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Kinematics of Mechanical Systems: Fundamentals, Analysis and Synthesis

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This book is aimed to provide comprehensive and systematic knowledge of kinematic synthesis as developed up to date. Modern mechanical systems require advance kinematics knowledge to support mechanism design with sound theories and methods. The book includes not only the classical foundations of kinematic synthesis, but also the latest advances developed by the authors. Moreover, many examples are included to illustrate both methods and their supporting theory.

The focus is on systems of rigid bodies forming closed loops. The four-bar linkage, representing the foundations of mechanical systems,  is given due attention, in its three domains: planar, spherical, and spatial. The book contains six chapters, the first two covering fundamentals for kinematic synthesis, including qualitative synthesis. Chapters 3–5 describe, in full detail,  the function, motion, and path syntheses of single-dof linkages.  In the last chapter, the synthesis of single-dof complex linkages, including six-bar and ten-bar linkages, is introduced.

The book is suitable for graduate students of mechanical engineering, researchers of mechanism and robot design, and machine design engineers.

Author(s): Jorge Angeles, Shaoping Bai
Series: Mathematical Engineering
Publisher: Springer
Year: 2022

Language: English
Pages: 341
City: Cham

Preface
Contents
List of Figures
List of Tables
1 Introduction to Kinematic Synthesis
1.1 The Role of Kinematic Synthesis in Mechanical Design
1.2 Glossary
1.3 Kinematic Analysis Versus Kinematic Synthesis
1.3.1 A Summary of Systems of Algebraic Equations
1.4 Algebraic and Computational Tools
1.4.1 The Two-Dimensional Representation of the Cross Product
1.4.2 Algebra of 2 times 22times2 Matrices
1.4.3 Algebra of 3 times 33times3 Matrices
1.4.4 Linear-Equation Solving: Determined Systems
1.4.5 Linear-Equation Solving: Overdetermined Systems
1.5 Nonlinear-Equation Solving: The Determined Case
1.5.1 The Newton–Raphson Method
1.6 Overdetermined Nonlinear Systems of Equations
1.6.1 The Newton–Gauss Method
1.7 Packages Relevant to Linkage Synthesis
References
2 The Qualitative Synthesis of Kinematic Chains
2.1 Notation
2.2 Background
2.3 Kinematic Pairs
2.3.1 The Π Kinematic Pair
2.4 Graph Representation of Kinematic Chains
2.5 Groups of Displacements
2.5.1 Displacement Subgroups
2.6 Kinematic Bonds
2.7 The Chebyshev–Grübler–Kutzbach–Hervé Formula
2.7.1 Trivial Chains
2.7.2 Exceptional Chains
2.7.3 Paradoxical Chains
2.8 Applications to Robotics
2.8.1 The Synthesis of Robotic Architectures and Their Drives
References
3 Linkage Synthesis for Function Generation
3.1 Introduction
3.2 Input-Output (IO) Functions
3.2.1 Planar Four-Bar Linkages
3.2.2 The Denavit-Hartenberg Notation
3.2.3 Spherical Four-Bar-Linkages
3.2.4 Spatial Four-Bar-Linkages
3.3 Exact Synthesis
3.3.1 Planar Linkages
3.3.2 Spherical Linkages
3.3.3 Spatial Linkages
3.4 Analysis of the Synthesized Linkage
3.4.1 Planar Linkages
3.4.2 Spherical Four-Bar Linkages
3.4.3 Spatial Four-Bar Linkages
3.5 Approximate Synthesis
3.5.1 The Approximate Synthesis of Planar Four-Bar Linkages
