Guide to kinematic theory for the analysis of spatial mechanisms and manipulators
Kinematics of General Spatial Mechanical Systems is an effective and proficient guide to the kinematic description and analysis of the spatial mechanical systems such as serial manipulators, parallel manipulators and spatial mechanisms. The author highlights the analytical and semi-analytical methods for solving the relevant equations and considers four main elements: The mathematics of spatial kinematics with the necessary theorems, formulas and methods; The kinematic description of the links and joints including the rolling contact joints; Writing the kinematic chain and loop equations for the systems to be analyzed; and Solving these equations for the unspecified variables both in the forward and inverse senses together with the multiplicity and singularity analyses.
Comprehensive in scope, the book covers topics ranging from rather elementary subjects such as spatial mechanisms with single degree of freedom to more advanced topics such as serial manipulators including redundant and deficient ones, parallel manipulators, and non-holonomic spatial cam mechanisms that involve rolling without slipping motions. The author presents an effective and accessible symbolic manipulation method making it possible to obtain neat and transparent expressions that describe the systems showing all the kinematic details. Such expressions readily lead to analytical or semi-analytical solutions. They also facilitate the identification and analysis of the multiplicities and singularities.
This all-time beneficial book:
- Provides an easy-to-use systematic formulation method that is applicable to all sorts of spatial machanisms and manipulators
- Introduces a symbolic manipulation method, which is effective and straightforward to use, so that kinematic relationships can be simplified by using all the special geometric features of the system
- Offers an accessible format that uses a systematic and easy-to-conceive notation which has proven successful
- Presents content written by an author who is a renowned expert in the field
- Includes an accompanying website
Written for academicians, students, engineers, computer scientists and any other people working in the area of spatial mechanisms and manipulators, Kinematics of General Spatial Mechanical Systems provides a clear notation, formulation, and a logical approach to the topic and offers a fresh presentation of challenging material.
Table of Contents
Preface xv
Acknowledgments xix
List of Commonly Used Symbols, Abbreviations, and Acronyms xxi
About the Companion Website xxvii
1 Vectors and Their Matrix Representations in Selected Reference Frames 1
1.1 General Features of Notation 1
1.2 Vectors 2
1.2.1 Definition and Description of a Vector 2
1.2.2 Equality of Vectors 2
1.2.3 Opposite Vectors 3
1.3 Vector Products 3
1.3.1 Dot Product 3
1.3.2 Cross Product 3
1.4 Reference Frames 4
1.5 Representation of a Vector in a Selected Reference Frame 6
1.6 Matrix Operations Corresponding to Vector Operations 7
1.6.1 Dot Product 7
1.6.2 Cross Product and Skew Symmetric Cross Product Matrices 8
1.7 Mathematical Properties of the Skew Symmetric Matrices 9
1.8 Examples Involving Skew Symmetric Matrices 10
1.8.1 Example 1.1 10
1.8.2 Example 1.2 11
1.8.3 Example 1.3 11
2 Rotation of Vectors and Rotation Matrices 13
2.1 Vector Equation of Rotation and the Rodrigues Formula 13
2.2 Matrix Equation of Rotation and the Rotation Matrix 15
2.3 Exponentially Expressed Rotation Matrix 16
2.4 Basic Rotation Matrices 16
2.5 Successive Rotations 17
2.6 Orthonormality of the Rotation Matrices 18
2.7 Mathematical Properties of the Rotation Matrices 20
2.7.1 Mathematical Properties of General Rotation Matrices 20
2.7.2 Mathematical Properties of the Basic Rotation Matrices 22
2.8 Examples Involving Rotation Matrices 22
2.8.1 Example 2.1 22
2.8.2 Example 2.2 23
