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6. Computing Capture Tubes.

Author

Luc Jaulin, Daniel Lopez, Vincent Le Doze, Stéphane Le Ménec, Jordan Ninin, Gilles Chabert, Mohamed Saad Ibn Seddik, and Alexandru Stancu

Published
2014
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28. Towards a Generic Interval Solver for Differential-Algebraic CSP

Author

Victor Reyes, Bertrand Neveu, Luc Jaulin, Gilles Chabert, Simon Rohou, Alexandre Goldsztejn, Abderahmane Bedouhene, Gilles Trombettoni, Lab-STICC_ENSTAB_CID_PRASYS, Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance (Lab-STICC), Institut Mines-Télécom [Paris] (IMT)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-École Nationale d'Ingénieurs de Brest (ENIB)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)-Université Bretagne Loire (UBL)-Institut Mines-Télécom [Paris] (IMT)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-École Nationale d'Ingénieurs de Brest (ENIB)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)-Université Bretagne Loire (UBL), Laboratoire d'Informatique Gaspard-Monge (LIGM), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, IRT Jules Vernes, Laboratoire des Sciences du Numérique de Nantes (LS2N), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Agents, Apprentissage, Contraintes (COCONUT), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Simonis, Helmut, and ANR-16-CE33-0024,CONTREDO,Intervalles et contracteurs pour les systèmes dynamiques(2016)

Subjects
Physics, 050101 languages & linguistics, 05 social sciences, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 02 engineering and technology, Interval (mathematics), Solver, Interval arithmetic, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], Tree (data structure), [SPI]Engineering Sciences [physics], 0202 electrical engineering, electronic engineering, information engineering, Constraint programming, Initial value problem, 020201 artificial intelligence & image processing, 0501 psychology and cognitive sciences, Boundary value problem, Algorithm, Differential algebraic equation, Interval Analysis, Differential Algebraic Equations
Abstract

International audience; In this paper, we propose an interval constraint programming approach that can handle the differential-algebraic CSP (DACSP), where an instance is composed of real and functional variables (also called dynamic variables or trajectories) together, and differential and/or “static” numerical constraints among those variables. Differential-Algebraic CSP systems can model numerous real-life problems occurring in physics, biology or robotics. We introduce a solver, built upon the Tubex and Ibex interval libraries, that can rigorously approximate the set of solutions of a DACSP system. The solver achieves temporal slicing and a tree search by splitting trajectories domains. Our approach provides a significant step towards a generic interval CP solver for DACSP that has the potential to handle a large variety of constraints. First experiments highlight that this solver can tackle interval Initial Value Problems (IVP), Boundary Value Problems (BVP) and integro-differential equations.

Published
2020
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29. A three-step methodology for dimensional tolerance synthesis of parallel manipulators

Author

Stéphane Caro, Alexandre Goldsztejn, Gilles Chabert, Institut de Recherche en Communications et en Cybernétique de Nantes (IRCCyN), Mines Nantes (Mines Nantes)-École Centrale de Nantes (ECN)-Ecole Polytechnique de l'Université de Nantes (EPUN), Université de Nantes (UN)-Université de Nantes (UN)-PRES Université Nantes Angers Le Mans (UNAM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Informatique de Nantes Atlantique (LINA), Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), and Mines Nantes (Mines Nantes)

Subjects
0209 industrial biotechnology, Mathematical optimization, Mechanical Engineering, Parallel manipulator, Bioengineering, 02 engineering and technology, Workspace, Upper and lower bounds, parametric Kantorovich theorem, [SPI.AUTO]Engineering Sciences [physics]/Automatic, [SPI.MECA.GEME]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanical engineering [physics.class-ph], Computer Science Applications, Nonlinear programming, Nonlinear system, 020901 industrial engineering & automation, Mechanics of Materials, parallel manipulators, Kantorovich theorem, 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, 020201 artificial intelligence & image processing, Certified tolerance synthesis, Global optimization, Mathematics, Parametric statistics
Abstract

