Nonlinear static modeling of a tip-tilt-piston micropositioning stage comprising leaf-spring flexure hinges

Abstract

Nonlinear static modeling is crucial for the design of stages with large travel ranges. However, few studies have investigated complex spatial compliant mechanisms. The present study proposes an optimization algorithm based on substructure constraint conditions to formulate the nonlinearity of the force–displacement characteristic of a tip-tilt-piston stage comprising leaf-spring flexure hinges. First, the nonlinear force–displacement characteristics of the compound basic parallelogram mechanism are derived using an optimization algorithm based on two constraint conditions (I and II). Second, the nonlinear static modeling of the tip-tilt-piston stage is conducted based on the modeling results of the compound basic parallelogram mechanism. The stage is divided into three parts, and force analyses are conducted for all three parts. The vertical displacement of the compound basic parallelogram mechanism in part 1 and the rotational angle of the rotational plate in part 2 are calculated. Subsequently, the force–displacement characteristics of the tip-tilt-piston stage are obtained based on a third constraint condition (III). A comparison of the finite element analysis results and the theoretical calculation indicates less than 4% errors. In the experimental tests conducted on the proposed stage and four compound basic parallelogram mechanisms, the displacements were evidently larger than those calculated using the finite element analysis. Therefore, a weight coefficient of the axial force w is introduced in the theoretical calculation to solve the problem of the large deviations between the experimental results and the finite element analysis results. When w is set to 1, the theoretical results are in good agreement with the finite element analysis results; when w is set to 0.05 for the tip-tilt-piston stage and 0.15 for compound basic parallelogram mechanisms, the theoretical results are consistent with the experimental results (less than 8.5% errors).