Christophe Jermann

Decomposition of Geometric Constraint Systems: a Survey

Signiflcant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial flelds like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition techniques that transform a large geometric constraint system into a set of smaller ones. In this paper, we propose a survey of the decomposition techniques for geometric constraint problems a . We classify them into four categories according to their modus operandi, establishing some similarities between methods that are traditionally separated. We summarize the advantages and limitations of the difierent approaches, and point out key issues for meeting industrial requirements such as generality and reliability.

Global Optimization and Constraint Satisfaction

Constraint Satisfaction.- Efficient Pruning Technique Based on Linear Relaxations.- Inter-block Backtracking: Exploiting the Structure in Continuous CSPs.- Accelerating Consistency Techniques and Pronys Method for Reliable Parameter Estimation of Exponential Sums.- Global Optimization.- Convex Programming Methods for Global Optimization.- A Method for Global Optimization of Large Systems of Quadratic Constraints.- A Comparison of Methods for the Computation of Affine Lower Bound Functions for Polynomials.- Using a Cooperative Solving Approach to Global Optimization Problems.- Global Optimization of Convex Multiplicative Programs by Duality Theory.- Applications.- High-Fidelity Models in Global Optimization.- Incremental Construction of the Robots Environmental Map Using Interval Analysis.- Nonlinear Predictive Control Using Constraints Satisfaction.- Gas Turbine Model-Based Robust Fault Detection Using a Forward - Backward Test.- Benchmarking on Approaches to Interval Observation Applied to Robust Fault Detection.

A Constraint Programming Approach for Solving Rigid Geometric Systems

This paper introduces a new rigidification method -using interval constraint programming techniques- to solve geometric constraint systems. Standard rigidification techniques are graph-constructive methods exploiting the degrees of freedom of geometric objects. They work in two steps: a planning phase which identifies rigid clusters, and a solving phase which computes the coordinates of the geometric objects in every cluster. We propose here a new heuristic for the planning algorithm that yields in general small systems of equations. We also show that interval constraint techniques can be used not only to efficiently implement the solving phase, but also generalize former ad-hoc solving techniques. First experimental results show that this approach is more efficient than systems based on equational decomposition techniques.

Interval-based projection method for under-constrained numerical systems

This paper presents an interval-based method that follows the branch-and-prune scheme to compute a verified paving of a projection of the solution set of an under-constrained system. Benefits of this algorithm include anytime solving process, homogeneous verification of inner boxes, and applicability to generic problems, allowing any number of (possibly nonlinear) equality and inequality constraints. We present three key improvements of the algorithm dedicated to projection problems: (i) The verification process is enhanced in order to prove faster larger boxes in the projection space. (ii) Computational effort is saved by pruning redundant portions of the solution set that would project identically. (iii) A dedicated branching strategy allows reducing the number of treated boxes. Experimental results indicate that various applications can be modeled as projection problems and can be solved efficiently by the proposed method.

A branch and prune algorithm for the computation of generalized aspects of parallel robots

Parallel robots enjoy enhanced mechanical characteristics that have to be contrasted with a more complicated design. In particular, they often have parallel singularities at some poses, and the robots may become uncontrollable, and could even be damaged, in such configurations. The computation of the connected components in the set of nonsingular reachable configurations, called generalized aspects, is therefore a key issue in their design. This paper introduces a new method, based on numerical constraint programming, to compute a certified enclosure of the generalized aspects. Though this method does not allow counting their number rigorously, it constructs inner approximations of the nonsingular workspace that allow commanding parallel robots safely. It also provides a lower-bound on the exact number of generalized aspects. It is moreover the first general method able to handle any parallel robot in theory, though its computational complexity currently restricts its usage to robots with three degrees of freedom. Finally, the constraint programming paradigm it relies on makes it possible to consider various additional constraints (e.g., collision avoidance), making it suitable for practical considerations.

Certified Parallelotope Continuation for One-Manifolds

Starting from an initial solution, continuation methods efficiently produce a sequence of points on a manifold typically defined as the solution set of an underconstrained system of equations. They have a wide range of applications ranging from curve plotting to polynomial root-finding by homotopy. However, classical methods cannot guarantee that the returned points all belong to the same connected component of the manifold, i.e., they may jump from one component to another. Trying to overcome this issue has given birth to several sophisticated heuristics on the one hand and to guaranteed methods based on rigorous computations on the other hand. In this paper we introduce a new rigorous predictor corrector continuation method based on interval computations. Its novelty lies in the fact that it uses parallelotopes as defined in A. Goldsztejn and L. Granvilliers, A new framework for sharp and efficient resolution of NCSP with manifolds of solutions, Constraints, 15 (2010), pp. 190--212, to enclose consecuti...

On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach

The global resolution of constrained non-linear bi-objective optimization problems (NLBOO) aims at covering their Pareto-optimal front which is in general a one-manifold in

Inter-block backtracking: exploiting the structure in continuous CSPs

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