Types of self-motions of planar Stewart Gough platforms
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We show that the self-motions of general planar Stewart Gough platforms can be characterized in the complex extension of the Euclidean 3-space by the movement of three platform points in planes orthogonal to the planar base (3-point Darboux motion) and a simultaneous sliding of three planes orthogonal to the planar platform through points of the base (3-plane Mannheim motion). Based on this consideration, we prove that all one-parametric self-motions of a general planar Stewart Gough platform can be classified into two types (type I DM and type II DM, where DM abbreviates Darboux Mannheim). We also succeed in presenting a set of 24 equations yielding a type II DM self-motion that can be computed explicitly and that is of great simplicity seen in the context of self-motions. These 24 conditions are the key for the complete classification of general planar Stewart Gough platforms with type II DM self-motions, which is an important step in solving the famous Borel Bricard problem.