Multi-Body Kinematics and Dynamics with Lie Groups

Author： Chevallier Dominique Paul;Lerbet Jean

Publisher： Elsevier Science‎

Publication year： 2017

E-ISBN: 9780081023570

P-ISBN(Paperback): 9781785482311

Subject： O152.5 Lie group

Keyword： Energy technology & engineering

Language： ENG

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Description

Multi-body Kinematics and Dynamics with Lie Groups explores the use of Lie groups in the kinematics and dynamics of rigid body systems.

The first chapter reveals the formal properties of Lie groups on the examples of rotation and Euclidean displacement groups. Chapters 2 and 3 show the specific algebraic properties of the displacement group, explaining why dual numbers play a role in kinematics (in the so-called screw theory). Chapters 4 to 7 make use of those mathematical tools to expound the kinematics of rigid body systems and in particular the kinematics of open and closed kinematical chains. A complete classification of their singularities demonstrates the efficiency of the method.

Dynamics of multibody systems leads to very big computations. Chapter 8 shows how Lie groups make it possible to put them in the most compact possible form, useful for the design of software, and expands the example of tree-structured systems.

This book is accessible to all interested readers as no previous knowledge of the general theory is required.

Presents a overview of the practical aspects of Lie groups based on the example of rotation groups and the Euclidean group
Makes it clear that the interface between Lie groups methods in mechanics and numerical calculations is very easy
Includes theoretical results that have appeared in scientific articles

Chapter

Front Cover

Multi-Body Kinematics and Dynamics with Lie Groups

Contents

List of Notations

Introduction

1. The Displacement Group as a Lie Group

1.1. General points

1.2. The groups O(E) and SO(E) as Lie groups

1.3. The group U of normalized quaternions

1.4. Cayley transforms

1.5. The displacement group as a Lie group

1.6. Conclusion

1.7. Appendix 1: The algebra of quaternions

1.8. Appendix 2: Lie subalgebras and ideals of D

2. Dual Numbers and “Dual Vectors” in Kinematics

2.1. The Euclidean module D over the dual number ring

2.2. Dualization of a real vector space

2.3. Dual quaternions

2.4. Differential calculus in Δ-modules

3. The “Transference Principle”

3.1. On the meaning of a general algebraic transference principle

3.2. Isomorphy between the adjoint group D* and SO(Ê)

3.3. Regular maps

3.4. Extensions of the regular maps from U to SO(E)

4. Kinematics of a Rigid Body and Rigid Body Systems

4.1. Introduction

4.2. Kinematics of a rigid body

4.3. The position space of a rigid body

4.4. Relations to the models of bodies

4.5. Changes of frame in kinematics

4.6. Graphs and systems subjected to constraints

4.7. Kinematics of chains

5. Kinematics of Open Chains, Singularities

5.1. The mathematical picture of an open chain

5.2. Singularities of a kinematic chain

5.3. Examples: Singularities of open kinematic chains with parallel axes

5.4. Calculations of the successive derivatives of f

5.5. Transversality and singularities of a product of exponential mappings

6. Closed Kinematic Chains: Mechanisms Theory

6.1. Geometric framework and regular case

6.2. Exhaustive classification of the local singularities of mechanisms

6.3. Singular mechanisms with degree of mobility one

6.4. Concrete examples and calculations

7. Dynamics

7.1. Changes of frame in dynamics, objective magnitudes

7.2. The inertial mass of a rigid body

7.3. The fundamental law of dynamics

8. Dynamics of Rigid Body Systems

8.1. Systems subjected to constraints

8.2. The principles of dynamics for multibody systems

8.3. Tree-structured systems

8.4. Complement: Lagrange’s form of the virtual power of the inertial forces

8.5. Appendix: The subspaces n(s) and m(s) associated with the Lie subalgebras of D

Bibliography

Index

Back Cover

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