On the control of complementary-slackness juggling mechanical systems

International audienceThis paper studies the feedback control of a class of complementary-slackness hybrid mechanical systems. Roughly, the systems we study are composed of an uncontrollable part (the "object") and a controlled one (the "robot"), linked by a unilateral constraint and an impact rule. A systematic and general control design method for this class of systems is proposed. The approach is a nontrivial extension of the one degree-of-freedom (DOF) juggler control design. In addition to the robot control, it is also useful to study some intermediate controllability properties of the object's impact Poincaré mapping, which generally takes the form of a non-linear discrete-time system. The force input mainly consists of a family of dead-beat feedback control laws, introduced via a recur-sive procedure, and exploiting the underlying discrete-time structure of the system. The main goal of this paper is to highlight the role of various physical and control properties characteristic of the system on its stabilizability properties and to propose solutions in certain cases

Feedback Control of Impact Dynamics: the Bouncing Ball Revisited

peer reviewedWe study the the design of a tracking controller for the popular bouncing ball model: the continuous-time actuation of a table is used to control the impacts of the table with a bouncing ball. The proposed control law uses the impact times as the sole feedback information. We show that the acceleration of the table at impact plays no role in the stability analysis but is an important parameter for the robustness of the feedback system to model uncertainty, in particular to the uncertainty on the coefficient of restitution

On the Control of a One Degree-of-Freedom Juggling Robot

International audienceThis paper is devoted to the feedback control of a one degree-of-freedom (dof) juggling robot, considered as a subclass of mechanical systems subject to a unilateral constraint. The proposed approach takes into account the whole dynamics of the system, and focuses on the design of a force input. It consists of a family of hybrid feedback control laws, that allow to stabilize the object around some desired (periodic or not) trajectory. The closed-loop behavior in presence of various disturbances is studied. Despite good robustness properties, the importance of good knowledge of the system parameters, like the restitution coefficient, is highlighted. Besides its theoretical interest concerning the control of a class of mechanical systems subject to unilateral constraints, this study has potential applications in non-prehensile manipulation, extending pushing robotic tasks to striking-and-pushing tasks

Some perspectives on the analysis and control of complementarity systems

International audienceThis paper is devoted to presenting controllability and stabilizability issues associated to a class of nonsmooth dynamical systems, namely complementarity dynamical systems. The main existing results are summarized, and some possible research directions are provided. Convex analysis and complementarity problems are claimed to be the main analysis tools for control related studies. This paper mainly focuses on mechanical applications

Modeling, analysis and control of robot-object nonsmooth underactuated Lagrangian systems: A tutorial overview and perspectives

International audienceSo-called robot-object Lagrangian systems consist of a class of nonsmooth underactuated complementarity Lagrangian systems, with a specific structure: an "object" and a "robot". Only the robot is actuated. The object dynamics can thus be controlled only through the action of the contact Lagrange multipliers, which represent the interaction forces between the robot and the object. Juggling, walking, running, hopping machines, robotic systems that manipulate objects, tapping, pushing systems, kinematic chains with joint clearance, crawling, climbing robots, some cable-driven manipulators, and some circuits with set-valued nonsmooth components, belong this class. This article aims at presenting their main features, then many application examples which belong to the robot-object class, then reviewing the main tools and control strategies which have been proposed in the Automatic Control and in the Robotics literature. Some comments and open issues conclude the article

Motion planning and control methods for nonprehensile manipulation and multi-contact locomotion tasks

Many existing works in the robotic literature deal with the problem of nonprehensile dynamic manipulation. However, a unified control framework does not exist so far. One of the ambitious goals of this Thesis is to contribute to identify planning and control frameworks solving classes of nonprehensile dynamic manipulation tasks, dealing with the non linearity of their dynamic models and, consequently, with the inherited design complexity. Besides, while passing through a number of connections between dynamic nonprehensile manipulation and legged locomotion, the Thesis presents novel methods for generating walking motions in multi-contact situations

Trajectory tracking in switched systems: an internal model principle approach: the elliptical billiard system as a benchmark for theory

