Dimensions of Copeland-Erdos Sequences

The base-$k$ {\em Copeland-Erd\"os sequence} given by an infinite set $A$ of positive integers is the infinite sequence \CE_k(A) formed by concatenating the base-$k$ representations of the elements of $A$ in numerical order. This paper concerns the following four quantities. The {\em finite-state dimension} \dimfs (\CE_k(A)), a finite-state version of classical Hausdorff dimension introduced in 2001. The {\em finite-state strong dimension} \Dimfs(\CE_k(A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of \dimfs(\CE_k(A)) satisfying \Dimfs(\CE_k(A)) \geq \dimfs(\CE_k(A)). The {\em zeta-dimension} \Dimzeta(A), a kind of discrete fractal dimension discovered many times over the past few decades. The {\em lower zeta-dimension} \dimzeta(A), a dual of \Dimzeta(A) satisfying \dimzeta(A)\leq \Dimzeta(A). We prove the following. \dimfs(\CE_k(A))\geq \dimzeta(A). This extends the 1946 proof by Copeland and Erd\"os that the sequence \CE_k(\mathrm{PRIMES}) is Borel normal. \Dimfs(\CE_k(A))\geq \Dimzeta(A). These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in $[0,1]$ satisfying the four above-mentioned inequalities.Comment: 19 page

Force estimation in a piezoelectric cantilever using the Inverse-Dynamics-Based UIO technique.

International audienceThis paper presents the estimation of the force applied by a piezocantilever dedicated to micromanipulation/ microassembly. Relative to previous works, the presented method avoids the reliance on the force dynamics on the characteristics of the microobjects. Furthermore, the estimation is a closed-loop kind technique so that convergency can be ensured efficiently. To perform these, we consider the force at the tip of a piezocantilever as an unknown input and we use an Unknown Input Observation technique. We especially use the Inverse-Dynamics-Based UIO technique because it is well suited for a piezocantilever model. The experiments show that the performances of the observer are convenient for micromanipulation/ microassembly tasks

Representation of problems during the conceptual design: A roadmap from functional to physical domains

International audienceConsidering that the representation of problems is a key step in the design of new technical systems, we propose a model of representation of problems. By integrating the problem solving between the functional modelling and the physical specification of the future system, we present a model enabling to build a bridge between these two domains. We will present in this article the model of representation of problems and the heuristics we use to instantiate this model. The use of one the heuristic will be illustrate by a study case

Hybrid Force/Position control applied to automated guiding tasks at the Microscale.

International audienceForce feedback control constitutes a promising solution for achieving fully automated micro-assembly. In this paper, we report the study of force feedback control used during guiding task. An experimental setup with two sensorized fingers is proposed. It enables to grasp a micropart of 2 mm x 50 μm x 50 μm in size and to estimate the lateral contact force. A simulator is developed to take into account the gripping forces evolution during a lateral perturbation. External hybrid force/position control is chosen for performing automated guiding tasks. Experimental results demonstrate the ability of the implemented controller and suitable design of microsystems

New bottom-up algorithm for assembly plan generation : opportunities for micro-factory design.

International audienceThis paper discusses a new approach dedicated to assembly plan generation, called "bottom-up algorithm". It is compared to the traditional "top-down approach", usually used to perform this stage of the design process of the assembly systems for "macro-products". We explore why this new algorithm is better adapted for designing a microassembly system. The case of watch assembly plans generation is described through the both approaches and the obtained results are compared

Force Tracking Impedance Control with Unknown Environment at the Microscale.

International audienceA new method to estimate the environment parameters is proposed in order to perform force tracking in impedance control despite the presence of an unknown environment. In impedance force tracking, the location of the environment relative to the robot and the stiffness of the environment should be known. The proposed method estimates the environment location and stiffness using only force and position measurements. The study is done for microscale taking into consideration microscale specificities, especially pull-off force. The impedance control formulation is tested experimentally in a contact transition scenario consisting of a compliant microforce sensor mounted on a microrobotic positioner, and three compliant microstructures with different stiffness. A traditional double mass-spring-damper model of the overall robot is employed to develop the closed-loop impedance control

Hinf deflection control of a unimorph piezoelectric cantilever under thermal disturbance.

International audienceThe effect of the temperature variation on a unimorph piezoelectric cantilever is studied. Its influences on the thermal expansion, the piezoelectric constant, the transient part and the creep are experimentally analyzed. Afterwards, a Hinf controller is synthesized in order ot reject the thermal disturbance and to reach performances required in micromanipulation. Finally, the closed-loop experiments end the paper

Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilevers.

International audienceThe feedforward compensation of nonlinearities, i.e. hysteresis and creep, and unwanted vibrations in micromanipulators is presented in this paper. The aim is to improve the general performances of piezocantilevers dedicated to micromanipulation/ microassembly tasks. While hysteresis is attenuated using the Prandtl-Ishlinskii inverse model, a new method is proposed to decrease the creep phenomenon. As no model inversion is used, the proposed method is simple and easy to implement. Finally, we employ an input shaping technique to reduce the vibration of the piezocantilevers. The experimental results show the efficiency of the feedforward techniques and their convenience to the micromanipulation/microassembly requirements

Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever.

International audiencePiezocantilevers are commonly used for the actuation of micromechatronic systems. These systems are generally used to perform micromanipulation tasks which require high positioning accuracy. However, the nonlinearities, i.e. the hysteresis and the creep, of piezoelectric materials and the influence of the environment (vibrations, temperature change, etc.) create difficulties for such a performance to be achieved. Various models have been used to take into account the nonlinearities but they are often complex. In this paper, we study a one degree of freedom piezoelectric cantilever. For that, we propose a simple new model where the hysteresis curve is approximated by a quadrilateral and the creep is considered to be a disturbance. To facilitate the modelling, we first demonstrate that the dynamic hysteresis of the piezocantilever is equivalent to a static hysteresis, i.e. a varying gain, in series with a linear dynamic part. The obtained model is used to synthesize a linear robust controller, making it possible to achieve the performances required in micromanipulation tasks. The experimental results show the relevance of the combination of the developed model and the synthesized robust H∞ controller