to an assembly feature "f'on part A. On the other hand, in Fig. 5-3 part A is joined
to part B by making a surface on one part coincident with a surface on the other. The figure 53(a) and 5-4(a) show apparently identical nominal situations, while the figure 5-3(b) and 5-4(b)
show apparently identical varied situations. However, in Figure 5-3, we cannot tell the cause of
the variation because it does not contain a separate and coordinated group of surfaces called an
assembly feature. All sources of error must therefore be attributed to mis-located surfaces, and
all surfaces are treated identically. In fact, in some CAD systems we cannot even tell if the error
is on part A or on part B. In Figure 5-4, we can represent the fact that the entire feature on part A
is misoriented. Alternatively, we can represent mis-manufacture of the feature leading to its
having one misoriented surface. In fact, every kind of error that could occur in practice can be
represented individually and unambiguously. This is a huge advantage when analyzing
variations.
Some of the variation analysis techniques can be used both for top-down approach and bottomup approach. (e.g. the vectorial tolerancing and matrix transform based approach can be used
both in the case of top-down and bottom-up methods).
5.4 Contribution Analysis for Location of Parts in an Assembly:
This section presents a new method of evaluating sensitivity of the location of a part in an
assembly due to variation in location of any assembly feature in the assembly. This method
utilizes the information contained in the constraint representation of the assembly itself to
perform the analysis. This section is organized in the following way. First sub-section presents
the approach of this method. Second sub-section describes the method itself. Third sub-section
presents two examples. Fourth sub-section presents the facts of this method.
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5.4.1 Approach of Modeling Variation in Assembly Feature Location:
The variation in the location of an assembly feature can propagate to assembly level only if it is
along a direction which is constrained by the assembly feature. If the variation is along a
direction which the assembly feature does not constrain, it will not propagate to the assembly
level. Example: The variation in x-location of slot will not have any effect if the slot length is
designed properly because the slot does not pass the constraints in x-direction. However, the
variation in y-location of slot will propagate to the assembly level (at least as far as the next part
mated to part feature) because the slot does pass the constraint to other contacting part in ydirection (see Fig. 5-5).
y
X
Fig. 5-5: Pin in a Slot Assembly Feature
This approach models variation as an additional degree of freedom. For example, the variation in
the y-location of the slot can be modeled by an additional degree of freedom. The additional
degree of freedom changes the twist-matrix of the feature. The twist-matrix will now have an
additional row corresponding to the new degree of freedom modeling the variation. Variation in
the location can be modeled either as a translational degree of freedom or as a rotational degree
of freedom.
5.4.2 Sensitivity in the Part Location to the Variation in Assembly Feature Location:
This sub-section proposes a method to find the sensitivity in the location of a part to the variation
in the location of an assembly feature. This method is valid for three dimensional assemblies.
There are four phases in the method:
1. Modeling the Variation:
The variation can be modeled as a translational or as a rotational degree of freedom. The shift
in the location of an assembly feature in any direction should be modeled as the translational
degree of freedom in that direction. The rotational degree of freedom should be used to
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model the angular variation in the location of assembly feature. The degree of freedom which
models the variation shall be called variational degree of freedom of the assembly feature.
2. Performing the Motion Analysis:
The motion analysis can be used to check what new degrees of freedom a part attains due to
the additional degree of freedom introduced at an assembly feature for modeling the
variation. Motion analysis was presented in section 4.1.3 (in fourth chapter). Motion analysis
will reveal the new directions along which the part in question can move. The results of the
motion analysis can be divided in the following two categories:
a. Rotational Motion due to variation
In case of rotational motion one will have to find the center of the rotation. It can be
found using the information in the results of the motion analysis itself. The
translational part of the results will furnish this information. It shall be explained
further in the examples.
b. Translational Motion due to variation
If the results of the motion analysis predict translational motion, it is straightforward
to explain. The part being analyzed shall be able to translate in this direction due to
the variation in the location of assembly feature.
3. Finding the Chain from the Part to be analyzed to the Assembly Feature:
The objective of this analysis is to find the sensitivity of the location of a part to variation in
the location of an assembly feature. Motion (both magnitude and direction) of the part (being
analyzed) due to the unit motion along the variational degree of freedom of the assembly
feature shall be called the sensitivity of the location of the part to the variation in the location
(along a particular direction) of assembly feature. Motion of the part can be rotational and in
this case the center of the rotation, axis of rotation and the magnitude of the angular velocity
due to the unit motion in variational degree of freedom of the assembly feature need to be
calculated. Motion of the part can be translational and in this case only the direction of the
translation and the magnitude of the translation need to be known. The sensitivity of the
location of a part to the variation in the location of an assembly feature is dependent upon the
configuration of the assembly.
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Motion analysis tells what direction the part can move due to the variation in the location (in
a particular direction) of an assembly feature. However, motion analysis does not give the
information about the magnitude of the motion. The magnitude of the motion can be found
out by analyzing a chain of parts starting from the part being analyzed and terminating at any
part which connects to the assembly feature in question (on which the variation is being
modeled). This chain may have other parts and assembly features in between. The chain can
be identified from a part-feature diagram. Part-feature diagram is another representation of
DFC and it was introduced in section 4.1.2 (in fourth chapter). In motion analysis also a
similar approach of identifying the chain of parts was adopted to find the relationship
between the magnitudes of dependent degrees of freedom (see section 4.1.3.3 in fourth
chapter). The purpose of finding the chain in case of contribution analysis is to find the
magnitude of the motion of a part due to a given magnitude of relative motion in an assembly
feature. As in case of motion analysis, multiple chains may exist and all such chains should
be found out. While finding the chains the nodes corresponding to the fixed part (the part
which is grounded in the physical mechanism) and the nodes corresponding to all assembly
features connected to the fixed part should be avoided in the middle of the chain. The
grounded part does not move due to variation in the location of any assembly feature. Other
parts move due to variation in location of assembly features.
Example:
Fig. 5-6(a) shows a part-feature diagram. Lets assume that variation needs to be modeled on
assembly feature Ri and the sensitivity of the location of part L4 to this variation needs to be
found out. So, chain starting from L4 and terminating at a part that connects to assembly
feature RI needs to be found out. Assembly feature Ri connects to part Li and L2. Li is the
grounded part hence it should be avoided. Fig. 5-6(b),(c) show the two chains which are
possible from IA to L2. There is no other chain possible from L2 to IA which also avoid the
fixed part node (Li) and nodes corresponding to assembly features that connect to fixed part
(R4).
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Ll= Fixed Link
RI
R2
R3
R4
R5
L1= Fixed Link
L1= Fixed Link
L2
RI
L2
R1
L2
L3
L4
R2
L3
R2
R3
L4
R3
L3
1A
L5
R4
R5
L5
R4
R5
L6
L6
L5
L6
(b)
(a)
(c)
Fig. 5-6: Multiple Chains on Part-Feature Diagram
Use of Chain for Contribution Analysis:
Parts on the chain share the points on the successive origins of the co-ordinate frames of the
assembly features. If an assembly feature connects two parts there exist two points on the
origin of the co-ordinate frame of the assembly feature belonging to either of the parts (or
their imaginary extensions). Fig. 5-7 shows an assembly feature. There exist two points 01
and 02 belonging to part-1 and part-2 respectively both lying on the origin of the co-ordinate
frame of the assembly feature (0). These points will have same velocity components along
the constrained direction of the assembly feature.
Assembly Feature
Y Co-ordinate Frame
y
0 1
x
02
4x
Slot
0
Pin
Part-I
Part-2
Part-i
Part-2
Fig. 5-7: Velocity Components at the Origin of Assembly Feature
1. Finding the Sensitivity:
The magnitude of the motion on the part to be analyzed due to unit magnitude of motion in
the variational degree of freedom can be found by analyzing the chain of parts found in the
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previous step. Variational degree of freedom models variation on the assembly feature. One
needs to assign unit magnitude of motion to the variational degree of freedom. One will have
to identify the new degrees of freedom introduced by the variation in all the parts which are
on the chain using motion analysis.
Multiple Chains and Existence of a Chain that can be used for Analysis:
If multiple chains are found in the previous step it is possible that more than one chain can be
used for finding the magnitude of the motion of the part to be analyzed due to unit magnitude
of motion along variational degree of freedom. It is also possible that some chains cannot be
used for this purpose. This can happen if the part in the middle of the chain is such that it has
not gained any new degrees of freedom due to introduction of variational degree of freedom
at the assembly feature in question.
If the new degree of freedom of the part to be analyzed is due to the variational degree of
freedom there exist at least one chain which can be used for finding the magnitude of the
motion along the new degree of freedom of the part due to a given magnitude of motion
along the variational degree of freedom. It can be understood in the following way.
Variational degree of freedom models the variation in the location of an assembly feature in a
particular direction. If the part to be analyzed attains a new degree of freedom due to this
variation it is certain that the location of this assembly feature affects the location of the part
being analyzed along the new degree of freedom. Alternatively, the relative motion along the
variational degree of freedom must induce motion in the part being analyzed along the new
degree of freedom. There must be a chain of parts connected by assembly features
responsible for transferring the motion from the assembly feature in question to the part
being analyzed.
Process of Analysis:
The following example (Fig. 5-8) shall explain the process of finding the magnitude of the
motion of a part due to a given magnitude of motion along the variational degree of freedom.
Though this example is a two-dimensional one there is nothing specific in this process to
dimensionality of the problem. The similar process can be used for a three-dimensional
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problem as well. Fig. 5-8(a) shows a five-bar structure. Fig. 5-8(b) shows the part-feature
diagram of the assembly. The problem may be to find the sensitivity of the location of link
L4 due to variation in the location of assembly feature RI (in x-direction). Variation will be
modeled as an extra translational degree of freedom in assembly feature R1. Variation in the
location of assembly feature RI will cause some motion of link L4. The direction of the
motion can be found out using the motion analysis. To find the magnitude of the motion of
link L4 due to the variation in location of assembly feature R1, one needs to find a chain
from link L4 to link L2 (link L2 connects to assembly feature Ri). Link L2 needs to be
assumed as fixed part for the purpose of finding a chain from link L4 to L2. A chain from L4
to L2 is L4-L3-L2. This chain is shown in Fig. 5-8(b). Note that, this is the only chain from
L4 to L2 which avoids fixed part node and nodes corresponding to the assembly features
connected to fixed part. Now the new degrees of freedom of L4, L3 and L2 need to be found
out (due to the variation in the assembly feature RI). Motion analysis is used to find the
degrees of freedom of these parts. Motion analysis shall reveal that all these parts have one
rotational degree of freedom. The next step should be finding the respective center of rotation
for each of these parts. The center of rotation can be found out using the information from the
result of motion analysis itself. This process shall be explained later in the solved example.
Unit motion along the variational degree of freedom of assembly feature shall be unit
translation along x-direction because variation along x-direction is being modeled. One can
find the magnitude of the motion of link L2 (i.e. the magnitude of its angular velocity) due to
unit motion at assembly feature R1 because complete velocity (both direction and magnitude)
at a point on this link (coinciding with the origin of the co-ordinate frame of assembly feature
RI) along with the possible directions of motion for this link (i.e. axis of rotation and the
location of axis of rotation) in this case are known. Once the motion for link L2 is known,
one can find the complete velocity of a point on this link that coincides with the origin of coordinate frame of assembly feature R2. Now one can find the magnitude of the motion of link
L3 (i.e. the magnitude of its angular velocity) because complete velocity (both direction and
magnitude) at a point on this link (coinciding with the origin of the co-ordinate frame of
assembly feature R2) along with the possible directions of motion for this link (i.e. axis of
rotation and its location) are known. This information in turn may be used to derive the
complete velocity of a point on this link that coincides with the origin of co-ordinate frame of
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assembly feature R3. Now one can find the magnitude of the motion for link L4 (i.e. the
magnitude of its angular velocity) through a similar process as used for link L2 or L3.
R2
L1= Fixed Link
R3
L3
L2
y
L5
R1
R4
x
RI
L2
R2
R3
L3
IA
R4
L5
Li
(b)
(a)
Fig. 5-8: A Five-Bar Linkage
5.4.3 Examples:
1:
y
4L
~h
4~
Fig. 5-9: Two Plates
Fig. 5-9 presents an example of the two plates being constrained with respect to each other by
two assembly features. The first assembly feature is "a pin in a hole" which allows rotation about
the axis of the pin (z-axis). The second assembly feature is "a pin in a slot" which allows the
rotation about the axis of the pin (z-axis) and translation along the direction of the slot (x-axis).
The global co-ordinate frame is at the center of the "pin in a hole" assembly feature. The analysis
of this example is presented as follows:
The twist-matrix of the first assembly feature:
T1= [0 0 10 0 0];
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The twist-matrix of the second assembly feature:
T2= [0 01 0 -4 0; 0 0 0 10 0];
Suppose, the variation in the y-location of the slot need to be modeled. The modified twistmatrix of the slot assembly feature:
T2'= [0 01 0 -4 0; 0 0 0 1 00; 0 0 0 010];
Motion analysis will produce the following result for the upper plate (part-2) regarding its
degrees of freedom with respect to the lower plate (part-1).
T = [0 0 10 0 0];
The results of the motion analysis predict that the upper plate will rotate about z-axis if the slot
moves along y-direction. Since, the results have pure rotation (translational components are zero)
the center of the rotation will be same as the global co-ordinate frame. The process of finding the
chain from the part to be analyzed to a part that connects to the assembly feature in question is
trivial in this case. The assembly feature connects to the part being analyzed. The sensitivity of
the location of the upper plate to the variation in the y-location of the slot shall be given by the
magnitude of the angular velocity.
(5-1)
V = R xQ
Where;
x = Vector cross product
R = Vector from the center of the rotation to the origin of the assembly feature coordinate frame.
V = Unit velocity along variational degree of freedom. Variational degree of freedom
models variation along y-location of slot.
Q = Angular Velocity of Upper Plate
Let's revisit the whole analysis with global co-ordinate system placed somewhere else (at the left
corner of the plate, please see Fig. 5-9). This shall explain how one can find the center of the
rotation from the results of motion analysis if the global co-ordinate frame is not the center of
124
rotation as well (in other words, when the translational components are not zero in the results of
motion analysis).
