Abstract
The virtual joint method (VJM) cannot calculate the strain energy stored in each rod. In order to solve the problem, a modeling method of the elastostatic stiffness was investigated for the UP/UPS parallel manipulators (PMs), taking the example of the 6-SPS PM. The modeling method was based on screw theory, Castigliano’s theorem, and strain energy (where U, P, and S, respectively, denote universal, prismatic, and spherical joints). First, the actuator and constraint wrenches of the mechanism were obtained by screw theory. Second, compact limb stiffness matrices were obtained in terms of strain energy and Castigliano’s second theorem. Finally, analytic expressions for the overall stiffness matrix of the mechanism and the amplitudes of the actuator force were obtained by adopting the virtual work principle and the balance equation for the mobile platform. All relative errors between the results of the analytical model and the finite element model are below 2%, which validates the effectiveness of the elastostatic stiffness model. The virtual work index was adopted to evaluate the stiffness performance of the mechanism, and the results show that the stiffness is not only related to position and orientation but also closely related to the directions of external loads. It is also demonstrated that the method has general adaptability for the stiffness analysis for the US/UPS PMs, laying the foundation for further reasonable dynamic design and optimization of such manipulators.
Highlights
In order to solve the problem of the fact that the virtual joint method cannot calculate the strain energy stored in each rod, the main content of this paper is to investigate a modeling method for elastostatic stiffness analysis for the UP/UPS parallel manipulators (PMs), taking the example of the 6-SPS PM, which is based on the screw theory, Castigliano’s theorem, and strain energy method. rough calculating the strain energy stored in each rod, the stiffness matrices of each rod and PMs can be obtained, which have a clear physical meaning and concise expression
A modeling method is investigated for elastostatic stiffness analysis for the UP/UPS PMs, taking the example of the 6-SPS PM, which is based on the screw theory, Castigliano’s theorem, and strain energy method
E global stiffness matrix was derived. e virtual work index was adopted to evaluate the stiffness of the 6-SPS PM, and the VWI was used to measure the deformation of the 6SPS PM in resisting external loads in specific directions. e main conclusions of the study are as follows: (1) the theoretical method used in this paper is much simplified and has many limitations, the results show that the relative errors of the theoretical model and finite element model are less than 2%, which verifies the correctness of the analytical model
Summary
PMs with closed kinematic structures perform well in terms of accuracy, rigidity, and payload capacity and show high potentials to deal with numerous tasks [1,2,3]. erefore, the detailed analyses of PMs are extremely important, including the workspace [4], kinematics [5], dynamics [6], elastostatics [7], and motion/force transmissibility [8]. Adam et al analyzed and evaluated the stiffness matrix of a double Stewart PM truss adopting an analytical method [28]. Since the rod of two-force only subjected to tension and compression, the virtual joint method is used to analyze the elastostatic stiffness performance of PMs [32, 33]. In order to solve the problem of the fact that the virtual joint method cannot calculate the strain energy stored in each rod, the main content of this paper is to investigate a modeling method for elastostatic stiffness analysis for the UP/UPS PMs, taking the example of the 6-SPS PM, which is based on the screw theory, Castigliano’s theorem, and strain energy method.
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