Nonlinear Static Modeling of a Large Range XY-Nanopositioning System

Abstract High-precision and high-speed requirements in flexure-based positioning systems require stringent dynamic performance. In the presence of geometric non-linearities, which are relevant in large stroke flexure mechanisms, a standard approach is to linearize the system dynamics about an operating point and then design suitable controllers. Therefore, the first step in this subsequent dynamic analysis, is to have static equilibrium conditions of the flexure mechanism that can serve as the operating point. With this motivation, the objective of this paper is to propose a nonlinear static model for a specific XY flexure mechanism capable of decoupled motion along two orthogonal axes. This XY flexure mechanism is of practical use in high-precision high-speed positioning systems. Initially, the generalized coordinates of the flexure mechanism are chosen from the various displacement coordinates of the system. The strain energy and generalized forces of the flexure depend not only on the generalized coordinates but also on other displacement coordinates. This complicates the application of the principle of virtual work. To address this complexity, we utilize the fact that other displacement components of the system implicitly depend on the generalized coordinates due to nonlinear constraints arising from the geometric nonlinearities associated with beam arc-length conservation. By considering this implicit dependence when applying derivatives in the principle of virtual work, we can derive the nonlinear equilibrium equations governing the static behavior of the flexure mechanism. This approach for arriving at a static model for the XY flexure is then generalized into a step-by-step procedure, capable of deriving the governing nonlinear static equations of a diverse class of flexure mechanisms. A physical understanding of the system is then employed to solve these equations to obtain the operating point for uniaxial loading. Resulting expressions are then used in a subsequent analysis with perturbation expansions to obtain closed-form, highly accurate solutions for the operating point of the XY flexure under biaxial loading. The results are validated through numerical simulations and Finite Element Analysis.