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Dynamics of Multibody Systems Robert E. Roberson Richard Schwertassek Dynamics of Multibody Systems With 100 Figures Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Contents PART I. INTRODUCTION 1 1. Multibody Systems 3 1.1 Origins 3 1.1.1 Rise of Rotational Mechanics 3 1.1.2 Rotation in Technology 5 1.1.3 Status: mid-Twentieth Century 9 1.2 Multibody System Dynamics 11 1.2.1 Multibody Renaissance 11 1.2.2 Examples of Applications 13 1.2.3 Goals of Multibody System Investigation 21 1.2.4 Multibody Formalisms and Computer Codes .... 24 1.3 References 36 2. Mathematical Preliminaries 38 2.1 Terminology and Notation 38 2.1.1 Frames and Coordinates 38 2.1.2 Vectors and Cartesian Tensors 42 2.1.3 Matrices 45 2.1.4 Arrays of Vectors and Dyadics 46 2.2 Vectors, Dyadics and their Matrix Representations 47 2.2.1 Basic Operations 47 2.2.2 Transformations 50 2.2.3 Cartesian Tensors 53 2.3 References 56 PART II. KINEMATICS OF A RIGID BODY 57 3. Location and Orientation 59 3.1 Introduction 59 3.1.1 Degrees of Freedom 59 3.1.2 Rotation 61 XI 3.2 Direction Cosine Matrices 63 3.2.1 Representation of General Rotation 63 3.2.2 Infinitesimal Rotation 67 3.3 Angle Representations of Rotation 68 3.3.1 Elementary Rotations 68 3.3.2 Euler Angles 70 3.3.3 Tait-Bryan Angles 73 3.4 Other Representations of Rotation 75 3.4.1 Euler Parameters 75 3.4.2 Euler-Rodrigues Parameters 77 3.4.3 Six-Parameter Methods 77 3.5 References 78 4. Velocity 79 4.1 Foundations 79 4.1.1 Translational Velocity 79 4.1.2 Angular Velocity 80 4.1.3 Velocity Variables 82 4.2 Angular Velocity and Parameter Derivatives 83 4.2.1 Angle Parametrizations 83 4.2.2 Euler and Euler-Rodrigues Parameters 84 4.3 Relative Derivatives 85 4.3.1 General Form 85 4.3.2 Transformations of Velocity and Acceleration ... 86 5. Kinematical Equations of Motion 89 5.1 Unconstrained Motion 89 5.1.1 Variables 89 5.1.2 Kinematical Differential Equations 90 5.2 Constrained Motion 93 5.2.1 Constraints . 93 5.2.2 Modes of Motion 97 5.2.3 State Space Representation of Kinematics 103 5.2.4 Examples 104 5.3 References 125 PART III. DYNAMICS OF A RIGID BODY 127 6. Physical Preliminaries 129 6.1 Mass Geometry 129 6.1.1 Mass and Mass Moments 129 6.1.2 Transformations of the Inertia Matrix 132 XII 6.2 Momentum 133 6.2.1 Linear Momentum 133 6.2.2 Angular Momentum 134 6.3 Force and Torque .136 6.3.1 Basics 136 6.3.2 Systems of Forces . 139 6.4 References 141 7. Dynamical Equations 142 7.1 Basic Forms 142 7.1.1 Laws of Newton and Euler 142 7.1.2 Center of Mass as a Reference Point 143 7.2 Unconstrained Motion 144 7.2.1 Matrix Representation 144 7.2.2 Accelerated Reference Frames 147 7.2.3 State-Space Equations 150 7.2.4 Generalized Coordinates, Velocities and Forces . 152 7.3 Constrained Motion 154 7.3.1 Separation of Forces and Torques by Modes .... 154 7.3.2 State Space Equations and Constraint Forces and Torques 155 7.3.3 Examples 157 7.3.4 Other Methodologies 166 7.4 References 172 PART IV. MULTIBODY SYSTEMS 173 8. Foundations 175 8.1 Basic Multibody Notation 175 8.1.1 Primitive Equations of Motion 175 8.1.2 Gross Motion and Reference Frames 177 8.1.3 Quantities Related to the Interaction of Bodies . 