International audience; Computing the maximal pose error given an upper bound on model parameters uncertainties, called perturbations in this paper, is challenging for parallel robots, mainly because the direct kinematic problem has several solutions, which become unstable in the vicinity of parallel singularities. In this paper, a local uniqueness hypothesis that allows safely computing pose error upper bounds using nonlinear optimization is proposed. This hypothesis, together with a corresponding maximal allowed perturbation domain and a certified crude pose error upper bound valid over the complete workspace, will be proved numerically using a parametric version of Kantorovich theorem and certified nonlinear global optimization. Then, approximate linearizations are used in order to determine approximated tolerances reaching a prescribed maximal pose error over a given workspace. Those tolerances are finally verified using optimal pose error upper bounds, which are computed using global optimization techniques. Two illustrative examples are studied in order to highlight the contributions of the paper.

Published
2016
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30. A parametric Kantorovich theorem with application to tolerance synthesis

Author

Alexandre Goldsztejn, Stéphane Caro, Gilles Chabert, Optimisation Globale et Résolution Ensembliste (OGRE), Laboratoire des Sciences du Numérique de Nantes (LS2N), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Robots and Machines for Manufacturing, Society and Services (RoMas), Centre National de la Recherche Scientifique (CNRS), Département Automatique, Productique et Informatique (IMT Atlantique - DAPI), and IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique)

Subjects
global optimization, Kantorovich theorem, tolerance synthesis, [SPI.MECA.GEME]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanical engineering [physics.class-ph], [SPI.AUTO]Engineering Sciences [physics]/Automatic
Abstract

International audience; We propose a parametric Kantorovich theorem, which will achieve these two tasks. The idea is to compute worst case Kantorovich constants with respect to parameters q using a branch and bound algorithm dedicated to nonlinear nonsmooth global optimization. A rigorous first order model of the dependence with respect to parameters p is used to enforce the convergence of the error upper-bound. Details about these developments can be found in [1]. We provide here a different point of view, in particular emphasizing the reason why not using the interval Newton operator, in spite of its theoretical superiority on Kantorovich theorem in this context [2].

Published
2018

31. Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation

Author

Gilles Chabert, Alexandre Goldsztejn, Optimisation Globale et Résolution Ensembliste (OGRE), Laboratoire des Sciences du Numérique de Nantes (LS2N), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Département Automatique, Productique et Informatique (IMT Atlantique - DAPI), IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-École Centrale de Nantes (ECN), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - Faculté des Sciences et des Techniques, and Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Lyapunov function, 0209 industrial biotechnology, MathematicsofComputing_NUMERICALANALYSIS, 02 engineering and technology, Lyapunov exponent, exponentially stable fixed point, Fixed point, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, 020901 industrial engineering & automation, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Stability theory, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Nonlinear systems, 0202 electrical engineering, electronic engineering, information engineering, Interval analysis, Lyapunov equation, interval analysis Contribution, Electrical and Electronic Engineering, Lyapunov redesign, Control-Lyapunov function, Mathematics, [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, Nonlinear non-smooth systems, 020208 electrical & electronic engineering, Mathematical analysis, Lyapunov optimization, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], Control and Systems Engineering, symbols, domain of attraction
Abstract

Publié dans Automatica, Volume 100, February 2019, Pages 371-377; International audience; The Lyapunov equation allows finding a quadratic Lyapunov function for an asymptotically stable fixed point of a linear system. Applying this equation to the linearization of a nonlinear system can also prove the exponential stability of its fixed points. This paper proposes an interval version of the Lyapunov equation, which allows investigating a given Lyapunov candidate function for non-smooth nonlinear systems inside an explicitly given neighborhood, leading to rigorous estimates of the domain of attraction (EDA) of exponentially stable fixed points. These results are developed in the context of uncertain systems. Experiments are presented, which show the interest of the approach including with respect to usual approaches based on sum-of-squares for the computation of EDA.

