Interval Constraint Solving for Camera Control and Motion Planning

TL;DR

This paper introduces interval arithmetic-based numerical methods for soundly solving nonlinear inequalities and tuning problems, demonstrating their application in camera control and motion planning with promising experimental results.

Contribution

It presents faster, sound numerical methods using interval arithmetic for solving nonlinear inequalities and tuning problems, with applications in camera control and motion planning.

Findings

Interval methods are faster than classical symbolic approaches.

The methods effectively handle the guaranteed tuning problem for one-dimensional perturbations.

Experimental prototype demonstrates practical camera motion modeling.

Abstract

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.