Dual-Number Methods in Kinematics, Statics

This well-organized book uses 3x3 coordinate-transformation matrices and 3-element vectors with dual-number elements to analyze the mechanics of mechanism, robots, and other mechanical systems.

Chapter 1 -- Definitions

1

(8)

The Dual Number

1

(2)

The Dual Angle

3

(1)

Trigonometric Functions and Identities of Dual Angles

3

(1)

Vectors

4

(2)

Free Vectors

4

(1)

Bound Vectors

5

(1)

Dual Vectors

5

(1)

The Equation of Transformation

6

(2)

Exercises -- Definitions

8

(1)

Chapter 2 -- Coordinate Transformation

9

(32)

Relative Displacements and Orientations

9

(1)

Screw Motion

9

(1)

Screw Motions With Respect to the Coordinate Axes

9

(8)

Screw Motion With Respect to the X-axis

10

(4)

Example

14

(1)

Screw Motion With Respect to the Y-axis

15

(1)

Screw Motion With Respect to the Z-axis

16

(1)

Motion of a Vector Through a Dual Angle

17

(1)

Example

17

(1)

Inverse Transformation

18

(2)

Example

20

(1)

Successive Transformations

20

(2)

Example

21

(1)

Roll-Pitch-Yaw About the Fixed Axes

22

(2)

Example

23

(1)

ZYX Euler Angles

24

(4)

Relative Transformation

28

(1)

Example

28

(1)

ZYZ Euler Angles

29

(1)

Screw Motion With Respect to a General Unit Vector

30

(3)

Determination of the Dual Angle and Axis of a Given Screw Motion

33

(7)

The Case of a General Motion

33

(3)

The Case of Pure Translation

36

(1)

The Case of a Half-Turn Screw Motion

37

(2)

Example

39

(1)

Exercises -- Coordinate Transformation

40

(1)

Chapter 3 -- Matrix Modeling of Joints and Links

41

(10)

Definitions

41

(1)

Frame Location

42

(1)

Joint-Link Transformations

43

(2)

Links with Ball Joints

45

(2)

Links with Plane Joints

47

(1)

Other Conventions

48

(1)

Exercises -- Matrix Modeling of Joints and Links

49

(2)

Chapter 4 -- Displacement Analysis

51

(40)

Condition of Loop Closure

51

(1)

Planar Four-Bar Mechanism

52

(2)

Spherical Four-Bar Mechanism

54

(2)

Cardan Joint

56

(3)

General Spatial Four-Bar Mechanism

59

(7)

Output Angular Displacement Theta(4)

61

(2)

Translational Displacement s(4)

63

(1)

Intermediate Rotation Theta(3) and Translation s(3)

64

(1)

Intermediate Rotation Theta(2) and Translation s(2)

64

(2)

Swash-Plate Mechanism

66

(3)

Robotic Systems (Open-Loop Mechanisms)

69

(5)

Example

72

(2)

Generalized Slider-Crank Mechanism

74

(1)

Uicker-Denavit-Hartenberg Solution Scheme

75

(5)

Planar Slider-Crank Mechanism

80

(2)

Origin-Displacement Equation

82

(7)

Application to Calculation of Translational Displacements

84

(5)

Exercises -- Displacement Analysis

89

(2)

Chapter 5 -- Velocity Analysis

91

(30)

Basic Concepts

91

(5)

Dual Velocity of Points on the Screw Axis

91

(1)

Dual Velocity of Points Not on the Screw Axis

92

(2)

Screw Axis of a Dual Velocity

94

(1)

Vector Locating the Screw Axis Given Dual Velocity at a Point

95

(1)

Transformation-Matrix Methods

96

(4)

Velocities at Different Points in a Rigid Body

96

(1)

Velocities Expressed in Different Frames

96

(1)

Example

97

(3)

Relative Velocity

100

(4)

Example

101

(3)

Planar Four-Bar Mechanism

104

(5)

Spatial Four-Bar Mechanism

109

(3)

Generalized Slider-Crank Mechanism

112

(4)

Planar Slider-Crank Mechanism

116

(3)

Exercises -- Velocity Analysis

119

(2)

Chapter 6 -- Static Analysis

121

(18)

Dual Force

121

(1)

Example

121

(1)

Example

121

(1)

Equilibrium

122

(1)

Static Equilibrium of a General Link

123

(1)

Spatial Four-Bar Mechanism

124

(5)

Planar Four-Bar Linkage

129

(3)

Planar Slider-Crank Mechanism

132

(4)

Robotic Manipulator

136

(2)

Exercises -- Static Analysis

138

(1)

Chapter 7 -- Dynamic Analysis

139

(40)

Acceleration

139

(1)

Dual Momentum

140

(9)

Center of Mass and Radii of Gyration

149

(1)

Inertial Dual Force

150

(1)

Dynamic Equilibrium of a Link

151

(1)

Dynamic Analysis of Mechanisms

152

(1)

Planar Four-Bar Mechanism

153

(8)

Planar Slider-Crank Mechanism

161

(7)

Spatial Four-Bar Mechanism

168

(9)

Exercises -- Dynamic Analysis

177

(2)

Chapter 8 -- Dual-Number Programming

179

(30)

The Utility of Dual-Number Programming

179

(1)

The Duplex Header File

180

(1)

RCCC, Spatial-Mechanism Displacement-Analysis Program

181

(10)

The Ideal Cardan Joint

186

(4)

The Pin-and-Block Cardan Joint

190

(1)

Statement List, Duplex.hpp

191

(7)

Statement List, RCCC.cpp

198

(11)

Bibliography

209

(6)

Index

215

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Dual-Number Methods in Kinematics, Statics and Dynamics

Summary

Table of Contents

Supplemental Materials