Modeling, identification and control of a metrological Atomic Force Microscope with a 3DOF stage

Atomic Force Microscopes (AFMs) are widely used for the investigation of samples at nanometer scale. In this paper, we present the modeling, the identification and the control of a metrological AFM. The metrological AFM is used for the calibration of transfer standards for commercial AFMs. Therefore, the focus of the presented work is on scanning accuracy rather than on scanning speed. The contribution of this paper is the combination of 3 degree-of-freedom (DOF) control, including position feedforward, with an AFM with fixed cantilever and a piezo-stack driven stage. The amount of coupling between all DOFs is assessed by a non-parametric MIMO identification of the AFM. Since the dynamics appear to be decoupled in the frequency range of interest, feedback controllers are designed using loopshaping techniques for each DOF separately. Position feedforward is added to the stage in x and y direction, which improves the tracking performance by a factor two. The controlled stage is able to track scanning profiles within the sensor bound of 5 nm. With the proposed control method, the metrological AFM can produce images of the transfer standards with a sensor bound of 2 nm. Furthermore, real-time imaging of the sample is possible without the need for a-posteriori image correction. Finally, it is shown that the proposed control method almost completely compensates the hysteresis in the system

Modeling and compensation of asymmetric hysteresis in a piezo actuated metrological AFM

The manipulation of samples in atomic force microscopes (AFMs) is often performed using piezoelectric actuators. In this paper, a metrological AFM with a 3 degree-of-freedom (DOF) stage driven by piezo-stack actuators is considered. The piezo actuators exhibit hysteresis, which can change the system dynamics and/or acts as a non-linear disturbance on the system. This deteriorates the performance of the AFM. The 3 DOF stage exhibits asymmetric hysteresis, which is modeled by extending the Coleman-Hodgdon model. The asymmetry includes a scan range dependent offset and an asymmetry between the trace and retrace directions. Non-linear multi-variable optimization is employed to derive the optimal generic model for all scan ranges. The proposed extended Coleman-Hodgdon model describes the asymmetric hysteresis over all scan ranges with an accuracy of 97%. Based on the model, a feedforward compensation method is developed. Experiments on the metrological AFM show that the application of the hysteresis feedforward largely improves the scanning accuracy