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Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this approach is limited as it is based on parameter space discretization and therefore cannot be considered a proof of feasibility. To overcome this limitation, set-membership methods based on interval analysis, guaranteed integration, and contractor programming have proven their efficiency. Guaranteed integration algorithms can predict the possible trajectories of a system initialized in a given set in the form of tubes of trajectories. The contractor programming consists in removing the trajectories violating predefined constraints from a system’s tube of possible trajectories. Our contribution consists in merging both approaches to allow for the usage of differential constraints in a contractor programming framework. We illustrate our method through examples related to robotics. We also released an open-source implementation of our algorithm in a unified library for tubes, allowing one to combine it with other constraints and increase the number of possible applications.

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.

International audience; This paper deals with the localization problem of a robot in an environment made of indistinguishable landmarks, and assuming the initial position of the vehicle is unknown. This scenario is typically encountered in underwater applications for which landmarks such as rocks all look alike. Furthermore, the position of the robot may be lost during a diving phase, which obliges us to consider unknown initial position. We propose a deterministic approach to solve simultaneously the problems of data association and state estimation, without combinatorial explosion. The efficiency of the method is shown on an actual experiment involving an underwater robot and sonar data.

In this paper we present a reliable method to verify the existence of loops along the uncertain trajectory of a robot, based on proprioceptive measurements only, within a bounded-error context. The loop closure detection is one of the key points in simultaneous localization and mapping (SLAM) methods, especially in homogeneous environments with difficult scenes recognitions. The proposed approach is generic and could be coupled with conventional SLAM algorithms to reliably reduce their computing burden, thus improving the localization and mapping processes in the most challenging environments such as unexplored underwater extents. To prove that a robot performed a loop whatever the uncertainties in its evolution, we employ the notion of topological degree that originates in the field of differential topology. We show that a verification tool based on the topological degree is an optimal method for proving robot loops. This is demonstrated both on datasets from real missions involving autonomous underwater vehicles and by a mathematical discussion.

This paper presents a new approach to bounded-error state estimation involving time uncertainties. For a given bounded\ud observation of a continuous-time non-linear system, it is assumed that neither the values of the observed data nor their\ud acquisition instants are known exactly. For systems described by state-space equations, we prove theoretically and demonstrate\ud by simulations that the proposed constraint propagation approach enables the computation of bounding sets for the systems’\ud state vectors that are consistent with the uncertain measurements. The bounding property of the method is guaranteed even\ud if the system is strongly non-linear. Compared with other existing constraint propagation approaches, the originality of the\ud method stems from our definition and use of bounding tubes which enable to enclose the set of all feasible trajectories inside\ud sets. This method makes it possible to build specific operators for the propagation of time uncertainties through the whole\ud trajectory. The efficiency of the approach is illustrated on two examples: the dynamic localization of a mobile robot and the\ud correction of a drifting clock.

International audience; Localization for underwater robots remains a challenging issue. Typical sensors, such as Global Navigation Satellite System (GNSS) receivers, cannot be used under the surface and other inertial systems suffer from a strong integration drift. On top of that, the seabed is generally uniform and unstructured, making it difficult to apply Simultaneous Localization and Mapping (SLAM) methods to perform localization. Reliable Robot Localization presents an innovative new method which can be characterized as a raw-data SLAM approach. It differs from extant methods by considering time as a standard variable to be estimated, thus raising new opportunities for state estimation, so far underexploited. However, such temporal resolution is not straightforward and requires a set of theoretical tools in order to achieve the main purpose of localization. This book not only presents original contributions to the field of mobile robotics, it also offers new perspectives on constraint programming and set-membership approaches. It provides a reliable contractor programming framework in order to build solvers for dynamical systems. This set of tools is illustrated throughout this book with realistic robotic applications.

National audience; Ces dernières décennies, la robotique mobile s’est développée largement dans des milieux structurés et déjà cartographiés. Dans desenvironnements inconnus et non-structurés, comme les planètes lointaines, les volcans, des grottes profondes, des zones irradiées, desveines karstiques, les bâtiments en feu, les fonds marins, la robotique change de nature.Non seulement, elle devient indispensable car l’humain peut difficilement intervenir de façon sécurisée, mais souvent l’opérateur ne peut plus aider les robots par téléopération. Ces derniers doivent alors posséder un maximum d’autonomie et d’intelligence afin de pouvoir accomplir leur mission. On parle alors de robotique exploratoire car le robot doit cartographier son environnement, prendre des décisions, se localiser et être capable de revenir.

This paper proposes a new method for guaranteed integration of state equations. Within this framework, the variables of interest are trajectories submitted to both arithmetic and differential equations. The approach consists in formalizing a problem thanks to a constraint network and then apply these constraints to sets of trajectories. The contribution of the paper is to provide a reliable framework to enclose the solutions of these differential equations. Its use is shown to be simple, more general and more competitive than existing approaches dealing with guaranteed integration, especially when applied to mobile robotics. The flexibility of the developed framework allows to deal with non-linear differential equations or even differential inclusions built from datasets, while considering observations of the states of interest. An illustration of this method is given over several examples with mobile robots. A contractor-based approach is proposed for guaranteed integration of state equations.The framework is based on the use of tubes as envelopes of feasible trajectories.A dedicated differential contractor is provided to deal with dynamical systems.The use of tubes is well suited for multi-differential and non-linear equations.An illustration of this approach is given over several mobile robotics examples.