181 8.2 Graph Characterization of System Topology 184 8.2.1 Graph Fundamentals 184 8.2.2 Matrices Associated with a Graph 188 8.2.3 Multibody System Graph 199 8.3 References 201 9. Formalisms 202 9.1 Survey on Methodologies 202 9.1.1 Eulerian Approaches 202 9.1.2 Lagrangian Approaches 208 9.2 Summary Description of Present Formalism 210 XIII 9.2.1 Reformulation of Primitive Dynamical Equations . 210 9.2.2 Selection of Dependent Variables . 212 9.2.3 Data to Describe the System 214 9.2.4 Derivation of State Space Equations 221 9.3 References 227 10. Kinematics 229 10.1 Position 229 10.1.1 Formulation of Problems 229 10.1.2 Path Vectors : . 230 10.1.3 Barycentric Vectors 233 10.1.4 Unification of Results 239 10.1.5 Orientation Matrices 240 10.2 Velocity and Acceleration 241 10.2.1 Velocity 241 10.2.2 Velocity Derivatives 244 10.3 Constraint Equations 245 10.3.1 Classification of Constraints 245 10.3.2 Joint Constraints and Branch Orientation .... 248 10.3.3 Consistency Conditions in Kinematic Circuits . 248 10.3.4 Circuit Connection Vectors 252 10.4 Kinematical Differential Equations 255 10.4.1 Unreduced System Equations 255 10.4.2 Incorporation of Constraints 256 10.5 References 258 11. Dynamics 259 11.1 Unreduced System Equations 259 11.1.1 Translation 259 11.1.2 Rotation 265 11.1.3 Generalized Inertia Dyadics 271 11.1.4 External and Inertial Forces and Torques 274 11.1.5 Matrix Equations 275 11.1.6 Properties of Generalized Inertia Matrix 282 11.1.7 Problem Types 284 11.2 Systems with Tree Structure 287 11.2.1 Separation by Modes 287 11.2.2 State Space Equations 289 11.2.3 Constraint Forces and Torques 291 11.3 Systems with Closed Circuits 292 11.3.1 Constraints across Primary Joints 292 11.3.2 State-Space Equations 293 11.3.3 Choice of Spanning Tree 298 11.4 References J 300 XIV 12. Special Topics 301 12.1 Forces and Torques 301 12.1.1 Dry Friction 301 12.1.2 Interaction and Feedback Control 305 12.1.3 Gravitational Effects on Multibody Systems . 307 12.2 Application to Robotics 311 12.2.1 Multibody Robot Models 311 12.2.2 Recursive Computation of Inverse Dynamics . 317 12.3 References :- 328 13. Linearized Equations 329 13.1 Introduction 329 13.1.1 Collection of Basic Equations 329 13.1.2 Small Variables Representing Relative Motion . 330 13.1.3 Absolute and Relative Motion .332 13.2 Kinematics 335 13.2.1 Unreduced Equations of Motion 335 13.2.2 Consistency Conditions 336 13.2.3 Constraint Equations . 337 13.2.4 State-Space Equations 341 13.3 Dynamics 342 13.3.1 Introduction 342 13.3.2 Generalized Inertia Matrix 343 13.3.3 Interactions 344 13.3.4 External and Inertia! Action 345 13.3.5 Matrices of Consistency Conditions 349 13.3.6 Unreduced System Equations 351 13.3.7 State-Space Representation 354 13.3.8 System Parameters 363 13.4 References 364 14. Computer Simulation 365 14.1 Introduction 365 14.1.1 Requirements on Multibody Software 365 14.1.2 General Orginization of Programs 369 14.2 User-Oriented Input of Data . 373 14.2.1 Generalities 373 14.2.2 System Data and Libraries 378 14.2.3 Feasibility of System Configuration 380 14.3 Generation of System Equations 381 14.3.1 Summary of Equations 381 14.3.2 Recursive Computational Schemes 387 14.3.3 Graph Algorithms 394 XV 14.4 Numerical Integration 400 14.4.1 Formulation of Problems 400 14.4.2 Some Solution Techniques 402 14.4.3 Solution of Partially Reduced Equations 408 14.5 References 410 APPENDIX A. Notational Summary 412 APPENDIX B. List of Symbols 416 APPENDIX C. Bibliography 442 INDEX 455.
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