Polynomial Parameterizations in Robust Control and Visual Servoing

Abstract

This talk describes some recent results that we have proposed in control and robotics based on the use of polynomial parameterizations. In particular, in the first part of the talk we consider the problem of establishing robust stability of linear systems affected by polytopic parametric uncertainty. For this problem we propose the class of homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs) which is motivated by the property that a polytope of matrices is stable if and only there exists a HPD-QLF. The main result is a condition for determining the sought HPD-QLF which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation of homogeneous matricial forms and the Lyapunov matrix equation. In the second part of the talk we consider the problem of realizing visual servoing for robot manipulators taking into account constraints such as camera visibility and obstacle avoidance while minimizing a cost function such as spanned image area and trajectory length. To solve this problem we propose a path-planning scheme based on polynomial parameterizations of the camera trajectory in the six dimensional space obtained through the introduction of a parameter-dependent object reconstruction and an extension of the Euler parameters. This scheme allows one to consider all possible camera trajectories and establish their feasibility through simple computations.