# 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 r**otations 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**.