The twist-matrix of the first assembly feature:
T1= [0 0 12 -2 0];
The twist-matrix of the second assembly feature:
T2= [0 0 1 2 -6 0; 0 0 0 10 0];
Suppose again the variation in the y-location of the slot need to be modeled. The modified twistmatrix of the slot assembly feature:
T2'= [0 01 2 -6 0; 0 0 0 1
;0 0 0 0 10];
Motion analysis will produce the following result for the upper plate (part-2) regarding its
degrees of freedom with respect to the lower plate (part-1).
T = [0 0 12 -2 0];
The results of the motion analysis predict that the upper plate will rotate about z-axis if the slot
moves along y-direction. The results predict that there will be a rotation about z-axis which also
causes some motion at the origin of the global co-ordinate frame. Hence, the origin of the global
co-ordinate frame is not the center of the rotation. The center of rotation can be found by the
following generic formula (which is similar to equation no. (5-1) but the explanation of the
symbols is different):
(5-2)
R=Rx
Where;
x = Vector cross product
V = Translational component of the Motion Analysis Results
=
Rotational component of the Motion Analysis Results
R = Location of the center of the rotation with respect to the part co-ordinate frame
After finding the center of the rotation, one can apply the equation no. (5-1) to find the angular
variation resulting from the unit variation in the location of the assembly feature.
125
2.:
R2
L3
L1= Fixed Link
R3
R1
I L
i L5
',,
10
R4{
0--
R2
L2
L3
R3
IA
R4
L5
Li
0777)
W
MW
10
(b)
(a)
(c)
Fig. 5-10: Variation in the Five-Bar Linkage
Fig. 5-10(a) presents a five-bar structure. The problem can be to find the sensitivity of the
location of Link "L4" to the variation in the location (in x-direction) of the assembly feature
"Ri". Fig. 5-10(b) shows the feature-part diagram of the structure shown in Fig. 5-10(a). Refer
to section 4.1.2 for details about the technique of making a part-feature diagram.
The twist-matrix of the first assembly feature:
R1= [0 0 10 0 0];
The twist-matrix of the second assembly feature:
R2= [0 0 1 10 -2 0];
The twist-matrix of the third assembly feature:
R3= [0 0 1 10 -8 0];
The twist-matrix of the fourth assembly feature:
R4= [0 0 10 -10 0];
The variation in the x-location of the first pin & hole assembly feature (R1) shall be modeled as
translational degree of freedom along x-direction. The modified twist-matrix of the assembly
feature:
R1'= [0 0 1 0 00; 0 0 0 10 0];
126
Now, the motion analysis for each link needs to be done. Motion analysis will produce the
following results:
The degrees of freedom of the link "L2" when link "Li" is fixed link:
T2 = [0 0 1 12.5 0 0]
The degrees of freedom of the link "L3" when link "Li" is fixed link:
T3 = [0 0 10 -10 0]
The degrees of freedom of the link "IA" when link "Li" is fixed link:
T4 = [0 0 10 -10 0]
The degrees of freedom of the link "L5" when link "Li" is fixed link:
T5= [0 0 10 -10 0]
Now, the center of rotations for each link needs to be found out using equation no. (5-2). This
equation is used to find the center of rotation of each part for all of its rotational degrees of
freedom.
This analysis gives the center of rotations for all the links:
For L2: (0 12.5 0)
For L3, L4 and L5: (10 0 0)
Now, a chain needs to be found out from link "L4" to link "L2" (link "L2" connects to assembly
feature "Ri") because the objective is to find the sensitivity in the location of link "L4" to
variation in the location (along x-direction) of assembly feature "RI". Fig. 5-10(b) shows such a
chain. The chain starts from part-node "LA" and terminates at part-node "L2". Fig. 5-10(c) shows
this chain in the physical mechanism.
It needs to be found out what motion is induced at link "L4" due to unit motion along the
variational degree of freedom of assembly feature "Ri". Variational degree of freedom models
the variation along x-direction. Lets assume that assembly feature "RI" has a unit magnitude
along the x-direction to model the variation. Unit motion of assembly feature "Ri" in negative xdirection will generate an angular velocity in the link "L2". Using equation no. (5-1) it can be
found out that the angular velocity is:
Q2 = -(1/12.5)z
127
This angular velocity shall induce a velocity at a point coinciding with the origin of the coordinate frame of assembly feature "R2". The magnitude of this velocity can again be found out
using equation (5-1). It is:
-(1/12.5)*(2.5x + 2y)
Since, the center of rotation for link "L3" has already been calculated. The motion of the origin
of the co-ordinate frame of assembly feature "R2" can be used to calculate the magnitude of the
angular velocity of this link as well again using the equation no. (5-1). The angular velocity of
link "L3" is:
Q3 = (1/50)z
This angular velocity shall in turn induce a velocity at a point coinciding with the origin of the
co-ordinate frame of assembly feature "R3". The magnitude of this velocity can again be found
out using equation (5-1). It is:
-(1/50)*(10x + 2y)
This motion of the origin of the co-ordinate frame of assembly feature "R3" shall induce an
angular velocity in link "L4". This angular velocity is computed again by the use of equation (51). The angular velocity of link "L4" is:
Q4 = (0.02)z
So the sensitivity of angular position of "L4" with respect to unit variation in the location of
assembly feature "RI" is -0.02 (i.e. the unit translation of assembly feature "RI" along positive
x-axis will cause -0.02 z angular rotation about the center point (10 0 0) in link "L4").
5.4.4 Facts of Contribution Analysis:
It is possible that additional degree of freedom modeling the variation in the location of an
assembly feature does not give rise to a new degree of freedom in a part. This can happen in the
following two cases:
1. If the location of the assembly feature in question does not affect the location of the part
being analyzed. Naturally, this means that sensitivity of the part location to the location
of the assembly feature in question is zero.
2. If there are over-constraints in the assembly it is possible that a part attains no new degree
of freedom due to the variational degree of freedom. In this case the parts physically
connected to the assembly feature in question need to be analyzed for over-constraints.
128
Over-constraints may cause problem of assemblability. Seventh chapter presents the
classification of assemblies and it discusses different type of over-constraints.
5.5 Summary:
The top-down approach requires a connective feature-based assembly model. The assembly
model reflects design intent. The mating conditions among assembly features are decided by the
design team before the detailed design of parts. On the other hand, the bottom-up approach
attempts to identify the design intent from the collage of parts.
Both of the approaches may employ similar techniques (matrix transforms, vectorial loops) to
represent the part locations. However, the differences become obvious when variation and
contribution analysis is performed. The top-down approach will be more successful in
identifying the source of variation whereas it will be hard to find the source of variation in case
of the bottom-up approach.
A new technique to perform the contribution analysis using the constraint representation of DFC
can be used to find sensitivities of part locations to variations in the locations of assembly
features. This analysis uses only the constraint information in the DFC and the nominal
dimensions about location of assembly features. Currently, this method finds the sensitivity of
the location of a part to the variation in the location (in a particular direction) of only one
assembly feature. The method needs to be further developed to find the sensitivity of the location
of part to the simultaneous variations in multiple assembly features.
The next chapter shall focus on the effect of variation in the size dimensions of assembly features
resulting into uncertainty in the locations of parts. The form tolerance shall also be covered under
tolerance on size dimensions.
129
130
Chapter 6: Uncertainty due to Design-in-Clearance
In the previous chapter, the effect of variation in the location of assembly features over assembly
level dimensions was studied. The manufacturing imperfections also cause the variation in the
shape and size of the assembly features. The shape variations (or form variations) are assumed to
be absorbed by size variations. The form variations can be related to the size variations. [Cogun,
1991] developed a correlation between deviations in form and size tolerances. This correlation
was applicable for work-pieces from nominal sizes to 1000 mm. [Osanna, 1979] related surface
roughness to size tolerances.
Size dimensions are important in case of bi-directional assembly features. Unidirectional
assembly features do not have any size dimensions. They have only location dimensions (e.g. A
planar surface (on one part) providing constraints to another planar surface (on some other part)
cannot have a size dimension). Normally, all assembly features which provide bi-directional
constraints are built with a clearance on their bi-directionally constrained dimension. This
clearance shall be referred as design-in-clearance henceforth. Essentially, design-in-clearance is
empty space between two parts. One can place a tolerance on design-in-clearance and it will
have variation. Design-in-clearance introduces the uncertainty in the location of assembly
feature. Design-in-clearance is generally used to satisfy the fit requirements. If it is used to
relieve the over-constraints in the assembly it may become necessary to find and analyze
multiple tolerance chains. This chapter presents a method to analyze uncertainty in the location
of a part due to design-in-clearance on assembly features.
This chapter has been organized in the following fashion. First section proposes a way of
analyzing the effect of design-in-clearance on the part locations in the context of top-down
design process. Second section presents the current approach or the approach supported by the
bottom-up design process for analyzing design-in-clearance or the variations associated with
size. Third section compares the proposed approach of the top-down design process and the same
of the bottom-up design process. Fourth section presents the summary of this chapter.
131
6.1 Design-in-clearance and Size Variations in Top-down Design Process:
An appropriate clearance is assigned to assembly features where the assembly feature halves
might not fit together in a desired way if the clearance is not appropriate (e.g. one needs an
appropriate amount of clearance on a peg and hole assembly feature for a desired type of fit).
This clearance on assembly feature is referred as design-in-clearance in this chapter. Design-inclearance introduces the uncertainty in the location of one part with respect to the other.
In case of properly constrained assemblies, the part locations can be defined in terms of the
locations of assembly features unambiguously because there exist unique chain of mates for the
location of a part. Over-constrained assemblies may have multiple tolerance chains for some of
the assembly level dimensions. Design-in-clearance may ensure assemblability. However, it may
become necessary to find and analyze the multiple tolerance chains.
It need to be figured out when design-in-clearance will just introduce the location uncertainty,
when it may become necessary to find the multiple tolerance chains, how the effect of design-inclearance on location uncertainty can be analyzed and how the multiple tolerance chains can be
analyzed. This section of the chapter attempts to analyze all these issues. First sub-section
presents how the design-in-clearance introduces location uncertainty. Second sub-section
addresses the issue of multiple tolerance chains in presence of design-in-clearance. Third subsection presents the method of modeling location uncertainty due to design-in-clearance. Fourth
sub-section presents how the uncertainty in the part locations due to design-in-clearance can be
analyzed.
6.1.1 Design-in-clearance and Uncertainty in the Location of Assembly Features:
Assembly features constrain one part with respect to other in certain degrees of freedom. The
word "constraint" is related to location. If a part is constrained in a particular direction, this
means that its location is known in that direction. This location may not be stable. Some
assembly features like a lap joint between two plates create unidirectional constraint. While such
assembly features can locate a part exactly along the directions of constraints, the location
1Fourth chapter presented the constraint analysis of assemblies. Constraint analysis divides assemblies into three
categories: Properly, Under- and Over-constrained. This shall be discussed in detail in next chapter which discusses
the assembly classification in detail.
132
provided by them is not stable. Additional assembly features ("contacts") for stabilizing the
locations are required. The block in Fig. 6-1 is constrained along the X-direction but its location
is not stable. There are some very common assembly fixtures that provide unidirectional
constraint during assembly process. Kinematically speaking, unidirectional features do not have
the load bearing capacity in every direction.
On the other hand, assembly features realizing bi-directional constraints do not need any
additional stabilizers but they cannot locate the parts exactly. Either such an assembly feature
will have design-in-clearance or there will be deformation due to interference. In both the
situations, a zone of uncertainty will be associated with the location of the parts. A bi-directional
feature constrains the block of Fig. 6-2 in the X-direction. It does not need any other stabilizer.
Fig. 6-2(a) shows the nominal configuration of the feature. However, the feature will always end
up either as shown in Fig. 6-2(b) or Fig. 6-2(c). Features realizing bi-directional constraints such
as holes are very common. A pin in a hole is bi-directionally constrained along the two mutually
perpendicular directions in its radial plane. Bi-directionally constrained directions do have loadbearing capacity.
Yt-
X
Fig. 6-1: Unidirectional Constraint
S eDimension
(a)
(b)
(c)
Fig. 6-2: Bi-directional Constraint
Assembly features realizing unidirectional constraints will have tolerances only on their location
because the issue of clearance simply does not arise. Assembly features realizing bi-directional
constraints will have tolerance on the size dimension (Fig. 6-2) as well apart from the tolerance
133
on their location. This is an important difference between assembly features carrying
unidirectional and bi-directional constraints.
Bi-directional assembly features do not need any extra stabilizer along the bi-directionally
constrained directions (bi-directional assembly features also have some directions which are
unidirectionally constrained for assemblability). For example, the peg is inserted into the hole
along the direction of the axis of the peg. This direction is not bi-directionally constrained (see
Appendix B for the list of bi-directionally constrained directions for various assembly features).
Design-in-clearance on bi-directional assembly features can be decided based upon the type of
required fit and the tolerance on feature size dimension only, if the assembly is properly
constrained. Design-in-clearance can be decided on all the features of assembly shown in Fig. 63 by just knowing the tolerances on the size dimensions of respective assembly features because
this is a properly constrained assembly. If the assembly is over-constrained, the design-inclearance needs to take into consideration the tolerances on location of other features as well.
Design-in-clearance on the features of assembly shown in Fig. 6-4 needs to take in to
consideration the tolerances on link lengths as well as the tolerance on pin size and hole size
because this is an over-constrained assembly.
Two Tolerance Chains
Unique Tolerance Chain
Fig. 6-4: Over-Constrained Assembly
Fig. 6-3: Properly Constrained Assembly
6.1.2 Design-in-Clearance and Multiple Tolerance Chains:
For an over-constrained assembly, the tolerance chains are ambiguous at the nominal level itself.
Moreover, over-constraints may cause the problem of assemblability as well. To avoid the
problems of assemblability, one way is to put design-in-clearance on the bi-directional assembly
134
features (if available in the assembly). The presence of design-in-clearance may make the
physical assembly possible but the multiple tolerance chains due to over-constraints must be
analyzed if the variation in the location of parts needs to be kept with certain specifications. Fig.
