A mathematical approach for analyzing ultra precision positioning system with compliant mechanism

A monolithic and symmetric structure is good for reducing mechanical noise, such as thermal and compliant errors. Design and analysis of six-degree freedom ultra precision positioning system with compliant mechanism is described in this paper. An inclined bridge displacement amplifier is developed to achieve out-of-plane motions. A mechanism to amplify displacement is used to overcome short travel range of piezoelectric actuator. Mathematical model is derived using Castiglianoʹs second theorem to express the relation between displacement and force applied at the end point of flexure hinge. Using vector analysis, a relative displacement of flexure hinge is calculated. A stiffness matrix of proposed system is derived using least action principle. Three-dimensional modeling and simulation of the proposed system with flexure hinge mechanism is carried out to compare the results with that obtained by stiffness matrix which is derived by hand. Translational differences of two methods are 28% in x-axis and 29% in y-axis and rotational differences are17% in x-axis, 23% in y-axis and 13% in z-axis.