Brogliato Nonsmooth Mechanics

Models, Dynamics and Control
3rd Auflage 2016
ISBN: 978-3-319-28664-8
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 657 Seiten

Reihe: Communications and Control Engineering

Now in its third edition, this standard reference is a comprehensive treatment of nonsmooth mechanical systems refocused to give more prominence to issues connected with control and modelling. It covers Lagrangian and Newton-Euler systems, detailing mathematical tools such as convex analysis and complementarity theory. The ways in which nonsmooth mechanics influence and are influenced by well-posedness analysis, numerical analysis and simulation, modelling and control are explained. Contact/impact laws, stability theory and trajectory-tracking control are given detailed exposition connected by a mathematical framework formed from complementarity systems and measure-differential inclusions. Links are established with electrical circuits with set-valued nonsmooth elements as well as with other nonsmooth dynamical systems like impulsive and piecewise linear systems.
Nonsmooth Mechanics (third edition) retains the topical structure familiar from its predecessors but has been substantially rewritten, edited and updated to account for the significant body of results that have emerged in the twenty-first century-including developments in:
the existence and uniqueness of solutions;
impact models;
extension of the Lagrange-Dirichlet theorem and trajectory tracking; and
well-posedness of contact complementarity problems with and without friction.
Many figures (both new and redrawn to improve the clarity of the presentation) and examples are used to illustrate the theoretical developments. Material introducing the mathematics of nonsmooth mechanics has been improved to reflect the broad range of applications interest that has developed since publication of the second edition. The detail of some mathematical essentials is provided in four appendices.
With its improved bibliography of over 1,300 references and wide-ranging coverage, Nonsmooth Mechanics (third edition) is sure to be an invaluable resource for researchers and postgraduates studying the control of mechanical systems, robotics, granular matter and relevant fields of applied mathematics.'The book's two best features, in my view are its detailed survey of the literature... and its detailed presentation of many examples illustrating both the techniques and their limitations... For readers interested in the field, this book will serve as an excellent introductory survey.'Andrew Lewis in Automatica'It is written with clarity, contains the latest research results in the area of impact problems for rigid bodies and is recommended for both applied mathematicians and engineers.'
Panagiotis D. Panagiotopoulos in Mathematical Reviews'The presentation is excellent in combining rigorous mathematics with a great number of examples... allowing the reader to understand the basic concepts.'
Hans Troger in Mathematical Abstracts<

Bernard Brogliato is Senior Researcher at INRIA Grenoble, France, where he founded and leads the team BIPOP. He published more than 70 journal articles in the fields of systems and Control, Solid Mechanics, and Applied Mathematics, as well as 5 monographs. His research interests are in non-smooth dynamical systems (mechanical systems with constraints, impacts, friction, electrical circuits with non-smooth components, sliding-mode control, optimal control with state constraints), and dissipative systems. He was an Associate Editor for Automatica, and the chairman of two Euromech Colloquia dedicated to Impact Mechanics. He coordinated the FP5 European project SICONOS (2 million euros fundings, 13 partners), and two projects funded by the French National Agency for Scientific Research, on multiple impacts and discrete-time sliding mode control.

Autoren/Hrsg.