2.8.3 Example 2.3 24
2.8.4 Example 2.4 24
2.9 Determination of the Angle and Axis of a Specified Rotation Matrix 25
2.9.1 Scalar Equations of Rotation 25
2.9.2 Determination of the Angle of Rotation 26
2.9.3 Determination of the Axis of Rotation 26
2.9.4 Discussion About the Optional Sign Variables 29
2.10 Definition and Properties of the Double Argument Arctangent Function 29
3 Matrix Representations of Vectors in Different Reference Frames and the Component Transformation Matrices 31
3.1 Matrix Representations of a Vector in Different Reference Frames 31
3.2 Transformation Matrices Between Reference Frames 32
3.2.1 Definition and Usage of a Transformation Matrix 32
3.2.2 Basic Properties of a Transformation Matrix 33
3.3 Expression of a Transformation Matrix in Terms of Basis Vectors 34
3.3.1 Column-by-Column Expression 34
3.3.2 Row-by-Row Expression 34
3.3.3 Remark 3.1 35
3.3.4 Remark 3.2 35
3.3.5 Remark 3.3 36
3.3.6 Example 3.1 36
3.4 Expression of a Transformation Matrix as a Direction Cosine Matrix 37
3.4.1 Definitions of Direction Angles and Direction Cosines 37
3.4.2 Transformation Matrix Formed as a Direction Cosine Matrix 38
3.5 Expression of a Transformation Matrix as a Rotation Matrix 38
3.5.1 Correlation Between the Rotation and Transformation Matrices 38
3.5.2 Distinction Between the Rotation and Transformation Matrices 39
3.6 Relationship Between the Matrix Representations of a Rotation Operator in Different Reference Frames 40
3.7 Expression of a Transformation Matrix in a Case of Several Successive Rotations 40
3.7.1 Rotated Frame Based (RFB) Formulation 41
3.7.2 Initial Frame Based (IFB) Formulation 41
3.8 Expression of a Transformation Matrix in Terms of Euler Angles 42
3.8.1 General Definition of Euler Angles 42
3.8.2 IFB (Initial Frame Based) Euler Angle Sequences 42
3.8.3 RFB (Rotated Frame Based) Euler Angle Sequences 43
3.8.4 Remark 3.4 44
3.8.5 Remark 3.5 44
3.8.6 Remark 3.6: Preference Between IFB and RFB Sequences 45
3.8.7 Commonly Used Euler Angle Sequences 45
3.8.8 Extraction of Euler Angles from a Given Transformation Matrix 46
3.9 Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices 51
3.9.1 Position of a Point Expressed in Different Reference Frames 51
3.9.2 Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships 52
3.9.3 Affine Coordinate Transformation Between Two Reference Frames 53
3.9.4 Homogeneous Coordinate Transformation Between Two Reference Frames 54
3.9.5 Mathematical Properties of the Homogeneous Transformation Matrices 55
3.9.6 Example 3.2 58
4 Vector Differentiation Accompanied by Velocity and Acceleration Expressions 63
4.1 Derivatives of a Vector with Respect to Different Reference Frames 63
4.1.1 Differentiation and Resolution Frames 63
4.1.2 Components in Different Differentiation and Resolution Frames 64
4.1.3 Example 65
4.2 Vector Derivatives with Respect to Different Reference Frames and the Coriolis Transport Theorem 66
4.2.1 First Derivatives and the Relative Angular Velocity 66
4.2.2 Second Derivatives and the Relative Angular Acceleration 68
4.3 Combination of Relative Angular Velocities and Accelerations 70
4.3.1 Combination of Relative Angular Velocities 70
4.3.2 Combination of Relative Angular Accelerations 71
4.4 Angular Velocities and Accelerations Associated with Rotation Sequences 71
4.4.1 Relative Angular Velocities and Accelerations about Relatively Fixed Axes 71
4.4.2 Example 72
4.4.3 Angular Velocities Associated with the Euler Angle Sequences 74
4.5 Velocity and Acceleration of a Point with Respect to Different Reference Frames 77
4.5.1 Velocity of a Point with Respect to Different Reference Frames 77
4.5.2 Acceleration of a Point with Respect to Different Reference Frames 78
4.5.3 Velocity and Acceleration Expressions with Simplified Notations 79
5 Kinematics of Rigid Body Systems 81
5.1 Kinematic Description of a Rigid Body System 82
5.1.1 Body Frames and Joint Frames 82
