Optimal Design of Smoothed Raster Scan Trajectory for Repetitive Control Based High-Speed Atomic Force Microscopy Imaging

Due to the uniform sampling pattern and concise trajectory generation, the raster scan has become the mainstream scan mode for atomic force microscopes (AFMs). However, the high-frequency components of the triangular trajectory and the stair trajectory for generating the raster scan tend to motivate the lightly damped resonance of the nano-positioner insides AFMs and thus severely limit the imaging speed of AFMs. To handle this issue, a novel smoothed raster scan trajectory generation method is proposed in this article via replacing the non-imaging path of both fast axis and slow axis with an optimally designed smoothed transition trajectory. A universal mathematical expression of the smoothed raster scan trajectory is developed, following by the detailed frequency spectral analysis. The analysis results reveal that the inner model of the smoothed slow-axis trajectory is the sum of two cascaded integrators and a series of sinusoidal internal models distributed at the scanning frequency along with its harmonic frequencies. As a result, an embedded repetitive control (RC) scheme is adopted to achieve the high-bandwidth and high-accuracy tracking for both fast-and slow-axis trajectories so as to contribute to a high-speed raster scanning. Comprehensive trajectory tracking experiments and imaging experiments are performed on a commercial AFM. The experimental results show that, under the scanning frequency of 100 Hz, the root-mean-square tracking error of the proposed smoothed trajectory is reduced from 47.1 nm (fast axis) and 7.5 nm (slow axis) of the conventional trajectory to 6.9 nm (fast axis) and 3.8 nm (slow axis) with the same RC schemes, and maximum tracking error of the same smoothed slow-axis trajectory is reduced from 387.7 nm with PI scheme and 138.2 nm with PI+PDOB scheme to 13.0 nm with the RC scheme, which verify the effectiveness and advancement of the proposed trajectory smoothing method and the developed control scheme.
Note to Practitioners
—Raster trajectory implemented via fast-and slow-axis coordinated motion has been widely used within modern precision manufacturing and measuring systems such as the atomic force microscope, coordinate measuring machine, laser or ion-beam machine tool. The high-accuracy and high-speed tracking of the raster trajectory is the basic requirement for these sophisticated systems to achieve the intended function. However, the trajectory mutation of the raster trajectory induces large amounts of high-frequency trajectory components for each servo axis and thus brings a significant challenge for its high-performance motion control. In this article, a novel smoothed raster scan trajectory generation method is developed to reduce the high-frequency components of the conventional raster trajectory and thus facilitate its trackable performance on the basis of the frequency spectral analysis of the smoothed raster scan trajectory. The spectral analysis results reveal for the first time that the frequency spectral function of the smoothed slow-axis trajectory is the linear combination of those of two cascaded integrator and a periodic trajectory. This conclusion motivates the usage of a simple yet efficient embedded repetitive control (RC) scheme by integrating a proportional-integral (PI) controller with a repetitive controller for the high-performance tracking of both fast and slow axes whilst suppressing their cross-coupling errors without extra compensators. The effectiveness of the proposed smoothed raster scan trajectory as well as the developed embedded RC scheme have been preliminarily verified via high-speed AFM imaging. This development can also be applied on other systems involved in the high-speed tracking of the raster trajectory.