6-5 shows an over-constrained assembly of two plates being constrained with respect to each
other by two peg and hole assembly features. The x-location of the plate is ambiguous. It may be
decided by either of the peg and hole assembly features. It should be noted that the problem of
multiple tolerance chains is due to over-constraint and not due to design-in-clearance. Overconstraint is a property of the nominal design itself and over-constraints can be identified by
constraint analysis presented in fourth chapter.
x
(a)
(b)
(c)
Fig. 6-5: Over-Constrained Assembly
In case of properly constrained assemblies, the tolerance chains are unambiguous. Though, the
design-in-clearance does add to the variation or uncertainty in the assembly level dimensions but
this uncertainty can be analyzed by modifying the uncertainty-matrix representing the
uncertainty in the co-ordinate transform of the assembly feature (it shall be explained in the next
sub-section). Fig. 6-6 shows a properly constrained assembly of two plates being constrained
with respect to each other one "peg and hole" and one "pin in slot". The x-location of the plate is
unambiguous with in a limit. It is always decided by the peg and hole assembly feature.
(a)
(b)
Fig. 6-6: Properly Constrained AssemblyI
135
(c)
6.1.3 Modeling Uncertainty in Assembly Feature Location due to Design-in-Clearance:
Third chapter illustrated that the assembly features are composed of two sets of surfaces (one on
either of the two parts which constitute the assembly features). Each assembly feature will have
an ideal configuration where the clearance is not provided. However, in reality clearance will
always be provided. This design-in-clearance causes the uncertainty in the location of one part
with respect to another. Fig. 6-7 shows various configurations of a square peg in a square-hole
assembly feature. Lets assume that square-hole is on part-1 and square-peg is on part-2. Fig. 67(a) shows the ideal configuration with no clearance. In this case, the location of part-2 will be
exactly same as that of part-1. Fig. 6-7(b,c,d,e) show the configurations with clearance. In these
cases, the location of part-2 will have some uncertainty with respect to the location of part-1.
Uncertainty introduced by design-in-clearance will be proportional to the amount of clearance.
No Clearance
B
Clearance
Clearance
Clearance
(b)
(c)
(d)
Clearance
C
(a)
(e)
Fig. 6-7: Square Peg in Square Hole
Manufacturing variations are modeled as tolerances on the location of assembly feature. The
uncertainty created in the location of assembly features due to design-in-clearance can also be
modeled in a similar way. Adding design-in-clearance in an assembly feature will amount to
adding a zone of uncertainty to the co-ordinate transform of the assembly feature.
Example: The uncertainty due to clearance on square-peg in square-hole assembly feature can be
modeled with the help of following uncertainty-matrix:
0
6Z 0
SX
-6z
0
0
0
0
1
sy
0
0
0
0
1
136
x and 6y represent the uncertainty in x and y directions. Oz represents the uncertainty in the
angular position. Oz is a function of
x and 6y. The matrix representation can represent
uncertainty in three dimensions. It is important to make distinction between the error-matrix
modeling manufacturing variations and the uncertainty-matrix modeling design-in-clearance.
6.1.4 Analyzing Uncertainty in the Location of Parts due to Design-in-Clearance:
Several researchers have worked on optimizing design-in-clearance on assembly features.
[Desrochers and Riviere, 1997] presented a method to represent the clearances in form of 4X4
matrices. This article modeled the clearance as tolerance zones. The 4X4 matrix transforms were
used to compute the variation in part locations. This article did not differentiate between the
manufacturing variations and uncertainty due to design-in-clearance. [Ngoi and Min, 1999]
presented a method of allocating optimum clearances and tolerances in an assembly using the
interaction requirements in the assembly. The method presented in this article assigns extremely
loose tolerances first and then the tolerances are tightened without violating the interaction
requirements in the assembly. This is not an analysis method. [Tischler and Samuel, 1999]
presented a method to predict the slop in the general spatial linkages due to design-in-clearances
on assembly joints. The primary focus of this article was on identifying the joints in a
mechanism which contribute most to the slop in a mechanism.
This chapter makes a distinction between manufacturing variations and uncertainty due to
design-in-clearance. Previous sub-section presented how the uncertainty on the location of
assembly features due to design-in-clearance can be modeled in terms of 4X4 matrices. This
section presents how the effect of design-in-clearance on part location can be analyzed. In order
to analyze the location of a part in an assembly it is important to have a chain of mates (tolerance
chain) which passes constraints from fixed part (or fixture) to the part in question. Third chapter
emphasized that these chains of mates are constructed by the design team following a top-down
approach. On the other hand, the chains of mates need to found out from solid models of parts in
case of bottom-up approach. Fifth chapter gave appropriate references to techniques for
analyzing the effect of variation in location of assembly features on assembly level dimensions.
In a top-down approach, the effect of variation in the location of assembly features is analyzed
137
2
Error transform represent the
by modeling the manufacturing variations as error transforms2.
variation in the location of assembly features. Error transforms are inserted in the chains of mates
to compute the variation in an assembly level dimension. Similarly, the effect of uncertainty on
the location of assembly features due to design-in-clearance can also be analyzed by inserting the
uncertainty matrices in the chain of mates.
For example, the 4X4 uncertainty matrixes can be used to analyze the uncertainty in the location
of a part on top of a stack-up. Successive parts have co-ordinate transforms which represent the
location of the part in the co-ordinate frame of the previous part. Uncertainty matrices can be
associated with these transforms. Uncertainty in the location of the top part can be analyzed both
in worst-case and statistical sense. The statistical information may be more useful. In general, the
uncertainty created due to the design-in-clearance is handled depending upon the constraint
properties of assemblies. In case of properly constrained assemblies, the tolerance chains are
unambiguous. For an over-constrained assembly, the tolerance chains are ambiguous at the
nominal level itself. Multiple tolerance chains due to over-constraints must be analyzed to
calculate the effect of design-in-clearance on the part locations.
6.1.4.1 Analysis of Design-in-clearance in Properly Constrained Assemblies:
Design-in-clearance is primarily used in properly constrained assemblies to satisfy the fit
requirements. Sometimes measurement process can also be used during assembly to provide the
final location to the parts. Assembly features may be providing just rough locations. The
measurement process provides the constraints to the part being located and the assembly features
just become contact 3 . The role of design-in-clearance in such cases become to make sure that the
assembly feature remains contact and does not become mate. Example: Fig. 6-6 shows a properly
constrained assembly of two plates. It may be possible that a measurement process determines
the location of top plate with respect to the bottom and the assembly features are used only to
provide rough location to the top plate during assembly process. Fig. 6-8(a) shows a chart
2 Assembly
features are represented by co-ordinate transforms which give the location of assembly features with
respect to the corresponding part co-ordinate frames. An error transforms models the variation in the location of an
assembly feature due to manufacturing variations.
3 Contacts provide the stability to already established location. Assembly features like Over-sized bolt stabilize the
location of the part and thus act as "contacts".
138
regarding how the design-in-clearance and measurement process during assembly may affect the
properly constrained assemblies.
Over-Constrained I
Properly Constrained
Design-in-Clearance
used for fit requirements
Measuring
Process used
to locate the
parts
No Measuring
Process used
to locate the
parts
No uncertainty
due to designin-clearance
remains.
Uncertainty in
location can be
modeled using
matrix-based
approach
Design-in-Clearance
used for assemblability
Measuring
Process used
to locate the
parts
No Measuring
Process used
to locate the
parts
Clearance should
make sure that
assembly feature
is contact and not
mate.
Multiple
tolerance
chains exist
If the design cannot be
modified now, all the
tolerance chain needs to be
analyzed for assembly level
dimensions or the statistical
simulation of variation in the
assembly
(a)
(b)
Fig. 6-8: Design-in-Clearance in Over- and Properly Constrained Assemblies
If no measurement process is being used to locate the part it becomes necessary to analyze the
effect of the uncertainty in the location of assembly features on the part location. Though, it is
possible to analyze the uncertainty in the location of a part in worst-case sense but it will make
more sense to describe the uncertainty statistically. Location of the part can be expressed as
function of the location of assembly features (chain of mates shall be used for this purpose)
unambiguously in case of properly constrained assemblies. Statistical distributions can be
139
associated with the size dimensions and also with all other location dimensions to simulate the
effect of manufacturing variations. Uncertainty in the location of a part can be simulated
statistically. This process shall be illustrated with the help of the following example:
6.1.4.1.1 Statistical Simulation of Uncertainty in Properly Constrained Assemblies:
Lets consider the assembly of two plates shown in Fig. 6-9. Design-in-Clearance is provided on
both of the assembly features in this assembly. Lets assume that hole and slot are on the bottom
plate that is fixed. The top plate has two pins (pegs). First pin (pegl) corresponds to the "peg &
hole" assembly feature and the second pin (peg2) corresponds to the "pin in slot" assembly
feature. The global co-ordinate frame is attached to the center of the hole on bottom plate. The
co-ordinate frame of the top plate is attached to the center of Pegi. The dimensions are shown in
Fig. 6-9. Variation in the x-location of the top plate and variation in its angular location due to
design-in-clearance shall be analyzed. This assembly is properly constrained in XY plane.
There are two assembly features between top plate and bottom plate. There exist a unique
function for location of top plate with respect to the bottom plate. The "peg & hole" assembly
feature decides x-location of the top pate. The angular location of the top plate is given by the
following formula:
0=
(Yp -Yp2)/
(6-1)
Lp
Yp,= Y-location of Pegi;
Yp2= Y-location of Peg2;
Lp= Distance between Peg1 & Peg2;
This formula is valid only when (Yp, -Yp2 ) is small compared to Lp.
140
8,
0
Top Plate
Bottom Plate
Fig. 6-9: Properly Constrained Assembly
Diameter of the hole in "peg & hole" assembly feature (Dhl): 20.1
Diameter of the peg in "peg & hole" assembly feature (Dpi): 19.8
Width of the slot in "pin in slot" assembly feature (Sw): 20.1
Diameter of the peg in "pin in slot" assembly feature (Dp2 ): 19.8
Distance between two pegs (Lp): 100
If it is assumed that the two assembly features shown in Fig. 6-9 are the only assembly features
that decide the location of top plate with respect to the bottom plate x-location of top plate shall
have an uncertainty. The uncertainty in x-location shall be equal to the amount of design-inclearance on "peg & hole" assembly feature. In general there will be a statistical distribution
associated with the dimensions. Lets assume that all the dimensions can be described by a
normal distribution and the values of different dimensions are the mean values. The symbols for
standard deviations of different dimensions are as follows:
Standard deviation of Dhl
G hl
Standard deviation of Dp1= G pi
Standard deviation of S,= a w
Standard deviation of Dp2=
G p2
Standard deviation of L= (5,
141
Statistical simulation for uncertainty in the location variables (x and 0) is required. The mean,
variance and other statistical quantities for uncertainty in the location of the top plate can be
found only by simulation. The process of statistical simulation is as follows:
1. First step in the simulation draws all the dimensions based upon their associated
statistical distributions.
2. The second step calculates the maximum and minimum of the quantity being simulated
(x or 0). This step essentially, generates the range of the uncertainty for a particular set of
dimensions. For example, equation no. (6-1) should be used to find the maximum and
minimum of angular location of top plate for a given set of dimension. The difference
between maximum and minimum of angular location represents uncertainty for a given
set of dimensions.
These two steps should be repeated a large number of times. This shall produce a simulation for
maximum and minimum of a quantity. These two simulations can be used to predict the
statistical properties of the uncertainty in the location of a part. Uncertainty is defined as the
difference between maximum and minimum of x-location for a given set of dimensions. It is
important to include interference conditions when formulating simulation for any quantity. In
this case interference can happen only if the diameter of the peg is larger than that of the hole or
diameter of the pin is larger than width of the slot. These are the interference conditions.
For the nominal dimensions given in Fig. 6-9 MATLAB was used to setup simulation (details
can be found in Appendix B). Standard deviation of the distance between pegi and peg2 is
assumed to be 0.20 and the same for rest of the dimensions is assumed to be 0.07. The
manufacturing variations causing y-shift in the location of the slot and y-shift in the location of
peg2 with respect to peg-1 are ignored in the simulation to highlight the effect of uncertainty.
Though, one can include them in the simulation without adding any complexity. Fig. 6-10 shows
the plots for maximum and minimum of x-location of top plate for different runs of simulation.
Fig. 6-11 shows the plots for maximum and minimum of 0 -location of top plate for different
runs of simulation. The darker dots correspond to maximum of 0 and lighter dots correspond to
minimum of 0. Each run of the simulation refers to a given set of dimensions. The statistical
properties of the results about uncertainty in location variables of top plate are as follows:
142
Mean of uncertainty in x-location = 0.3006; Mean of uncertainty in 0 -location = -1.2330e-006
Standard deviation of uncertainty in x-location = 0.0993
Standard deviation of uncertainty in 0 -location = 0.0014
Fig. 6-10: Uncertainty in X-location (Properly Constrained Assembly)
Fig. 6-11: Uncertainty in 0 -location (Properly Constrained Assembly)
143
6.1.4.2 Analysis of Design-in-clearance in Over-Constrained Assemblies:
Design-in-clearance is used to increase/ensure assemblability in addition to satisfy the fit
requirements in over-constrained assemblies. In case of over-constrained assemblies as well, the
role of measurement process is important. It is possible to have large design-in-clearance on
assembly features and use measurement process to locate the parts. Design-in-clearance is used
to make sure that the assembly feature remains contact and does not become mate. Example: Fig.
6-5 shows two plates constrained by two peg & hole assembly features. It is an over-constrained
assembly. However, if the design-in-clearance is large enough both the assembly feature may
become contacts and a measurement process may be deciding the location of top plate with
respect to the bottom plate. Fig. 6-8(b) shows a chart regarding how the design-in-clearance and
measurement process during assembly may affect the over-constrained assemblies. Fig. 6-12
shows some examples of over-constrained assemblies.
If no measurement process is being used to locate the part (in other words the part is overconstrained even after recognizing all contacts) it becomes necessary to find the multiple
tolerance chains and then one need to analyze them all for uncertainty in the part location. The
location of a part may become an ambiguous function in over-constrained assemblies. The
manufacturing variations in the location of assembly features may decide which set of part level
dimensions will decide the location of the part. Depending upon the manufacturing variations
different sets of assembly features may decide part location in different degrees of freedom. Fig.
