Robot stiffness theory reconsideration based on Schur complement eigenvalues: Extension to GSP dynamic stiffness evaluation

• Published in: Mechanism and machine theory 2023-04, Vol.182, p.105257, Article 105257
• Main Authors: Portman, V.T., Chapsky, V.S.
• Format: Article
• Language: English
• Subjects: Dynamic stiffness; Eigenvalues; Robots; Schur complement; Stiffness evaluation; Stiffness matrix
• ISSN: 0094-114X; 1873-3999
• DOI: 10.1016/j.mechmachtheory.2023.105257

•A Schur complement method is extended to the robot dynamic stiffness evaluation.•Dynamic stiffness values are defined through the Schur complement eigenvalues.•Natural frequencies vanish the Schur complement eigenvalues, and vice versa.•Frequency-dependent stiffness values allow the synthesis of performance indices.•Stiffness-related indices are used to compare the four analyzed GSP versions. The known robot stiffness evaluation algorithm based on the Schur complements’ eigenvalues (SCE) is refined and its application is extended to the dynamic stiffness problem. For the refinement, the Schur complements synthesis is reduced to standard transformations of the stiffness matrix into the triangular block matrices. In these terms, the Schur complements of the robot dynamic stiffness matrix are formulated for steady-state forced sinusoidal vibrations of a damping-free system. The interrelationships between the forced-frequency-dependent SCE and natural frequencies are established: the frequencies resulting in the SCE zero values are proven to be identical to the natural frequencies, and vice versa. The SCE of the dynamic stiffness matrix defines the range and extremes of the dynamic stiffness values that allow the building of the translational and rotational stiffness ellipsoids and formulation of the stiffness-related robot performance indices. In four application examples, the SCE-based dynamic stiffness values of the Gough-Stewart platform (GSP) on weightless supports are defined. The definitions allow the formulation of the dynamic stiffness-related performance indices which are used for quantitative evaluation and comparison of the GSP designs.