Algebraic Geometry for parallel and serial manipulators

The proposed research will focus on investigating serial and parallel manipulators, two of the main families of mechanical devices which are largely used in industrial applications. The main technique that will be used in this study is algebraic geometry, namely the study of polynomials equations. Although the community of researchers in mathematics and kinetics has been studying serial and parallel manipulators for many years, several basic and important questions still lack an answer. For example, at the moment a complete classification of serial manipulators constituted of six rods forming a closed chain and admitting one degree of freedom is not known. Similarly, we do not know how to characterize parallel manipulators that are formed by two rigid bodies connected by six rods via spherical joints and admitting one degree of freedom. In recent years, methods from real and complex algebraic geometry have been successfully applied to this kind of problems, and led to advances in both algorithmic and theoretical aspects. In particular, the classification of both parallel and serial manipulators with five rods that have one additional degree of freedom has been accomplished. In this research project we want to apply old and new techniques of algebraic geometry to still unsolved questions in kinematics, such as some (partial) characterization of parallel manipulators with more than six rods, or the understanding of those that admit infinitely many rods, keeping the same degree of freedom. From a more computational point of view, we want to investigate whether a better knowledge of the geometry of the objects involved can be translated into better algorithms to solve synthesis of such mechanisms. The success of this research project would provide better understanding of fundamental objects in kinematics, and strengthen the connection between algebraic geometry and more applied fields, in the hope that the massive amount of results that are available to algebraic geometers can be used effectively for the resolution of problems in kinematics.

The goal of the project ``Algebraic Geometry for parallel and serial manipulators'' was to apply techniques from algebraic geometry (the branch of mathematics that studies systems of polynomial equations) to the investigation of mechanical devices constituted of two rigid bodies connected by rods (parallel manipulators) or of bars connected consecutively by rotational joints (serial manipulators), in particular regarding their classification. Most of the attention was dedicated to devices that exhibit higher mobility than expected. Although these questions have risen interest in the last 70 years mainly due to the connection to robotics, their roots date back to at least two centuries; the desired contribution of this project was of theoretical kind, trying to shed some light on problems that still elude our understanding. Algebraic geometry has already been used for this purpose, but the peculiarity of this project was to try approaches involving tools that were not used in the existing literature. On one hand, together with our collaborators, we obtained new results about parallel manipulators, mainly concerning their classification, and we were able to shed new lights on old results. More specifically, we could discover that the mobility of a classical example of mechanisms, Bricard's flexible octahedra, can be explained by letting emerge a combinatorial structure on ``extremal'' configurations, and we could describe all the possible configuration curves of a certain family of parallel manipulators with six legs. Moreover, the understanding of a particular way of encoding configurations of points on the sphere up to rotations opened to us the way to results about rigid and flexible graphs on the sphere (or equivalently about mechanisms with rotational joints whose axes all pass through a common point); this technique may hopefully be used in the future to investigate other similar mechanisms. Unfortunately, we cannot say the same concerning serial manipulators, since the ideas and methods that we adopted did not turn out to be able to effectively attack the problems that we investigated as we hoped from the beginning. However, the techniques that we used proved to be useful also in nearby areas, as rigidity theory or the study of flexibility of polyhedra. These results were obtained sometimes by applying well-known tools in algebraic geometry that were, to our knowledge, not adopted before, and actually we believe that a very relevant part of our contribution has been to manipulate the original problems so that they become amenable to these classical techniques. Therefore, we are hopeful that this process can be repeated again for other similar questions, so that we can draw fully from the wealth of results that more ``abstract'' branches of mathematics have developed and that can lead to new perspectives on the area of kinematics and flexibility theory.