6-12 shows two assemblies which are over-constrained. Fig. 6-12(b) shows an assembly of
where a plate is being located by three fixture pins. The plate has one hole and two slots. This
assembly is over-constrained. Its location is an ambiguous function of the location of assembly
features. For example, the angular location may be decided by the "peg & hole" assembly feature
and either of the "slot in pin" assembly features. The following approach is proposed to analyze
multiple tolerance chains in over-constrained assemblies:
6.1.4.2.1
Statistical Simulation of Uncertainty in Over-Constrained Assemblies:
It is possible to generate all the tolerance chains for an assembly level dimension and analyze
them all for uncertainty in the location. The problem of generating the contact states of
144
assemblies is well researched. [Chen and Hwang, 1992] presented a motion planner for
manipulators. The task of motion planning involves generating the contact states of different
solids. [Dakin and Popplestone, 1993] presented the algorithm to generate all the contact states
for a narrow-clearance assembly. The different contact states will correspond to different
tolerance chains. [Liu and Popplestone, 1994] presented another group theoretic approach to
generate the surface contacts between solids. Using these approaches, one can identify the
different tolerance chains in an assembly. However, these methods may be computationally
expensive for complex assemblies.
After generating all tolerance chains, it is possible to generate multiple functions for the location
of part. Fig. 6-13 shows some of the different configurations of the over-constrained assembly
shown in 6-12(a). The assembly shown in Fig. 6-12(a) has two parts. Part-1 has one hole and one
slot. There are two corresponding fixture pins. Part-1 is also touching with part-2 as shown in the
figure. Part-2 has one hole and corresponding fixture pin. There is another fixture locator to set
angular location of part-2. One can get different functions for the location of part-1 by the
multiple tolerance chains. Fig. 6-13(a) shows that the x-location of part-1 can be decided by part2. Fig. 6-13(b) shows that the x-location of part-1 can be decided by fixture pin corresponding to
the "peg & hole" assembly feature. Multiple functions for the location of the part correspond to
different configurations
and different configuration represent different tolerance chains.
Manufacturing variations decide which configuration will prevail.
1
0
CIE)
(b)
(a)
Fig. 6-12: Ambiguous Tolerance Chains for Over-Constrained Assemblies
145
FEl
21x-location
x
(a)
(b)
Fig. 6-13: Ambiguous Tolerance Chains for Over-Constrained
Conceptually, it is possible to form statistical simulation for over-constrained assemblies as well.
It shall have following steps:
1. Draw all dimensions from their respective statistical distributions.
2. Find which configuration of the assembly prevails.
3. Pick the function for the location of the part to be analyzed which corresponds to the
configuration identified in previous step.
4. Find the maximum and minimum of the location variable of the interest.
5. Repeat previous steps sufficient number of times.
This procedure shall give the statistical distribution of the maximum and minimum of a quantity
and it gives information about the statistical properties of the uncertainty in the location of the
part. The main difficulty lies in step-2. It is hard to analytically relate the set of dimensions to the
configuration for a general assembly of arbitrary complexity. It is important to mention that
interference conditions need to be included in this proposed approach as well. Use of some
numerical approach to solve the difficulty of step-2 may produce a useful approach for predicting
uncertainty in a general assembly. This is still an active research area and further work is
required. The proposed approach in this section is explained next with the help of a simple overconstrained assembly. It will be hard to set up the simulation model for complex assemblies.
Example:
Fig. 6-14 shows the assembly of two plates that has two pin and hole assembly features. Both the
assembly features have design-in-clearance on them. Lets assume that both holes are on the
bottom plate that is fixed. The top plate has two pins (pegs). First pin (peg1) corresponds to the
assembly feature "A" and the second pin (peg2) corresponds to the assembly feature "B".
146
Similarly, the bottom plate has two holes. First hole corresponds to the assembly feature "A" and
the second hole corresponds to the assembly feature "B". The global co-ordinate frame is
attached to the center of the hole corresponding to assembly feature "A" on bottom plate. The
co-ordinate frame of the top plate is attached to the center of Pegi. The dimensions are shown in
Fig. 6-14. Here, uncertainty in the x-location of the top plate due to design-in-clearance shall be
analyzed. This assembly is over-constrained in XY plane.
The assembly may not be possible if the design-in-clearance on the assembly features are not
appropriate. Moreover, it cannot be said definitely whether assembly feature "A" or "B" decide
the x-location of the top plate with respect to the bottom plate. Therefore, there exist multiple
tolerance chains for the x-location of the top plate with respect to the bottom plate.
'21
19.8198
A
1 .1
B2
80)
ToD Plate
Bottom Plate
Fig. 6-14: Over-Constrained Assembly
Diameter of the hole in assembly feature "A" (Dhl): 20.1
Diameter of the peg in assembly feature "A" (Dpi): 19.8
Diameter of the hole in assembly feature "B"
Diameter of the peg in assembly feature "B"
(Dh2):):
(Dp2):
Distance between two pegs (L): 100
Distance between two holes (Lh): 100
147
20.1
19.8
Xmin
Xmin
(a)
(b)
Xnax
Xm-ax
(c)
(d)
Fig. 6-15: Multiple Tolerance Chains
The minimum and maximum of x-location may be decided by either of the assembly features. It
is shown in Fig. 6-15.
Xnin = Minimum of the x-location of top plate;
Xmax = Maximum of the x-location of top plate;
Xmin will be decided by assembly feature "A" if (see Fig. 6-15(a)):
(Lh ±
0.5 * Dhl
Xmin will
0.5 * Dh2) <= (4L + 0.5 * Dpi -0.5 * Dp2)
(6- 2)
be decided by assembly feature "B" if (see Fig. 6-15(b)):
(Lh -0.5* Dhl -0.5* Dh2) <= (4L -0.5 * Dpi -0.5 * Dp2)
(6- 3)
Xmax will be decided by assembly feature "A" if (see Fig. 6-15(c)):
(Lh -0.5* Dhl-0.5* Dh2) <= ( L-0.5* Dpi -0.5 * Dp2)
(6-4)
Xmax will be decided by assembly feature "B" if (see Fig. 6-15(d)):
(Lh
-0.5 * Dhl+0.5* Dh2)
<=
(L -0.5 * Di + 0.5 * Dp2)
148
(6- 5)
The statistical distribution of x-location of top plate can be simulated because it is easy to map
the set of dimensions to the configuration of assembly. The mean, variance and other statistical
quantities can be found by simulation. It is important to include interference conditions when
formulating simulation for any quantity. In this case interference can happen if the diameter of
any of the pegs is larger than that of corresponding holes or the distance between the centers of
two holes do not match (within a limit) with distance between centers of two pegs. More
precisely, the interference conditions are:
(Lh
-0.5 * Dhl -0.5 * Dh2)
(Lh
+0.5 * Dhl+0.5 * Dh2)
>
(4L -0.5 * Dpi -0.5 * Dp2)
(6-6)
(4 +0.5* Di +0.5* Dp2)
(6-7)
(Dhl)
(Dpi)
(6- 8)
(Dh2)
(Dp2)
(6- 9)
For the nominal dimensions given in Fig. 6-14 MATLAB was used to setup simulation (details
can be found in Appendix B). Standard deviation of the distance between pegi and peg2 is
assumed to be 0.04. The standard deviation of distance between hole1 and hole2 is also assumed
to be 0.04. Standard deviation for rest of the dimensions is assumed to be 0.07. The
manufacturing variations causing y-shift in the location of the hole2 with respect to hole1 and yshift in the location of peg2 with respect to pegi are ignored in the simulation to highlight the
effect of uncertainty. Though, one can include them in the simulation without adding any
complexity. Fig. 6-16 shows the plots for maximum and minimum of x-location for different
runs of simulation. Each run of the simulation refers to a given set of dimensions. The statistical
properties of the results about uncertainty in the x-location of top plate are as follows:
Mean of uncertainty in x-location = 0.2667
Standard deviation of uncertainty in x-location = 0.0860
149
Fig. 6-16: Uncertainty in X-location (Over-Constrained Assembly)
6.2 Size Tolerance in Bottom-Up Design Process:
Traditionally, size tolerance is considered as a key to achieve the fit requirements. In a bottomup approach the size tolerances may be assigned using the following techniques:
1. Standards for Fit-requirements
There are standards for assigning tolerance on size dimensions according to the fit
requirements (e.g. running fit, sliding fit etc.). These standards do not consider the problem
of uncertainty in the location of different parts in the assembly. Decision about tolerance on
the size dimension of an assembly feature is taken in isolation without considering the
constraint structure of the assembly.
2. Geometric Dimensioning and Tolerancing (GD&T)
GD&T also called true position tolerancing, was developed to deal with solid objects and to
avoid the difficulties associated with dimensions that are only good for making drawings
[Foster, 1979; Meadows, 1995]. GD&T is used both for deciding the tolerance on location
and design-in-clearance on assembly features. In essence, the goal of GD&T is to define each
150
part so that it will assemble interchangeably with any example of its intended mate 100% of
the time in spite of unavoidable variations in each part's dimensions, and to provide an
unambiguous way of inspecting these parts individually to ensure that this goal will be
achieved [Meadows,
1995, page 5]. GD&T accomplishes this with its more careful
specification of three-dimensional shape. Much of the logic behind GD&T reflects the use of
gages to determine if parts meet specifications. The size of a cylinder is measured by a gage
that fits over its entire length. The hole in this gage is the maximum allowed diameter of the
cylinder. If the cylinder is bent then the gage may not function, even though the cylinder's
diameter is always within specifications. Thus the cylinder must be straight and round when
its diameter is as large as allowed. Similarly for a hole, a plug gage the same depth as the
hole is used. The hole must be straight and round when its diameter is as small as allowed. A
common term for biggest cylinder and smallest hole is "maximum material condition,"
abbreviated MMC. Rule #1 in GD&T states that the feature must have perfect form at MMC.
This protects the ability of gages to function. Corresponding to the method for determining
size at MMC is the method for determining size at least material condition (LMC). For a
cylinder, this would consist of a caliper that would check two opposing points anywhere on
the cylinder. There is no requirement for perfect shape at LMC. These measuring methods of
parts are not entirely satisfactory. For example, calipers are not noted for repeatability. Also,
as the cylinder gets longer with respect to its diameter, it must be straighter for the same
deviation from perfect diameter, or else the gage will not go on all the way. GD&T does not
provide a way to ascertain the uncertainty on the part locations due to clearance on the
assembly features.
The main focus in Computer Aided Tolerancing (CAT) tools remains on variation anlysis for
finding the variation in the location of a part due to manufacturing variations. The CAT tools
have limited capability regarding the analysis of uncertainty in the location of a part due to
design-in-clearance on assembly features. One can model clearance (or gap) in CAT tools as a
function of other part dimensions and the gap can be analyzed. However, the uncertainty in the
location of a part which arise from clearances at multiple assembly features cannot be analyzed.
151
6.3 Comparison between the Size Variation Analysis Approach of TopDown Method and that of Bottom-Up Method:
In a bottom-up approach, role of the size tolerance and hence design-in-clearance on assembly
features often is not well understood. The main emphasis of GD&T is to assign tolerances on the
part for 100% interchangeability. GD&T does not provide a method for determining the
uncertainty in the part location due to clearance on assembly features. CAT tools provide
functionality to analyze gaps and clearances as function of other dimensions. However, the
problem of finding uncertainty in the location of a part in presence of design-in-clearance on
multiple assembly features cannot be solved these tools. The size tolerances are often allocated at
the individual part level without clearly identifying the significance of the design-in-clearance on
assembly level dimensions.
The proposed method of finding uncertainty in the location of a part due to design-in-clearance
on assembly features makes a distinction between the uncertainty in the part location and the
variation in the part location due to manufacturing variations in the location of assembly
features. This method can be used to analyze the impact of design-in-clearance on assembly level
dimensions.
6.4 Summary:
This chapter presented a method to analyze the uncertainty in the part locations due to design-inclearance on assembly features. Design-in-clearance on assembly features can be modeled as
uncertainty in the matrix transform associated with the assembly feature co-ordinate system. The
method proposed a simulation approach to derive the statistical properties of the uncertainty in
the location of a part. Manufacturing variations on all the dimensions are considered in the
simulation. Solved examples for a properly constrained assembly and an over-constrained
assembly are presented.
The traditional approach of allocating size tolerances overlooks the impact which design-inclearance can make on assembly level dimensions. GD&T only presents a solution for worstcase problem. Moreover, GD&T is a part-centric tolerance allocation technique and still it is not
152
very much compatible with the top-down design process. CAT tools also cannot analyze the
uncertainty in part location due to clearance at multiple assembly features.
The next chapter shall present the classification of assemblies based upon their constraint
properties. The attributes of the constraint properties shall be discussed in detail in this chapter.
153
154
Chapter 7: Classification of Mechanical Assemblies'
In the world of real designs, simple classifications of mechanical assemblies that classify
assemblies in two or three classes do not help designers much because these classifications
present a black and white picture. No assembly can be judged good or bad without looking at the
functional requirements. Any classification of assembly must relate the context of the problem to
itself. A classification based upon constraint structure of the assembly may categorize the
assemblies into three classes (under-, over-, or properly-constrained). However, it does not help
designers because they don't know what is good or bad in the context of the problem at hand.
This chapter presents
a comprehensive
classification of mechanical
assemblies.
The
classification is based on the constraint structure of the assembly but it takes into account the
nature of the parts (rigid or flexible), it lists out what analysis tools one should use in order to
analyze the different assemblies, and it presents what different types of variations are possible
for different assemblies. This classification defines the characteristics of different types of
assemblies and most importantly it identifies the possible mistakes that designers may commit.
The classification presented in this chapter is primarily based upon the motion and constraint
analyses of assembly. Motion and constraint analyses were presented in the fourth chapter.
