Nonsmooth Mechanics (2nd Ed., 2nd ed. 1999. Softcover reprint of the original 2nd ed. 1999) Models, Dynamics and Control Communications and Control Engineering Series
Langue : Anglais
Auteur : Brogliato Bernard
Thank you for opening the second edition of this monograph, which is devoted to the study of a class of nonsmooth dynamical systems of the general form: ::i; = g(x,u) (0. 1) f(x, t) 2: 0 where x E JRn is the system's state vector, u E JRm is the vector of inputs, and the function f (-, . ) represents a unilateral constraint that is imposed on the state. More precisely, we shall restrict ourselves to a subclass of such systems, namely mechanical systems subject to unilateral constraints on the position, whose dynamical equations may be in a first instance written as: ii= g(q,q,u) (0. 2) f(q, t) 2: 0 where q E JRn is the vector of generalized coordinates of the system and u is an in put (or controller) that generally involves a state feedback loop, i. e. u= u(q, q, t, z), with z= Z(z, q, q, t) when the controller is a dynamic state feedback. Mechanical systems composed of rigid bodies interacting fall into this subclass. A general prop erty of systems as in (0. 1) and (0. 2) is that their solutions are nonsmooth (with respect to time): Nonsmoothness arises primarily from the occurence of impacts (or collisions, or percussions) in the dynamical behaviour, when the trajectories attain the surface f(x, t) = O. They are necessary to keep the trajectories within the subspace = {x : f(x, t) 2: O} of the system's state space.
1 Distributional model of impacts.- 1.1 External percussions.- 1.2 Measure differential equations.- 1.2.1 Some properties.- 1.2.2 Additional comments.- 1.3 Systems subject to unilateral constraints.- 1.3.1 General considerations.- 1.3.2 Flows with collisions.- 1.3.3 A system theoretical geometric approach.- 1.3.4 Descriptor variable systems.- 1.4 Changes of coordinates in MDEs.- 1.4.1 From measure to Carathéodory systems.- 1.4.2 Decoupling of the impulsive effects (commutativity conditions).- 1.4.3 From measure to Filippov’s differential equations: the Zhuravlev-Ivanov method.- 2 Approximating problems.- 2.1 Simple examples.- 2.1.1 From elastic to hard impact.- 2.1.2 From damped to plastic impact.- 2.1.3 The general case.- 2.2 The method of penalizing functions.- 2.2.1 The elastic rebound case.- 2.2.2 A more general case.- 2.2.3 Uniqueness of solutions.- 3 Variational principles.- 3.1 Virtual displacements principle.- 3.2 Gauss’ principle.- 3.2.1 Additional comments and studies.- 3.3 Lagrange’s equations.- 3.4 External impulsive forces.- 3.4.1 Example: flexible joint manipulators.- 3.5 Hamilton’s principle and unilateral constraints.- 3.5.1 Introduction.- 3.5.2 Modified set of curves.- 3.5.3 Modified Lagrangian function.- 3.5.4 Additional comments and studies.- 4 Two bodies colliding.- 4.1 Dynamical equations of two rigid bodies colliding.- 4.1.1 General considerations.- 4.1.2 Relationships between real-world and generalized normal di-rections.- 4.1.3 Dynamical equations at collision times.- 4.1.4 The percussion center.- 4.2 Percussion laws.- 4.2.1 Oblique percussions with friction between two bodies.- 4.2.2 Rigid body formulation: Brach’s method.- 4.2.3 Additional comments and studies.- 4.2.4 Rigid body formulation: Frémond’s approach.- 4.2.5 Dynamical equations during the collision process: Darboux-Keller’s shock equations.- 4.2.6 Stronge’s energetical coefficient.- 4.2.7 3 dimensional shocks- Ivanov’s energetical coefficient.- 4.2.8 A third energetical coefficient.- 4.2.9 Additional comments and studies.- 4.2.10 Multiple micro-collisions phenomenon: towards a global coef-ficient.- 4.2.11 Conclusion.- 4.2.12 The Thomson and Tait formula.- 4.2.13 Graphical analysis of shock dynamics.- 4.2.14 Impacts in flexible structures.- 5 Multiconstraint nonsmooth dynamics.- 5.1 Introduction. Delassus’ problem.- 5.2 Multiple impacts: the striking balls examples.- 5.3 Moreau’s sweeping process.- 5.3.1 General formulation.- 5.3.2 Application to mechanical systems.- 5.3.3 Existential results.- 5.3.4 Shocks with friction.- 5.4 Complementarity formulations.- 5.4.1 General introduction to LCPs and Signorini’s conditions.- 5.4.2 Linear Complementarity Problems.- 5.4.3 Relationships with quadratic problems.- 5.4.4 Linear complementarity systems.