Brogliato, Bernard

Inhaltsverzeichnis

1;Preface;8
1.1;Acknowledgments;1
2;Contents;14
3;Notation;21
4;1 Impulsive Dynamics and Measure Differential Equations;23
4.1;1.1 Impulsive Forces;23
4.2;1.2 Measure Differential Equations (MDEs);29
4.2.1;1.2.1 A First Class of MDEs;30
4.2.2;1.2.2 A Second Class of MDEs: ODEs Driven by Measure Inputs;33
4.2.3;1.2.3 Further Reading;37
4.2.4;1.2.4 A Third Class of MDEs: ODEs with State Jump Mappings;38
4.2.5;1.2.5 Further Reading;40
4.3;1.3 Systems Subject to Unilateral Constraints;41
4.3.1;1.3.1 General Considerations;41
4.3.2;1.3.2 Flows with Collisions (Vibro-Impact Systems);48
4.3.3;1.3.3 Unilaterally Constrained Systems: A Geometric Approach;56
4.3.4;1.3.4 Bilaterally Constrained Mechanical Systems and Impulsive Dynamics;60
4.4;1.4 Changes of Coordinates in MDEs;61
4.4.1;1.4.1 From Measure to Carathéodory Systems;61
4.4.2;1.4.2 Decoupling of the Impulsive Effects (Commutativity Conditions);64
4.4.3;1.4.3 From Unilaterally Constrained Mechanical Systems to Filippov's Differential Inclusions: the Zhuravlev--Ivanov Method;66
5;2 Viscoelastic Contact/Impact Rheological Models;72
5.1;2.1 Simple Examples;73
5.1.1;2.1.1 From Elastic to Hard Impact;73
5.1.2;2.1.2 From Damped to Plastic Impact;76
5.1.3;2.1.3 The General Case;77
5.2;2.2 Viscoelastic Contact Models and Restitution Coefficients;87
5.2.1;2.2.1 Linear Spring-Dashpot;87
5.2.2;2.2.2 Nonlinear Elasticity and Viscous Friction: Simon-Hunt-Crossley and Kuwabara-Kono Dissipations;89
5.2.3;2.2.3 Conclusions;98
5.3;2.3 Viscoelastic Models with Dry Friction Elements: Viscoelasto-Plastic Models;99
5.3.1;2.3.1 Conclusions and Further Reading;103
5.4;2.4 Penalizing Functions in Mathematical Analysis;104
5.4.1;2.4.1 The Elastic Rebound Case;104
5.4.2;2.4.2 The Case with Dissipation (Linear Viscous Friction);105
5.4.3;2.4.3 Uniqueness of Solutions;110
5.4.4;2.4.4 Further Existence and Uniqueness Results;113
5.5;2.5 Some Comments on Compliant Models;114
6;3 Variational Principles;115
6.1;3.1 Virtual Displacements, Velocities, and Accelerations Principles;115
6.1.1;3.1.1 The ``Classical'' Presentation;115
6.1.2;3.1.2 Using Variational and Quasi-Variational Inequalities Formalisms;118
6.2;3.2 A Coordinate Invariance Principle;122
6.2.1;3.2.1 Perfect Constraints;123
6.3;3.3 Gauss' Principle;124
6.3.1;3.3.1 Further Reading;125
6.4;3.4 Lagrange Dynamics;127
6.4.1;3.4.1 External Impulsive Forces;127
6.4.2;3.4.2 Example: Flexible Joint Manipulators;128
6.5;3.5 Hamilton's Principle and Unilateral Constraints;130
6.5.1;3.5.1 Hamilton's Principle Without Impacts;130
6.5.2;3.5.2 Hamilton's Principle With Impacts;131
6.5.3;3.5.3 Modified Set of Curves;135
6.5.4;3.5.4 Modified Lagrangian Function;139
6.5.5;3.5.5 Additional Comments and Studies;142
7;4 Two Rigid Bodies Colliding;146
7.1;4.1 Dynamical Equations of Two Rigid Bodies Colliding;146
7.1.1;4.1.1 General Considerations;146
7.1.2;4.1.2 The Local Kinematics;148
7.1.3;4.1.3 The Gap Function;151
7.1.4;4.1.4 The Two-Body System Dynamics;154
7.1.5;4.1.5 Dynamical Equations and Energy Loss at Collision Times;156
7.1.6;4.1.6 The Percussion Center;161
7.2;4.2 Restitution Laws;162
7.2.1;4.2.1 Elastoplastic Impacts and Restitution Coefficients;166
7.2.2;4.2.2 Adhesive Effects;177
7.2.3;4.2.3 Beyond Hertz: Conformal Contact Models;181
7.2.4;4.2.4 Conditions for Quasistatic Impacts;183
7.2.5;4.2.5 Incorporating Friction Effects;187
7.2.6;4.2.6 Conclusions;190
7.2.7;4.2.7 Material Parameters: Some Values;191
7.3;4.3 Impacts with Friction;192
7.3.1;4.3.1 Simple Examples;192
7.3.2;4.3.2 Kinematic CoR: Brach's Method;207
7.3.3;4.3.3 Additional Comments and Studies;212
7.3.4;4.3.4 Kinematic CoR: Frémond's approach;217