Motion analysis is a well-researched area. However, constraint analysis has received less
attention than it deserves (see section 4.1.1 in fourth chapter for previous work in the area of
motion and constraint analyses). There are significant differences regarding what is properly
constrained and what is not, among different research communities. The CAD community (e.g.,
[Thomas, 1991]) says that a part or an assembly is properly constrained if it has geometric
consistency. However, according to screw theory, a part in an assembly is over-constrained if
more than one constraint are trying to locate the part in same degree of freedom. According to
screw theory, a part in an assembly is properly constrained if exactly all six of its degrees of
freedom are constrained, no more and no less. This is why several designs are called properly
constrained by the CAD systems while they will be categorized as over-constrained by screw
1This chapter is based on article [Whitney, Shukla and Von-Praun, 2001].
155
theory. e.g. a part with two pegs in a part with two corresponding holes. CAD systems do not
care if the designer keeps on adding constraints to the assembly as long as the new constraints
are geometrically compatible (even though new constraints cause over-constraints as defined by
screw theory).
Unlike the simple classifications of mechanical assemblies, the classification presented in this
chapter also presents the difference between "design mistakes" and "design intent". This
classification is intended to serve as a guide to the designers. It will force designers to decide
whether they have made a design decision to achieve an assembly level requirement or whether
their decision may cause problems in achievement of certain assembly level requirements. This
chapter is organized in the following way. First section presents the past work in the area of
assembly classifications. Second section presents the classification of mechanical assemblies.
Third section presents a summary of the chapter.
7.1 Previous Work:
Traditionally, researchers have divided mechanical assemblies into two groups (Mechanisms and
Structures). Both of the groups have been further classified into sub-groups depending upon
functional criteria, sometimes other criteria (no. of parts, type of joints, industry etc.) as well.
[Dutta and Woo, 1995] presented another classification of assemblies that was based on their
complexity. He divided assemblies into two groups - Parallel and Sequential. Parallel assemblies
can be divided into sub-assemblies that can be assembled separately. [Mantripragada and
Whitney, 1998] divided the assemblies into two groups based upon the use of fixtures in the
assembly process. As explained in the second chapter, Type-i's do not require assembly fixtures
while Type-2's do. This chapter presents a classification of assemblies that is based on their
constraint properties evaluated by Screw Theory (under-, over-, or properly constrained
assemblies). However, this classification is expanded upon to account for more subclasses.
7.2 Classification of Mechanical Assemblies:
Fig. 7-1 presents a simple assembly classification based upon the results of constraint analysis.
Fig. 7-2 contains more detail. The following discussion deals with named subsets of Fig. 7-2:
156
7.2.1 Under-Constrained Assemblies:
Under-Constraints may be required for functional reasons. E.g., mechanisms and linkages are
under-constrained. Under-constraints may appear in the results of constraint analysis because
some fixtures may not have been included in the constraint analysis. Alternatively, underconstraints may be due to improper choice of assembly features or due to improper configuration
of the same. If the under-constraint is the result of a mistake, it must be fixed.
Proper y onstrained
Und -Co trained
Needed for
DFC is Robust DFC is not
Robust.
to Variation
Over
Needs Fixture Ne
for Assembly
d for
n rained
Mis
e
u ction
Mis take
Needed for
*
Assembly
Redundant
Stress Present
(Zero Stress)*
During
Stress Added
Non-Zero
fter Assembly Assembly
Stress
*
Fig. 7-1: Simple Assembly Classification
7.2.2 Properly Constrained Assemblies:
Properly constrained assemblies need to be checked for allowed variations on the parts. If
properly constrained assemblies are robust, there will be a unique DFC that expresses the
designer's intent. Moreover, there will be unique tolerance chains for parts with respect to the
base part. One can use 4*4 matrices to do the tolerance analysis for variation on assembly level
dimensions. Methods presented by [Laperriere and Lafond, 1999] or by [Whitney et. al., 1994]
are appropriate.
157
If the DFC is non-robust then the assembly may become over-constrained due to allowed
variations. Some "contacts" become "mates" under variation. Such situations can be avoided by
increasing the clearance on the "contacts" or by redesigning them
7.2.3 Over-Constrained Assemblies:
Over-constraints may be categorized into three groups:
7.2.3.1 Over-Constraint Needed for Function: There might be several types of functional
requirements that can be achieved only with over-constraint assemblies.
a.
Over-Constraint due to a Single Feature: Functional requirements such as "press fits"
will require over-constrained assembly features. Such over-constraints do not affect the
overall DFC. The FEM analysis of whole assembly may not be required because the effect
of such over-constraints may be limited to the local region of the feature.
b.
Over-Constraint due to Combination of Features: Here again, one should understand the
context of design and the analysis.
i. Deliberate Introduction of Idealized Assembly Features in Constraint Analysis: The
designer may have done a rigorous constraint analysis ignoring all the clearances on
assembly features and assuming line fits. This assumption is the most optimistic for
defining the location of parts, but the constraint analysis phase will report numerous overconstraints. Only some of them may be due to mistakes. The designer can go over all the
over-constraints to identify the mistakes. Once the designer believes that all mistakes have
been taken care of, the constraint analysis can be re-done with real features that have
practical clearances. Introduction of clearances on assembly features will amount to
reduction in number of degrees of freedom constrained by the corresponding assembly
features and will require additional steps to be taken during the Variation Design Phase.
ii. Non-Unique DFC: If there are over-constraints in the design and the designer believes
that the design does not have any mistakes, there will be multiple tolerance chains, and a
conventional tolerance analysis will be impossible. Over-constraints may cause local
interference. (Local interference will result into local stress.) If the designer adds clearance
to avoid interference, a situation called redundant constraint can arise. So, this situation can
be further divided into two groups:
158
1.
No Stress in Assembly: Here, the clearance on assembly features is kept large
enough so that there is no local interference. Clearance gives rise to multiple kinematic
states in which different surfaces on the features of a part contact different surfaces on
the features of its mating part. Each kinematic state gives rise to a different DFC and
tolerance chain. One must do variation analysis using the 4*4 matrices for the multiple
tolerance chains to calculate the variation. One must also include clearance in the
variation analysis to ascertain the randomness associated with the locations of parts.
E.g. A part with two pegs in a part with two corresponding holes with sufficient
clearance. Methods for doing such analyses are presented in [Sacks and Joskowicz,
1998] and [Chase et. al., 1997].
2. Stress in Assembly: Here again, there are two cases: There are assemblies where the
stress is added deliberately after assembly by imposing relative movement of certain
parts, and there are assemblies where the stress is present as soon as the assembly
process is done or even during the assembly process.
a. Stress Added Deliberately: An example is a preloaded set of ball bearings. The
assembly is properly constrained and completely stress-free until the parts have
been assembled together. However, relative movement of certain parts then overconstrains the parts and adds the stress. In such cases, assembly features are
designed keeping these functional requirements in mind. If the assembly is already
over-constrained, it may have only little or no local stress. Movement of the parts
changes the configuration of mating assembly features that in turn introduces the
stress. FEM analysis will be required in this case to ascertain the location of parts.
An example is a valve in a guideway in an internal combustion engine. It is properly
constrained when open but over-constrained with little stress when it is closed.
b. Stress Present During or After the Assembly: These assemblies may not have
sufficient clearance on their assembly features, which causes local stress. E.g. A part
with two pegs in a part with two corresponding holes with insufficient clearance.
Alternatively, the parts may be flexible and they might be getting deformed during
the assembly process. E.g. Sheet metal assemblies.
159
Mechanical Assemblies
Over-Constrained
at Nominal
Dimensions
at Nominal
Dimensions
Needed fNeeded
f( rstake
Needed for
(Sheet Metal Parts (Remove Locators; Function
(Any Mechanism
it) Add
Clearance)
' with Slip Jo
or Linkage)
ints)Mates
Mistake
(Add Locators)
Over-Constraint Over- onstraint
due to One
due to
Over-Constrained
Feature
of
Combination
under V iations
Several Features
There is a
Mistake
Unique and
The FC is
(Enlarge Clearance
Permanent DFC
at "contacts";
Not nique
(PressFits)
(Press Fits)
Tighten tolerances)
The Design is
deliberately built with
Zero-Clearance
Features to help find
Mistakes
(Add Clearance to the
deliberate overconstraints after
removing the mistakes)
Little or No
Internal Stress
(Redundant
Constraint)
Substantial
Internal Stress
Enumerate
Kinematic States
The Stress is
added after
Asse bly
Traditiofal Variation
Analysis Applies and
Tr aditional Variation is needed to Verify
A nalysis (applied to Design and to Build
Each Assembly
all possible tolerance
(Pre-loaded Ball
chains) can find
Bearing Sets; Engine
Varied Locations
Valves)
nr
inh
f(Ace-i
Fixtures with
Clearance. Fourlegged Stools)
Needed for
Ass mbly
Mae4 arer
Properly
Constrained
C
Contacts are to
Support
provide
(
("Mates" before
"Contacts"; Sheet
Metal or Cloth
Parts: n-2-1
principle)
The Stress is
present as soon
as Assembly is
aAnished
(Sheet Metal
Assemblies)
There are
Multiple
Tolerance
C ns
Part Locati s are
known only after
Stress Analysis
Fig. 7-2: Classification of Assemblies
7.2.3.2 Over-Constraint Needed for Assembly: These situations arise when the parts are not
rigid enough and they require more support than what is required for a rigid part.
However, here designers should take the precaution of differentiating between "mates"
and "contacts". I.e. some fixture elements may be acting as locators while some other
may be just reinforcing the location. E.g. Sheet metal assemblies assembled with n-2-1
principle. In such situations, one should understand that assembly sequence would
160
become important, and all mates should be completed before any contacts are. There
might also be situations when the difference between the "mates" and "contacts" is not
apparent. These could result in mistakes.
7.2.3.3 Over-Constraint as Mistakes: Some over-constraints may be result of mistakes. It is the
designer's responsibility to check whether over-constraint may cause any problems in the
KC delivery.
7.3 Summary:
A classification of mechanical assemblies based upon their constraint structure has been
presented. The classification is based upon the constraint properties of assembly evaluated using
screw theory. The classification outlines the different reasons responsible for over-constraints in
assemblies. Designers need to identify the over-constraints (if any) from the given set of
possibilities.
The next chapter presents the guidelines for a top-down design procedure of mechanical
assemblies. Designers need to refer back and forth to the classifications presented in this chapter
while evaluating their design at various stages.
161
162
Chapter 8: Design Procedure and Detection of Mistakes'
One of the most difficult problems which designers of complex mechanical assemblies face
routinely is not to be aware of when they are committing mistakes. More and more dependence
of designers on the CAD systems further aggravates the problem of committing the mistake of
not giving proper attention to the kinematic structure of the assembly. This chapter outlines a
comprehensive design procedure that will help designers in organizing their product
development process. The design procedure puts together the generic design steps in a logical
order and it provides the information regarding which tool should be used to analyze assemblies
at what stage of design process. Essentially, the design procedure shall require designers to make
sure that they follow a top-down approach and they justify their design decisions. The tools and
techniques for evaluating the design are referenced appropriately in the presentation of design
procedure.
A classification of several design-techniques (functional build, statistical coordination, etc.) for
achieving the assembly tolerances along with the steps involved in these techniques is presented
next. The purpose of this classification is to facilitate the use of the design procedure and
assembly classification (presented in the previous chapter). The designers can layout the tasks
from the generic outline of the design process. The classification of assemblies presented in
previous chapter can be utilized in order to understand what type of problem they have at hand
and what sort of variation they will encounter. Depending upon the context of the problem, one
may decide upon choosing a particular design-technique as the most suitable for achieving the
assembly tolerances.
This chapter is organized in the following way. First section presents the outline of different
design stages. This outline of design stages is generic. Second section presents the classification
of different design-techniques that can be used to ensure that manufacturing delivers the parts
with required tolerance specifications. Third section presents the summary of the chapter.
'This chapter is based upon the article by [Whitney, Shukla and Von-Praun, 2001].
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8.1 Design Procedure:
This section presents a coherent scheme of how different design steps might take place in an
actual top-down design process. Some of these steps are already known to designers while others
are not that familiar with them. Datum Flow Chain (DFC) is used as the assembly modeling
technique for describing the various steps. Tools and techniques associated with the DFC
(described in the previous chapters) shall be required during different design stages. These tools
and techniques are referenced appropriately. The design procedure has been divided into three
phases, the nominal design phase, the constraint analysis phase and the variation design phase.
Fig. 8-1 presents all the steps that may be required in a top-down design process. These steps are
discussed in detail in this Section.
8.1.1 Nominal Design Phase:
The nominal design phase refers to the set of design activities that start with identification of key
characteristics and that culminate with a layout of the framework of the physical embodiment or
in other words the DFC of the design. Broadly speaking, the nominal design phase can be
divided in the following general steps:
8.1.1.1 Identification of Key Characteristics:
Top-down design process starts with identification of Key Characteristics (KCs). Defining KCs
will involve interpreting customer level requirements in terms of engineering requirements. For
example "reduction in noise level inside the car" is a customer level requirement. Sealing
between car doors and car body primarily defines the noise level inside the car. Hence, this
customer level requirement can be interpreted as "a limit on the gap between car door and car
body".
8.1.1.2 Selection of a conceptual framework of the design (Making a DFC):
After KC identification, designers need to think whether they will use features on the parts to
locate them with respect to one another (type-1) or they will use fixtures to locate the parts
during the assembly process (type-2). Typically, it may be very expensive to manufacture
assembly features with tight tolerances on very large parts like car body or aircraft fuselage. It
164
may be cost effective to use a set of precise fixtures and let them locate the assemblies. This
decision will affect the product architecture and the manufacturing system design.
The design team may start with several architectures which can potentially solve the design
problem at hand. Most of the design activity may be performed in form of discussions about
sketches or may be in form of contemplation in one's head. Often, legacy designs may provide
good starting points. Legacy designs may or may not be related to the same industry. Designers
may identify architecture and its DFC need to be drawn. Each KC should have its own DFC.
This task may require some discussion regarding major sub-assemblies in the proposed design
solution and approximate spatial relationships among them. Designers are definitely free to try
innovative locating schemes at this stage.
