## Geometric Fundamentals of RoboticsGeometric Fundamentals of Robotics provides an elegant introduction to the geometric concepts that are important to applications in robotics. This second edition is still unique in providing a deep understanding of the subject: rather than focusing on computational results in kinematics and robotics, it includes significant state-of-the art material that reflects important advances in the field, connecting robotics back to mathematical fundamentals in group theory and geometry. Key features: * Begins with a brief survey of basic notions in algebraic and differential geometry, Lie groups and Lie algebras * Examines how, in a new chapter, Clifford algebra is relevant to robot kinematics and Euclidean geometry in 3D * Introduces mathematical concepts and methods using examples from robotics * Solves substantial problems in the design and control of robots via new methods * Provides solutions to well-known enumerative problems in robot kinematics using intersection theory on the group of rigid body motions * Extends dynamics, in another new chapter, to robots with end-effector constraints, which lead to equations of motion for parallel manipulators Geometric Fundamentals of Robotics serves a wide audience of graduate students as well as researchers in a variety of areas, notably mechanical engineering, computer science, and applied mathematics. It is also an invaluable reference text. ----- From a Review of the First Edition: "The majority of textbooks dealing with this subject cover various topics in kinematics, dynamics, control, sensing, and planning for robot manipulators. The distinguishing feature of this book is that it introduces mathematical tools, especially geometric ones, for solving problems in robotics. In particular, Lie groups and allied algebraic and geometric concepts are presented in a comprehensive manner to an audience interested in robotics. The aim of the author is to show the power and elegance of these methods as they apply to problems in robotics." --MathSciNet |