7.3.5;4.3.5 First Order Impact Dynamics: Darboux-Keller's Shock Equations;219
7.3.6;4.3.6 The Energetic Coefficient of Restitution;236
7.3.7;4.3.7 Examples;241
7.3.8;4.3.8 Other Energetical Coefficients;244
7.3.9;4.3.9 Additional Comments and Studies;244
7.3.10;4.3.10 Multiple Microcollisions Phenomenon: Toward a Global Coefficient;246
7.3.11;4.3.11 Conclusion;250
7.3.12;4.3.12 The Thomson-and-Tait Formula;251
7.3.13;4.3.13 Graphical Analysis of the Shock Dynamics;252
7.4;4.4 Impacts in Flexible Structures;255
7.4.1;4.4.1 Multimodal Modeling Approach;255
7.4.2;4.4.2 Infinite Dimensional System Approach;257
7.4.3;4.4.3 Further Reading;258
7.5;4.5 General Comments;258
8;5 Nonsmooth Lagrangian Systems;260
8.1;5.1 Lagrange Dynamics with Multiple Constraints;260
8.1.1;5.1.1 Frictionless Bilateral Constraints: The Contact Problem;263
8.1.2;5.1.2 Frictionless Unilateral Constraints: The Contact Problem;266
8.1.3;5.1.3 Mixed Bilateral/Unilateral Frictionless Constraints: The Contact Problem;270
8.1.4;5.1.4 Singular Mass Matrix: From Singular Lagrange's to Singular Hamilton's Dynamics;273
8.2;5.2 Moreau's Sweeping Process;275
8.2.1;5.2.1 First-Order Sweeping Process;275
8.2.2;5.2.2 Second-Order Sweeping Process: Frictionless Mechanical Systems;277
8.2.3;5.2.3 Well-Posedness Results;296
8.2.4;5.2.4 Continuous Dependence on Initial Data;303
8.3;5.3 Coulomb's Friction;304
8.3.1;5.3.1 Coulomb's Friction Model;305
8.3.2;5.3.2 Coulomb--Moreau's Disk;307
8.3.3;5.3.3 De Saxcé's Associated Formulation;309
8.3.4;5.3.4 Coulomb's Friction at the Acceleration Level;311
8.3.5;5.3.5 Further Comments on Friction Models;312
8.3.6;5.3.6 Sweeping Process with Friction;313
8.3.7;5.3.7 Additional Comments and Studies;316
8.4;5.4 Complementarity Formulations;316
8.4.1;5.4.1 Two Bodies: Signorini's Conditions;317
8.4.2;5.4.2 Linear Complementarity Problem (LCP);318
8.4.3;5.4.3 Relationships with Quadratic Problems;321
8.4.4;5.4.4 Linear Complementarity Systems (LCS);323
8.4.5;5.4.5 Controllability of LCS;343
8.4.6;5.4.6 Observability and Observers for LCS;346
8.4.7;5.4.7 Complementarity Systems and Hybrid Dynamical Systems;346
8.5;5.5 The Contact Problem with Coulomb's Friction;348
8.5.1;5.5.1 Introduction;348
8.5.2;5.5.2 Dissipativity of the Constrained Lagrange Dynamics;349
8.5.3;5.5.3 Extension of the Results of Sects.5.1.1, 5.1.2, 5.1.3?;350
8.5.4;5.5.4 The Contact Problem for a Planar Particle;351
8.5.5;5.5.5 A Second Simple Mechanism with Friction;354
8.5.6;5.5.6 Non-Uniqueness of the Contact Force;358
8.5.7;5.5.7 Comments;360
8.6;5.6 Painlevé's Paradoxes: Sliding Rod Example;361
8.6.1;5.6.1 The Dynamics of Painlevé's Example;361
8.6.2;5.6.2 The Contact LCP;363
8.6.3;5.6.3 Analysis of the Dynamical Singularities;366
8.6.4;5.6.4 Further Reading;371
8.6.5;5.6.5 Conclusions;374
8.7;5.7 Numerical Simulation;375
8.7.1;5.7.1 Event-Driven Algorithms;375
8.7.2;5.7.2 Compliant Contact/Impact Models;376
8.7.3;5.7.3 Time-Stepping (Event-Capturing) Numerical Algorithms;377
9;6 Generalized Impact Laws and Multiple Impacts;390
9.1;6.1 Particular Features of Multiple Impacts;390
9.1.1;6.1.1 Some Specific Features of Multiple Impacts;391
9.1.2;6.1.2 Han-Gilmore's and Binary Collisions Models;398
9.1.3;6.1.3 Penalization at Contacts (Compliance);402
9.1.4;6.1.4 Multiplicity of Multiple Impacts;404
9.2;6.2 Kinematic Multiple-Impact Law (Generalized Newton);405
9.2.1;6.2.1 The Quasi-Lagrange Equations;405
9.2.2;6.2.2 The Kinetic Energy;409
9.2.3;6.2.3 The Contact Forces Power;411
9.2.4;6.2.4 Restitution Law for Frictionless Systems;413
9.2.5;6.2.5 Restitution Law with Tangential Effects;417
9.2.6;6.2.6 Tangential Restitution;421
