An analytical approach to the stiffness calculation of statically indeterminate 3-DOF delta parallelogram-type parallel robot

This paper introduces an analytical approach entailing a clear physical meaning for calculating the stiffness of the 3-DOF Delta parallel robot. By defining suitable linear and torsion springs, the stiffness of the robot components in different postures and the stiffness coefficients of the indeterminate parallelogram under conceivable wrenches are comprehensively obtained. Moreover, the stiffness coefficients for each limb chain are assigned, and then by considering the assigned stiffness coefficients and static equilibriums the indeterminate problem of 3-RUU DPRs is solved. By reformulating the forward kinematics which takes into account the deflected components and based on the elastic properties, the deflection in each manipulator component is calculated and the amount of moving platform twist is obtained. The proposed approach is validated by the FEA method and results reveal that the maximum error of the proposed method is 0.1489 degrees in the orientation and 0.1420 mm in translation compared to the FEA. Then to avoid the problem of different units of translation and orientation in indexing, the virtual work concept is used for assigning an index number.