Nonsmooth Mechanics: Models, Dynamics and Control

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 Caratheodory 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: Fremond'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 Painleve'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 Poincare map discontinuities. - 7.2.2 The stroboscopic Poincare 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.