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.

The dual-loop controller (DLC) with both inner damping and outer tracking controllers has demonstrated exceptional performance in high-speed control of piezo-actuated nano-positioners. However, the accurate low-order model of the plant is imperative for designing the damping controller, according to the conventional DLC design principle. This limits its application in controlling piezoelectric tube scanners (PTSs) with compound dynamics. To handle this problem, this study introduces a frequency domain method for designing the DLC based on the frequency response data of the PTS. This method mitigates issues related to modeling errors. Specifically, the Nyquist diagram is employed to provide the stability boundary for parameters determination. A constraint optimization problem is formulated to achieve a high bandwidth with a flat amplitude frequency response. And the differential evolution algorithm is then adopted to find an optimal solution. Experimental validation on a PTS comfirms the effectiveness of this frequency domain design method. The results indicate that the control bandwidth of the optimized DLC achieves 848 Hz for a PTS with the first resonant frequency of 702 Hz. The superiorities of the DLC designed by the proposed optimization method are also validated via comparative tracking experiments involving step and triangular trajectories.Note to Practitioners—Frequency response data (FRD)-based methods open a new perspective to the controller design and optimization. Unlike traditional methods relying on transfer function models, the FRD of the system is directly identified and employed for the controller design and optimization, eliminating the need for extensive modeling and system identification efforts, thereby reducing errors. This paper proposes an FRD-based method for the simultaneous optimization of the DLC to achieve a high control bandwidth with a flat amplitude frequency response. This method relieves the requisite of accurate low-order model of the plant in conventional DLC designs. A closed-loop control bandwidth that exceeds the first resonant frequency is obtained for a PTS with complex high-order dynamic models. The proposed method enriches the scope of the DLC for the high bandwidth control, extending its applications to the systems with compound dynamics, which is significant to the high-speed control tasks of nano-positioners, such as the atomic force microscope imaging.

The normal-stressed electromagnetic actuators (NSEAs) have emerged as a promising actuation technology for developing high-performance compliant nano-positioning stages. But less attention was devoted to the modeling of the multi degrees-of-freedom (DoF) NSEA-based stages. This paper aims to improve the modeling accuracy of the stages’ static and dynamic performances by introducing a novel electromagnetic-mechanical modeling method. Unlike the previous studies, the proposed modeling method considers the effects of the NSEAs’ negative stiffnesses along both its actuation direction and the vertical-to-actuation direction in the electromagnetic modeling step. Together with the model of the mechanisms, the coupled electromagnetic-mechanical model is analytically derived. As an application case, the working stroke and resonant frequency of a NSEA-based parallel-kinematic 2-DoF compliant nano-positioning stage are evaluated with the proposed model. It is demonstrated by numerical, simulation, and experimental studies, that the proposed modeling method is accurate for predicting the performances of the NSEA-based nano-positioning stages. This is significant for the future development and applications of NSEA-based mechatronic systems.Note to Practitioners—The multi-DoF NSEA-based nano-positioning stages are the promising choice to develop long stroke, high natural frequency nano-positioning stages. However, some inherent electromagnetic-mechanical coupling effects are ignored in previous study when modeling the multi-DoF NSEA-based stages, which leads to a significant prediction error. By considering the electromagnetic characteristics both along and vertical to actuation direction, this study proposed an electromagnetic-mechanical model for the 2-DoF NSEA-based stages. The proposed model shows a higher accuracy to evaluate the static and dynamic performances of the NSEA-based stages, which are beneficial to practical applications such as trajectories tracking.

