Enumeration of
Kinematic Structures
According to Function
Mechanism Design
© 2001 by CRC Press LLC
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This textbook introduces a systematic methodology for the creation and classifi-
cation of mechanisms. The approach is partly analytical and partly algorithmic. It
is based on the idea that, during the conceptual design phase, some of the functional
requirements of a desired mechanism can be transformed into structural characteris-
tics that can be employed for systematic enumeration of mechanisms. The kinematic
structure of a mechanism contains the essential information about which link is con-
nected to which other link by what type of joint. Using graph theory, combinatorial
analysis, and computer algorithms, kinematic structures of the same nature, i.e., the
same the number of degrees of freedom, type of motion (planar or spatial), and com-
plexity can be enumerated in an essentially systematic and unbiased manner. Then
each mechanism structure is sketched and evaluated with respect to the remaining
functional requirements. This results in a class of feasible mechanisms that can be
subject to dimensional synthesis, kinematic and dynamic analyses, design optimiza-
tion, and design detailing.
This textbook is organized as follows:
Chapter 1 provides a brief review of the design process and a systematic method-
ology for creation of mechanisms. Some terminologies related to the kinematics of
mechanism are defined. Mechanisms are classified according to the nature of motion
into planar, spherical, and spatial mechanisms.
© 2001 by CRC Press LLC
Chapter 2 is concerned withthebasic concepts of graph theory, which is essential for
structural analysis and structural synthesis of mechanisms. This material is extremely
important since the design methodology employs graphs to represent the mechanism
structure and mechanism structures are enumerated with the aid of graph theory.
Chapter 3 introduces several methods of representation of the kinematic structure
of mechanisms. The kinematic structure, which contains the essential information
about which link is connected to which other links by what types of joint, will be used
for enumeration of mechanisms.
Chapter 4 examines the structural characteristics of mechanisms. The correspon-
dence between graph and mechanism is established, from which several important

epicyclic transmission gear trains.
Prerequisites for readers of this textbook include the basic concepts of combinato-
rial analysis, graph theory, matrix theory, and the kinematics of mechanisms that are
usually taught at the undergraduate level. Thomas Edison said, “genius is one percent
© 2001 by CRC Press LLC
inspiration and ninety-nine percent perspiration.” Inspiration can occur more readily
when perspiration is properly directed and focused. The methodology presented in
this book is intended to help designers better organize the perspiration so that the
inspiration can take place early in the design process. For those who are willing to
try, the rewards should be well worth it.
The author wishes to express his sincere appreciation to Dr. Bernard Roth, his for-
mer Ph.D. advisor at Stanford University, and Dr. Ferdinand Freudenstein, Professor
Emeritus at Columbia University, for their lifelong advice and encouragement. A ma-
jor portion of the material presented in this textbook is derived from Dr. Freudenstein
and his former students’ research results. Others are taken from the author’s research
in collaboration with professional colleagues, Ting Liu and Roland Maki, and with his
former students, Sun-Lai Chang, Goutam Chatterjee, Dar-Zen Chen, Hsin-I Hsieh,
Chen-Chou Lin, Richard Stamper, and Farhad Tahmasebi. Their efforts are greatly
appreciated. Lastly, the author appreciates the patience and sacrifice of his family
members, Lung-Chu Tsai, Jule Ann Tsai, and David Jeanchung Tsai, over the past
few years while the textbook was being written.
Lung-Wen Tsai
Riverside, California
© 2001 by CRC Press LLC
The Author
Lung-Wen Tsai is a Presidential Chair Professor in the Department of Mechanical
Engineering at the University of California in Riverside. He obtained his B.S. degree
in mechanical engineering from the National Taiwan University in Taipei, Taiwan;
M.S. degree in engineering science from the State University of New York (SUNY)
in Buffalo, New York; and Ph.D. in mechanical engineering from Stanford University

1.4 Kinematic Chains, Mechanisms, and Machines
1.5 Kinematics of Mechanisms
1.6 Planar, Spherical, and Spatial Mechanisms
1.7 Kinematic Inversions
1.8 Summary
References
2 Basic Concepts of Graph Theory
2.1 Definitions
2.1.1 Degree of a Vertex
2.1.2 Walks and Circuits
2.1.3 Connected Graphs, Subgraphs, and Components
2.1.4 Articulation Points, Bridges, and Blocks
2.1.5 Parallel Edges, Slings, and Multigraphs
2.1.6 Directed Graph and Rooted Graph
2.1.7 Complete Graph and Bipartite
2.1.8 Graph Isomorphisms
2.2 Tree
2.3 Planar Graph
2.4 Spanning Trees and Fundamental Circuits
2.5 Euler’s Equation
2.6 Topological Characteristics of Planar Graphs
2.7 Matrix Representations of Graph
2.7.1 Adjacency Matrix
2.7.2 Incidence Matrix
2.7.3 Circuit Matrix
2.7.4 Path Matrix
2.8 Contracted Graphs
© 2001 by CRC Press LLC
2.9 Dual Graphs
2.10 Summary

4.10.4 Degree Code
4.11 Partially Locked Kinematic Chains
4.12 Summary
References
Exercises
5 Enumeration of Graphs of Kinematic Chains
5.1 Introduction
5.2 Enumeration of Contracted Graphs
5.3 Enumeration of Conventional Graphs
5.4 Atlas of Graphs of Kinematic Chains
© 2001 by CRC Press LLC
5.5 Summary
References
Exercises
6 Classification of Mechanisms
6.1 Introduction
6.2 Planar Mechanisms
6.2.1 Planar Linkages
6.2.2 Planar Geared Mechanisms
6.2.3 Planar Cam Mechanisms
6.3 Spherical Mechanisms
6.4 Spatial Mechanisms
6.4.1 Spatial One-dof Mechanisms
6.4.2 Spatial Multi-dof, Multiple-Loop Mechanisms
6.5 Summary
References
Exercises
7 Epicyclic Gear Trains
7.1 Introduction
7.2 Structural Characteristics

8.5.1 Structural Characteristics of Canonical Graphs
8.5.2 Enumeration of Canonical Graphs
8.5.3 Identification of Fundamental Circuits
8.5.4 Detection of Transfer Vertices
8.6 Atlas of Epicyclic Gear Transmission Mechanisms
8.7 Summary
References
Exercises
9 Robotic Mechanisms
9.1 Introduction
9.2 Parallel Manipulators
9.2.1 Functional Requirements
9.2.2 Structural Characteristics
9.2.3 Enumeration of Planar Parallel Manipulators
9.2.4 Enumeration of Spherical Parallel Manipulators
9.2.5 Enumeration of Spatial Parallel Manipulators
9.3 Robotic Wrist Mechanisms
9.3.1 Functional Requirements
9.3.2 Structural Characteristics
9.3.3 Enumeration of Three-dof Wrist Mechanisms
9.4 Summary
References
Exercises
A Solving m Linear Equations in n Unknowns
A.1 Solving One Equation in n Unknowns
A.2 Solving m Equations in n Unknowns
References
B Atlas of Contracted Graphs
C Atlas of Graphs of Kinematic Chains
D Atlas of Planar Bar Linkages