Stiffness Synthesis of a Variable Geometry Six-Degrees-of-Freedom Double Planar Parallel Robot

Abstract

In this paper, we address the stiffness synthesis problem of variable geometry double planar parallel robots. For a desired stiffness matrix, the free geometrical variables are calculated as a solution of a corresponding polynomial system. Since in practice the set of free geometrical variables might be deficient, the suggested solution addresses also the case where not all stiffness matrix elements are attainable. This is done through the use of Gröbner bases that determine the solvability of the stiffness synthesis polynomial systems and by transforming these systems into corresponding eigenvalue problems using multiplication tables. This method is demonstrated on a novel variable geometry double planar six-degrees-of-freedom robot having six free geometric variables. A solution of the double planar stiffness synthesis problem is obtained through decomposing its stiffness matrix in terms of the stiffness matrices of its planar units. An example of this procedure is presented in which synthesizing six elements of the robot's stiffness matrix is obtained symbolically and validated numerically yielding 384 real solutions.