Hexapods with Plane-Symmetric Self-Motions

Abstract

A hexapod is a parallel manipulator where the platform is linked with the base by six legs, which are anchored via spherical joints. In general, such a mechanical device is rigid for fixed leg lengths, but, under particular conditions, it can perform a so-called self-motion. In this paper, we determine all hexapods possessing self-motions of a special type. The motions under consideration are so-called plane-symmetric ones, which are the straight forward spatial counterpart of planar/spherical symmetric rollings. The full classification of hexapods with plane-symmetric self-motions is achieved by formulating the problem in terms of algebraic geometry by means of Study parameters. It turns out that besides the planar/spherical symmetric rollings with circular paths and two trivial cases (butterfly self-motion and two-dimensional spherical self-motion), only one further solution exists, which is the so-called Duporcq hexapod. This manipulator, which is studied in detail in the last part of the paper, may be of interest for the design of deployable structures due to its kinematotropic behavior and total flat branching singularities.