A Joint-Space Parametric Formulation for the Vibrations of Symmetric Gough-Stewart Platforms

Natural frequencies of a Symmetric Gough-Stewart Platform (SGSP) mechanically limit its bandwidth and precision e.g. in CNCs or optical collimation systems. Hence, the required vibrational behavior at the neutral configuration of an SGSP can be regarded as an essential property to be optimized. However, due to the complexity of its geometry, the analysis of the vibrational behavior, using analytical methods, is quite challenging and in the literature a complete joint-space formulation of SGSP vibrations has not yet been addressed. In this paper, we present an analytical and parametric formulation of this problem in the joint space. We parametrically formulate the Jacobian matrix, the linearized equations of motion and calculate the eigenvectors and eigenfrequencies in terms of the design variables of the system. The parametric model presented in this study can be directly employed for design, optimization and control of SGSPs. It is concluded that for SGSPs, the joint-space formulation gives additional insights to the modal properties complementing the Cartesian-space analysis.