UNIVERSITÉ DE NICE - SOPHIA ANTIPOLIS - UFR Sciences

In this thesis we are interested in a particular class of problems which frequently appear in robotics (and many other areas as chemistry, molecular biology, Computer-Aided Design (CAD), and aeronautics). They are systems of distance equations with uncertainties. Uncertain values mean values which are not exactly determined but are bounded by well-known limits. These values are represented as intervals, and frequently come from measurements. In a model, these values appear as existentially quantified parameters. Solving such a problem with uncertainties means to find a set of solutions taking into account these inaccuracies in order to obtain certified answers (in the way that no solution is lost). The aim of the works contained in this thesis is to solve systems of distance equations with uncertainties in their parameters as accurately as possible, combining techniques from Constraint Programming and Interval Analysis communities. A common approximation for the solutions for these types of problems is to replace parameters with interval values by real numbers, and to solve the problem without considering the inaccuracies. We show that this approximation is not convenient, especially when certified solutions are required (for example for safely reasons for a Surgical Robot). In a first phase, we propose a special Branch and Prune algorithm with conditional bisection which is able to compute a rough approximation of each continuum of solutions for a given problem. A rough approximation (a box) is not enough in all the cases, thus a sharp approximation (a set of boxes) describing continuous solution sets is often required. We show that this approximation must consider an inner box test in order to detect large parts of the search space containing only solutions to the problem. Using inner box tests not only reduces the number of generated boxes but also provides more information about the geometry of the solutions set. We propose and compare various inner box tests for distance equations with uncertainties. When a single solution point belonging to a continuum of solutions is given, an inner box around this point and totally included within the continuum of solutions may be very interesting for tolerance issues. For this reason we propose a strategy for building such a box based on theoretical results of Modal Interval Analysis combined with a well-known technique of Constraint Programming called