Self-motions of parallel manipulators associated with flexible octahedra

A Stewart Gough (SG) platform is a parallel manipulator consisting of a moving platform, which is connected via six Spherical-Prismatical-Spherical (SPS) legs with the base. Moreover a SG platform is called planar if the six platform anchor points mi (i = 1, . . . ,6) are located in a plane and if the corresponding base anchor points Mi are coplanar as well (see Fig. 1a). Note that a SG platform is controlled only by a variation of the six leg lengths (⇒ P-joints are active, S-joints are passive). If all P-joints are locked, a SG platform is in general rigid in one of its 40 possible assembly modes (cf. HUSTY [1], DIETMAIER [2]). But, it can even be the case that the manipulator can perform an n-parametric motion (n > 0), which is called self-motion. Moreover, all self-motions of SG platforms are solutions to the famous Borel Bricard problem (cf. BOREL [3], BRICARD [4], HUSTY [5], VOGLER [6]) which is still unsolved.