Sébastien Lagrange

On Sufficient Conditions of the Injectivity: Development of a Numerical Test Algorithm via Interval Analysis

This paper presents a new numerical algorithm based on interval analysis able to verify that a continuously differentiable function is injective. The efficiency of the method is demonstrated by illustrative examples. These examples have been treated by a C++ solver which is made available.

Analytical Solution of the Blind Source Separation Problem Using Derivatives

In this paper, we consider independence property between a random process and its first derivative. Then, for linear mixtures, we show that cross-correlations between mixtures and their derivatives provide a sufficient number of equations for analytically computing the unknown mixing matrix. In addition to its simplicity, the method is able to separate Gaussian sources, since it only requires second order statistics. For two mixtures of two sources, the analytical solution is given, and a few experiments show the efficiency of the method for the blind separation of two Gaussian sources.

Model-based approach for fault diagnosis using set-membership formulation

This paper describes a robust model-based fault diagnosis approach that enables to enhance the sensitivity analysis of the residuals. A residual is a fault indicator generated from an analytical redundancy relation which is derived from the structural and causal properties of the signed bond graph model. The proposed approach is implemented in two stages. The first stage consists in computing the residuals using available input and measurements while the second level leads to moving horizon residuals enclosures according to an interval consistency technique. These enclosures are determined by solving a constraint satisfaction problem which requires to know the derivatives of measured outputs as well as their boundaries. A numerical differentiator is then proposed to estimate these derivatives while providing their intervals. Finally, an inclusion test is performed in order to detect a fault upon occurrence. The proposed approach is well suited to deal with different kinds of faults and its performances are demonstrated through experimental data of an omni-directional robot.

A visibility information for multi-robot localization

This paper proposes a set-membership method based on interval analysis to solve the pose tracking problem for a team of robots. The originality of this approach is to consider only weak sensor data: the visibility between two robots. The paper demonstrates that with this poor information, without using bearing or range sensors, a localization is possible. By using this boolean information (two robots see each other or not), the objective is to compensate the odometry errors and be able to localize in an indoor environment all the robots of the team, in a guaranteed way. The environment is supposed to be defined by two sets, an inner and an outer characterizations. This paper mainly presents the visibility theory used to develop the method. Simulated results allow to evaluate the efficiency and the limits of the proposed algorithm.

COINC library: a toolbox for the network calculus: invited presentation, extended abstract

This talk will present the Scilab toolbox for Network Calculus computation. It was developed thanks to the INRIA ARC COINC project (COmputational Issue in Network Calculus see http://perso.bretagne.ens-cachan.fr/~bouillar/coinc/spip.php?rubrique1). This software library deals with the computation of ultimate pseudo-periodic functions. They are very useful to compute performance evaluation in network (e.g. Network Calculus) or in embedded system (Real Time Calculus).

Nonlinear Blind Parameter Estimation

This note deals with parameter estimation of nonlinear continuous-time models when the input signals of the corresponding system are not measured. The contribution of the note is to show that, with simple priors about the unknown input signals and using derivatives of the output signals, one can perform the estimation procedure. As an illustration, we consider situations where the simple priors, e.g., independence or Gaussianity of the unknown inputs, is assumed.

A numerical approach to compute the topology of the Apparent Contour of a smooth mapping from R2 to R2

A rigorous algorithm for computing the topology of the Apparent Contour of a generic smooth map is designed and studied in this paper. The source set is assumed to be a simply connected compact subset of the plane and the target space is the plane. Whitney proved that, generically, critical points of a smooth map are folds or cusps (Whitney, 1955). The Apparent Contour is the set of critical values, that is, the image of the critical points. Generically speaking, the Apparent Contour does not have triple points and double points are normal crossings (i.e. crossing without tangency). Each of those particular cases, cusp and normal crossing, is described in order to be rigorously handled by an interval analysis based scheme. The first step of the presented method provides an enclosure of those particular points. The second part of the designed method is a computation of a graph which is homeomorphic to the Apparent Contour. Edges of this graph are computed by testing connectivity of those particular points in the source space. This paper also defines a concept called portrait. Relations between this notion and the more classical notion of Apparent Contour are discussed.

Guaranteed interval analysis localization for mobile robots

This paper presents a set membership method (named Interval Analysis Localization (IAL)) to deal with the global localization problem of mobile robots. By using a LIDAR (LIght Detection And Ranging) range sensor, the odometry and a discrete map of an indoor environment, a robot has to determine its pose (position and orientation) in the map without any knowledge of its initial pose. In a bounded error context, the IAL algorithm searches a set of boxes (interval vector), with a cardinality as small as possible that includes the robot’s pose. The localization process is based on constraint propagation and interval analysis tools, such as bisection and relaxed intersection. The proposed method is validated using real data recorded during the CAROTTE challenge, organized by the French ANR (National Research Agency) and the French DGA (General Delegation of Armament). IAL is then compared with the well-known Monte Carlo Localization showing weaknesses and strengths of both algorithms. As it is shown in this paper with the IAL algorithm, interval analysis can be an efficient tool to solve the global localization problem. Graphical Abstract

Performance analysis of linear systems over semiring with additive inputs

This paper deals with the computation of a maximal flow in single input single output (max, +) linear systems. Assuming known a system composed of some subsystems - each one being described by a transfer function and some secondary inputs interfering with the principal flow on consecutive sub-systems, the computation of a maximal principal output is addressed. Transfer functions, inputs and outputs are represented by periodical series in a semiring of formal series, namely Nopfmindelta. Previously, it is shown that the Hadamard product of such series allows to compute the addition of inputs, and that this product is both residuated and dually residuated. These properties are used to compute the maximal principal output. An example concludes the paper and allows to illustrate the efficiency of the proposed approach.

LiDAR-only based navigation algorithm for an autonomous agricultural robot

The purpose of the work presented in this paper is to develop a general and robust approach for autonomous robot navigation inside a crop using LiDAR (Light Detection And Ranging) data. To be as robust as possible, the robot navigation must not need any prior information about the crop (such as the size and width of the rows). The developed approach is based on line extractions from 2D point clouds using a PEARL based method. In this paper, additional filters and refinements of the PEARL algorithm are presented in the context of crop detection. A penalization of outliers, a model elimination step, a new model search and a geometric constraint are proposed to improve the crop detection. The approach has been tested over a simulator and compared with classical PEARL and RANSAC based approaches. It appears that adding those modification improved the crop detection and thus the robot navigation. Those results are presented and discussed in this paper. It can be noticed that even if this paper presents simulated results (to ease the comparison with other algorithms), the approach also has been successfully tested using an actual Oz weeding robot, developed by the French company Naio Technologies.