Robots and Screw Theory Applications of Kinematics and Statics to Robotics

by Davidson, Joseph K.; Hunt, Kenneth H.

• ISBN13: 9780198562450
• ISBN10: 0198562454
• Format: Hardcover
• Copyright: 2004-06-03
• Publisher: Oxford University Press

Summary

Robots and Screw Theory describes the mathematical foundations, especially geometric, underlying the motions and force-transfers in robots. The principles developed in the book are used in the control of robots and in the design of their major moving parts. The illustrative examples and theexercises in the book are taken principally from robotic machinery used for manufacturing and construction, but the principles apply equally well to miniature robotic devices and to those used in other industries. The comprehensive coverage of the screw and its geometry lead to reciprocal screwsystems for statics and instantaneous kinematics. These screw systems are brought together in a unique way to show many cross-relationships between the force-systems that support a body equivalently to a kinematic serial connection of joints and links. No prior knowledge of screw theory is assumed. The reader is introduced to the screw with a simple planar example yet most of the book applies to robots that move three-dimensionally. Consequently, the book is suitable both as a text at the graduate-course level and as a reference book for theprofessional. Worked examples on every major topic and over 300 exercises clarify and reinforce the principles covered in the text. A chapter-length list of references gives the reader source-material and opportunities to pursue more fully topics contained in the text.

Author Biography

Joseph K. Davidson served as Associate Editor of the ASME Journal of Mechanisms, Transmissions, and Automation in Design from 1982-86. He also served one term as a member of the Executive Committee of the International Federation for the Theory of Machines and Mechanisms.
Davidson is a Fellow of the American Society of Mechanical Engineers.
Davidson is a member of the Robotics & Automation Society of the IEEE.
Kenneth H. Hunt served on the Honorary Editorial Advisory Board of the Pergamon journal Mechanism and Machine Theory from its inception in 1966 (as the Journal of Mechanisms) until his death in 2002. He was a Founding Executive Councillor of the International Federation for the Theory of Machines and Mechanisms, and he also was the Founding Chairman of the Australian National Committee for IFToMM.
Hunt is a Fellow of the American Society of Mechanical Engineers
Hunt is a Fellow of the Institution of Mechanical Engineers
Hunt is a Fellow of the Institution of Engineers, Australia
Hunt is a Fellow of the Australian Academy of Technological Sciences

