Ü.M. Selig

Geometrie Fundamentals of Robotics Second Edition

Springer

Contents

Preface 1 Introduction 1.1 Theoretical Robotics? 1.2 Robots and Mechanisms 1.3 Algebraic Geometry 1.4 Differential Geometry 2 Lie Groups 2.1 Defmitions and Examples 2.2 More Examples — Matrix Groups 2.2.1 The Orthogonal Group 0(n) 2.2.2 The Special Orthogonal Group SO(n) 2.2.3 The Symplectic Group Sp{2n, R) 2.2.4 The Unitary Group U(n) 2.2.5 The Special Unitary Group SU(n) 2.3 Homomorphisms 2.4 Actions and Products 2.5 The Proper Euclidean Group 2.5.1 Isometries 2.5.2 Chasles's Theorem 2.5.3 Coordinate Frames

vii 1 1 2 4 7 11 12 15 15 16 17 18 18 18 21 23 23 25 27

xii

Contents

3

Subgroups 3.1 The Homomorphism Theorems 3.2 Quotients and Normal Subgroups 3.3 Group Actions Again 3.4 Matrix Normal Forms 3.5 Subgroups of ££(3) 3.6 Reuleaux's Lower Pairs 3.7 Robot Kinematics

31 31 34 36 37 41 44 46

4

Lie 4.1 4.2 4.3 4.4

51 51 54 57 61 63 66 68 71 71 73 75 76 77 80 81

4.5

4.6 4.7 4.8

Algebra Tangent Vectors The Adjoint Representation Commutators The Exponential Mapping 4.4.1 The Exponential of Rotation Matrices 4.4.2 The Exponential in the Standard Representation of SE(3) 4.4.3 The Exponential in the Adjoint Representation of SE(3) Robot Jacobians and Derivatives 4.5.1 The Jacobian of a Robot 4.5.2 Derivatives in Lie Groups 4.5.3 Angular Velocity 4.5.4 The Velocity Screw Subalgebras, Homomorphisms and Ideals The Killing Form The Campbell-Baker-Hausdorff Formula

5

A Little Kinematics 5.1 Inverse Kinematics for 3-R Wrists 5.2 Inverse Kinematics for 3-R Robots 5.2.1 Solution Procedure 5.2.2 An Example 5.2.3 Singularities 5.3 Kinematics of Planar Motion 5.3.1 The Euler-Savaray Equation 5.3.2 The Inflection Circle 5.3.3 Ball's Point 5.3.4 The Cubic of Stationary Curvature 5.3.5 The Burmester Points 5.4 The Planar 4-Bar

85 85 89 89 92 94 98 101 103 104 105 106 108

6

Line Geometry 6.1 Lines in Three Dimensions 6.2 Plücker Coordinates 6.3 The Klein Quadric 6.4 The Action of the Euclidean Group

113 113 115 117 119

Contents

6.5

6.6 6.7 6.8

Ruled Surfaces 6.5.1 The Regulus 6.5.2 The Cylindroid 6.5.3 Curvature Axes Line Complexes Inverse Robot Jacobians Grassmannians

xiii

123 124 126 128 130 133 135

7 Representation Theory 7.1 Definitions 7.2 Combining Representations 7.3 Representations of 50(3) 7.4 50(3) Plethyism 7.5 Representations of SE(3) 7.6 The Principle of Transference

139 139 142 148 151 153 158

8

Screw Systems 8.1 Generalities 8.2 2-systems 8.2.1 The Case M2 8.2.2 The Case SO(2) x K 8.2.3 The Case 50(3) 8.2.4 The Case Hp x R2 8.2.5 The Case SE{2) 8.2.6 The Case SE{2) x l 8.2.7 The Case £E(3) 8.3 3-systems 8.3.1 The Case M3 8.3.2 The Case 50(3) 8.3.3 The Case SE(2) 8.3.4 The Case Hp x K2 8.3.5 The Case SE(2) x l 8.3.6 The Case SE(3) 8.4 Identification of Screw Systems 8.4.1 1-systems and 5-systems 8.4.2 2-systems 8.4.3 4-systems 8.4.4 3-systems 8.5 Operations on Screw Systems

