Direct position analysis in analytical form of the parallel manipulator that features a planar platform supported at six points by six planes

The paper presents the direct position analysis of the parallel manipulator that features and end-effector with six coplanar spherical joints constrained to lie on x-y sliding tables. By properly controlling the postures of the sliding tables, the end-effector is endowed with up to six degrees of freedom with respect to the frame. The direct position analysis, aimed at determining all positions of the end-effector consistent with a given set of sliding table postures, is solved in analytical form by first devising a system of eight nonlinear compatibility equations in a corresponding number of unknowns. By an original elimination procedure, a final eighth-order univariate polynomial equation is obtained whose eight roots represent as many assembly configurations for the analyzed manipulator. A numerical example is finally reported which confirms the new theoretical results.