The achievable performance with repetitive control is limited due to its inherent sensitivity to the frequency shift away from the intended periodic frequencies and the undesired gain amplification of the aperiodic disturbances. To address these limitations, the paper proposed a general parametric repetitive control (GPRC) method based on the IIR filter with the features of low-order and excellent magnitude responses to result in better tracking performance in diverse applications. By analyzing its sensitivity function, it is found that the design of GPRC can be converted to the explicit parametric design of an IIR high pass filter. The controller design process and the stability analysis are presented in detail. To show the effectiveness of GPRC, comparative experiments are conducted via tracking sinusoids, triangular trajectories and other complex trajectories with multi-frequency components. The experimental results show that, in contrast with the conventional repetitive control (CRC) and a modified repetitive control (MRC), the GPRC exhibits excellent robustness against the frequency shift and advanced performance at the aperiodic frequencies. The tracking results of the sinusoids show that the maximum tracking error obtained with GPRC for a frequency shift of 3 Hz decreases from$0.0259~\mu m$(CRC) and$0.2207~\mu m$(MRC) to$0.0101~\mu m$at the nominal frequency of 1000 Hz, demonstrating the merits of the proposed GPRC. Note to Practitioners—To enable automation systems, one of the crucial requirements is to track repetitive references with high precision. Although the normal repetitive control (RC) based schemes are successfully applied to improve the tracking accuracy of periodic trajectories, the existing RC schemes suffer from the problems of lower robustness against frequency shift and the unwanted gain amplification at the aperiodic frequencies due to Bode’s sensitivity integral. To overcome this problem, this paper proposes a novel general parametric repetitive control (GPRC) method via characterizing the loop properties quantitatively based on the IIR high pass filter design. Focusing on specific issues, the detailed variations to handle the errors at only the odd- and even-harmonics are also demonstrated. This framework leads to a flexible solution in practical implementations. The experimental validation on a piezo-actuated nanopositioning stage is comparatively presented in terms of tracking accuracy and rejection ability of the gain amplification at the aperiodic frequencies. With its flexibility and effectiveness, the proposed GPRC can be easily implemented in diverse applications.

This article presents a novel high-gain loop-shaping (HGLS) method for precision motion devices by introducing high gains into the control loop. The high gains can be generated via the inclusion of a low pass filter in a feedback manner, which could significantly improve the tracking accuracy for general trajectories with a simple structure. This control law also endows the superiority of robustness against model variations and external disturbances. To evaluate the performance of the proposed HGLS method, tracking and disturbance rejection experiments are conducted on a custom-designed piezo-actuated nanopositioner. The experimental results demonstrate the advancement of HGLS in terms of precision motion control, where the root-mean-squared error is reduced from 7.8 nm to 1.8 nm as compared with the high-gain proportional-integral controller with the same phase margin when tracking a random Non-Uniform Rational B-splines curve with large external disturbances. With its remarkable advantages of high tracking accuracy and simple structure, this method offers a practical solution for industrial automation applications for enhancing the control performance.Note to Practitioners—Advanced control methods are of great importance in meeting the tremendous requirements of automation in mechanical and electronic systems. Nonetheless, it is still challenging for a controller to simultaneously realize high control accuracy and strong robustness for general trajectories while retaining a simple structure. In this paper, a novel HGLS method is proposed. For this method, high control accuracy and strong robustness can be achieved by introducing high gains into the control loop, which can be generated by the inclusion of a simple low pass inside the closed-loop plant. A trade-off between the control gain and stability margin can be balanced only by tuning the parameters of the employed low-pass filter, and thus the implementation of the HGLS is rather simple. This method has no constraints on the form or order of the plant models and holds robustness against model variations. As compared with the high-gain proportionalintegral controller with the same phase margin, the control gain of the HGLS is much higher within the effective control bandwidth. Therefore, outstanding control performance can be achieved for general trajectories within this bandwidth. Due to the strong universality and simple structure of the HGLS, it is promising to be implemented in various industrial automation equipment, such as machine tools, multi-axis systems, robotics, scanners, etc.

This article presents a data-driven, Koopman operator-based modeling scheme for analyzing and predicting cross-coupling hysteresis effects of the piezoelectric tube scanners (PTSs) used in atomic force microscopes (AFMs). Such cross-coupling hysteresis effects between different PTS axes significantly reduce the positioning precision of AFMs. In contrast to most of the existing methods for PTS hysteresis, which involve complex nonlinear dynamics identification processes, the present study leverages the Koopman operator theory instead to treat the nonlinear hysteresis as a linear system. Therein, a Hankel extended dynamic mode decomposition (H-EDMD) algorithm is proposed to learn the finite-dimensional descriptions of the Koopman operator and the associated Koopman eigenspectrum. Moreover, the proposed H-EDMD even allows sparse sampling on the PTS systems, which is desirable in real industrial applications. Finally, extensive comparison experiments with a mainstream modified Prandtl-Ishlinskii model are conducted on an NTMDT Prima AFM to substantiate the effectiveness and superiority of the proposed H-EDMD method.

