Dynamics of Multibody Systems

I. Introduction.- 1. Multibody Systems.- 1.1 Origins.- 1.1.1 Rise of Rotational Mechanics.- 1.1.2 Rotation in Technology.- 1.1.3 Status: mid-Twentieth Century.- 1.2 Multibody System Dynamics.- 1.2.1 Multibody Renaissance.- 1.2.2 Examples of Applications.- 1.2.3 Goals of Multibody System Investigation.- 1.2.4 Multibody Formalisms and Computer Codes.- 1.3 References.- 2. Mathematical Preliminaries.- 2.1 Terminology and Notation.- 2.1.1 Frames and Coordinates.- 2.1.2 Vectors and Cartesian Tensors.- 2.1.3 Matrices.- 2.1.4 Arrays of Vectors and Dyadics.- 2.2 Vectors, Dyadics and their Matrix Representations.- 2.2.1 Basic Operations.- 2.2.2 Transformations.- 2.2.3 Cartesian Tensors.- 2.3 References.- II. Kinematics of a Rigid Body.- 3. Location and Orientation.- 3.1 Introduction.- 3.1.1 Degrees of Freedom.- 3.1.2 Rotation.- 3.2 Direction Cosine Matrices.- 3.2.1 Representation of General Rotation.- 3.2.2 Infinitesimal Rotation.- 3.3 Angle Representations of Rotation.- 3.3.1 Elementary Rotations.- 3.3.2 Euler Angles.- 3.3.3 Tait-Bryan Angles.- 3.4 Other Representations of Rotation.- 3.4.1 Euler Parameters.- 3.4.2 Euler-Rodrigues Parameters.- 3.4.3 Six-Parameter Methods.- 3.5 References.- 4. Velocity.- 4.1 Foundations.- 4.1.1 Translational Velocity.- 4.1.2 Angular Velocity.- 4.1.3 Velocity Variables.- 4.2 Angular Velocity and Parameter Derivatives.- 4.2.1 Angle Parametrizations.- 4.2.2 Euler and Euler-Rodrigues Parameters.- 4.3 Relative Derivatives.- 4.3.1 General Form.- 4.3.2 Transformations of Velocity and Acceleration.- 5. Kinematical Equations of Motion.- 5.1 Unconstrained Motion.- 5.1.1 Variables.- 5.1.2 Kinematical Differential Equations.- 5.2 Constrained Motion.- 5.2.1 Constraints.- 5.2.2 Modes of Motion.- 5.2.3 State Space Representation of Kinematics.- 5.2.4 Examples.- 5.3 References.- III. Dynamics of a Rigid Body.- 6. Physical Preliminaries.- 6.1 Mass Geometry.- 6.1.1 Mass and Mass Moments.- 6.1.2 Transformations of the Inertia Matrix.- 6.2 Momentum.- 6.2.1 Linear Momentum.- 6.2.2 Angular Momentum.- 6.3 Force and Torque.- 6.3.1 Basics.- 6.3.2 Systems of Forces.- 6.4 References.- 7. Dynamical Equations.- 7.1 Basic Forms.- 7.1.1 Laws of Newton and Euler.- 7.1.2 Center of Mass as a Reference Point.- 7.2 Unconstrained Motion.- 7.2.1 Matrix Representation.- 7.2.2 Accelerated Reference Frames.- 7.2.3 State-Space Equations.- 7.2.4 Generalized Coordinates, Velocities and Forces.- 7.3 Constrained Motion.- 7.3.1 Separation of Forces and Torques by Modes.- 7.3.2 State Space Equations and Constraint Forces and Torques.- 7.3.3 Examples.- 7.3.4 Other Methodologies.- 7.4 References.- IV. Multibody Systems.- 8. Foundations.- 8.1 Basic Multibody Notation.- 8.1.1 Primitive Equations of Motion.- 8.1.2 Gross Motion and Reference Frames.- 8.1.3 Quantities Related to the Interaction of Bodies.- 8.2 Graph Characterization of System Topology.- 8.2.1 Graph Fundamentals.- 8.2.2 Matrices Associated with a Graph.- 8.2.3 Multibody System Graph.- 8.3 References.- 9. Formalisms.- 9.1 Survey on Methodologies.