Table of Contents

Preface	p. xv
The Planar Serial Robot-arm	p. 1
Introduction	p. 1
Freedom of the End-effector	p. 2
The Instantaneous Centres in a Planar Robot-arm	p. 3
The 'Inverse Velocity-problem' Solved by Instantaneous Centres	p. 4
Instantaneous Kinematics and Static Equilibrium	p. 6
The 'Forward Velocity-problem' Solved by Instantaneous Centres	p. 7
Exercises 1A	p. 7
Velocities by Superposition	p. 8
The Linear Sliding Joint	p. 13
Torques at the Actuated Joints	p. 15
The Assembly-configurations of a Planar Robot-arm	p. 17
Exercises 1B	p. 19
Foreshadowing the Spatial Serial Robot-arm	p. 21
Describing the Screw	p. 23
The Screw in Mechanics	p. 23
The Screw in Statics	p. 23
The Screw in Instantaneous Kinematics	p. 25
Other Applications in Mechanics	p. 29
The Finite Twist	p. 30
Freedom and Constraint of a Rigid Body	p. 31
Twists, Wrenches, and Screws Summarized	p. 33
Exercises 2A	p. 34
Analysing the Screw	p. 35
Background	p. 35
Screw Coordinates	p. 36
The Coordinates	p. 36
Physical Interpretation of the Coordinates	p. 37
The Axis and Pitch of a Screw; Normalization of its Coordinates	p. 37
Homogeneity of Screw Coordinates	p. 39
A Line as the Join of Two Finite Points	p. 40
Exercises 3A	p. 42
Homogeneous Coordinates of a Point	p. 43
A Point in Projective Space	p. 44
A Line as the Join of Two Points	p. 47
Homogeneous Coordinates of a Plane	p. 47
A Line as the Meet of Two Planes	p. 49
Homogeneity, Dimensions, and Units	p. 52
Ray- and Axis-coordinate Orders for Screw Coordinates	p. 55
Duality and Lines	p. 55
Exercises 3B	p. 56
Transformations for Coordinates That Locate A Rigid Body	p. 59
Introduction	p. 59
Coordinates	p. 59
Coordinate Transformations for Two Dimensions	p. 60
Rotational Transformations with Points	p. 60
General Transformations with Points on Coplanar Laminae	p. 63
Determining from [A[superscript ij]] the Axis and Angle of Rotation	p. 66
Determining [A[superscript ij]] from the Axis and Angle of Rotation	p. 67
Transformations with Free Vectors and Planes	p. 69
General Rotational Transformations	p. 71
Successive Rotations	p. 71
Rotational Transformations with Screws, Lines, Wrenches, and Twists	p. 72
Interpretations of a Transformation	p. 73
The Active Interpretation and the Active Transformation	p. 74
Exercises 4A	p. 75
Coordinate Transformations for Three Dimensions	p. 77
The General Transformations with Points	p. 77
Transformations with Vectors and Planes	p. 79
General Transformations with Screws, Lines, Wrenches, and Twists	p. 80
The Finite Twist	p. 84
The Finite Twist and the Finite Screw	p. 84
The Pitch h and q-Pitch q of a Finite Twist or a Finite Screw	p. 85
Determining [A[superscript ij]] from a Finite Twist $[superscript ij] (q)	p. 87
Determining the Finite Twist $[superscript ij] (q) from [A[superscript ij]] and [$$[superscript ij]]	p. 88
Exercises 4B	p. 94
Linear Dependence: Reciprocity of Screws: Linear and Non-Linear Screw Systems	p. 97
Linear Dependence of Points and Planes	p. 97
The Linear Two-System of Screws	p. 99
Exercises 5A	p. 104
Linear Screw Systems	p. 105
The One-system	p. 105
The Two-system	p. 106
The Three-system	p. 107
The Four-system	p. 108
The Five-system	p. 109
The Six-system	p. 109
Systems that are Invariant with Finite Joint-displacements	p. 110
Exercises 5B	p. 110
Reciprocity of Screws	p. 113
A Rotating Body Acted on by a Force	p. 113
A Twisting Body Acted on by a Wrench	p. 116
Reciprocity and Linear Screw Systems	p. 119
Exercises 5C	p. 120
Linear and Non-linear Screw Systems	p. 121
Some Finite Displacements and Their Screw Systems	p. 123
The System of Finite Screws for the Twists that Displace a Point	p. 124
The System of Finite Screws for the Twists that Displace a Directed Line a	p. 125
