Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators (PMs), particularly those with over constrained subclosed loops, poses a challenge while ensuring numerical stability. This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis (MSA) and independent displacement coordinates (IDCs) extraction techniques. To begin, the closed-loop PM is transformed into an open-loop PM by eliminating constraints. A subassembly element is then introduced, which considers the flexibility of both rods and joints. This approach helps circumvent the numerical instability typically encountered with traditional constraint equations. The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix, utilizing multipoint constraint theory and singularity analysis, all unified within a single coordinate frame. Subsequently, the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix, alongside its elastostatic model. The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix, eliminating the need for additional equations. An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method. In conclusion, the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.