Many problems in robust control and motion planning can be reduced to either find a sound approximation of the solution space determined by a set of nonlinear inequalities, or to the guaranteed tuning problem'' as defined by Jaulin and Walter, which amounts to finding a value for some tuning parameter such that a set of inequalities be verified for all the possible values of some perturbation vector. A classical approach to solve these problems, which satisfies the strong soundness requirement, involves some quantifier elimination procedure such as Collins' Cylindrical Algebraic Decomposition symbolic method. Sound numerical methods using interval arithmetic and local consistency enforcement to prune the search space are presented in this paper as much faster alternatives for both soundly solving systems of nonlinear inequalities, and addressing the guaranteed tuning problem whenever the perturbation vector has dimension one. The use of these methods in camera control is investigated, and experiments with the prototype of a declarative modeller to express camera motion using a cinematic language are reported and commented.
The method presented by @cite_40 is strongly related to the one we present in the following, since they rely on usual interval constraint solving techniques to compute sound boxes for some constraint system. Starting from a seed that is known to belong to the solution space, they enlarge the domain of the variables around it in such a way that the new box computed is still included in the solution space. They do so by using local consistency techniques to find the points at which the truth value of the constraints change. Their algorithm is particularly well suited for the applications they target, the enlargement of tolerances. It is however not designed to solve the guaranteed tuning problem. In addition, it is necessary to obtain a seed for each connected subset of the solution space, and to apply the algorithm on each seed if one is interested in computing several solutions (e.g. to ensure representativeness of the samples).
@cite_40
1557280638
This paper introduces a new framework for extending consistent domains of numeric CSP. The aim is to offer the greatest possible freedom of choice for one variable to the designer of a CAD application. Thus, we provide here an efficient and incremental algorithm which computes the maximal extension of the domain of one variable. The key point of this framework is the definition, for each inequality, of an univariate extrema function which computes the left most and right most solutions of a selected variable (in a space delimited by the domains of the other variables). We show how these univariate extrema functions can be implemented efficiently. The capabilities of this approach are illustrated on a ballistic example.