8.1.1.3 Selection or construction of the assembly features (Realizing the DFC):
The output of the previous step shall be represented in terms of a DFC in a convenient symbolic
form. The next step is defining "mates" (assembly features) that will realize the constraints
between parts. Designers may choose an assembly feature from a library or they might design a
new one of their own. The DFC is a chain of mates. The assembly features realize the physical
relationships among parts. These assembly features need to be located with respect to the
corresponding parts. Chapter 3 presented how assembly features carry constraints and how one
can construct assembly features. Appendix D lists the constraints provided by some assembly
features.
The nominal design phase can be summarized in the following points:
" Identify the Key Characteristics that the assembly must deliver.
" Sketch the parts and draw a liaison diagram. Mark each KC on the liaison diagram by adding
a specially marked arc between the parts related by the KC.
" Tentatively classify the assembly as Type-1 or Type-2.
" Establish a tentative DFC for each KC, identifying possible constraint requirements between
parts (and fixtures if necessary). Mark which liaison diagram arcs would be mates and which
would be contacts.
165
"
Identify places where fixtures or measurements will be needed by noting the existence of
KCs between parts that are not joined by a chain of mates.
* Define a tentative set of features that can carry the desired constraint, consistent with
functional requirements on the features.
8.1.2 Constraint Analysis Phase:
The constraint analysis phase refers to the set of design activities that evaluate the DFC and
identify an appropriate sequence of assembling and locating the parts. Constraint analysis phase
evaluates nominal design and the appropriate corrections are made in the nominal design in this
stage. Designers sometimes overlook the importance of the nominal design phase. The primary
reason for this phenomenon is that CAD systems provide the functionality for variation analysis
but CAD systems do not have tools for evaluation of nominal design (i.e. constraint analysis).
CAD systems require the designers to jump to variation analysis as soon as they decide about
basic mating relationships among the parts and constraint analysis remains the most neglected
portion of the design process.
Constraint properties of the kinematic structure of the design are very important. There may be
enough scope of improvements at this stage itself by changing the mating relationships among
parts or by re-configuring them. Moreover, making changes at this stage of design doesn't cost
as much as it will after detailed part design and tolerance analysis or even later when the
problems are reported from the manufacturing plants. Broadly speaking, the constraint analysis
phase can be divided in the following general steps:
8.1.2.1 Motion & Constraint Analysis (Checking DFC):
Motion analysis reveals what motions are possible for different parts due to the configuration of
assembly features. The results of motion analysis can verify the required relative motion between
the parts (such as the crankshaft of an engine). These results can also bring any unexpected
motion to the designer in form of undesired under-constraints. It is recommended to include the
fixtures in the DFC and in the motion analysis so that the under-constraints are not reported due
to not including fixtures. Any other type of under-constraints will lead to random variations in
the assembly that may lead to non-delivery of certain KCs. Hence, such under-constraints must
166
be avoided by reorienting or relocating the assembly features. One might have to look for an
alternative choice of assembly features in order to avoid that.
Constraint analysis is rather more insightful than motion analysis. It gives the information about
all the degrees of freedoms for all the parts in an assembly which are being constrained by more
than one assembly feature. Over-constraints are the property of nominal design. Identification of
over-constraints does not require any information about tolerances on the assembly features.
Procedures for "motion" and "constraint" analyses are presented in fourth chapter.
If the assembly is reported as over-constrained, the designer should try to identify how the overconstraint is being caused. The design team needs to identify the over-constraints with one of the
several possibilities shown in the assembly classification presented in the previous (7 th) chapter.
Over-constraints may be required for functionality (preloaded ball bearing sets), for facilitating
assembly process (redundant supports for flexible parts), or they might be result of a design
mistake. If over-constraints are required for functionality, stress analysis may be required to
ensure that KCs are delivered. If over-constraints are required for assembly, the designer needs
to look for which assembly features are acting as "mates" and which others are acting as
"contacts". If the designer does not understand the underlying DFC, he/she may confuse the
"mates" and "contacts" and remove the over-constraint in an incorrect way.
8.1.2.2 Making corrections in DFC:
Over-constraint mistakes would either cause random variation or might introduce unwanted local
stresses. Both of these two phenomena must be avoided by reorienting or by relocating the
assembly features. Designers might have to resort to an alternative choice of assembly features.
Over-constraint causes the assembly to be non-robust to small variations in the parts. If overconstraints are required for functionality, design team may have to worry about some specific
design-techniques
to avoid interference
and assemblability problems
later on during
manufacturing (e.g. simultaneous machining, selective fitting etc.). These techniques are related
to achievement of tolerance specifications. These techniques shall be discussed in section-2 of
this chapter.
167
8.1.2.3 Identification and selection of assembly sequences:
The assembly sequences can be found out using the local constraint analysis. [DeFazio et. al.,
1993] presented an interactive system of finding assembly sequences. This method restricts
assembly sequences to the ones which allow only properly constrained sub-assemblies at various
stages of assembly process.
8.1.2.4 Detection of KC conflict:
After the designer has finally arrived at a properly constrained DFC it might be necessary to look
for KC conflicts. A KC conflict occurs if there are not enough degrees of freedom in the
assembly to permit the dimensions and tolerances of all KCs to be specified independently. It
can be detected by checking if more than one KC shares some part of the same DFC. [Whitney
et. al. 1999] presented an approach to detect multiple KC conflicts using the constraint analysis
by screw theory. If the assembly is type-2 it may be possible to find an assembly sequence that
relieves the KC conflict. However, if KC conflict is unavoidable, the designer needs to prioritize
the KCs. Designers also need to add "contacts" in the assembly if required for supporting the
locations fixed by "mates". Designers need to ensure that "contacts" do not affect the DFC by
adding unwanted constraints. They also must ensure that the assembly sequence is chosen in
such a fashion that "contacts" are closed after "mates".
The constraint analysis phase can be summarized in the following points:
" Examine these feature sets for over- or under-constraint, making necessary corrections.
" Identify geometrically feasible assembly sequences, utilizing local constraint knowledge
deduced from the features. If fixtures are part of the assembly process, identify only subsequences that utilize a single fixture, and string together such sub-sequences into a final
sequence.
* Restrict the assembly sequences to those that build fully constrained subassemblies and
which make all the mates on a part before any of its contacts.
* If the assembly contains several KCs, examine it for the possibility that there are not enough
degrees of freedom to adjust them independently or to achieve them within tolerances with
statistical independence. This occurs because more than one KC lays claim to the same
degrees of freedom on the same arc of a DFC. It sometimes occurs because the chosen
168
assembly sequence achieves the KCs all at once. Possibly another assembly sequence can
achieve them one at a time, relieving the conflict. Otherwise, either the conflict must be
accepted by prioritizing the KCs, the KCs must be redefined, or major changes to the features
must be considered.
8.1.3 Variation Design Phase:
The variation design phase refers to the evaluation process which checks the robustness of the
DFC to part level variations. However, the variation design phase is not simply about only
variation analysis. The Variation Design Phase is mainly about checking whether the intended
kinematic structure in the DFC remains preserved despite the variation in the part dimensions. Of
course, the variation analysis which finds the variation in assembly level dimensions due to
variation in part level dimensions is an important part of variation design phase. The variation
design phase is divided in the following general steps:
8.1.3.1 Checking Robustness of the DFC:
The DFC should not have any undesired under- or over-constraints. The under- and overconstraints should be permitted only if they are required for functionality of the assembly. If the
DFC is properly constrained, it needs to be checked whether the DFC stays properly constrained
under variation. This is to ensure that some "contacts" do not become "mates" under allowed
variations. If this does happen, designer needs to redefine the clearances on contacts or might
have to resort to redefining some of the assembly features. If the assembly requires use of
fixtures then there can be a scope of changing the assembly sequence and satisfying more KCs.
Similarly, one may decide to change the DFC at this stage as well.
8.1.3.2 Allocating tolerances to the KCs and to the Mates:
First of all the design team needs to agree upon certain tolerances on the KCs. These tolerances
reflect the allowed variation on the assembly level dimensions representing different KCs. The
designer needs to allocate the tolerances on the location of assembly features. The design-inclearance on the assembly features (if applicable) is also an important design issue and designers
need to assign design-in-clearance according to fit requirements and keeping in mind that designin-clearance also contributes towards variation in assembly level dimensions.
169
If the KC conflict is encountered during the constraint analysis phase and the KCs have been
prioritized. The designer will have to add clearance on lower priority KCs or the tolerance
specifications on it may have to be lowered.
8.1.3.3 Variation and Contribution Analysis:
Finally, the designer needs to perform a conventional variation analysis for each KC to ensure
that it is being delivered a high enough percentage of time. If this is not being achieved, the
designer may have to think about some innovative ways like coordinated machining, use of
fixtures instead of features on parts (type-2 instead of type-1) etc. (Different techniques to
achieve tolerance specifications will be discussed in section-2). Contribution analysis finds the
sensitivity of variation in any assembly level dimension to the variation in the location of
assembly features. Contribution analysis shall help the designers in allocating the tolerances on
the location of assembly features or even further reconfiguring them. A new technique to
perform contribution analysis is presented in the 5t chapter. This technique requires information
only about the constraint structure of the DFC. It does not require any information about the
tolerances on the location of assembly features.
The variation analysis has two components in it. Variation in assembly level dimensions due to
variation in the location of assembly features is one component. The techniques to perform
variation analysis due to variation in location of assembly features are presented in
5 th
chapter.
Variation due to design-in-clearance on assembly features is also important. Techniques to
analyze the uncertainty due to design-in-clearance are presented in 6 h chapter.
The variation design phase can be summarized in the following points:
" Examine each arc in each DFC to determine if variation in the size and location of a feature,
mate, or contact could alter the DFC. Improve the design, tolerances, or clearances related to
these items until the DFC is robust against such variations.
" Analyze the ability of the candidate DFC, feature set, fixtures, and sequence to deliver the
KC(s) by performing a 3D variation analysis of each DFC. Extend the analysis over chains of
fixtures if necessary, being careful to include any datum transfers that occur between
fixtures. If the KC cannot be delivered with the required accuracy or frequency, then some
170
portion of the design must be repaired, starting with the assembly sequence and fixtures, if
any, and retreating to different DFCs and features if nothing else works. Possibly the
assembly cannot be made as a Type-1 and will have to be re-designated as a Type-2. Then
the whole process begins again.
Define Key Characteristics
Nominal
Design
Phase
Declare the Assembly
Type- I or Type-2
Draw a Datum Flow Chain
for Each KC
[Check: a chain of mates from one
end of the KC to the other]
Define Mates
Create Features
Ensure that Mates
Create Proper
Check for K
Conflict
Pioritizig
Conflicting KCs
[Check: In Type-2 assemblies, the
DFC may pass through fixtures]
V
Define an
Assembly
Sequence that
Try another
Assembly
Sequence
& Makes Mates
Before Contacts
Fig. 8-1: Design Process Chart
171
Constraint
Analysis
Anas
De me Contactsif they add overconstraint, then
ensure that it
does not affect
the DFCs
8.2 Meeting Assembly Tolerances:
Once the design specifications are finalized, the next big step is to design the assembly for
variation. Fig. 8-2 presents the classification of the three schemes that can be used to achieve the
required tolerances on parts.
This classification is based on the idea of coordination from
economics. Coordination is the activity required to see that different but related activities are
done so that the relational requirements are met. The simplest example is to make parts at
different suppliers and have them assemble randomly and interchangeably at final assembly.
Fig. 8-2 classifies techniques of achieving tolerance specifications according to three kinds of
coordination.
8.2.1
Deterministic Coordination:
Here, required tolerances are so tight that it is uneconomical to make the parts interchangeable.
One can use "fitting", "simultaneous machining", "selective assembly" or "functional build" to
achieve assemblability. Functional build is applicable only when adjustment of tools and dies is
allowed. These adjustments shift the nominal dimensions from their means but variations are
driven out from the production process by process improvement. One may use Cpk data on parts
after making adjustments to ensure that the new process mean does not shift any more and that
the variation remains small. Documenting mean shifts on design after making adjustments is a
recommended practice. If one uses only Cp data on parts and the adjustment of the mean is not
documented, it may lead to problems in diagnosis of problems later on.
8.2.2
Statistical Coordination:
Here, the probability of meeting the tolerances is kept high enough and parts are interchangeable.
This is the most popular technique to meet the tolerances on parts and hence the quality
requirements over the whole assembly. The process is monitored using the Cpk data so that
process mean remains close to nominal values and variation also remains within tolerance
specifications. If these conditions are met, then variation at the assembly level can be attributed
to the variation around the mean in each part. This gives rise to the familiar root sum square
(RSS) method of estimating assembly-level tolerances. In the simple case where each part's
variation contributes equally to assembly-level variation, assembly error grows with the square
root of the number of parts in the tolerance chain and each part can be assigned a tolerance equal
172
to the assembly-level tolerance divided by the square root of the number of parts in the chain. If
the process Cpk is more than one only sample inspection will be required; otherwise 100%
inspection will be required. High enough Cpk ensures higher probability of meeting the
requirements and smaller sample size. Failure to monitor the process using Cpk data may lead to
undocumented mean shift, which invalidates the RSS method.
>US
g
Statistical
Coordination
No
Coordination
Statistical
Tolerancing
Worst Case
Tolerancing
Tool & Die
Net 3uild
Net Build
Part Tol =
Assy Tol/ 4N
Use SPC & Cpk
on parts to keep
Process Mean =
Nominal
to
ean
&
irts
Assembly
Errors Grow
with 4N
Cpk > 1:
Sample Insp.
Part Tol =
Assy Tol/ N
Assembly
Errors Grow
with N
Failur e to use
SPC Ieads to
undoct imented
Mea n Shift
Cpk < 1:
100% Insp.
Fig. 8-2: Classification of Techniques of
Achieving Tolerance Specifications. (N is the
number of parts in the tolerance chain.)
173
100%
Inspection
8.2.3
No Coordination:
One can ensure 100% interchangeability among parts and 100% likelihood of meeting the
assembly-level tolerances by using a worst-case tolerancing scheme. GD&T methods can be
used to define worst-case tolerances on parts. (However, it should be noted that GD&T would
not help in identifying the worst-cases in complex assemblies with multiple parts.) In the simple
case where each part's variation contributes equally to assembly-level variation, assembly error
grows linearly with the number of parts in the tolerance chain and each part must be assigned a
tolerance equal to the assembly-level tolerance divided by the number of parts in the chain. Here
again, failure to monitor the process using statistical Cpk data may lead to undocumented mean
shift.
