Vibrations and Dynamic Isotropy in Hexapods - Analytical Studies

Abstract: The present work was initiated based on an industrial demand for designing a high-bandwidth hexapod of an advanced large optical telescope. In this dissertation, we have generalized this industrial problem to fully-parametric models of the hexapod vibrations as well as analytical studies on dynamic isotropy in parallel robots, which can be directly used in any hexapod application. Hexapods (also known as Gough-Stewart Platforms, GSPs), being the most widely used type of 6-DOF parallel robots, are employed in numerous modern applications. This work firstly establishes a comprehensive and fully-parametric model for the vibrations in hexapods at symmetric configurations. We have developed three models presenting the Cartesian-space formulation and the joint-space formulation of the hexapod vibrations as well as a refined model taking also the inertia of the struts into account. It is noteworthy that such complete analytical models were not available in the literature prior to the present work. In particular, it is for the first time that the inertia of the struts is being taken into account with a full-analytical approach. Kinematics and accordingly the Jacobian of hexapods are developed parametrically. The equations of motion are formulated and linearized based on a Lagrangian dynamics approach. Inertia, stiffness and damping matrices are also parametrically formulated. The eigenvectors and eigenfrequencies are then established in both the Cartesian and joint spaces. By introducing the inertia of the struts, despite the apparent symmetric geometry, the equivalent inertia matrix in the Cartesian space turns out to be a non-diagonal matrix. In addition, the decoupled vibrations are analytically investigated where it is shown that the consideration of the strut inertia may lead to significant changes of the decoupling conditions. Furthermore, for a reference hexapod, the vibrational behavior with respect to different design variables are systematically studied. The problem of dynamic isotropy, as an optimal design solution for hexapods, is also addressed in this dissertation. Dynamic isotropy is a condition in which all eigenfrequencies of a robot are equal. This is a powerful tool in order to obtain dynamically optimized architectures for parallel robots. We analytically present the conditions of dynamic isotropy in hexapods with and without the consideration of the strut inertia. Dynamic isotropy in hexapods is bound to a constraint related to the inertia properties (classical isotropic constraint), which makes it practically impossible to obtain dynamic isotropy by the standard hexapod architecture. To overcome this limitation, we have extended our study on hexapods to a general study on platforms supported elastically by three nodal joints in 6 DOFs. This method establishes dynamically isotropic solutions for kinematic designs with 3-2-1 and 2-2-2 arrangements. We have shown how effectively this method can be used for obtaining a generalized class of hexapods in order to eliminate the classical isotropic constraint. Accordingly, we have proposed a novel architecture of a generalized hexapod which is dynamically isotropic for a wide range of inertia properties. Finally, we have further extended our work on dynamic isotropy to analytical studies which are not necessarily limited to hexapods. Firstly, we have shown how dynamic isotropy can be achieved with the same approach in other parallel robots. A 3-DOF Gantry Tau robot is presented as an example. Then, the following two novel concepts in dynamic isotropy are introduced and developed in this dissertation: • Analytical index of dynamic isotropy: the analytical index of dynamic isotropy is a measure that mathematically combines all the eigenfrequencies and analytically represents the closeness of a robot to complete dynamic isotropy. It is a powerful tool in order to obtain dynamically isotropic architectures. Furthermore, when there exist geometrical/inertial constraints that make it impossible to accomplish complete dynamic isotropy, this analytical index can also be used for obtaining near-dynamic-isotropy design solutions. • Optimal dynamic isotropy: the presented results show that dynamically isotropic architectures are not unique in hexapods. Therefore, we have mathematically introduced a novel concept in order to find an optimal dynamically isotropic architecture.