- 9.1.1 Eulerian Approaches.- 9.1.2 Lagrangian Approaches.- 9.2 Summary Description of Present Formalism.- 9.2.1 Reformulation of Primitive Dynamical Equations.- 9.2.2 Selection of Dependent Variables.- 9.2.3 Data to Describe the System.- 9.2.4 Derivation of State Space Equations.- 9.3 References.- 10. Kinematics.- 10.1 Position.- 10.1.1 Formulation of Problems.- 10.1.2 Path Vectors.- 10.1.3 Barycentric Vectors.- 10.1.4 Unification of Results.- 10.1.5 Orientation Matrices.- 10.2 Velocity and Acceleration.- 10.2.1 Velocity.- 10.2.2 Velocity Derivatives.- 10.3 Constraint Equations.- 10.3.1 Classification of Constraints.- 10.3.2 Joint Constraints and Branch Orientation.- 10.3.3 Consistency Conditions in Kinematic Circuits.- 10.3.4 Circuit Connection Vectors.- 10.4 Kinematical Differential Equations.- 10.4.1 Unreduced System Equations.- 10.4.2 Incorporation of Constraints.- 10.5 References.- 11. Dynamics.- 11.1 Unreduced System Equations.- 11.1.1 Translation.- 11.1.2 Rotation.- 11.1.3 Generalized Inertia Dyadics.- 11.1.4 External and Inertial Forces and Torques.- 11.1.5 Matrix Equations.- 11.1.6 Properties of Generalized Inertia Matrix.- 11.1.7 Problem Types.- 11.2 Systems with Tree Structure.- 11.2.1 Separation by Modes.- 11.2.2 State Space Equations.- 11.2.3 Constraint Forces and Torques.- 11.3 Systems with Closed Circuits.- 11.3.1 Constraints across Primary Joints.- 11.3.2 State-Space Equations.- 11.3.3 Choice of Spanning Tree.- 11.4 References.- 12. Special Topics.- 12.1 Forces and Torques.- 12.1.1 Dry Friction.- 12.1.2 Interaction and Feedback Control.- 12.1.3 Gravitational Effects on Multibody Systems.- 12.2 Application to Robotics.- 12.2.1 Multibody Robot Models.- 12.2.2 Recursive Computation of Inverse Dynamics.- 12.3 References.- 13. Linearized Equations.- 13.1 Introduction.- 13.1.1 Collection of Basic Equations.- 13.1.2 Small Variables Representing Relative Motion.- 13.1.3 Absolute and Relative Motion.- 13.2 Kinematics.- 13.2.1 Unreduced Equations of Motion.- 13.2.2 Consistency Conditions.- 13.2.3 Constraint Equations.- 13.2.4 State-Space Equations.- 13.3 Dynamics.- 13.3.1 Introduction.- 13.3.2 Generalized Inertia Matrix.- 13.3.3 Interactions.- 13.3.4 External and Inertial Action.- 13.3.5 Matrices of Consistency Conditions.- 13.3.6 Unreduced System Equations.- 13.3.7 State-Space Representation.- 13.3.8 System Parameters.- 13.4 References.- 14. Computer Simulation.- 14.1 Introduction.- 14.1.1 Requirements on Multibody Software.- 14.1.2 General Orginization of Programs.- 14.2 User-Oriented Input of Data.- 14.2.1 Generalities.- 14.2.2 System Data and Libraries.- 14.2.3 Feasibility of System Configuration.- 14.3 Generation of System Equations.- 14.3.1 Summary of Equations.- 14.3.2 Recursive Computational Schemes.- 14.3.3 Graph Algorithms.- 14.4 Numerical Integration.- 14.4.1 Formulation of Problems.- 14.4.2 Some Solution Techniques.- 14.4.3 Solution of Partially Reduced Equations.- 14.5 References.- Appendix A. Notational Summary.- Appendix B. List Of Symbols.- Appendix C. Bibliography.