Schrödinger Fellowship J4253

The Schrödinger Programme, funded by the Austrian Science Fund (FWF), gives researchers the possibility to spend a period of research time outside Austria, followed by a return phase in an Austrian institution.

In my case, I spent two years (2019-2020) at the International School for Advanced Studies (SISSA) in Trieste (Italy). In mid February 2021, I started by return phase at the Radon Institute for Computational and Applied Mathematics (RICAM) in Linz.

AIM

The goal of the project Erwin Schrödinger Fellowship J4253 is to use techniques from algebraic geometry in order to study parallel and serial manupulators, in particular those that exhibit unexpected mobility. Though diverse communities of researchers have been investigating this topic for many years, many fundamental problems are still open or only partially solved: among others, the complete characterization of mechanisms displaying exceptional mobility (namely, mobility higher than the one expected for a general machine of the same kind), the recognition of rigidity, or the design of algorithms for the synthesis of devices interpolating a given sequence of poses.

PROBLEM DESCRIPTION

Parallel manipulators are mechanical devices constituted of two rigid bodies, called the base and the platform, that are connected by rod, called legs. Each leg is anchored to the base and to the platform via spherical joints, namely joints that allow any rotation around them. It is known that parallel manipulators with six legs, also called hexapods or Stewart-Gough platforms, are in general rigid: there are only finitely many possible positions of the platform with respect to the base. Hence, it is of interest to understand which hexapods are mobile; in its full generality, this question does not have an answer yet.

Our aim is to obtain necessary and sufficient conditions on the geometry of these mobile objects. The tools we adopt are suitable also for investigating rigidity and mobility of parallel manipulators with infinitely many legs, or polyhedra.

Serial manipulators are, instead, mechanisms consisting of rods connected via rotational joints, namely joints that allow rotations around a fixed axis. Many mathematical aspects of the possible mobility of serial manipulators are still unknown: for example, we do not possess a classification of those closed serial chains (namely, serial manipulators that form a loop) that are mobile and are constituted of six joints (chains constituted of more than six chains are always mobile).

TECHNIQUES

To attack these problems, we adopt techniques from algebraic geometry.
Algebraic geometry is the study of solutions of systems of polynomial equations by means of algebra and geometry. Most of parallel and serial manipulators' configuration spaces can be modelled via polynomial equations and so the tools of algebraic geometry can be used to understand the geometry of mobile and rigid mechanisms.

RESULTS

This is a list of the results obtained so far:

- M. Gallet, G. Grasegger, J. Schicho, Counting realizations of Laman graphs on the sphere.
- M. Gallet, G. Grasegger, J. Legerský, J. Schicho, On the existence of paradoxical motions of generically rigid graphs on the sphere.
- M. Gallet, G. Grasegger, J. Legerský, J. Schicho, Combinatorics of Bricard's octahedra.
- M. Gallet, G. Grasegger, J. Legerský, J. Schicho, Zero-sum cycles in flexible polyhedra.
- H.-C. Graf von Bothmer, M. Gallet, J. Schicho, Hexapods with a small linear span.
- M. Gallet, J. Schicho, A new line-symmetric mobile infinity-pod.
- M. Gallet, G. Grasegger, J. Legerský, J. Schicho, Zero-sum cycles in flexible non-triangular polyehdra.