Our research area is the accurate control of polyarticulated systems. A lot of these systems uses only polygonal or elliptic part trajectories in order to quickly calculate interpolated curves without extraneous oscillations. In this aim, we have investigated the creation of smooth trajectories using L1-norm minimization methods which preserve shapes and don't produced extraneous oscillations. Because of the importance of calculus time in our research context, we have developed a five point window algorithm which allows us to quickly calculate cubic interpolation curves using algebraic solution. We have proved that this algorithm keeps the properties of global methods and thus can be used in the process command of the systems. From this basic algorithm, we have developed other methods in the area of interpolation and approximation. In the context of curve interpolation, we have used our basic algorithm to create smoothest curves like Pythagorean Hodograph ones from Rida T. Farouki's methods or Ck-continuous polynomial curves by iterating the basic algorithm. For the surface case, we have just apply our basic algorithm in four main directions to construct a bicubic interpolation surface. This method have been successfully tested for image manipulation such resampling, rotating or warping. With all these interpolation methods using basic algebraic algorithm, we have shown that the results are efficient for shape preserving with a cheap calculus time. In the area of approximation, we have developed also a five point window method which is solved by primal affine algorithm. The aim is to keep linear calculus time over the number of points in the data sets. Our approximating method can be iteratively used over a data set in order to obtain a smoothest curve and give also good results from noisy data points.

FIRST SPEAKER: Eric NYIRI is an assistant professor of computer science at the Ecole Nationale Superieure des Arts et Metiers (ENSAM) ParisTech center in Lille, France, where he conducts research in the Information and System Sciences Laboratory. Between 2000 and 2004, E. Nyiri was a member of a multidisciplinary team that developed models for the milling and control of inductor prototypes for the VULCAIN engine in collaboration with SNECMA Motors. Subsequently, he worked on trajectory optimization for milling and, in 2006, he joined the 'six-axis milling robot' project in partnership with the Stäubli and EADS companies. Interpolation and approximation by L1-norm-based polynomial splines, called L1 splines, has been the focus of his work in this area. Prof. Nyiri designed a local algorithm for calculating the coefficients of L1 interpolating splines that reduces the computing time by orders of magnitude with no negative effect on accuracy.

SECOND SPEAKER: Since September 2008, Olivier GIBARU has been a professor of applied mathematics at the  Ecole Nationale Superieure des Arts et Metiers  (ENSAM) ParisTech center in Lille, France, where he conducts research in the Information and System Sciences Laboratory. He was previously associate professor at ENSAM Lille from 1998 to 2008. Prof. Gibaru's research activities are in computer aided geometric design for high-precision, multi-axis systems and robots for the automotive and aeronautic industries and atomic force microscopes for nanotechnology. With the goal of improving the performance of robots to achieve a level of accuracy close to that achieved by machine tools, Prof. Gibaru and colleagues at ENSAM Lille developed geometric trajectories of the end-effector of the robot based on quintic C²-continuous splines calculated by minimizing the L1 norm (rather than the traditional L2 norm) of the second derivative of the spline. This method prevents Gibb's phenomenon, which improves the operation of the robots. Calculation of the coefficients of the spline is a nonlinear program which Prof. Gibaru and colleagues solve by a new, efficient local strategy. Along with this research on path planning, Prof. Gibaru works on real-time estimation algorithms to calculate the parameters of the dynamic model of the robot using algebraic-derivative methods. These algorithms are fast (real-time being the goal), deterministic (noise considered to be fast fluctuation) and non-asymptotic (finite-time convergence).