Reliable bounding of the implicitly defined sets with applications to robotics

Abstract

The article considers approximation of solution sets to indeterminate systems for non-linear equations. We developed a method to obtain inner and outer approximations of such sets. The method uses interval analysis techniques and relies on the Krawczyk operator and its modification based on the Baumann bicentered interval extension. We developed software that efficiently constructs and visualizes the computed approximations of the solution sets for systems of non-linear equations. The software is implemented in Python programming language and is available for free download and use. Finding the solution set of indeterminate systems of equations finds at least one important application in practice: bounding the workspace of a robotic manipulator. We perform experiments for the 2-RPR robot and evaluate the tightness of the obtained approximations. As expected the Krawczyk bicentered method noticeably improves the quality of approximations as compared with the classical Krawczyk operator.