Stiffness Synthesis of a Variable Geometry Planar Robot

This paper addresses the problem of task-based stiffness synthesis of a variable geometry three DOF (Degrees Of Freedom) planar robot. The synthesis considers the case where the robot has a limited number of free geometric parameters and constant actuator stiffness coefficients. This defines twenty problems of stiffness synthesis, in which, three parameters of the stiffness matrix are controlled according to task requirements. These problems are modeled as systems of polynomials in the free geometric parameters of the robot’s base platform. Using Gröbner bases, the solubility of these polynomial systems is characterized. It is shown that arbitrary desired values of the Cartesian stiffness elements (kxx and kyy) are unattainable when only the geometry of the base platform is variable. An example of synthesizing three stiffness elements of the planar robot is solved and shown to have at most 48 solutions in the complex plane. In a numerical case study, sixteen real solutions are obtained, of which only eight are nonsingular.