The System of Finite Screws for the Twists that Displace a Point on a Directed Line	p. 128
Commutativity and Sequential Finite Twists	p. 130
Exercises 5D	p. 131
Spatial Serial Robot-Arms	p. 134
Introduction	p. 134
Some Typical Six-actuator Arms	p. 134
A Gantry Arm	p. 139
Axes of the Actuated Joints and the Jacobian	p. 140
Det [J] and Special Configurations	p. 141
The Reciprocal Screw at a Special Configuration	p. 144
The Ubiquity of Special Configurations	p. 144
The Inverse of the Jacobian	p. 147
[J superscript -1] and Special Configurations	p. 148
The Gantry Arm with an 'Offset Roll-pitch-roll' Wrist	p. 148
The 'Pitch-yaw-roll' Wrist	p. 151
The Spherical '3-Roll Wrist'	p. 153
Other Wrist Designs	p. 155
Exercises 6A	p. 156
The CM T[superscript 3]-566 Arm (Elbow Manipulator)	p. 157
The Forward and Inverse Rate-problems	p. 159
Special Configurations: Individual Conditions	p. 160
Transversals and Reciprocal Screws	p. 160
Special Configurations: Combinations of Conditions	p. 162
A Unimate PUMA Arm	p. 164
A Manipulator with Rotary Joints in Just Three Directions	p. 167
General Features of Special Configurations	p. 168
Workspace	p. 170
Geometrical Constructions	p. 171
Configurations of a Robot-arm when B is at the Boundary	p. 172
Transversals and Reciprocal Screws in Workspace Identification	p. 176
Influence of Excursion-limits at the Joints	p. 177
Subspaces within the Reachable Point-workspace	p. 178
Workspaces of Reference Planes and Lines on the End-effector	p. 178
Five-actuator Arms	p. 178
Exercises 6B	p. 182
Control	p. 186
Joint Control and Cartesian Control	p. 186
Closing the Feedback Loop on the Task	p. 186
Wrench Control and Hybrid Control	p. 187
Torques (Forces) at the Joints of a Six-actuator Arm	p. 188
Exercises 6C	p. 191
The Assembly-Configurations of Serial Robot-Arms	p. 193
Introduction	p. 193
Placement of Cartesian Coordinate Frames on Links	p. 194
Forward and Inverse Kinematics for Position	p. 194
The Scalar Equation a cos [phi] + b sin [phi] = c	p. 196
The Assembly-configurations of Six-actuator Robot-arms	p. 197
A Gantry Arm	p. 198
The CM T[superscript 3]-566 Arm (Elbow Manipulator)	p. 200
A Unimate PUMA Arm	p. 204
The Inverted CM T[superscript 3]-566 Arm with an Equivalent Spherical Joint	p. 208
A Five-actuator Arm	p. 211
Exercises 7A	p. 214
Six-actuator Robot-arms with Generally Placed Axes	p. 215
A Standard Placement of Cartesian Coordinate Frames on Links	p. 217
The Fundamental Equations	p. 219
Two Alternative Methods	p. 223
The Motoman-V6 Robot-arm	p. 224
Continuation Methods	p. 225
Robot-arms with Closed-form Solutions	p. 226
Exercises 7B	p. 227
In-Parallel Actuation I: Simple and Direct	p. 229
Introduction	p. 229
The 6-6 Fully In-parallel Manipulator	p. 230
The Bricard-Borel Phenomena	p. 231
Assembly Configurations	p. 234
Special Configurations and Other Limitations: Generalities	p. 235
The Octahedral Manipulator: Geometry	p. 240
Polyhedra and Cauchy's Theorem	p. 242
Assembly-configurations and Concavity	p. 242
Exercises 8A	p. 244
Transitory Kinematic Equivalence: Serial versus In-parallel	p. 245
The General 'Canonical' Wrench-applicator and the Unactuated Screw-support	p. 247
Series-parallel Comparisons	p. 250
The Wrench-applicator for a Pure Couple	p. 253
The Wrench-applicator for a Pure Force	p. 254
Some Variants of Wrench-applicators	p. 255
Exercises 8B	p. 256
Statics and Kinematics of Fully In-parallel Robots	p. 258
Charts of Analogues	p. 260
The Octahedral Manipulator: Proportions and Configurations	p. 265
The Datum Configuration	p. 266
Departures From the Datum Configuration	p. 270
A Substitution for the Double-spherical Joints	p. 272
Separation of the Double-spherical Joints	p. 273
Actuation of Force-applicators	p. 276
