Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizableand $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback, only depending on $(A,B)$ and the parameters of the persistent excitation, which globally exponentially stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.

}, keywords = {Robotics}, author = {A. Chaillet and Y. Chitour and A. Lor and M. Sigalotti} } @booklet {CHCHLOSI07, title = {Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases}, howpublished = {Submitted to Mathematics of Control, Signals and Systems}, year = {2007}, abstract = {Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizable and $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback, only depending on $(A,B)$ and the parameters of the persistent excitation, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.

}, keywords = {Robotics}, author = {A. Chaillet and Y. Chitour and A. Lor and M. Sigalotti} } @conference {DOUBLEINT, title = {On the PE stabilization of time-varying systems: open questions and preliminary answers}, booktitle = {Proc. IEEE Int. Conf. on Decision and Control}, year = {2005}, month = {December}, pages = {6847{\textendash}6852}, address = {Sevilla, Spain}, abstract = {We address the following fundamental question: given a double integrator and a linear control that stabilizes it exponentially, is it possible to use the {\em same} control input in the case that the control input is multiplied by a time-varying term? Such question has many interesting motivations and generalizations: 1) we can pose the same problem for an input gain that depends on the state and time hence, a specific persistency of excitation property for nonlinear systems must be imposed; 2) the stabilization {\textendash}with the same method{\textendash} of chains of integrators of higher order than two is fundamentally more complex and has applications in the stabilization of driftless systems; 3) the popular backstepping method stabilization method for systems with non-invertible input terms. The purpose of this note is two-fold: we present some open questions that we believe are significant in time-varying stabilization and present some preliminary answers for simple, yet challenging case-studies.

}, keywords = {Robotics}, author = {A. Lor and A. Chaillet and G. Besancon and Y. Chitour} } @article {MCB04, title = {Reachability and Steering of Rolling Polyhedra: A Case Study in Discrete Nonholonomy}, journal = {IEEE Trans. on Automatic Control}, volume = {49}, number = {5}, year = {2004}, month = {May}, pages = {710-726}, abstract = {Rolling a ball on a plane is a standard example of nonholonomy reported in many textbooks, and the problem is also well understood for any smooth deformation of the surfaces. For non-smoothly deformed surfaces, however, much less is known. Although it may seem intuitive that nonholonomy is conserved (think e.g. to polyhedral approximations of smooth surfaces), current definitions of {\textquoteleft}{\textquoteleft}nonholonomy{\textquoteright}{\textquoteright} are inherently referred to systems described by ordinary differential equations, and are thus inapplicable to such systems. \{\.I}n this paper we study the set of positions and orientations that a polyhedral part can reach by rolling on a plane through sequences of adjacent faces. We provide a description of such reachable set, discuss conditions under which the set is dense, or discrete, or has a compound structure, and provide a method for steering the system to a desired reachable configuration, robustly with respect to model uncertainties. \\Based on ideas and concepts encountered in this case study, and in some other examples we provide, we turn back to the most general aspects of the problem and investigate the possible generalization of the notion of (kinematic) nonholonomy to non-smooth, discrete, and hybrid dynamical systems. To capture the essence of phenomena commonly regarded as {\textquoteleft}{\textquoteleft}nonholonomic{\textquoteright}{\textquoteright}, at least two irreducible concepts are to be defined, of {\textquoteleft}{\textquoteleft}internal{\textquoteright}{\textquoteright} and {\textquoteleft}{\textquoteleft}external{\textquoteright}{\textquoteright} nonholonomy, which may coexist in the same system. These definitions are instantiated by examples.

}, keywords = {Embedded Control, Robotics}, author = {A. Bicchi and Y. Chitour and A. Marigo} } @conference {MCB97, title = {Manipulation of polyhedral parts by rolling}, booktitle = {Proc. IEEE Int. Conf. on Robotics and Automation}, year = {1997}, keywords = {Hybrid and Embedded Control Systems, Nonlinear Control Systems, Quantized Control Systems}, author = {A. Marigo and Y. Chitour and A. Bicchi} } @inbook {CMPB97, title = {Rolling Polyhedra on a Plane: Analysis of the Reachable Set}, booktitle = {Algorithms for Robotic Motion and Manipulation}, year = {1997}, pages = {277-286}, publisher = {A. K. Peters}, organization = {A. K. Peters}, address = {Wellesley, MA, U.S.A.}, keywords = {Embedded Control, Robotics}, author = {Y. Chitour and A. Marigo and D Prattichizzo and A. Bicchi}, editor = {J.P. Laumond and M. Overmars} } @conference {BCMP96, title = {Dexterity through Rolling: Towards Manipulation of Unknown Objects}, booktitle = {Proc. Third Int. Symp. on Methods and Models for Automation and Robotics}, year = {1996}, address = {Miedzyzdroje, Poland}, keywords = {Nonlinear Control Systems, Robot Hands}, author = {A. Bicchi and Y. Chitour and A. Marigo and D Prattichizzo} } @inbook {CMPB96, title = {Reachability of Rolling Parts}, booktitle = {Advances in Robotics: The ERNET Perspective}, year = {1996}, pages = {51-60}, publisher = {World Scientific Publisher Corp.}, organization = {World Scientific Publisher Corp.}, address = {Singapore}, keywords = {Haptics, Robotics}, author = {Y. Chitour and A. Marigo and D Prattichizzo and A. Bicchi}, editor = {C. Bonivento and C. Melchiorri and H. Tolle} }