9.2.7;6.2.7 Comments;421
9.3;6.3 Energetic-CoR Multiple-Impact Law;422
9.3.1;6.3.1 Presentation of the LZB Impact Dynamics;423
9.3.2;6.3.2 Applications and Validations;426
9.3.3;6.3.3 Comparison of Different Multiple Impact Mappings;431
9.4;6.4 Further Reading;432
9.4.1;6.4.1 Kinetic Restitution (Poisson);432
9.4.2;6.4.2 Kinematic Restitution (Newton and Moreau);433
9.4.3;6.4.3 Other Approaches;433
10;7 Stability of Nonsmooth Dynamical Systems;435
10.1;7.1 Stability of Measure Differential Equations;435
10.1.1;7.1.1 Stability of Impulsive ODEs;435
10.1.2;7.1.2 Stability of Measure Driven ODEs (MDEs);437
10.1.3;7.1.3 Additional Comments and Studies;438
10.2;7.2 Stability of the Discrete Dynamic Equations;439
10.2.1;7.2.1 The Bouncing-Ball with Fixed Obstacle;440
10.2.2;7.2.2 Lyapunov Stability of Discrete-Time Systems;443
10.3;7.3 Impact Oscillators;444
10.3.1;7.3.1 Existence of Periodic Trajectories;444
10.3.2;7.3.2 Further Reading;448
10.3.3;7.3.3 Comments on the Poincaré Impact Map Stability Analysis;450
10.3.4;7.3.4 Other Studies on Stability;454
10.3.5;7.3.5 Bouncing-Ball with Moving Base;455
10.3.6;7.3.6 Additional Comments and Studies;456
10.4;7.4 Grazing or C-Bifurcations;458
10.4.1;7.4.1 The Stroboscopic Poincaré Map Discontinuities;460
10.4.2;7.4.2 The Stroboscopic Poincaré Map Around Grazing-Motions;463
10.4.3;7.4.3 Further Comments and Studies;465
10.5;7.5 Complementarity Lagrangian Systems: Stability of Fixed Points;466
10.5.1;7.5.1 The Dynamical System;467
10.5.2;7.5.2 The Stability Analysis;469
10.5.3;7.5.3 Dissipativity Properties;472
10.5.4;7.5.4 Further Reading and Comments;475
10.5.5;7.5.5 Global Finite-Time Stability via the Zhuravlev-Ivanov Transformation;478
10.6;7.6 Stabilization of Impacting Systems: From Compliant to Rigid Models;480
10.6.1;7.6.1 System's Dynamics;480
10.6.2;7.6.2 Lyapunov Stability Analysis;482
10.6.3;7.6.3 Analysis of Quadratic Stability Conditions for Large Stiffness Values;483
10.6.4;7.6.4 A Stiffness-Independent Convergence Analysis;487
10.7;7.7 Stability of Linear Complementarity Systems;491
10.8;7.8 Further Reading;493
11;8 Trajectory Tracking Feedback Control;495
11.1;8.1 Trajectory Tracking: Rigid-Joint Rigid-Body Systems;495
11.1.1;8.1.1 Basic Concepts;497
11.1.2;8.1.2 Controller Design;503
11.1.3;8.1.3 Tracking Control Framework;505
11.1.4;8.1.4 Design of the Desired Contact Force During Constraint Phases;508
11.1.5;8.1.5 Strategy for Takeoff at the End of Constraint Phases ?kJ;510
11.1.6;8.1.6 Closed-Loop Stability Analysis;512
11.1.7;8.1.7 Illustrative Examples;513
11.1.8;8.1.8 Proof of Lemma 8.1;517
11.1.9;8.1.9 Proof of Theorem 8.1;521
11.2;8.2 Short Bibliography;524
11.3;8.3 Trajectory Tracking: Flexible-Joint Rigid-Link Systems;526
11.3.1;8.3.1 Basic Concepts;527
11.3.2;8.3.2 Tracking Control Framework;530
11.3.3;8.3.3 Desired Contact Force During Constraint Phases;533
11.3.4;8.3.4 Strategy for Takeoff at the End of Constraint Phases ?2k+1Bk;535
11.3.5;8.3.5 Closed-Loop Stability Analysis;536
11.3.6;8.3.6 Illustrative Example;537
11.3.7;8.3.7 Proof of Proposition 8.7;541
11.3.8;8.3.8 Proof of Lemma 8.2;542
11.3.9;8.3.9 Proof of Lemma 8.3;542
11.3.10;8.3.10 Proof of Theorem 8.2;544
11.4;8.4 A Unified Point of View;547
11.5;8.5 Further Results;547
11.5.1;8.5.1 Experimental Control of the Transition Phase;547
11.5.2;8.5.2 Juggling Robots Analysis and Control;549
11.5.3;8.5.3 Mechanisms with Joint Clearance;550
11.5.4;8.5.4 Observability and State Observers;551
12;Erratum to: Nonsmooth Mechanics;553
12.1;References;562
13;Appendix A Distributions, Measures, Functionsof Bounded Variations;564
14;Appendix B Elements of Convex Analysis;575
15;References;591
16;Index;647