Sistemi dinamici caratterizzati dall'interazione tra dinamiche continue e discrete sono detti sistemi ibridi. Un sistema switched è un particolare sistema ibrido costituito da una famiglia di sottosistemi a tempo continuo e da una legge che ne regola le transizioni. Questi sistemi hanno numerose applicazioni nel controllo di sistemi meccanici, nell'industria automobilistica e aeronautica, nel controllo del traffico aereo, nell'elettronica di potenza, etc. Questa tesi sarà incentrata sul problema dell'inseguimento asintotico di traiettoria per sistemi switched. Nella prima parte, il problema di inseguimento è stato propriamente definito e risolto prendendo in esame il sistema biliardo ellittico. Al fine di definire una classe di traiettorie di riferimento ammissibili per il sistema biliardo un problema di pianificazione di traiettoria è stato approntato e risolto attraverso l'utilizzo di risultati della teoria dei polinomi non negativi e tecniche LMI. Il problema di inseguimento in presenza di incertezze nei parametri del sistema è stato considerato e risolto sia nel caso di feedback dallo stato che dalla sola posizione. Nella seconda parte della tesi i risultati ottenuti per il sistema biliardo sono stati generalizzati per una classe di sistemi switched con dinamica lineare in ogni modo operazionale, mappe di reset lineari e dimensione dello spazio di stato possibilmente variabile tra i vari modi. In tutti i casi la strategia di controllo proposta è basata su una versione discontinua del principio del modello interno.Dynamical systems that are described by an interaction between continuous and discrete dynamics are called hybrid systems. Their evolution is generally given by equations of motion containing mixtures of logic, discrete-valued or digital dynamics, and continuous-variable or analog dynamics. A switched system is a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. These systems have numerous applications in control of mechanical systems, automotive industry, aircraft and air traffic control, switching power converters, and many others. This thesis will focus on the problem of asymptotic trajectory tracking for switched systems. First, the tracking control problem is properly stated and solved for a controlled elliptical billiard system. In order to find an admissible class of reference trajectories inside the billiards a motion planning problem has been solved by using results from the theory of non-negative polynomials and LMIs techniques. The trajectory tracking problem in presence of uncertainties on the plant parameters has been also considered and solved in both cases of state-feedback and output-feedback. In the second part, the results obtained for the billiard system are generalized for a class of switched systems having linear dynamics in each operating mode, linear reset maps and possible nonuniform state space among the different modes. In all cases the proposed control strategy is based on a dynamic compensator, whose state is subject to discontinuities and whose structure is based on a nonsmooth version of the internal model principle

Variability, Symmetry, and Dynamics in Human Rhythmic Motor Control

How humans and other animals control rhythmic behaviors, and locomotion in particular, is one of the grand challenges of neuroscience and biomechanics. And yet remarkably few studies address the fundamental control-systems modeling of locomotor control. This thesis attempts to address several pieces of this grand challenge through the development of experimental, theoretical, and computational tools. Specifically, we focus our attention on three key features of human rhythmic motor control, namely variability, symmetry, and dynamics. Variability: Little is known about how haptic sensing of discrete events, such as heel-strike in walking, in rhythmic dynamic tasks enhances behavior and performance. In order to discover the role of discrete haptic cues on rhythmic motor control performance, we study a virtual paddle juggling behavior. We show that haptic sensing of a force impulse to the hand at the moment of ball-paddle collision categorically improves performance over visual feedback alone, not by regulating the rate of convergence to steady state, but rather by reducing cycle-to-cycle variability. Symmetry: Neglecting evident characteristics of a system can certainly be a modeling convenience, but it may also produce a better statistical model. For example, the dynamics of human locomotion is frequently treated as symmetric about the sagittal plane for modeling convenience. In this work, we test this assumption by examining the statistical consequences of neglecting (or not) bilateral asymmetries in the dynamics of human walking. Indeed, we show that there are statistically significant asymmetries in the walking dynamics of healthy participants (N=8), but that by ignoring these asymmetries and fitting a symmetric model to the data, we arrive at a more consistent and predictive model of human walking. Dynamics: Rhythmic hybrid dynamic behaviors can be observed in a wide variety of biological and robotic systems. Analytic (white-box) modeling tools of such systems are limited to the case when we have a full (and preferably simple) mathematical model that can accurately describe the system dynamics. In contrast, data-driven (block-box) system identification methods have the potential to overcome this fundamental limitation and could play a critical role in describing and analyzing the dynamics of rhythmic behaviors based on experimental data. And yet few tools exist for identifying the dynamics of rhythmic systems from input--output data. In this context, we propose a new formulation for identifying the dynamics of rhythmic hybrid dynamical systems around their limit-cycles by using discrete-time harmonic transfer functions