Published
2018
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32. Computing Capture Tubes

Author

Vincent Doze, Gilles Chabert, Luc Jaulin, Jordan Ninin, Mohamed Saad Ibnseddik, Daniel Lopez, Alexandru Stancu, Stéphane Le Ménec, Lab-STICC_ENSTAB_CID_IHSEV, Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance (Lab-STICC), École Nationale d'Ingénieurs de Brest (ENIB)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Télécom Bretagne-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Université européenne de Bretagne - European University of Brittany (UEB)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS)-École Nationale d'Ingénieurs de Brest (ENIB)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Télécom Bretagne-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Université européenne de Bretagne - European University of Brittany (UEB)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS), Pôle STIC_OSM, École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne), Lab-STICC_ENSTAB_CID_PRASYS, École Nationale d'Ingénieurs de Brest (ENIB)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS)-Université Bretagne Loire (UBL)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-École Nationale d'Ingénieurs de Brest (ENIB)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS)-Université Bretagne Loire (UBL)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT), University Of Manchester Institute (UMIST/Department of Process Integration), University of Manchester [Manchester], MBDA France (MBDA), MBDA France, IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-École Nationale d'Ingénieurs de Brest (ENIB)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)-Université Bretagne Loire (UBL)-Institut Mines-Télécom [Paris] (IMT)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), and Institut Mines-Télécom [Paris] (IMT)-École Nationale d'Ingénieurs de Brest (ENIB)-École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)-Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)-Université Bretagne Loire (UBL)

Subjects
Nonholonomic system, business.industry, Differential equation, Computer science, Mobile robot, Robotics, Capture tube Contractors Interval arithmetic Robotics Stability, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Computer Science::Robotics, Control theory, Trajectory, Robot, Initial value problem, Artificial intelligence, [MATH]Mathematics [math], Differential (infinitesimal), business
Abstract

International audience; Many mobile robots such as wheeled robots, boats, or plane are described by nonholonomic differential equations. As a consequence, they have to satisfy some differential constraints such as having a radius of curvature for their trajectory lower than a known value. For this type of robots, it is difficult to prove some properties such as the avoidance of collisions with some moving obstacles. This is even more difficult when the initial condition is not known exactly or when some uncertainties occur. This paper proposes a method to compute an enclosure (a tube) for the trajectory of the robot in situations where a guaranteed interval integration cannot provide any acceptable enclosures. All properties that are satisfied by the tube (such as the non-collision) will also be satisfied by the actual trajectory of the robot.

Published
2016
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33. Contrainte de non-chevauchement entre objets décrits par des inégalités non-linéaires

Author

Ignacio Salas, Alexandre Goldsztejn, Gilles Chabert, Theory, Algorithms and Systems for Constraints (TASC), Laboratoire d'Informatique de Nantes Atlantique (LINA), Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Département informatique - EMN, Mines Nantes (Mines Nantes)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematical optimization, Sweep Algorithm, packing problem, Hybrid algorithm (constraint satisfaction), Binary constraint, interval arithmetic, Solver, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG], Minkowski addition, Constraint (information theory), Intersection, Constraint logic programming, Constraint programming, Minkowski Sum, Algorithm, Non-Overlapping Constraint, Non-Linear Constraints, Mathematics
Abstract