Other Possible Separation Arrangements for Double-spherical Joints	p. 277
An Actuated Reciprocal Connection	p. 279
Cognate Octahedral Manipulators	p. 282
Exercises 8C	p. 283
Special Configurations: Further Observations	p. 286
A Case Study	p. 286
Series-parallel Comparisons	p. 290
Exercises 8D	p. 291
In-Parallel Actuation II: Combinations with Serial Devices	p. 293
Introduction	p. 293
Two Composite Robots	p. 294
The Force-applicator: Some Variants in Six-actuator Robots	p. 297
Mobility, Connectivity, and Over-constraint	p. 300
The General Mobility Criterion	p. 300
Connectivity C[subscript ij]	p. 301
One Class of Over-constrained Devices	p. 303
Exercises 9A	p. 303
The Adjustable Tripod as a Manipulator	p. 304
Structure, Mobility, and Kinematic Substitutions	p. 305
Performance and Proportions of the Tripod	p. 306
Exercises 9B	p. 310
Generalized Reciprocal Connections: Some Derived Robots	p. 310
Three-freedom Planar-motion Robots	p. 311
Homokinetic Shaft Couplings for Parallel Shafts	p. 314
Two Planar In-parallel Robots	p. 316
The Planar In-parallel Robot with Three Linear Actuators	p. 316
A Planar In-parallel Robot with Three Rotary Actuators	p. 318
Exercises 9C	p. 321
Homokinetic Coupling Robots and a Derivative	p. 322
A Translatory Robot Based on a Homokinetic Coupling	p. 322
The Three Translatory Freedoms of the DELTA Robot	p. 326
The Inverse Kinematics for Position of Composite and Planar In-parallel Robots	p. 328
The Planar In-parallel Robot with Three Linear Actuators	p. 329
A Planar In-parallel Robot with Three Rotary Actuators	p. 330
A Coupling Robot and the Translatory Freedoms of the DELTA Robot	p. 331
Two Over-constrained Translatory Manipulators	p. 334
Exercises 9D	p. 337
Redundant Robotic Systems	p. 338
Introduction	p. 338
Kinematic Redundancy	p. 338
Pseudoinverse Control	p. 339
The Coordinates of a Screw and the Jacobian [J]	p. 340
The Pseudoinverse of [J] and Other Solutions to eqns (10.3)	p. 341
Solutions to eqns (10.3) by Augmenting [J]	p. 345
Comparison of [J superscript #] to [J superscript -1]	p. 346
The Control of a Four-axis Spherical Wrist	p. 348
Overspeeding in the Three-axis Orthogonal Spherical Wrist	p. 348
Pseudoinverse Control of the Four-axis Orthogonal Spherical Wrist	p. 350
Redundant Serial Arms with Rotary Joints in Just Four Directions	p. 354
Actuator-torques (Forces) at the Joints of Redundant Serial Arms	p. 356
Exercises 10A	p. 357
Statically Redundant Robots and Manipulators	p. 359
Screw Systems at Localized Contacts	p. 360
The Equilibrating and Interacting Force Fields	p. 361
Frictional Contacts	p. 363
The Jacobian of Force-components for Frictional Contacts	p. 365
The Pseudoinverse Solution and the Equilibrating System	p. 367
The Frictional Grasp of a Disc	p. 369
Optimization of a Grasp Using Interacting Systems of Forces	p. 371
Exercises 10B	p. 374
Static Stability in Legged Vehicles	p. 376
Introduction	p. 376
Wheeled and Legged Vehicles	p. 376
Margin of Static Stability	p. 378
The Principle of Normalized Virtual Power	p. 378
Other Measures for Margin of Stability	p. 380
Application to General Locations of the Contacts	p. 381
Four Contacts with the Ground	p. 381
Three Contacts with the Ground	p. 383
Comparison with a Horizontal Projection	p. 383
Virtual Power Used in Control	p. 384
A Display for Margin of Static Stability	p. 384
The Rectangular Display	p. 385
Three Contacts with the Ground	p. 386
Conclusion	p. 387
Exercises 11A	p. 387
Some Useful Expressions for Lines	p. 389
The Screw as a Point in Projective Five-Space	p. 402
The Finite Twist and Eduard Study's Coordinates	p. 407
Computer File for Chapter 10	p. 417
Answers to Exercises	p. 422
References	p. 436
Index	p. 448
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