5.1.2 Kinematic Chains, Kinematic Branches, and Kinematic Loops 83
5.1.3 Joints or Kinematic Pairs 83
5.2 Position Equations for a Kinematic Chain of Rigid Bodies 84
5.2.1 Relative Orientation Equation Between Successive Bodies 85
5.2.2 Relative Location Equation Between Successive Bodies 85
5.2.3 Orientation of a Body with Respect to the Base of the Kinematic Chain 85
5.2.4 Location of a Body with Respect to the Base of the Kinematic Chain 86
5.2.5 Loop Closure Equations for a Kinematic Loop 86
5.3 Velocity Equations for a Kinematic Chain of Rigid Bodies 87
5.3.1 Relative Angular Velocity between Successive Bodies 87
5.3.2 Relative Translational Velocity Between Successive Bodies 88
5.3.3 Angular Velocity of a Body with Respect to the Base 89
5.3.4 Translational Velocity of a Body with Respect to the Base 89
5.3.5 Velocity Equations for a Kinematic Loop 90
5.4 Acceleration Equations for a Kinematic Chain of Rigid Bodies 90
5.4.1 Relative Angular Acceleration Between Successive Bodies 91
5.4.2 Relative Translational Acceleration Between Successive Bodies 92
5.4.3 Angular Acceleration of a Body with Respect to the Base 92
5.4.4 Translational Acceleration of a Body with Respect to the Base 93
5.4.5 Acceleration Equations for a Kinematic Loop 93
5.5 Example 5.1 :A Serial Manipulator with an RRP Arm 94
5.5.1 Kinematic Description of the System 94
5.5.2 Position Analysis 95
5.5.3 Velocity Analysis 100
5.5.4 Acceleration Analysis 103
5.6 Example 5.2 :A Spatial Slider-Crank (RSSP) Mechanism 106
5.6.1 Kinematic Description of the Mechanism 106
5.6.2 Loop Closure Equations 108
5.6.3 Degree of Freedom or Mobility 109
5.6.4 Position Analysis 110
5.6.5 Velocity Analysis 119
5.6.6 Acceleration Analysis 122
6 Joints and Their Kinematic Characteristics 125
6.1 Kinematic Details of the Joints 125
6.1.1 Description of a Joint as a Kinematic Pair 125
6.1.2 Degree of Freedom or Mobility of a Joint 126
6.1.3 Number of Distinct Joints Between Two Rigid Bodies 126
6.1.4 Classification of the Joints 127
6.2 Typical Lower Order Joints 128
6.2.1 Single-Axis Joints 128
6.2.2 Universal Joint 130
6.2.3 Spherical Joint 131
6.2.4 Plane-on-Plane Joint 132
6.3 Higher Order Joints with Simple Contacts 132
6.3.1 Line-on-Plane Joint 132
6.3.2 Point-on-Plane Joint 133
6.3.3 Point-on-Surface Joint 133
6.4 Typical Multi-Joint Connections 134
6.4.1 Fork-on-Surface Joint 134
6.4.2 Triangle-on-Surface Joint 136
6.5 Rolling Contact Joints with Point Contacts 138
6.5.1 Surface-on-Surface Joint 138
6.5.2 Curve-on-Surface Joint 144
6.5.3 Curve-on-Curve Joint 147
6.6 Rolling Contact Joints with Line Contacts 148
6.6.1 Cone-on-Cone Joint 148
6.6.2 Cone-on-Cylinder Joint 155
6.6.3 Cone-on-Plane Joint 157
6.6.4 Cylinder-on-Cylinder Joint 161
6.6.5 Cylinder-on-Plane Joint 164
6.7 Examples 167
6.7.1 Example 6.1: An RRRSP Mechanism 167
6.7.2 Example 6.2: A Two-Link Mechanism with Three Point-on-Plane Joints 171
6.7.3 Example 6.3: A Spatial Cam Mechanism 174
6.7.4 Example 6.4: A Spatial Cam Mechanism That Allows Rolling Without Slipping 177
7 Kinematic Features of Serial Manipulators 185
7.1 Kinematic Description of a General Serial Manipulator 185
7.2 Denavit-Hartenberg Convention 186
7.3 D-H Convention for Successive Intermediate Links and Joints 187
7.3.1 Assignment and Description of the Link Frames 187
7.3.2 D-H Parameters 188
7.3.3 Relative Position Formulas Between Successive Links 189
7.3.4 Alternative Multi-Index Notation for the D-H Convention 189
7.4 D-H Convention for the First Joint 190
7.5 D-H Convention for the Last Joint 193
7.6 D-H Convention for Successive Joints with Perpendicularly Intersecting Axes 195
7.7 D-H Convention for Successive Joints with Parallel Axes 195
7.8 D-H Convention for Successive Joints with Coincident Axes 197
8 Position and Motion Analyses of Generic Serial Manipulators 199
8.1 Forward Kinematics 201
8.2 Compact Formulation of Forward Kinematics 202