National audience; Packing 2D objects in a limited space is an ubiquitous problem with many academic and industrial variants. In any case, solving this problem requires the ability to determine where a first object can be placed so that it does not intersect a second, previously placed, object. This subproblem is called the non-overlapping constraint. The complexity of this non-overlapping constraint depends on the type of objects considered. It is simple in the case of rectangles. It has also been studied in the case of polygons. This paper proposes a numerical approach for the wide class of objects described bynon-linear inequalities. Our goal here is to calculate the non-overlapping constraint, that is, to describe the set of all positions and orientations that can be assigned to the first object so that intersection with the second one is empty. This is done using a dedicated branch & bound approach. We first show that the non-overlapping constraint can be cast into a Minkowski sum, even if we take into account orientation. We derive from this an innercontractor, that is, an operator that removes from the current domain a subset of positions and orientations that necessarily violate the non-overlapping constraint. This inner contractor is then embedded in a sweeping loop, a pruning technique that was only used with discrete domains so far. We finally come up with a branch & bound algorithm that outperforms the generic state-of-the-art solver Rsolver.; Le placement d'objets 2D dans un espace limité est un problème omniprésent aussi bien sur le plan académique qu'industriel. Quel que soit le contexte, la résolution de ce problème exige la capacité de pouvoir déterminer où un premier objet peu être placé de telle façon qu'il ne chevauche pas un second objet, précédemment placé. Ce sous-problème s'appelle la contrainte de non-chevauchement. La complexité de cette contrainte de non-chevauchement dépend du type d'objets considérés. Elle est simple dans le cas de rectangles. Elle a également été étudiée dans le cas de polygones. Cet article propose une approche numérique pour la classe générale des objets décrits par des inégalités non-linéaires. Notre objectif ici est de calculer la contrainte de non-chevauchement, c'est à dire, de décrire l'ensemble de toutes les positions et orientations qui peuvent être attribuées au premier objet de telle sorte que l'intersection avec le second soit vide. Nous nous basons sur un algorithme de branch & prune dédié. Nous montrons d'abord que la contrainte de non-chevauchement, équivaut à une somme de Minkowski, même lorsque l'orientation est prise en compte. Nous en déduisons un contracteur intérieur, c'est à dire, un opérateur qui élimine du domaine courant un sous-ensemble de positions et orientations qui violent nécessairement la contrainte de non-chevauchement. Ce contracteur intérieur est intégré dans une boucle de sweep, une technique utilisée jusqu'ici uniquement pour les domaines discrets. Nous aboutissons ainsi à un algorithme de branch & prune présentant de bien meilleures performances que Rsolver, outil de référence pour la résolution de contraintes quantifiées en domaines continus.

Published
2014
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34. Certified Calibration of a Cable-Driven Robot Using Interval Contractor Programming

Author

Julien Alexandre Dit Sandretto, David Daney, Gilles Chabert, Gilles Trombettoni, Constraints solving, optimization and robust interval analysis (COPRIN), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Agents, Apprentissage, Contraintes (COCONUT), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Laboratoire d'Informatique de Nantes Atlantique (LINA), Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN), Federico Thomas, Alba Perez Gracia, Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), and Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Interval model, 0209 industrial biotechnology, Engineering, Robot calibration, business.industry, Cable-driven robots, Control engineering, 02 engineering and technology, Certification, Interval arithmetic, Computer Science::Robotics, 020901 industrial engineering & automation, Control theory, Outlier, Calibration, 0202 electrical engineering, electronic engineering, information engineering, Parallel architecture, Cable driven, Robot, Interval analysis, [INFO.INFO-RB]Computer Science [cs]/Robotics [cs.RO], 020201 artificial intelligence & image processing, business
Abstract

From the 6th International Workshop on Computational Kinematics (CK), May 2013, Barcelona, Spain.; International audience; In this paper, an interval based approach is proposed to rigorously identify the model parameters of a parallel cable-driven robot. The studied manipulator follows a parallel architecture having 8 cables to control the 6 DOFs of its mobile platform. This robot is complex to model, mainly due to the cable behavior. To simplify it, some hypotheses on cable properties (no mass and no elasticity) are done.An interval approach can take into account the maximal error between this model and the real one. This allows us to work with a simplified although guaranteed interval model. In addition, a specific interval operator makes it possible to manage outliers. A complete experiment validates our method for robot parameter certified identification and leads to interesting observations.