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### Contents

Introduction | 1 |

12 Robots and Mechanisms | 2 |

13 Algebraic Geometry | 4 |

14 Differential Geometry | 7 |

Lie Groups | 10 |

21 Definitions and Examples | 12 |

22 More Examples Matrix Groups | 15 |

222 The Special Orthogonal Group SOn | 16 |

84 Identification of Screw Systems | 183 |

842 2systems | 184 |

843 4systems | 188 |

844 3systems | 189 |

85 Operations on Screw Systems | 193 |

Clifford Algebra | 196 |

91 Geometric Algebra | 199 |

92 Clifford Algebra for the Euclidean Group | 206 |

223 The Symplectic Group Sp2nℝ | 17 |

224 The Unitary Group Un | 18 |

24 Actions and Products | 21 |

25 The Proper Euclidean Group | 23 |

252 Chasless Theorem | 25 |

253 Coordinate Frames | 27 |

Subgroups | 30 |

32 Quotients and Normal Subgroups | 34 |

33 Group Actions Again | 36 |

34 Matrix Normal Forms | 37 |

35 Subgroups of SE3 | 41 |

36 Reuleauxs Lower Pairs | 44 |

37 Robot Kinematics | 46 |

Lie Algebra | 50 |

42 The Adjoint Representation | 54 |

43 Commutators | 57 |

44 The Exponential Mapping | 61 |

441 The Exponential of Rotation Matrices | 63 |

442 The Exponential in the Standard Representation of SE3 | 66 |

443 The Exponential in the Adjoint Representation of SE3 | 68 |

45 Robot Jacobians and Derivatives | 71 |

452 Derivatives in Lie Groups | 73 |

453 Angular Velocity | 75 |

454 The Velocity Screw | 76 |

46 Subalgebras Homomorphisms and Ideals | 77 |

47 The Killing Form | 80 |

48 The CampbellBakerHausdorff Formula | 81 |

A Little Kinematics | 85 |

52 Inverse Kinematics for 3R Robots | 89 |

522 An Example | 92 |

523 Singularities | 94 |

53 Kinematics of Planar Motion | 98 |

531 The EulerSavaray Equation | 101 |

532 The Inflection Circle | 103 |

533 Balls Point | 104 |

534 The Cubic of Stationary Curvature | 105 |

535 The Burmester Points | 106 |

54 The Planar 4Bar | 108 |

Line Geometry | 112 |

62 Pliicker Coordinates | 115 |

63 The Klein Quadric | 117 |

64 The Action of the Euclidean Group | 119 |

65 Ruled Surfaces | 123 |

651 The Regulus | 124 |

652 The Cylindroid | 126 |

653 Curvature Axes | 128 |

66 Line Complexes | 130 |

67 Inverse Robot Jacobians | 133 |

68 Grassmannians | 135 |

Representation Theory | 139 |

72 Combining Representations | 142 |

73 Representations of SO3 | 148 |

74 SO3 Plethyism | 151 |

75 Representations of SE3 | 153 |

76 The Principle of Transference | 158 |

Screw Systems | 163 |

82 2systems | 167 |

821 The Case ℝ² | 169 |

823 The Case SO3 | 170 |

825 The Case SE2 | 171 |

827 The Case SE3 | 172 |

83 3systems | 175 |

831 The Case ℝ³ | 176 |

834 The Case Hp x ℝ² | 177 |

93 Dual Quaternions | 210 |

94 Geometry of Ruled Surfaces | 214 |

A Little More Kinematics | 221 |

1012 Points | 222 |

1013 Lines | 223 |

102 Euclidean Geometry | 224 |

1022 Meets | 225 |

1023 JoinsThe Shuffle product | 226 |

1024 PerpendicularityThe Contraction | 228 |

103 Piepers Theorem | 231 |

1032 The T³ Robot | 234 |

1033 The PUMA | 238 |

The Study Quadric | 241 |

112 Linear Subspaces | 245 |

1122 3planes | 246 |

1123 Intersections of 3planes | 248 |

1124 Quadric Grassmannians | 250 |

113 Partial Flags and Projections | 252 |

114 Some Quadric Subspaces | 255 |

115 Intersection Theory | 256 |

1151 Postures for General 6R Robots | 262 |

1152 Conformations of the 63 Stewart Platform | 264 |

1153 The Tripod Wrist | 266 |

1154 The 66 Stewart Platform | 267 |

Statics | 271 |

122 Forces Torques and Wrenches | 272 |

123 Wrist Force Sensor | 274 |

124 Wrench at the EndEffector | 276 |

125 Gripping | 278 |

126 Friction | 283 |

Dynamics | 286 |

132 Robot Equations of Motion | 292 |

1322 Serial Robots | 293 |

1323 Change in Payload | 296 |

134 Lagrangian Dynamics of Robots | 300 |

1341 EulerLagrange Equations | 301 |

1342 Derivatives of the Generalised Inertia Matrix | 303 |

1343 Small Oscillations | 304 |

135 Hamiltonian Dynamics of Robots | 306 |

136 Simplification of the Equations of Motion | 309 |

1362 Ignorable Coordinates | 312 |

1363 Decoupling by Coordinate Transformation | 316 |

Constrained Dynamics | 321 |

1411 Dynamics of Tree and Star Structures | 323 |

1412 Link Velocities and Accelerations | 324 |

1413 Recursive Dynamics for Trees and Stars | 325 |

142 Serial Robots with EndEffector Constraints | 327 |

1422 Constrained Dynamics of a Rigid Body | 330 |

1423 Constrained Serial Robots | 331 |

143 Constrained Trees and Stars | 333 |

1432 Parallel Mechanisms | 334 |

144 Dynamics of Planar 4Bars | 336 |

145 Biped Walking | 340 |

146 The Stewart Platform | 343 |

Differential Geometry | 348 |

152 Mobility of Overconstrained Mechanisms | 355 |

153 Controlling Robots Along Helical Trajectories | 360 |

154 Hybrid Control | 363 |

1542 Constraints | 364 |

1543 Projection Operators | 365 |

1544 The Second Fundamental Form | 369 |

373 | |

383 | |