Computational Algebraic Geometry with Applications to Robotics and Control Theory - May 2024

Title

Computational Algebraic Geometry with Applications to Robotics and Control Theory.

Speakers

Antonio Tornambè, Laura Menini, Corrado Possieri, Tor Vergata University of Rome

When and Where

• May 2nd, 6th, 9th, 13th, 20th, 27th 2024, from 15:00 to 17:00 (12 hours, 6 lectures)
  

All lectures in sala Riunioni (Ingegneria della Informazione building, ground floor, room AT 07)

Main Topics

The purpose of the course is to show how the methods of algebraic geometry can be used to solve various problems in the fields of robotics and control theory, including in the last topic both dynamic systems analysis and control design. It is not assumed that the student already knows algebraic geometry; only a basic knowledge of calculus and linear algebra is required. The course is organized to be self-contained, and many concepts are illustrated by examples; in particular, it will be shown how calculations can be performed by MACAULAY2, a CAS specialized in algebraic geometry. Algorithms presented by algebraic geometry provide methods for “exactly solving” systems of polynomial equalities and inequalities, which are very useful for the design of control laws and observers.

The first few lectures will be devoted to specifying exactly what is meant by “exact solution” in the framework of algebraic geometry. First, a concise but rigorous exposition of all the basics of algebraic geometry is given. As for robotics and control theory, it is assumed that the reader is familiar with the fundamentals of dynamical systems theory and some standard control design tools, such as the basic Lyapunov stability theorems and, as for linear systems, the structural properties and design principles of observers.

The second part of the course will be devoted to applications: Solution of Polynomial Systems, Inverse Kinematics of Robot Arms, Observer Design, Linearization by Immersion, and Motion Planning.