8.3 Detailed Formulation of Forward Kinematics 203
8.4 Manipulators with or without Spherical Wrists 205
8.5 Inverse Kinematics 207
8.6 Inverse Kinematic Solution for a Regular Manipulator 208
8.6.1 Regular Manipulator with a Spherical Wrist 208
8.6.2 Regular Manipulator with a Nonspherical Wrist 211
8.7 Inverse Kinematic Solution for a Redundant Manipulator 212
8.7.1 Solution by Specifying the Variables of Certain Joints 212
8.7.2 Solution by Optimization 213
8.8 Inverse Kinematic Solution for a Deficient Manipulator 214
8.8.1 Compromise in Orientation in Favor of a Completely Specified Location 214
8.8.2 Compromise in Location in Favor of a Completely Specified Orientation 215
8.9 Forward Kinematics of Motion 215
8.9.1 Forward Kinematics of Velocity Relationships 215
8.9.2 Forward Kinematics of Acceleration Relationships 216
8.10 Jacobian Matrices Associated with the Wrist and Tip Points 218
8.11 Recursive Position, Velocity, and Acceleration Formulations 220
8.11.1 Orientations of the Links 220
8.11.2 Locations of the Link Frame Origins 221
8.11.3 Locations of the Mass Centers of the Links 221
8.11.4 Angular Velocities of the Links 221
8.11.5 Velocities of the Link Frame Origins 222
8.11.6 Velocities of the Mass Centers of the Links 222
8.11.7 Angular Accelerations of the Links 222
8.11.8 Accelerations of the Link Frame Origins 222
8.11.9 Accelerations of the Mass Centers of the Links 223
8.12 Inverse Motion Analysis of a Manipulator Based on the Jacobian Matrix 223
8.12.1 Inverse Velocity Analysis of a Regular Manipulator 224
8.12.2 Inverse Acceleration Analysis of a Regular Manipulator 225
8.13 Inverse Motion Analysis of a Redundant Manipulator 225
8.13.1 Inverse Velocity Analysis 225
8.13.2 Inverse Acceleration Analysis 228
8.14 Inverse Motion Analysis of a Deficient Manipulator 229
8.15 Inverse Motion Analysis of a Regular Manipulator Using the Detailed Formulation 230
8.15.1 Inverse Velocity Solution 230
8.15.2 Inverse Acceleration Solution 231
9 Kinematic Analyses of Typical Serial Manipulators 233
9.1 Puma Manipulator 233
9.1.1 Kinematic Description According to the D-H Convention 234
9.1.2 Forward Kinematics in the Position Domain 235
9.1.3 Inverse Kinematics in the Position Domain 237
9.1.4 Multiplicity Analysis 240
9.1.5 Singularity Analysis in the Position Domain 242
9.1.6 Forward Kinematics in the Velocity Domain 244
9.1.7 Inverse Kinematics in the Velocity Domain 245
9.1.8 Singularity Analysis in the Velocity Domain 247
9.2 Stanford Manipulator 250
9.2.1 Kinematic Description According to the D-H Convention 250
9.2.2 Forward Kinematics in the Position Domain 251
9.2.3 Inverse Kinematics in the Position Domain 253
9.2.4 Multiplicity Analysis 254
9.2.5 Singularity Analysis in the Position Domain 255
9.2.6 Forward Kinematics in the Velocity Domain 255
9.2.7 Inverse Kinematics in the Velocity Domain 256
9.2.8 Singularity Analysis in the Velocity Domain 257
9.3 Elbow Manipulator 258
9.3.1 Kinematic Description According to the D-H Convention 259
9.3.2 Forward Kinematics in the Position Domain 260
9.3.3 Inverse Kinematics in the Position Domain 262
9.3.4 Multiplicity Analysis 264
9.3.5 Singularity Analysis in the Position Domain 266
9.3.6 Forward Kinematics in the Velocity Domain 269
9.3.7 Inverse Kinematics in the Velocity Domain 269
9.3.8 Singularity Analysis in the Velocity Domain 271
9.4 Scara Manipulator 273
9.4.1 Kinematic Description According to the D-H Convention 273
9.4.2 Forward Kinematics in the Position Domain 274
9.4.3 Inverse Kinematics in the Position Domain 275
9.4.4 Multiplicity Analysis 277
9.4.5 Singularity Analysis in the Position Domain 278
9.4.6 Forward Kinematics in the Velocity Domain 279
9.4.7 Inverse Kinematics in the Velocity Domain 279
9.4.8 Singularity Analysis in the Velocity Domain 280
9.5 An RP2R3 Manipulator without an Analytical Solution 281
9.5.1 Kinematic Description According to the D-H Convention 282
