Unified formulation for the stiffness analysis of spatial mechanisms

Abstract

This paper presents a complete stiffness analysis of spatial mechanisms. Links flexibility is modeled through structural elements while joints are inherently considered by means of kinematic relations including their degree of freedoms (dofs) and degree of constraints (docs). Actuation stiffness can be included as well as flexibility of some docs can be added in a selective way leaving the remaining rigid. Preloaded joints can be also modeled including joint wrenches. The Condensed Stiffness Matrix (CSM) of an elementary kinematic chain composed of a flexible two-node element and a spatial joint is derived using a robust mathematical formulation based on partitioned matrices and condensation techniques. The CSMs are then combined to find the global stiffness matrix through techniques coming from the structural analysis. The proposed method solves some critical issues of other formulations providing possibility to work with redundant legs (or joints) of fully- and over-constrained PKMs, inherent use of joints without resorting to Lagrangian multipliers, ability to exploit positive-semidefinite joint stiffness matrices without causing a rank-deficient global stiffness matrix; selective inclusion of stiffness in joints along dofs and docs. Finally, three examples to show the potentiality of the method for different applications of robotics are described.