8.3 Summary:
This chapter presented a coherent scheme of design steps forming a design procedure for a topdown design process of mechanical assemblies. The design procedure consists of three different
phases, the nominal design phase, the constraint analysis phase and the variation analysis phase.
Each of the design phases refers to different set of design activities.
The nominal design phase relates the key characteristics to the kinematic structure of the design.
The constraint properties are analyzed during the constraint analysis phase. The nominal design
can be updated depending upon the results of constraint analysis. Hence, the constraint analysis
phase may also be understood as a part of the nominal design phase. The tolerances on KCs (i.e.
accepted or allowed variations on KCs) are decided according to customer requirements in
variation design phase. The robustness of the nominal design (DFC) and the assembly level
variations are analyzed to ensure that the agreed upon variations in the KCs shall be met.
This chapter also presented a set of design-techniques for achieving the tolerance specifications.
The next chapter shall present the research issues of the future work and the conclusion of this
thesis.
174
Chapter 9: Conclusion
9.1 Review and Contribution:
The fundamental challenge of the product development process is to combine engineering detailspecific dimensions, assembly dimensions, part dimensions, materials etc. into a coherent whole.
Designing mechanical assemblies is one of the most important pieces of product development
(especially for automobile, aircraft, machine-equipments, industries). This thesis presented how
mechanical assemblies can be designed in a top-down design way along with several tools and
techniques which can be helpful in analyzing the assembly at various steps of design. Fig. 9-1
shows the steps of the top-down design process and Fig. 9-2 shows the steps of a bottom-up
design process. The steps shown in highlighted boxes were covered in this thesis. The new tools
for motion analysis, constraint analysis and contribution analysis were presented. The top-down
design process was compared with the bottom-up design process all along the presentation.
Chapter 2 presented the methodology of DFC which ensures that customer requirements drive
the assembly architecture and the detailed part design comes after that, once the context of the
part in the assembly is known.. DFC expresses designer's logical intent concerning how parts are
to be related to each other geometrically to deliver the KCs repeatedly. The approach of a
bottom-up design process is also compared with that of the DFC method. The design teams tend
to jump to detailed level part design without evaluating the concepts thoroughly. The main
drivers for the bottom-up design process are rudimentary properties of CAD systems as far as
conceptual level design is concerned and some business reasons (carryover designs, inflexible
manufacturing systems).
Chapter 3 presented how assembly features can be built and used. This chapter presented a
method to construct assembly features using the basic surfaces. Screw theory is used to represent
the constraints. Assembly features realize the constraint structure represented by the design team
through DFC as their intent of design. It is also presented how mating surfaces are identified and
grouped together in case of bottom-up approach. In bottom-up design process, parts are designed
individually and the chain of mates is often found through automatic constraint detection
175
techniques. TTRS' is one such technique that finds the chain of mates from 3D solid models. It
has been shown that TTRS is inadequate, in its current form, in identifying the chains of mating
surfaces and in representing some of the relative motions among parts.
Chapter 4 presents the motion and constraint analyses of DFC using screw theory. It is also
presented how CAD systems do motion and limited constraint analysis of assemblies constituted
by fully designed parts. A method of motion analysis has been presented in this chapter. This
method implements Waldron's series and parallel law properly for general assemblies.
Waldron's series and parallel law alone are not sufficient to analyze the problems with cross
coupling. The method of motion analysis outlines a procedure to perform detailed kinematic
analysis to determine the degrees of freedom of a part in cross-coupled situations. This method
of motion analysis has been compared with Konkar's method to highlight the contribution of this
research. A method of finding over-constraints has also been presented. A method of
systematically finding over-constraints associated with a part has also been presented in this
chapter. All over-constraints can be found when detailed kinematic analysis is not required in
motion analysis. When detailed kinematic analysis become necessary for motion analysis, exact
information about only some of the over-constraints may be found. Qualitative information about
other over-constraints can be found in these cases too. Some of the over-constraints may appear
extraneous. However, if this technique is combined with information about assembly sequence it
can become extremely beneficial for a designer. The designer can find over-constraints affecting
key characteristics at each sub-assembly station and he/she can take decisions accordingly. There
cannot be more than six over-constraints when only one part is added to a sub-assembly. So, if
one is analyzing over-constraints when one part is being added at a time it will not be tedious to
decide which over-constraint may affect assembly level requirements.
Chapter 5 presented the effect of location variation. The top-down approach requires a
connective feature-based assembly model. The assembly model reflects design intent. The
mating conditions among assembly features are decided by the design team before the detailed
design of parts. On the other hand, the bottom-up approach attempts to identify the design intent
1TTRS is a technique to identify chain of mates from CAD solid models. It is used for tolerance analysis. It is just
one such technique. Not every bottom-up method shall use TTRS. TTRS is picked for the purpose of comparison
only.
176
from the collage of parts. Both of the approaches may employ similar techniques (matrix
transforms, vectorial loops) to represent the part locations. However, the differences become
obvious when variation and contribution analysis is performed. The top-down approach will be
more successful in identifying the source of variation whereas it will be hard to find the source of
variation in case of the bottom-up approach. This chapter presented a new technique to perform
contribution analysis using the constraint representation of DFC. This technique can be used to
find sensitivities of part locations to variations in the locations of assembly features. This
analysis uses the constraint information in the DFC and the information about nominal locations
of assembly features.
Chapter 6 presented a method to analyze the uncertainty in the part locations due to design-inclearance on assembly features. Design-in-clearance on assembly features can be modeled as
uncertainty in the matrix transform associated with the assembly feature co-ordinate system. This
chapter proposed a simulation approach to derive the statistical properties of the uncertainty in
the location of a part. Solved examples for a properly constrained assembly and an overconstrained assembly are presented. Manufacturing variations on all the dimensions are
considered in the simulation. The traditional approach of allocating design-in-clearance on an
assembly feature considers this as a local problem that can be solved by determining the type of
fit on that particular assembly feature. However, the variation in size-dimensions needs to be
linked with the tolerance chains. GD&T presents the worst-case solution for the problem of
assigning design-in-clearance on assembly features. Moreover, GD&T is a part-centric tolerance
allocation technique and still it is not very much compatible with the top-down design process.
Chapter 7 presents the classification of assemblies based upon the properties of their constraint
structure. The classification outlines the different reasons responsible for over-constraints in
assemblies. Designers need to identify the over-constraints (if any) from the given set of
possibilities.
Chapter 8 presented a coherent scheme of design steps forming a design procedure for a topdown design process of mechanical assemblies. The design procedure consists of three different
phases, the nominal design phase, the constraint analysis phase and the variation analysis phase.
177
Each of the design phases refers to different set of design activities. The nominal design phase
relates the key characteristics to the kinematic structure of the design. The constraint properties
are analyzed during the constraint analysis phase. The nominal design can be updated depending
upon the results of constraint analysis. Hence, the constraint analysis phase may also be
understood as a part of the nominal design phase. The tolerances on KCs (i.e. accepted or
allowed variations on KCs) are decided according to customer requirements in variation design
phase. The robustness of the nominal design (DFC) and the assembly level variations are also
analyzed in this phase to ensure that the agreed upon variations in the KCs shall be met.
Top-Down Design Process
Bottom-Up Design Process
Customer
Requirements
Requirements
Concepts
Concepts
Datum Flow
Chain (DFC)
Detailed
Level Part
The Assembly
Features
Customer
How do they
constrain?
I1
Mating
Surfaces
How are they
built?
Constraint
Changing &
Unchanging
Directions
Kinematic
Loop (TTRS)
Analysis
Due to Changes
in Shape & Size
Propagation
of Variation
}
Variation
naIs
Analysis
Due to Changes
in Location
Detailed Level
Part Design
Due to Changes
in Location
Due to Changes
in Shape & Sizej
Customer
Requirements
Fig. 9-2
Fig. 9-1
178
9.2 Scope for Future Research:
Further software development that will support the functionality of DFC and incorporate the
motion analysis, constraint analysis, variation analysis, contribution analysis etc. can be useful
for commercial purposes. DFC can be used as a tool for symbolic representation of assemblies
that enables various analysis techniques especially in the early stages of design. DFC essentially
is a feature-based assembly model. The method of creating new features also needs to be
supported.
Apart from that, still more research is required in the area of extending analysis of overconstraints, probably integrating the assembly sequences with motion and constraint analysis,
using the results of contribution analysis for tolerance allocation and understanding design-inclearance The cross coupling among paths may induce dependency among the degrees of
freedom of paths. Detailed kinematic analysis solves the problem of finding mobility of a part in
such situations. However, the method of constraint analysis presented in this thesis cannot
generate quantitative information about all over-constraints when detailed kinematic analysis is
required for motion analysis. Though qualitative information about all over-constraints can be
generated in these cases too. Cross coupling may not introduce over-constraints in all cases and
the method presented in this thesis can identify those cases. However, the method of constraint
analysis needs to be extended to find quantitative information about all over-constraints for all
assemblies.
Information about assembly sequences can also be used while evaluating over-constraints. In
fact, an engineer can check for over-constraints as each part is added in to the sub-assembly
starting from assembly of two parts. If over-constraints at each sub-assembly stage are such that
they do not affect key characteristics of that particular sub-assembly station it will be a good
situation. However, if some over-constraints do affect the KC at any stage engineer need to think
about alternative fixture design or alternative choices of assembly features. So, there is a
possibility of combining assembly sequence optimization with the method of constraint analysis
which can be of significant industrial importance.
179
Variation analysis follows constraint analysis. This thesis lays a framework regarding when the
variation analysis can be performed using DFC. The graphical technique used to identify the
paths for motion and constraint analysis can be used to identify the multiple tolerance chains in
case of over-constrained assemblies. Contribution analysis presented in this thesis takes input
from DFC (regarding the constraints represented by screw models of assembly features). More
research may be needed to use the information of contribution analysis for variation analysis
modules. Over-constrained assemblies will have multiple tolerance chains. Performing the
variation analysis on multiple tolerance chains in presence of clearance on assembly features also
requires further research.
Design-in-clearance relieves over-constraint but it introduces uncertainty in the part locations.
More research is needed for the problem of deciding optimum design-in-clearance for overconstrained assemblies. Alternatively, the requirement on maximum clearance may offer a
tolerance design problem.
Significant efforts may be required to integrate this proposed design system with existing CAD
systems. The proposed new system will be required to use the legacy CAD data with minimal
efforts on the part of users.
180
References:
=
Ahuja D. V. and Coons S. A., "Geometry for Construction and Display", IBM Systems
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188
Appendix A:
Surface-to-Surface Contacts been used in this chapter. The surfaces have been divided in the
following categories:
Any Surface, Helical Surface, Surface of Revolution, Cylindrical Surface,
PlanarSurface, SphericalSurface.
The following table lists out the type of contacts possible between interiors of two basic surfaces.
Table A-1: Surface-to-Surface Contacts
Helical
Any
Any
"
e
*
Helical
Surface of
Revolution
Point
Line
Surface
e
Point
Line]
Surface
9
Line
*
*
*
Surface of
Revolution
Cylinder
*
*
*
*
*
Point
Line
Surface
Point
*
Point
*
Point
Point
Line
*
*
*
Point
Line
*
Point
Line
*
Line
*
*
Point
Line
0
Surface
*
Point
S
Point
Surface
*
*
Point
Surface
*
*
*
*
Point
P
Line
L
Surface
*
*
Planar
Spherical
e
Point
0 Line
Cylinder
Spherical
Point
Line
Surface
Point
Line
Surface
Point
* Line
9 Surface
*
Planar
Surface
e
*
Some of the entries in the table are in italics because these entries are possible only if any surface is locally
matching to the contacting surface (e.g. If a helical surface comes in contact with an any surface the contact area can
be a surface only if the any surface is locally a matching helical surface.
189
Different contacts between two basic surfaces can provide different relative degrees of freedom
(DOFs). The summary of these possibilities for contacts between different basic surfaces is as
follows:
Any Surface with:
1. Any Surface:
a. Unidirectional Contact:
Surface Contact: It may have at most two free DOFs. (May be zero or one)
i.
ii.
Line2 Contact: It may have at most three free DOFs. (May be zero, one or two)
Point Contact: It will constrain one translation.
iii.
Any
Any
An...-''
Any
Any
ny
Fig. A-1: Any Surface with Any Surface
b. Bi-directional Contact:
Surface Contact: It will constrain all six DOFs. It shall reduce to line or point
i.
contact in real situation if the clearance is provided on the joint.
Line Contact: It may have one (may be zero) DOF. It shall reduce to point contact
ii.
in real situation if the clearance is provided on the joint.
2. Helical Surface:
a. Unidirectional Contact:
Point Contact: Only a point contact is possible between "any surface" and helical
i.
surface. If this is not the case "any surface" is locally a matching helical surface.
Fig. A-2: Any Surface with Helical Surface
b. Bi-directional Contact:
Bi-directional contact is not possible between "any surface" and helical surface. If
i.
this is not the case "any surface" is locally a matching helical surface.
2 The
line contact refers to topologically one-dimensional contact. The contact area will be a one-dimensional entity
(e.g. curve, straight line).
190
3. Surface of Revolution:
a. Unidirectional Contact:
i.
Point Contact: Only a point contact is possible between "any surface" and "Surface
of Revolution". If this is not the case "any surface" is locally a matching "Surface
of Revolution
'
,
%
%
Fig. A-3: Any Surface with Surface of Revolution
b. Bi-directional Contact:
i.
Bi-directional contact
is not possible between "any surface" and "Surface of
Revolution". If this is not the case "any surface" is locally a matching "Surface of
Revolution".
4. Cylindrical Surface:
a. Unidirectional Contact:
i.
Point Contact: Only a point contact is possible between "any surface" and
cylindrical surface. If this is not the case "any surface" is locally a matching
cylinder.
Fig. A-4: Any Surface with Cylindrical Surface
b. Bi-directional Contact:
i.
Bi-directional contact is not possible between "any surface" and cylindrical surface.
If this is not the case "any surface" is locally a matching cylinder.