Published
2014
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35. Upper Bounding in Inner Regions for Global Optimization under Inequality Constraints

Author

Bertrand Neveu, Gilles Trombettoni, Gilles Chabert, Ignacio Araya, Departamento de Informatica [Valparaíso, Chile], Universidad Tecnica Federico Santa Maria [Valparaiso] (UTFSM), Constraints solving, optimization and robust interval analysis (COPRIN), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe CEP, Modèles Discrets pour les Systèmes Complexes (Laboratoire I3S - MDSC), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), École des Ponts ParisTech (ENPC), Laboratoire d'Informatique Gaspard-Monge (LIGM), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout-Centre National de la Recherche Scientifique (CNRS), imagine [Marne-la-Vallée], Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout-Centre National de la Recherche Scientifique (CNRS)-Centre Scientifique et Technique du Bâtiment (CSTB), Laboratoire d'Informatique de Nantes Atlantique (LINA), Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Theory, Algorithms and Systems for Constraints (TASC), Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Département informatique - EMN, Mines Nantes (Mines Nantes)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Agents, Apprentissage, Contraintes (COCONUT), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM), Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM)-Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM)-Centre Scientifique et Technique du Bâtiment (CSTB), Mines Nantes (Mines Nantes), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Département informatique - EMN, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)

Subjects
021103 operations research, Control and Optimization, Branch and bound, Applied Mathematics, global optimization, 0211 other engineering and technologies, upper bounding, 02 engineering and technology, Interval (mathematics), [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], Management Science and Operations Research, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Space (mathematics), Upper and lower bounds, Computer Science Applications, Combinatorics, Bounding overwatch, intervals, branch and bound, Convex polytope, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Point (geometry), Global optimization, Mathematics
Abstract

International audience; In deterministic continuous constrained global optimization, upper bounding the objective function generally resorts to local minimization at several nodes/iterations of the branch and bound. We propose in this paper an alternative approach when the constraints are inequalities and the feasible space has a non-null volume. First, we extract an inner region , i.e., an entirely feasible convex polyhedron or box in which all points satisfy the constraints. Second, we select a point inside the extracted inner region and update the upper bound with its cost. We describe in this paper two original inner region extraction algorithms implemented in our interval B&B called IbexOpt. They apply to nonconvex constraints involving mathematical operators like +,x,power,sqrt,exp,log,sin. This upper bounding shows very good performance obtained on medium-sized systems proposed in the COCONUT suite.

Published
2012
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36. The Conjunction of Interval AMONG Constraints

Author

Sophie Demassey, Gilles Chabert, Theory, Algorithms and Systems for Constraints (TASC), Laboratoire d'Informatique de Nantes Atlantique (LINA), Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Département informatique - EMN, Mines Nantes (Mines Nantes)-Inria Rennes – Bretagne Atlantique, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Constraint (information theory), Combinatorics, Discrete mathematics, Variable (computer science), Generalization, Interval (mathematics), Type (model theory), [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Filter (higher-order function), Time complexity, Satisfiability, Mathematics
Abstract

International audience; An AMONG constraint holds if the number of variables that belong to a given value domain is between given bounds. This paper focuses on the case where the variable and value domains are intervals. We investigate the conjunction of AMONG constraints of this type. We prove that checking for satisfiability -- and thus, enforcing bound consistency -- can be done in polynomial time. The proof is based on a specific decomposition that can be used as such to filter inconsistent bounds from the variable domains. We show that this decomposition is incomparable with the natural conjunction of \textsc{Among} constraints, and that both decompositions do not ensure bound consistency. Still, experiments on randomly generated instances reveal the benefits of this new decomposition in practice. This paper also introduces a generalization of this problem to several dimensions and shows that satisfiability is NP-complete in the multi-dimensional case.

