Stewart Gough platforms with self-motions

This project is devoted to Stewart Gough (SG) platforms with self-motions. To recall, a SG platform is a parallel manipulator, consisting of a moving platform, which is connected via six spherical-prismatic-spherical legs with the base, where only the prismatic joints are active. If the geometry of the SG platform and the six leg lengths are given, then the manipulator is in general rigid, but under particular conditions it can perform an n-parametric motion (n > 0), which is called self-motion. All self-motions of SG manipulators are also solutions to the still unsolved problem posed by the French Academy of Science for the Prix Vaillant of the year 1904, which reads as follows: "Determine and study all displacements of a rigid body in which distinct points of the body move on spherical paths." As the papers of Borel and Bricard were awarded prizes, although they only presented partial solutions, the problem is also known as Borel Bricard (BB) problem. It is well known, that SG platforms, which are singular (infinitesimal mobile or shaky) in every possible configuration, possess self-motions in each pose. These SG platforms are called architecturally singular and their designs are already well studied. Therefore, we are only interested in non-architecturally singular SG platforms with self-motions. Until now, only few examples of this type are known, as their computation is a very complicated task. In recent publications of the author, it was shown that based on the geometric-kinematic approach of redundancy, remarkable new results can be achieved in this field. The main aim of the project is the systematic determination, investigation and classification of SG platforms with self-motions, using the above mentioned redundancy property; i.e. we attach additional legs to the manipulator without changing its direct kinematics and singularity surface. Moreover, we want to prove a fundamental theorem on the BB problem, which was given by Duporcq 1898 without a proof. A further aim of the project, which is of great interest for mechanical and constructional engineers, is the definition of a geometrically meaningful index, evaluating how far a SG design is away from an architecturally singular design (which has a self-motion in each pose). In addition, we also want to determine the "best designs" of SG platforms with respect to this index. As SG platforms can be seen as a certain generalization of octahedra, we also want to think about the following self-suggesting question: Do there exist non-trivial geometric invariants of SG platforms under their self-motions, like it is the case for flexible polyhedra (total mean curvature and volume keep constant during the flex)? The submitted project, which belongs to basic research (BB problem) with direct application in robotics (SG platforms with self-motions), will be conducted at the research unit Differential Geometry and Geometric Structures of the Institute of Discrete Mathematics and Geometry (TU Vienna), under the guidance of Georg Nawratil, who conceived and formulated all parts of the project proposal.

A Stewart Gough (SG) platform is a parallel manipulator, consisting of a moving platform, which is connected via six spherical-prismatic-spherical legs with the base, where only the prismatic joints are active. If the geometry of the SG platform and the six leg lengths are given, then the hexapod is in general rigid, but under particular conditions it can perform an n-parametric motion (n > 0), which is called self-motion. It is well known, that SG platforms, which are singular (infinitesimal mobile or shaky) in every possible configuration, possess self-motions in each pose. These SG platforms are called architecturally singular and their designs are already well studied. Therefore the main aim of this project was the systematic determination, investigation and classification of non- architecturally singular SG platforms with self-motions. To do so, we adapted the basic idea of bonds originally introduced for the study of over- constrained closed chains with rotational joints for self-motions of parallel SG manipulators. Based on this bond theory we were able to obtain the following results: 1. Determination of all hexapods of SG type with mobility 2 or higher. This also includes a complete listing of pentapods with 2-dimensional self-motions. 2. Construction of mobile hexapods possessing a configuration curve of maximal degree by using liaison technique stemming from the theory of algebraic curves. 3. It was already known that the maximal finite number of points running on sphere is 20. We proved that self-motions of the resulting icosapods have to be line-symmetric motions. Moreover, we gave the first real example by using results on spectrahedra obtained in convex algebraic geometry. 4. We achieved novel results for congruent/equiform SG platforms with self-motions as well as mobile point-symmetric hexapods. Without using the theory of bonds, we characterized all SG platforms possessing pure translational self-motions and plane-symmetric self-motions, respectively. In addition we studied a famous theorem about rigid-body motions with spherical trajectories, which was stated by Ernest Duporcq in 1898 without giving a rigorous proof. We demonstrated that this theorem is not correct and presented a revised version of it. Moreover, we clarified the connection between Duporcq`s theorem and architecturally singular SG platforms. Finally, we were also able to come up with a novel motion (circular Darboux 2-motion) where all points of the Euclidean 4-space have circular paths.