- 5.4.5 Additional comments and studies.- 5.5 The Painlevé’s example.- 5.5.1 Lecornu’s frictional catastrophes.- 5.5.2 Conclusions.- 5.5.3 Additional comments and bibliography.- 5.6 Numerical analysis.- 5.6.1 General comments.- 5.6.2 Integration of penalized problems.- 5.6.3 Specific numerical algorithms.- 6 Generalized impacts.- 6.1 The frictionless case.- 6.1.1 About “complete” Newton’s rules.- 6.2 The use of the kinetic metric.- 6.2.1 The kinetic energy loss at impact.- 6.3 Simple generalized impacts.- 6.3.1 2-dimensional lamina striking a plane.- 6.3.2 Shock of a particle against a pendulum.- 6.4 Multiple generalized impacts.- 6.4.1 The rocking block problem.- 6.5 General restitution rules for multiple impacts.- 6.5.1 Introduction.- 6.5.2 The rocking block example continued.- 6.5.3 Additional comments and studies.- 6.5.4 3-balls example continued.- 6.5.5 2-balls.- 6.5.6 Additional comments and studies.- 6.5.7 Summary of the main ideas.- 6.5.8 Collisions near singularities: additional comments.- 6.6 Constraints with Amontons-Coulomb friction.- 6.6.1 Lamina with friction.- 6.7 Additional comments and studies.- 7 Stability of nonsmooth dynamical systems.- 7.1 General stability concepts.- 7.1.1 Stability of measure differential equations.- 7.1.2 Stability of mechanical systems with unilateral constraints.- 7.1.3 Passivity of the collision mapping.- 7.1.4 Stability of the discrete dynamic equations.- 7.1.5 Impact oscillators.- 7.1.6 Conclusions.- 7.2 Grazing orC-bifurcations.- 7.2.1 The stroboscopic Poincaré map discontinuities.- 7.2.2 The stroboscopic Poincaré map around grazing-motions...- 7.2.3 Further comments and studies.- 7.3 Stability: from compliant to rigid models.- 7.3.1 System’s dynamics.- 7.3.2 Lyapunov stability analysis.- 7.3.3 Analysis of quadratic stability conditions for large stiffness values.- 7.3.4 A stiffness independent convergence analysis.- 8 Feedback control.- 8.1 Controllability properties.- 8.2 Control of complete robotic tasks.- 8.2.1 Experimental control of the transition phase.- 8.2.2 The general control problem.- 8.3 Dynamic model.- 8.3.1 A general form of the dynamical system.- 8.3.2 The closed-loop formulation of the dynamics.- 8.3.3 Definition of the solutions.- 8.4 Stability analysis framework.- 8.5 A one degree-of-freedom example.- 8.5.1 Static state feedback (weakly stable task).- 8.5.2 Towards a strongly stable closed-loop scheme.- 8.5.3 Dynamic state feedback.- 8.6ndegree-of-freedom rigid manipulators.- 8.6.1 Integrable transformed velocities.- 8.6.2 Examples.- 8.6.3 Non-integrable transformed velocities: general case.- 8.6.4 Non-integrable transformed velocities: a strongly stable scheme.- 8.7 Complementary-slackness juggling systems.- 8.7.1 Some examples.- 8.7.2 Some controllability properties.- 8.7.3 Control design.- 8.7.4 Further comments.- 8.8 Systems with dynamic backlash.- 8.9 Bipedal locomotion.- A Schwartz’ distributions.- A.1 The functional approach.- A.2 The sequential approach.- A.3 Notions of convergence.- B Measures and integrals.- C Functions of bounded variation in time.- C.1 Definition and generalities.- C.2 Spaces of functions of bounded variation.- C.3 Sobolev spaces.- D Elements of convex analysis.
In addition to having served (1991 – 2001) as Chargé de Recherche at CNRS, and as, now, Directeur de Recherche at INRIA, Bernard Brogliato is an Associate Editor for Automatica (since 2001) a reviewer for Mathematical Reviews and writes book reviews for ASME Applied Mechanics Reviews. He has served on the organising and other committees of various European and international conferences sponsored by an assortment of organizations, most prominently, the IEEE. He has been responsible for examining the PhD and Habilitation theses of 16 students and takes an active part in lecturing at summer schools in several European countries. Doctor Brogliato is the director of SICONOS (a European project concerned with Modelling, Simulation and Control of Nonsmooth Dynamical Systems) which carries funding of €2 million.
This monograph contains the latest research results in the area of impact problems for rigid bodies
It contains a complete overview of the main problems of interest in this field
This edition has been expanded to include new references and examples
Date de parution : 10-2012
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