Classification of all self-motions of the original Stewart-Gough platform

Abstract

The ‘original’ Stewart-Gough (S-G) platform is a parallel 6-6 mechanism with both bases isosceles triangles cut at vertices. All six points of the upper base are connected by telescopic legs with six points of the lower base by spherical joints. If the length of all six legs remains constant then the S-G platform is stiff (a structure), i.e. it has no movability in general. Exceptions from this rule are called self-motions of the S-G platform and they can appear only in very special circumstances. They are known as Borel-Bricard motions, because they have been studied by Borel and Bricard at the beginning of this century [Borel E., Mémoire sur les déplacements à trajectoires sphériques. Mémoires présentés par divers savants, Paris, 1908, 33(2), 1–128. Bricard R., Mémoire sur les déplacements à trajectoires sphériques. Journal de l'École Polytechnique, 1906, 11(2), 1–96 1,2. Some classes of Borel-Bricard motions are known, but their classification is far from complete. In this paper we have determined all self-motions of the ‘original’ S-G platform. As a byproduct new types of Borel-Bricard motions have been discovered.During this task one has to use a computer for three different purposes. At first algebraic equations of self-motions have to be determined. For this purpose we have used Maple V on a workstation. The use of a workstation is necessary, because we had to handle expressions having up to 20,000 terms. After equations of the self-motion have been derived one has to visualize the result, which means to plot trajectories of points of the platform and corresponding positions of the upper base. Here we have two possibilities. If the motion can be parameterized, we can plot trajectories by a suitable graphical system. If the motion cannot be parameterized (this can happen because it is algebraic of order up to 8), we have to compute numerically some locations of the upper base of the platform and then to interpolate them. We did not go tot he last step because this would be beyond the scope of the paper. As a result we have shown that self-motions of the ‘original’ S-G platform can be translatory motions, pure rotations, generalized screw motions with fixed axis, spherical four-bar mechanisms or more general space motions. One of the special cases of the general self-motion is a space motion with three circular trajectories, which is not spherical. This motion yields a new spatial mechanism, which is a spatial analogy of the well known planar or spherical four-bar. It seems that Bricard already knew about the existence of such a motion.