9.5.2 Forward Kinematics in the Position Domain 282
9.5.3 Inverse Kinematics in the Position Domain 283
9.5.4 Multiplicity Analysis 285
9.5.5 Singularity Analysis in the Position Domain 287
9.5.6 Forward Kinematics in the Velocity Domain 287
9.5.7 Inverse Kinematics in the Velocity Domain 287
9.5.8 Singularity Analysis in the Velocity Domain 289
9.6 An RPRPR2 Manipulator with an Uncustomary Analytical Solution 290
9.6.1 Kinematic Description According to the D-H Convention 290
9.6.2 Forward Kinematics in the Position Domain 291
9.6.3 Inverse Kinematics in the Position Domain 293
9.6.4 Multiplicity Analysis 297
9.6.5 Singularity Analysis in the Position Domain 298
9.6.6 Forward Kinematics in the Velocity Domain 298
9.6.7 Inverse Kinematics in the Velocity Domain 299
9.6.8 Singularity Analysis in the Velocity Domain 301
9.7 A Deficient Puma Manipulator with Five Active Joints 303
9.7.1 Kinematic Description According to the D-H Convention 303
9.7.2 Forward Kinematics in the Position Domain 304
9.7.3 Inverse Kinematics in the Position Domain 305
9.7.3.1 Solution in the Case of Fully Specified Tip Point Location 305
9.7.3.2 Solution in the Case of Fully Specified End-Effector Orientation 307
9.7.4 Multiplicity Analysis in the Position Domain 307
9.7.4.1 Analysis in the Case of Fully Specified Tip Point Location 307
9.7.4.2 Analysis in the Case of Fully Specified End-Effector Orientation 308
9.7.5 Singularity Analysis in the Position Domain 308
9.7.5.1 Analysis in the Case of Fully Specified Tip Point Location 308
9.7.5.2 Analysis in the Case of Fully Specified End-Effector Orientation 309
9.7.6 Forward Kinematics in the Velocity Domain 310
9.7.7 Inverse Kinematics in the Velocity Domain 310
9.7.7.1 Solution in the Case of Fully Specified Tip Point Velocity 310
9.7.7.2 Solution in the Case of Fully Specified End-Effector Angular Velocity 311
9.7.8 Singularity Analysis in the Velocity Domain 312
9.7.8.1 Analysis in the Case of Fully Specified Tip Point Velocity 312
9.7.8.2 Analysis in the Case of Fully Specified End-Effector Angular Velocity 313
9.8 A Redundant Humanoid Manipulator with Eight Joints 313
9.8.1 Kinematic Description According to the D-H Convention 313
9.8.2 Forward Kinematics in the Position Domain 315
9.8.3 Inverse Kinematics in the Position Domain 316
9.8.4 Multiplicity Analysis 323
9.8.5 Singularity Analysis in the Position Domain 326
9.8.6 Forward Kinematics in the Velocity Domain 328
9.8.7 Inverse Kinematics in the Velocity Domain 328
9.8.8 Singularity Analysis in the Velocity Domain 333
9.8.9 Consistency of the Inverse Kinematics in the Position and Velocity Domains 335
10 Position and Velocity Analyses of Parallel Manipulators 341
10.1 General Kinematic Features of Parallel Manipulators 343
10.2 Position Equations of a Parallel Manipulator 347
10.3 Forward Kinematics in the Position Domain 351
10.4 Inverse Kinematics in the Position Domain 359
10.5 Velocity Equations of a Parallel Manipulator 368
10.6 Forward Kinematics in the Velocity Domain 371
10.7 Inverse Kinematics in the Velocity Domain 377
10.8 Stewart-Gough Platform as a 6UPS Spatial Parallel Manipulator 384
10.8.1 Kinematic Description 384
10.8.2 Position Equations 386
10.8.3 Inverse Kinematics in the Position Domain 387
10.8.4 Forward Kinematics in the Position Domain 389
10.8.5 Velocity Equations 396
10.8.6 Inverse Kinematics in the Velocity Domain 397
10.8.7 Forward Kinematics in the Velocity Domain 398
10.9 Delta Robot: A 3RS2S2 Spatial Parallel Manipulator 402
10.9.1 Kinematic Description 402
10.9.2 Position Equations 404
10.9.3 Independent Kinematic Loops and the Associated Equations 407
10.9.4 Inverse Kinematics in the Position Domain 410
10.9.5 Forward Kinematics in the Position Domain 412
10.9.6 Velocity Equations 417
10.9.7 Inverse Kinematics in the Velocity Domain 418
10.9.8 Forward Kinematics in the Velocity Domain 420
Bibliography 423
Index 425
