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.

中文翻译：

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US / UPS并联机械手的静态静刚度分析

虚拟关节法（VJM）无法计算存储在每个杆中的应变能。为了解决该问题，以6-SPS PM为例，研究了UP / UPS并联机械手（PM）的弹力刚度建模方法。建模方法基于螺丝理论，Castigliano定理和应变能（其中U，P和S分别表示万向节，棱柱节和球形节）。首先，通过螺丝理论获得了机构的致动器和约束扳手。其次，根据应变能和Castigliano第二定理获得了紧凑的肢体刚度矩阵。最后，采用虚拟工作原理和移动平台的平衡方程，得到机构整体刚度矩阵和执行机构力振幅的解析表达式。分析模型和有限元模型的结果之间的所有相对误差都在2％以下，这证明了弹性静刚度模型的有效性。通过虚拟工作指标对机构的刚度性能进行评估，结果表明刚度不仅与位置和方向有关，而且与外部载荷的方向密切相关。还证明了该方法对于US / UPS PM的刚度分析具有一般适应性，为进一步合理地设计和优化此类机械手奠定了基础。分析模型和有限元模型的结果之间的所有相对误差都在2％以下，这证明了弹性静刚度模型的有效性。通过虚拟工作指标对机构的刚度性能进行评估，结果表明刚度不仅与位置和方向有关，而且与外部载荷的方向密切相关。还证明了该方法对于US / UPS PM的刚度分析具有一般适应性，为进一步合理地设计和优化此类机械手奠定了基础。分析模型和有限元模型的结果之间的所有相对误差都在2％以下，这证明了弹性静刚度模型的有效性。通过虚拟工作指标对机构的刚度性能进行评估，结果表明刚度不仅与位置和方向有关，而且与外部载荷的方向密切相关。还证明了该方法对于US / UPS PM的刚度分析具有一般适应性，为进一步合理地设计和优化此类机械手奠定了基础。通过虚拟工作指标对机构的刚度性能进行评估，结果表明刚度不仅与位置和方向有关，而且与外部载荷的方向密切相关。还证明了该方法对于US / UPS PM的刚度分析具有一般适应性，为进一步合理地设计和优化此类机械手奠定了基础。采用虚拟工作指标评价机构的刚度性能，结果表明刚度不仅与位置和方向有关，而且与外部载荷的方向密切相关。还证明了该方法对于US / UPS PM的刚度分析具有一般适应性，为进一步合理地设计和优化此类机械手奠定了基础。