Nonsmooth Mechanics

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.

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

Autorentext

Bernard Brogliato was born in 1963, graduated from Ecole Normale Supérieure de Cachan (France), Mechanical Engineering Dept., Ph.D. and Habilitation degree in Automatic Control from Grenoble Institute of Technology in 1991 and 1995 respectively. He is Senior Researcher at INRIA Grenoble Rhône-Alpes. Research interests: non-smooth dynamical systems (analysis, control and observation, numerics), impact and contact mechanics, digital sliding-mode control. Wrote about 90 articles in international journals in the fields of Systems and Control, Mechanical Engineering, and Applied Mathematics. Also authored and co-authored five monographs. He was Associate Editor for Automatica (1999-2008), is Associate Editor for Nonlinear Analysis Hybrid Systems, and ASME Journal of Computational and Nonlinear Dynamics. Rogelio Lozano was born in Monterrey Mexico, on July 12, 1954. He received the B.S. degree in electronic engineering fromthe National Polytechnic Institute of Mexico in 1975, the M.S. degree in electrical engineering from Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico in 1977, and the Ph.D. degree in automatic control from Laboratoire d'Automatique de Grenoble, France, in 1981. He joined the Department of Electrical Engineering at CINVESTAV, Mexico, in 1981 where he worked until 1989. He was Head of the Section of Automatic Control from June 1985 to August 1987. He has held visiting positions at the University of Newcastle, Australia, from November 1983 to November 1984, NASA Langley Research Center VA, from August 1987 to August 1988, and Laboratoire d'Automatique de Grenoble, France, from February 1989 to July 1990. Since 1990 he is a CNRS (Centre National de la Recherche Scientifique) Research Director at University of Technology of Compiègne, France. He was Associate Editor of Automatica in the period 1987-2000. He is associate Editor of the Journal of Intelligentand Robotics Systems since 2012 and Associate Editor in the Int. J. of Adaptive Control and Signal Processing since 1988. Bernhard Maschke is Professor of Automatic Control of the University Claude Bernard of Lyon, Villeurbanne, France since 2000. The main streamline of his research is the nonlinear and passivity-based control of complex physical systems. He is one of the main initiators of the Port Hamiltonian formalism which bases the modelling, simulation and control of complex physical systems on network theory and thermodynamic theory. He has used this formalism for complex spatial mechanisms and in the mechatronic context of automotive applications and more recently to chemical enginneering processes and various multi-physics and multi-scale systems such as an adsorption process, a fuel cell or an Ion Polymer Metal Composite. Olav Egeland is a graduate of the Norwegian University of Science and Technology (NTNU), where he is professor of production automation. He was at Marine Cybernetics AS 2004-2011 as co-founder. He has received the Automatica Prize Paper Award and the Outstanding Paper Award of IEEE Trans. Control Systems Technology, and has been Associate Editor of IEEE Trans. Automatic Control and European Journal of Control. His research is on modelling, simulation and control of mechanical systems with applications to robotics and offshore systems.

Klappentext

This significantly enlarged and expanded second edition, focuses on nonsmooth finite dimensional mechanical systems, subject to unilateral constraints. It contains a complete overview of the main problems in the areas of mathematics, mechanics, stability and control, which are of interest in this field. The following topics are discussed in detail and are illustrated with examples: Short dynamics; Multiple impacts; Feedback control; Moreau's sweeping process; Complimentary formulations. With a comprehensive bibliography of over a thousand references, this book is an indispensable source of information on the subject treated.

Zusammenfassung

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 (-, . 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.

Inhalt
1 Distributional model of impacts.- 1.1 External percussions.- 1.2 Measure differential equations.- 1.3 Systems subject to unilateral constraints.- 1.4 Changes of coordinates in MDEs.- 2 Approximating problems.- 2.1 Simple examples.- 2.2 The method of penalizing functions.- 3 Variational principles.- 3.1 Virtual displacements principle.- 3.2 Gauss' principle.- 3.3 Lagrange's equations.- 3.4 External impulsive forces.- 3.5 Hamilton's principle and unilateral constraints.- 4 Two bodies colliding.- 4.1 Dynamical equations of two rigid bodies colliding.- 4.2 Percussion laws.- 5 Multiconstraint nonsmooth dynamics.- 5.1 Introduction. Delassus' problem.- 5.2 Multiple impacts: the striking balls examples.- 5.3 Moreau's sweeping process.- 5.4 Complementarity formulations.- 5.5 The Painlevé's example.- 5.6 Numerical analysis.- 6 Generalized impacts.- 6.1 The frictionless case.- 6.2 The use of the kinetic metric.- 6.3 Simple generalized impacts.- 6.4 Multiple generalized impacts.- 6.5 General restitution rules for multiple impacts.- 6.6 Constraints with Amontons-Coulomb friction.- 6.7 Additional comments and studies.- 7 Stability of nonsmooth dynamical systems.- 7.1 General stability concepts.- 7.2 Grazing orC-bifurcations.- 7.3 Stability: from compliant to rigid models.- 8 Feedback control.- 8.1 Controllability properties.- 8.2 Control of complete robotic tasks.- 8.3 Dynamic model.- 8.4 Stability analysis framework.- 8.5 A one degree-of-freedom example.- 8.6ndegree-of-freedom rigid manipulators.- 8.7 Complementary-slackness juggling systems.- 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 ofbounded 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.