This paper presents a general equivalent approach to solve the stiffness modeling or load-deformation problem of non-redundant parallel mechanisms. Based on the principal axes decomposition of structure compliance matrices, an equivalent six-degrees-of-freedom (6-DOFs) serial mechanism is established to approximate the load-deformation behavior of each flexible link in the mechanism. Hence, each limb of the parallel mechanism can be equivalent to a serial redundant rigid body mechanism with passive elastic joints, and the load-deformation problem can be transformed to the equilibrium configuration calculation of the equivalent mechanism. The main advantage of the proposed method is that the robotic kinematics and statics, rather than the elastic mechanics, can be directly adopted to solve the equilibrium configuration of the parallel mechanism under external load. Besides, a closed form solution of the corresponding deformation can be obtained, which can be solved by the gradient-based searching algorithm. Therefore, the final deformation will no longer be linear to the external load, which makes this method more accurate and more suitable for the deformation prediction and compensation in real industrial working conditions. In order to verify the effectiveness and correctness of this method, a 3PRRU parallel manipulator will be introduced as an example, to compare the load-deformation results with the finite element analysis (FEA) simulation and matrix calculation methods, so the nonlinearity feature can be shown in an intuitive manner.
Journal of Mechanisms and Robotics Open Issues