Dynamic Isotropy in 6-DOF Kinematically Constrained Platforms by Three Elastic Nodal Joints

The principle of kinematic design has a wide range of applications e.g. from optical mirror mounts to parallel robots. Despite the importance of dynamic isotropy in the optimization of dynamic performance, a thorough analysis of dynamic isotropy in different kinematic arrangements has not yet been addressed in the literature. Dynamic isotropy, leading to equal eigenfrequencies, is a powerful optimization measure. In this paper, we present fully-parametric solutions for obtaining dynamic isotropy in general 3D platforms kinematically constrained by three elastic nodal joints in 6 DOFs. It is analytically shown that there exist two possible kinematic arrangements which are described by 3-2-1 and 2-2-2 kinematic node spaces. Both kinematic arrangements are studied with respect to their Jacobian formulation, Jacobian singularity and stiffness decoupling. It is proven that decoupling of stiffness matrices and accordingly dynamic isotropy for both kinematic arrangements are possible. Subsequently, conditions concerning geometry, stiffness and inertia in order to obtain dynamic isotropy are parametrically established. Finally, it is numerically demonstrated that the presented formulation is general enough even for being directly used, as a novel and efficient approach, in order to design dynamically isotropic 6-6 Gough–Stewart platforms (6-6 hexapods).