Geometric Fundamentals of Robotics

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Springer, Jan 7, 2005 - Computers - 398 pages
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Geometric 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.

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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
References
373
Index
383

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