3.5.2 The Approximate Synthesis of Spherical Linkages
3.5.3 The Approximate Synthesis of Spatial Linkages
3.6 Linkage Performance Evaluation
3.6.1 Planar Linkages: Transmission Angle and Transmission Quality
3.6.2 Spherical Linkages: Transmission Angle and Transmission Quality
3.6.3 Spatial Linkages: Transmission Angle and Transmission Quality
3.7 Design Error Versus Structural Error
3.7.1 Minimizing the Structural Error
3.7.2 Branch-Switching Detection
3.7.3 Introducing a Massive Number of Data Points
3.8 Synthesis Under Mobility Constraints
References
4 Motion Generation
4.1 Introduction
4.2 Planar Four-Bar Linkages
4.2.1 Dyad Synthesis for Three Poses
4.2.2 Dyad Synthesis for Four Poses
4.2.3 Dyad Synthesis for Five Poses
4.2.4 Case Study: Synthesis of a Landing Gear Mechanism
4.2.5 The Presence of a P Joint in dyad Synthesis
4.2.6 Approximate Synthesis
4.3 Spherical Four-Bar Linkages
4.3.1 Dyad Synthesis for Three Attitudes
4.3.2 Dyad Synthesis for Four Attitudes
4.3.3 Dyad Synthesis for Five Attitudes
4.3.4 Spherical dyads with a P Joint
4.3.5 Approximate dyad Synthesis
4.3.6 Examples
4.4 Spatial Four-Bar Linkages
4.4.1 Geometric Constraints of CC and RC dyads
4.4.2 The Synthesis of the CC dyad
4.4.3 The Synthesis of the RC dyad
4.4.4 Synthesis of Four-Bar Linkages
4.4.5 A Semigraphical Solution of the Direction Equations
4.4.6 Solving the Linear Equations of the Moment Variables
4.4.7 Congruences of the Fixed and the Moving Axes
4.4.8 Examples
4.4.9 Summary
References
5 Trajectory Generation
5.1 Planar Linkages
5.1.1 Planar Path Generation with Prescribed Timing
5.1.2 Coupler-Curve Synthesis of Planar Four-Bar Linkages
5.2 Trajectory Generation for Spherical Four-Bar Linkages
5.2.1 Path Synthesis with Discrete Positions
5.2.2 Path Synthesis with Prescribed Timing
5.2.3 Examples
5.2.4 Summary
5.3 Path Generation for RCCC Linkages
5.3.1 A Generic RCCC Linkage
5.3.2 An Alternative Coordinate Frame
5.3.3 Constraint Equations of the CC and RC dyads
5.3.4 N-point Path Synthesis
5.3.5 Example
5.3.6 Summary
References
6 Synthesis of Planar Complex Linkages
6.1 Six-Bar Linkages
6.2 Synthesis of the Stephenson-III Linkage
6.2.1 Elimination of Motion Variables
6.3 Extension to Stephenson-I and II Linkages
6.3.1 Stephenson-II Linkage
6.3.2 Parameterized Coordinates of Point D
6.3.3 Synthesis Equations for the Stephenson-II Linkage
6.3.4 Stephenson-I Linkage
6.4 Synthesis in the Presence of Incomplete Data Sets
6.4.1 A Design Example
6.5 A 10-Bar Linkage
6.6 Synthesis for m Prescribed Poses
6.7 Solution of the 10-Posture Synthesis Equations
6.8 A Synthesis Example
References
Appendix A A Summary of Dual Algebra
A.1 Introduction
A.2 Definitions
A.3 Dual Algebra and Geometry
A.3.1 Link-Dimension Determination Based on Two Skew Lines
A.3.2 Normal Vector From a Point to a Line
A.4 Fundamentals of Rigid-Body Kinematics
A.4.1 Finite Displacements
A.4.2 Velocity Analysis
A.4.3 The Linear Invariants of the Dual Rotation Matrix
A.4.4 The Dual Euler–Rodrigues Parameters of a Rigid-Body Motion
A.5 The Dual Angular Velocity
A.6 Conclusions
Appendix B Coupler Curves
B.1 Planar Four-Bar Linkages
B.1.1 Special Cases
B.2 Spherical Four-Bar Linkages
B.2.1 Parametrized Coordinates
B.2.2 Coupler Curve of Spherical Four-Bar Linkages
B.3 Conclusions
Index