The Principal Axes Decomposition of Spatial Stiffness Matrices

Abstract

This paper presents an alternative decomposition of spatial stiffness matrices based on the concept of compliant axes. According to the congruence transformation of spatial stiffness, the coordinate-invariant aspects, which are referred to as the central principal components of the $6\times 6$ symmetric positive semidefinite matrices, can be derived uniquely. The proposed decomposition is free from the eigenvalue problems of the $6\times 6$ stiffness matrices so that both Plucker's ray and axis coordinates can be utilized to characterize the elastic system's force–deflection behavior. Hence, an arbitrary spatial stiffness matrix can be uniquely decomposed into two sets of orthogonal spring wrenches with finite and infinite pitches, respectively. The decomposed wrenches with finite pitches correspond to the stiffness’ wrench-compliant axes, along which linear deformations produce only wrenches parallel to them. As a result, three torsional and three screw springs are required, at the most, to realize a given spatial stiffness. Using the principal axes decomposition, some physical appreciations, such as the center of stiffness, the wrench-compliant axes, and the correspondence of compliance and stiffness, can be derived to reveal the inherent structure of spatial stiffness in an intuitive manner. In order to verify the effectiveness of the proposed method, two numerical examples are intensively studied with comparison to the eigenscrew decomposition. In addition, a potential application of the proposed stiffness decomposition method is also provided for the structural compliance modeling of flexible links in robot manipulators.