In these structures, it is common to treat the flexure hinges as flexible, and all the other components as rigid. In order to improve the performance, multiple flexure hinges are generally combined in various configurations, such as the parallelograms and the statically indeterminate symmetric (SIS) structures. In literature, many analytical and empirical models have been established for the compliance/stiffness of a single flexure hinge. Ī single flexure hinge can be treated as a revolute joint during micro- and nano-scale motions. Widely utilized flexure hinge profiles include circular and leaf. On the other hand, flexure-based mechanisms are capable of transmitting high-precision motions via the elastic deformations of the flexure hinges, making it ideal in building the transmission chains for PEAs. Many feedforward and feedback methodologies have been proposed to compensate for the hysteresis and creep nonlinearities of PEAs. However, PEAs suffer from the inherent hysteresis and creep nonlinearities. On the one hand, the shape of a PEA changes if charge or voltage is exerted, and thus generating sub-nanometer resolution actuation. The integrations of piezoelectric actuators (PEAs) and flexure-based mechanisms have been widely utilized in nano-positioning and manipulations. The established closed-form models are widely applicable in the design and optimization of planar flexure-based mechanisms. In this case, the established closed-form models can be utilized to calculate the bounds of the performance. As no effective method is currently available to measure or estimate the contact stiffness, it is impossible to precisely estimate the performance of the overall system. If PEAs are installed, the contact stiffness shows up in the models. Experimental results show that the estimation error on the output stiffness and first natural frequency can reach 2% and 1.7%, respectively. If no piezoelectric actuator (PEA) is installed, the influence of the contact stiffness can be eliminated. Experiments are conducted to verify the effectiveness of the established models. Theoretical analysis reveals that the lever’s compliance, the contact stiffness, and the load mass have significant influence on the static and dynamic performances of the system. This paper presents the closed-form modeling of an XY nano-manipulator consisting of statically indeterminate symmetric (SIS) structures using leaf and circular flexure hinges. The closed-form statics and dynamics modeling is difficult due to the complex topologies, the inevitable compliance of levers, the Hertzian contact interface, etc. Flexure-based mechanisms are widely utilized in nano manipulations.
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