Evaluation and optimization of dynamic stiffness values of the PKMs: Collinear stiffness value approach

Dynamic stiffness of the parallel-kinematics machines is considered and a new engineering index - dynamic collinear stiffness value (DynCSV) - is proposed. The DynCSV presents an extension of a concept derived by Portman for evaluation of the static stiffness values. The DynCSV depends on robot-configuration and inertia parameters and presents the function of the vibration frequency. Mathematically, the DynCSV is a quadratic form associated with the real symmetric matrix of the Newton-Euler equation. The minimal and maximal values of the DynCSV give natural criteria for evaluation, limitation, and optimization of dynamic systems of machines and robots. In the frequency range, where the dynamic stiffness matrix has a semipositive definition, the minimum DynCSV is applied to construct a PKM workspace satisfying preliminary given stiffness limits and compute a virtual protective barrier, keeping the PKM mechanism from approaching both singular configurations and resonance frequencies. As an application example, the DynCSV-based local and global stiffness features of the Gough-Stewart platform mechanism are simulated, investigated, and visualized.