Bernard Brogliato
Nonsmooth Mechanics
Models, Dynamics and Control
Paperback Engels 2012 2e druk 9781447111610
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Samenvatting
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.
Specificaties
ISBN13:9781447111610
Taal:Engels
Bindwijze:paperback
Aantal pagina's:552
Uitgever:Springer London
Druk:2
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Inhoudsopgave
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.
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