Published
2012

37. Sweeping with Continuous Domains

Author

Nicolas Beldiceanu, Gilles Chabert, Theory, Algorithms and Systems for Constraints (TASC), Laboratoire d'Informatique de Nantes Atlantique (LINA), Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)-Département informatique - EMN, Mines Nantes (Mines Nantes)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and David Cohen

Subjects
Constraint (information theory), Linear inequality, Mathematical optimization, Spacetime, Business rule, [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO], Fixed point, Algorithm, Symbolic processing, Integer (computer science), Interval arithmetic, Mathematics, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI]
Abstract

International audience; The geost constraint has been proposed to model and solve discrete placement problems involving multi-dimensional boxes (packing in space and time). The filtering technique is based on a sweeping algorithm that requires the ability for each constraint to compute a forbidden box around a given fixed point and within a surrounding area. Several cases have been studied so far, including integer linear inequalities. Motivated by the placement of objects with curved shapes, this paper shows how to implement this service for continuous constraints with arbitrary mathematical expressions. The approach relies on symbolic processing and defines a new interval arithmetic.

Published
2010
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38. Computing the Pessimism of Inclusion Functions

Author

Luc Jaulin, Gilles Chabert, Extraction et Exploitation de l'Information en Environnements Incertains (E3I2), École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne), Développement des Technologies Nouvelles (DTN), and Chabert, Gilles

Subjects
[INFO.INFO-AI] Computer Science [cs]/Artificial Intelligence [cs.AI], Infinite set, Inclusion (disability rights), media_common.quotation_subject, 010103 numerical & computational mathematics, Pessimism, 01 natural sciences, Image (mathematics), Interval arithmetic, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], Bounding overwatch, Error analysis, Applied mathematics, 0101 mathematics, media_common, Mathematics, Applied Mathematics, 010102 general mathematics, Function (mathematics), [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], Computational Mathematics, [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], Computer Science::Computer Vision and Pattern Recognition, Algorithm, Software
Abstract

WOS; International audience; “Computing the pessimism” means bounding the overestimation produced by an inclusion function. There are two important distinctions with classical error analysis. First, we do not consider the image by an inclusion function but the distance between this image and the exact image (in the set-theoretical sense). Second, the bound is computed over a infinite set of intervals. To our knowledge, this issue is not covered in the literature and may have a potential of applications. We first motivate and define the concept of pessimism. An algorithm is then provided for computing the pessimism, in the univariate case. This algorithm is general-purpose and works with any inclusion function. Next, we prove that the algorithm converges to the optimal bound under mild assumptions. Finally, we derive a second algorithm for automatically controlling the pessimism, i.e., determining where an inclusion function is accurate.

Published
2007

39. On the approximation of linear AE-solution sets

Author

Gilles Chabert, Alexandre Goldsztejn, Laboratoire d'Informatique de Nantes Atlantique (LINA), and Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Approximation theory, Mathematical analysis, Solution set, 010103 numerical & computational mathematics, 02 engineering and technology, State (functional analysis), Interval (mathematics), [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], System of linear equations, 01 natural sciences, Mathematical Operators, Set (abstract data type), 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, 020201 artificial intelligence & image processing, Set theory, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

When considering systems of equations, it often happens that parameters are known with some uncertainties. This leads to continua of solutions that are usually approximated using the interval theory. A wider set of useful situations can be modeled if one allows furthermore different quan- tifications of the parameters in their domains. In particu- lar, quantified solution sets where universal quantifiers are constrained to precede existential quantifiers are called AE- solution sets. A state of the art on the approximation of linear AE- solution sets in the framework of generalized intervals (in- tervals whose bounds are not constrained to be ordered in- creasingly) is presented in a new and unifying way. Then two new generalized interval operators dedicated to the ap- proximation of quantified linear interval systems are pro- posed and investigated.