5. Planar Surface:
a. Unidirectional Contact:
i.
Point Contact: Only a point contact is possible between "any surface" and a plane.
If this is not the case "any surface" is locally a plane.
Fig. A-5: Any Surface with Planar Surface
191
b. Bi-directional Contact:
Bi-directional contact is not possible with a planar surface. If this is not the case
i.
"any surface" is locally a plane.
6. Spherical Surface:
a. Unidirectional Contact:
Point Contact: Only a point contact is possible between "any surface" and spherical
i.
surface. If this is not the case "any surface" is locally a matching sphere.
Fig. A-6: Any Surface with Spherical Surface
b. Bi-directional Contact:
Bi-directional contact is not possible between "any surface" and spherical surface.
i.
If this is not the case "any surface" is locally a matching sphere.
Helical Surface with:
1. Helical Surface:
a. Unidirectional Contact:
Line Contact: It may result due to clearance on the bi-directional surface contact. It
i.
will have one rotational DOF and a translation along the rotation axis will be
coupled with it (Same as full surface contact). This joint will have an extra
translational DOF.
Point Contact: It will constrain one translation.
ii.
b. Bi-directional Contact:
Surface Contact: It will have one rotational DOF. Translational motion will be
i.
coupled with it. Bi-directionally constrained directions will have uncertainty if
clearance is provided on the joint.
V\/
Fig. A-7: Helical Surface with Helical Surface
192
2. Surface of Revolution:
a. Unidirectional Contact:
i.
Point Contact: Only a point contact is possible between helical surface and "Surface
of Revolution".
Fig. A-8: Helical Surface with Surface of Revolution
b. Bi-directional Contact:
i.
Bi-directional contact is not possible between helical surface and "Surface of
Revolution".
3. Cylindrical Surface:
a. Unidirectional Contact:
i.
Point Contact: Only a point contact is possible between helical surface and a
cylinder.
Fig. A-9: Helical Surface with Cylindrical Surface
b. Bi-directional Contact:
i.
Line Contact: It will have one translational and one rotational DOF. It may not be a
practical design. It may reduce to multiple point contacts.
4. Planar Surface:
a. Unidirectional Contact:
i.
Point Contact: Only a point contact is possible between helical surface and a plane.
Fig. A-10: Helical Surface with Planar Surface
b. Bi-directional Contact:
i.
Bi-directional contact is not possible with a planar surface.
193
5. Spherical Surface:
a. Unidirectional Contact:
Point Contact: Only a point contact is possible between helical surface and a sphere.
i.
Fig. A-11: Helical Surface with Spherical Surface
b. Bi-directional Contact:
Line Contact: It will constrain all the three translations. It will be unstable and it
i.
will reduce to point contact.
Surface of Revolution with:
1. Surface of Revolution:
a. Unidirectional:
Line Contact: It may have at most three DOFs. (May be zero, one or two)
i.
Point Contact: It will constrain only one translation.
ii.
b. Bi-directional:
Surface Contact: It will have one rotational DOF. It may become a line contact due
i.
to clearance.
Fig. A-12: Surface of Revolution with Surface of Revolution
2. Cylindrical Surface:
a. Unidirectional:
i. Line Contact: It may have at most three DOFs. (May be zero, one or two)
ii. Point Contact: It will constrain only one translation.
b. Bi-directional:
i. Line Contact: It will have one translational and one rotational DOF. It may not
be a practical design.
Fig. A-13: Surface of Revolution with Cylindrical Surface
194
3. Planar Surface:
a. Unidirectional:
i.
Line Contact: It will have two translations and two rotations.
ii.
Point Contact: It will constrain only one translation.
Fig. A-14: Surface of Revolution with Planar Surface
b. Bi-directional:
i.
Bi-directional contact is not possible with a planar surface.
4. Spherical Surface:
a. Unidirectional:
i.
Point Contact: It will constrain only one translation.
Fig. A-15: Surface of Revolution with Spherical Surface
b. Bi-directional:
i.
Line Contact: It will constrain all the three translations. It will be unstable and it
I
(D
will reduce to point contact.
Cylindrical Surface with:
Spherical
Surface(UdietnaLieCta)
with
A-16:C5ia SurfaceofRltn
| ~~ Fig.Surface:
1. Cylindrical
a. Unidirectional Contact:
i.
Line Contact: It will have one translational and one rotational DOF.
-g A1
i
195
ii.
Point Contact: It will constrain one translation.
FFig. A-17: Cylindrical Surface with Cylindrical Surface (Unidirectional Point Contact)
b. Bi-directional Contact:
i.
Surface Contact: It will have one translational and one rotational DOF. It may
reduce to line contact if clearance is provided on the line joint.
Fig. A-18: Cylindrical Surface with Cylindrical Surface (Bi-directional Contact)
2. Planar Surface:
a. Unidirectional Contact:
i.
Line Contact: It will have two translational and two rotational DOFs.
Fig. A-19: Cylindrical Surface with Planar Surface
b. Bi-directional Contact:
i.
Bi-directional contact is not possible with a planar surface.
3. Spherical Surface:
a. Unidirectional Contact:
i.
Point Contact: It will constrain one translation.
Cylinder
Sphere
Fig. A-20: Cylindrical Surface with Spherical Surface (Unidirectional Contact)
L-
196
i
b. Bi-directional Contact:
i.
Line Contact: It will constrain two translational DOFs. It may reduce to point
contact if clearance is allowed on the joint.
Cylinder
Sphere
Fig. A-21: Cylindrical Surface with Spherical Surface (Bi-directional Contact)
Planar Surface with:
1. Planar Surface:
a. Unidirectional Contact:
i.
Surface Contact: It will constrain one translational and two rotational DOFs.
Fig. A-22: Planar Surface with Planar Surface
b. Bi-directional Contact:
i.
Bi-directional contact is not possible between two planes.
2. Spherical Surface:
a. Unidirectional Contact:
i.
Point Contact: It will constrain one translation.
Fig. A-23: Planar Surface with Spherical Surface
b. Bi-directional Contact:
i.
Bi-directional contact is not possible with a planar surface.
197
Spherical Surface with:
1. Spherical Surface:
a. Unidirectional Contact:
i.
Point Contact: It will constrain one translation.
Fig. A-24: Spherical Surface with Spherical Surface (Unidirectional Contact)
b. Bi-directional Contact:
i.
Surface Contact: It will constrain three translations. It may become point contact if
clearance is allowed on the joint.
Fig. A-25: Spherical Surface with Spherical Surface (Bi-directional Contact)
"Surface-to-Edge" and "Edge-to-Edge" Contacts:
In assembly features the surfaces will have finite extent (i.e. surfaces will have edges, if
applicable). Sometimes, one or more pairs of surfaces in an assembly feature may be formed by a
surface on one part and an edge of a surface on the other part. Hence "surface-to-edge" contacts
and "edge-to-edge" contacts also become important. The "edge-to-edge" contacts are always
point contacts except when edges are adjacent to each other. However, both "line" and point
contacts are possible in case of "surface-to-edge" contacts.
The "surface-to-edge" contacts will normally be unidirectional. Bi-directional "surface-to-edge"
contacts are possible only if the profile of the edge is exactly matching to the contacting surface
and the contact is along a curve which is not in a plane. The wrench of the point contact for a
contact between an edge and a surface shall be given by the direction of the normal vector to the
tangent plane passing through the point of contact. Tangent plane is defined if the surface has
continuous second order partial derivatives around the point of contact.
Wrench Direction:
Normal to the tangent
plane passing through
the contact point
Any
Any
Fig. A-26: Wrench for a Point Contact between An Edge and A Surface
198
The following table presents all possible contacts between a surface and an edge of other surface:
Table A-2: Edge-to-Surface Contacts
Helical
Any
Any
Helical
Surface of
Revolution
0
e
Point
Line
*
*
*
Point
Line3
Point
Surface of
.reo
Planar
Cylinder
Revolution
Spherical
Point
*
Point
*
Point
*
Point
Line
9
Line
9
Line
*
Line
Point
*
Point
*
Point
* Point
* Line
Point
* Line
*
0Point
e
Point
Line
Point
* Line
0
Point
*
0
Point
Line
*
e
Point
Line
*
Line
e
Point
*
*
e
Cylinder
Planar
pec
of the entries in the table are in italics because these entries are possible only if any surface is locally
matching to the contacting edge or the edge of any surface is locally matching to the contacting surface.
3 Some
199
200
Appendix B:
Table B-1: Unidirectional and Bi-directional Degrees of Freedom of Assembly Features
S.No.
Feature
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Prismatic Peg in a Prismatic Hole
Plate Pin in Through Hole
Prismatic Slot, Prismatic Peg
Plate Slotted Pin Joint
Prismatic Slot, Round Peg
Round Peg in a Through or Blind Hole
Threaded Joint
Elliptical Ball and Socket
Plate-Plate Lap Joint
Spherical Joint
Oversize Hole
Elliptical Ball in Cylindrical Trough
Thin Rib, Plane Surface
Ellipsoid on Plane Surface
Spherical Ball in Cylindrical Trough
Peg in a Slotted Hole
Sphere on Plane Surface
201
Total No. of
Constrained
Dofs
6
5
5
4
4
4
5
4
3
3
3
3
2
2
2
2
1
Bi-directionally
Constrained
Dofs
5
4
4
2
2
4
5
4
0
3
0
3
0
0
2
2
0
1. Uncertainty in the Properly Constrained Assembly:
20.1
19.8
98
Toi Plate
Bottom Plate
Fig. B-1: Properly Constrained Assembly
Diameter of the hole in "peg & hole" assembly feature (Dhl): 20.1
Diameter of the peg in "peg & hole" assembly feature (Dp,): 19.8
Width of the slot in "pin in slot" assembly feature (Sw): 20.1
Diameter of the peg in "pin in slot" assembly feature (Dp2): 19.8
Distance between two pegs (Lp): 100
MATLAB Code for Simulation:
I = 0;j=O;
for i=1:10000
Dhl= 20.1 + 0.07*randn;
Dp1= 19.8 + 0.07*randn;
SW = 20.1 + 0.07*randn;
Dp2 = 19.8 + 0.07*randn;
Lp = 100 + 0.20*randn;
if ((Dpi >= D) I (Dp2 >= Sw))
1=1+1;
else
j=j+1;
a(j) = (DhI - Dpi + Sw - Dp2 )/ L,;
Xmin(j)= -0.5*( Dhl- Dp1);
Xmax(j) = 0.5*( Dhl- Dp1);
Tmin(j) = ((-0.5*( Dhl- Di))+(0.5*( Sw - Dp2 )))/ L,;
Tmax(j) = ((0.5*( Dhl- Dp))-(0.5*( Sw - Dp2 )))/ Lp;
end
end
202
I = Interference counter. I reached value 36 for this simulation. Hence, the probability of
interference with the given nominal dimensions and standard deviations is 0.36%.
Xmax(i) and Xmin(i) correspond to maximum and minimum values of x-location respectively
(for a given set of dimensions). Tmax(i) and Tmin(i) correspond to maximum and minimum
values of 0 -location respectively (for a given set of dimensions).
Uncertainty in x-location for a given set of dimensions = Xmax(i)-Xmin(i);
Uncertainty in 0 -location for a given set of dimensions = Tmax(i)-Tmin(i);
Mean of uncertainty in X-location = mean(Xmax-Xmin)
Standard deviation of uncertainty in X-location = sqrt(var(Xmax-Xmin))
Mean of uncertainty in 0 -location = mean(Tmax-Tmin)
Standard deviation of uncertainty in 0 -location = sqrt(var(Tmax-Tmin))
2. Uncertainty in Over-Constrained Assembly:
20 A .
8 19.8
120.
LO~A
S0
Top Plate
Bottom Plate
Fig. B-2: Over-Constrained Assembly
Diameter of the hole in assembly feature "A" (Dhl): 20.1
Diameter of the peg in assembly feature "A" (Dpi): 19.8
Diameter of the hole in assembly feature "B" (Dh2):): 20.1
Diameter of the peg in assembly feature "B" (Dp2 ): 19.8
Distance between two pegs (L): 100
Distance between two holes (Lh): 100
203
I
MATLAB Code for Simulation:
I= 0; j=0;
for i=1:10000
Dhl= 20.1 + 0.07*randn;
D, 1= 19.8 + 0.07*randn;
Dh2 = 20.1 + 0.07*randn;
Dp2 = 19.8 + 0.07*randn;
L= 100 + 0.04*randn;
Lh= 100 + 0.04*randn;
if ((Dpi >= D)
I (Dp2 >= Dh2)I
((Lp+0.5* Dp1 +0.5* Dp2 )>=( Lh+0.5* Dhl+0. 5 * Dh2))
((Lp-0.5* D, 1-0.5* Dp2 )<=( Lh-0.5* Dh1-0. 5 * Dh2)))
I=I+1;
else
j=j+1;
if (Lh+0.5* Dhl-0. 5 * Dh2)
<=
(Lp+0.5* D, 1-0.5* Dp2 )
Xmin(j) = -0.5*( Dhl- Dp1);
elseif (Lh-0.5* Dh1-0- 5 * Dh2) < (Lp-0.5* D, 1-0.5* Dp2 )
Xmin(j) = Lh-Lp-0.5*( Dh2- Dp2);
end
if (Lh-0.5* Dh1-0.5* Dh2) <= (Lp-0.5* Dp1 -0.5* Dp2 )
Xmax(j) = 0.5*( Dhl- Dpi);
elseif (Lh-0.5* Dhl+0-5*
Dh2)
< (Lp-0.5* Dp1+0.5* Dp2 )
Xmax(j) = Lh-Lp+0.5*( Dh2- Dp2 );
end
end
end
I = Interference counter. I reached value 32 for this simulation. Hence, the probability of
interference with the given nominal dimensions and standard deviations is 0.32%.
Xmax(i) and Xmin(i) correspond to maximum and minimum values of x-location respectively
(for a given set of dimensions).
Uncertainty in x-location for a given set of dimensions = Xmax(i)-Xmin(i);
Mean of uncertainty in X-location = mean(Xmax-Xmin)
Standard deviation of uncertainty in X-location = sqrt(var(Xmax-Xmin))
204