163 163 167 169 169 170 170 171 171 172 175 176 176 176 177 177 177 183 183 184 188 189 193

9

Clifford Algebra 9.1 Geometrie Algebra 9.2 Clifford Algebra for the Euclidean Group 9.3 Dual Quaternions 9.4 Geometry of Ruled Surfaces

197 199 206 210 214

xiv

Contents

10 A Little More Kinematics 10.1 Clifford Algebra of Points, Lines and Planes 10.1.1 Planes 10.1.2 Points 10.1.3 Lines 10.2 Euclidean Geometry 10.2.1 Incidence 10.2.2 Meets 10.2.3 Joins—The Shuttle product 10.2.4 Perpendicularity—The Contraction 10.3 Pieper's Theorem 10.3.1 Robot Kinematics 10.3.2 The T3 Robot 10.3.3 The PUMA

221 221 221 222 223 224 224 225 226 228 231 231 234 238

11 The Study Quadric 11.1 Study's Soma 11.2 Linear Subspaces 11.2.1 Lines 11.2.2 3-planes 11.2.3 Intersections of 3-planes 11.2.4 Quadric Grassmannians 11.3 Partial Flags and Projections 11.4 Some Quadric Subspaces 11.5 Intersection Theory 11.5.1 Postures for General 6-R Robots 11.5.2 Conformations of the 6-3 Stewart Platform 11.5.3 The Tripod Wrist 11.5.4 The 6-6 Stewart Platform

241 241 245 245 246 248 250 252 255 256 262 264 266 267

12 Statics 12.1 Co-Screws 12.2 Forces, Torques and Wrenches 12.3 Wrist Force Sensor 12.4 Wrench at the End-Effector 12.5 Gripping 12.6 Friction

271 271 272 274 276 278 283

13 Dynamics 13.1 Momentum and Inertia 13.2 Robot Equations of Motion 13.2.1 Equations for a Single Body 13.2.2 Serial Robots 13.2.3 Change in Payload 13.3 Recursive Formulation

287 287 292 292 293 296 296

Contents 13.4 Lagrangian Dynamics of Robots 13.4.1 Euler-Lagrange Equations 13.4.2 Derivatives of the Generalised Inertia Matrix 13.4.3 Small Oscillations 13.5 Hamiltonian Dynamics of Robots 13.6 Simplification of the Equations of Motion 13.6.1 Decoupling by Design 13.6.2 Ignorable Coordinates 13.6.3 Decoupling by Coordinate Transformation

xv 300 301 303 304 306 309 309 312 316

14 Constrained Dynamics 14.1 Trees and Stars 14.1.1 Dynamics of Tree and Star Structures 14.1.2 Link Velocities and Accelerations 14.1.3 Recursive Dynamics for Trees and Stars 14.2 Serial Robots with End-Effector Constraints 14.2.1 Holonomic Constraints 14.2.2 Constrained Dynamics of a Rigid Body 14.2.3 Constrained Serial Robots 14.3 Constrained Trees and Stars 14.3.1 Systems of Freedom 14.3.2 Parallel Mechanisms 14.4 Dynamics of Planar 4-Bars 14.5 Biped Walking 14.6 The Stewart Platform

321 321 323 324 325 327 327 330 331 333 333 334 336 340 343

15 Differential Geometry 15.1 Metrics, Connections and Geodesics 15.2 Mobility of Overconstrained Mechanisms 15.3 Controlling Robots Along Helical Trajectories 15.4 Hybrid Control 15.4.1 What is Hybrid Control? 15.4.2 Constraints 15.4.3 Projection Operators 15.4.4 The Second Fundamental Form

349 349 355 360 363 363 364 365 369

References

373

Index

383