Published
2006
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40. When Interval Analysis Helps Inter-Block Backtracking

Author

Gilles Trombettoni, Bertrand Neveu, Gilles Chabert, Constraints solving, optimization and robust interval analysis (COPRIN), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Algorithms, architectures, image analysis and computer graphics, Laboratoire d'Informatique Gaspard-Monge (LIGM), Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM)-Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM), Frederic Benhamou, and Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout-Centre National de la Recherche Scientifique (CNRS)

Subjects
Backtracking, Constrained optimization, 010103 numerical & computational mathematics, 02 engineering and technology, Solver, System of linear equations, 01 natural sciences, Search tree, Interval arithmetic, [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], Robustness (computer science), 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, 0101 mathematics, Algorithm, Mathematics, Sparse matrix
Abstract

International audience; Inter-block backtracking (IBB) computes all the solutions of sparse systems of non-linear equations over the reals. This algorithm, introduced in 1998 by Bliek et al., handles a system of equations previously decomposed into a set of (small) k × k sub-systems, called blocks. Partial solutions are computed in the different blocks and combined together to obtain the set of global solutions. When solutions inside blocks are computed with interval-based techniques, IBB can be viewed as a new interval-based algorithm for solving decomposed equation systems. Previous implementations used Ilog Solver and its IlcInterval library. The fact that this interval-based solver was more or less a black box implied several strong limitations. The new results described in this paper come from the integration of IBB with the interval-based library developed by the second author. This new library allows IBB to become reliable (no solution is lost) while still gaining several orders of magnitude w.r.t. solving the whole system. We compare several variants of IBB on a sample of benchmarks, which allows us to better understand the behavior of IBB. The main conclusion is that the use of an interval Newton operator inside blocks has the most positive impact on the robustness and performance of IBB. This modifies the influence of other features, such as intelligent backtracking and filtering strategies.

Published
2006
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41. Interval method for calibration of parallel robots: Vision-based experiments

Author

Yves Papegay, Nicolas Andreff, David Daney, Gilles Chabert, Constraints solving, optimization and robust interval analysis (COPRIN), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Laboratoire des sciences et matériaux pour l'électronique et d'automatique (LASMEA), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
0209 industrial biotechnology, Robot calibration, Calibration (statistics), Mechanical Engineering, Parallel manipulator, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, Experimental data, Bioengineering, 02 engineering and technology, Interval (mathematics), Kinematics, Computer Science Applications, Interval arithmetic, Set (abstract data type), 020901 industrial engineering & automation, Mechanics of Materials, 0202 electrical engineering, electronic engineering, information engineering, [INFO.INFO-RB]Computer Science [cs]/Robotics [cs.RO], 020201 artificial intelligence & image processing, Interval methods, Algorithm, Simulation, Mathematics
Abstract

International audience; This paper is a theoretical and experimental study of how interval arithmetic and analysis methods can be used to achieve (1) numerical certification of the kinematic calibration of parallel robots, and (2) a possible validation of the kinematic model used in calibration. First, a detailed description is given of our experimental device and vision-based measurement method. The usual calibration methods are then reviewed and applied to our experimental data set, yielding a motivation for numerical certification of the results. Next, interval calibration methods (which have already been described in a previous work) are also reviewed and applied to the data. Finally, the experimental results are discussed and interpreted.

Published
2006
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42. A generalized interval LU decomposition for the solution of interval linear systems

Author

Alexandre Goldsztejn, Gilles Chabert, Laboratoire d'Informatique de Nantes Atlantique (LINA), and Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Group (mathematics), 05 social sciences, Linear system, Zero (complex analysis), Context (language use), 010103 numerical & computational mathematics, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], 01 natural sciences, LU decomposition, law.invention, Interval arithmetic, Combinatorics, law, 0502 economics and business, Interval (graph theory), Multiplication, 0101 mathematics, 050203 business & management, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties allow one constructing a LU decomposition of a generalized interval matrix A: the two computed generalized interval matrices L and U satisfy A = LU with equality instead of the weaker inclusion obtained in the context of classical intervals. Some potential applications of this generalized interval LU decomposition are investigated.

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