Robot Modeling and Control - M. Vidyasagar

. .

Résumé :

Preface v

1 Introduction 1

1.1 Mathematical Modeling of Robots 5

1.1.1 Symbolic Representation of Robot Manipulators 5

1.1.2 The Configuration Space 5

1.1.3 The State Space 6

1.1.4 The Workspace 7

1.2 Robots as Mechanical Devices 7

1.2.1 Classification of Robotic Manipulators 8

1.2.2 Robotic Systems 10

1.2.3 Accuracy and Repeatability 10

1.2.4 Wrists and End Effectors 12

1.3 Common Kinematic Arrangements 13

1.3.1 Articulated Manipulator (RRR) 13

1.3.2 Spherical Manipulator (RRP) 14

1.3.3 SCARA Manipulator (RRP) 14

1.3.4 Cylindrical Manipulator (RPP) 15

1.3.5 Cartesian Manipulator (PPP) 15

1.3.6 Parallel Manipulator 18

1.4 Outline of the Text 18

1.4.1 Manipulator Arms 18

1.4.2 Underactuated and Mobile Robots 27

Problems 27

Notes and References 29

I The Geometry of Robots 33

2 Rigid Motions 35

2.1 Representing Positions 36

2.2 Representing Rotations 38

2.2.1 Rotation in the Plane 38

2.2.2 Rotations in Three Dimensions 41

2.3 Rotational Transformations 44

2.4 Composition of Rotations 48

2.4.1 Rotation with Respect to the Current Frame 48

2.4.2 Rotation with Respect to the Fixed Frame 50

2.4.3 Rules for Composition of Rotations 51

2.5 Parameterizations of Rotations 52

2.5.1 Euler Angles 53

2.5.2 Roll, Pitch, Yaw Angles 55

2.5.3 Axis-Angle Representation 57

2.5.4 Exponential Coordinates 59

2.6 Rigid Motions 61

2.6.1 Homogeneous Transformations 62

2.6.2 Exponential Coordinates for General Rigid Motions 65

2.7 Chapter Summary 65

Problems 67

Notes and References 73

3 Forward Kinematics 75

3.1 Kinematic Chains 75

3.2 The Denavit-Hartenberg Convention 78

3.2.1 Existence and Uniqueness 80

3.2.2 Assigning the Coordinate Frames 83

3.3 Examples 87

3.3.1 Planar Elbow Manipulator 87

3.3.2 Three-Link Cylindrical Robot 89

3.3.3 The Spherical Wrist 90

3.3.4 Cylindrical Manipulator with Spherical Wrist 91

3.3.5 Stanford Manipulator 93

3.3.6 SCARA Manipulator 95

3.4 Chapter Summary 96

Problems 96

Notes and References 99

4 Velocity Kinematics 101

4.1 Angular Velocity: The Fixed Axis Case 102

4.2 Skew-Symmetric Matrices 103

4.2.1 Properties of Skew-Symmetric Matrices 104

4.2.2 The Derivative of a Rotation Matrix 105

4.3 Angular Velocity: The General Case 107

4.4 Addition of Angular Velocities 108

4.5 Linear Velocity of a Point Attached to a Moving Frame 110

4.6 Derivation of the Jacobian 111

4.6.1 Angular Velocity 112

4.6.2 Linear Velocity 113

4.6.3 Combining the Linear and Angular Velocity Jacobians 115

4.7 The Tool Velocity 119

4.8 The Analytical Jacobian 121

4.9 Singularities 122

4.9.1 Decoupling of Singularities 123

4.9.2 Wrist Singularities 125

4.9.3 Arm Singularities 125

4.10 Static Force/Torque Relationships 129

4.11 Inverse Velocity and Acceleration 131

4.12 Manipulability 133

4.13 Chapter Summary 136

Problems 138

Notes and References 140

5 Inverse Kinematics 141

5.1 The General Inverse Kinematics Problem 141

5.2 Kinematic Decoupling 143

5.3 Inverse Position: A Geometric Approach 145

5.3.1 Spherical Configuration 146

5.3.2 Articulated Configuration 148

5.4 Inverse Orientation 151

5.5 Numerical Inverse Kinematics 156

5.6 Chapter Summary 158

Problems 160