Repetitive disturbances exist widely within the automation systems, which is one of the major issues that hinder the achievement of precision operations. Dedicated to mitigating these disturbances, a generalized periodic-disturbance observer (PDOB) using spectrum-selection filtering scheme is proposed in this paper with an application to a non-minimum phase system. The design process of the spectrum-selection filter that derives from the comb-like notch filter for the proposed PDOB is presented in detail. The variants of the proposed PDOB are also presented for the disturbances distributed only in odd- and even-harmonics. To achieve tracking of the desired trajectories, the proposed PDOB is combined in parallel with a baseline Proportional-Integral (PI) controller. The stability condition of the closed-loop system is derived to provide criteria for parameters selection. The experimental validation of the generalized PI+PDOB is conducted via real-time cross-coupling suppression in raster scanning of atomic force microscope (AFM), where the coupling induced root-mean-square tracking errors are reduced from 141.8$nm$to 2.4$nm$by employing the proposed PDOB for the scanning rate of 100 Hz. The results of convergence testing, raster scanning and AFM imaging are compared to illustrate the significant improvements achieved with the proposed PDOB. In addition, experimental results show that the tracking performance using PI+PDOB in the case of existing periodic disturbances resembles with that using PI control without periodic disturbances, which implies that the employment of the proposed PDOB does not interfere with the tracking-based control schemes for staircase trajectories, showing the advantages of the proposed PDOB.Note to Practitioners—Repetitive motions facilitate diverse advanced functions within the automation systems, including the scanning electron microscope, atomic force microscope, manipulators, and robotic systems. These operations would unavoidably introduce repetitive disturbances, hindering its achievable precision. One particular issue is the widely existent cross-coupling effect that results in these repetitive disturbances. In this paper, a generalized periodic-disturbance observer (PDOB) based on the spectrum-selection filtering scheme is developed to address the repetitive disturbances for real-time coupling suppression in raster scanning of Atomic Force Microscopy. Consequently, the offline learning procedures required for other learning based schemes can be avoided. Without loss of generalization, the design of PDOB is demonstrated with an application to non-minimum phase systems. A scheme of the varying parameter is also developed to achieve both fast convergence and better performance at rejecting the periodic disturbances. The detailed variations to handle the disturbances distributed only in odd- and even-harmonics are also demonstrated. Different from other disturbance observers, the inclusion of the spectrum-selection filtering scheme provides more flexibility in rejecting specific repetitive disturbances. In terms of tracking accuracy and the simple structure, this development can also be easily implemented to other systems that suffer from repetitive disturbances.

In this article, an intelligent feedforward prediction and compensation scheme to combine with a dual-loop high-bandwidth controller is proposed for high-speed and high-precision tracking controls of a nanopositioning stage. First, the dual-loop controller consisting of an inner loop damping and an outer loop tracking controller is developed with all the parameters optimized simultaneously, which could provide a control bandwidth over the first resonant frequency of the stage. Next, the Gaussian process machine learning model is employed to capture the dynamic characteristics of the tracking error of the dual-loop controlled plant. Then, a feedforward compensator is constructed to add a compensation term to the initial reference trajectory. Experimental investigations on a self-made piezoelectric-actuated stage validate the effectiveness of the intelligent tracking error prediction method and the excellent performance of the control strategy for high-precision tracking of high-frequency reference trajectories.

Due to the lightly damped resonance and intrinsic nonlinearities, it is difficult for the piezoactuated nanopositioning stage to realize high-bandwidth and high-accuracy control. To handle these limitations, in this work, a dual-loop control scheme based on state-feedback-based modal method is designed to both actively damp and stiffen the resonant mode and to suppress the effects of nonlinearities of the piezoactuated nanopositioning stage. In this scheme, the state-feedback-based modal controller is first designed in the inner loop to enlarge both the damping ratio and natural frequency of the first resonant mode. Then, a proportional–integral (PI) controller is utilized in the outer loop for eliminating the tracking errors caused by other disturbances and nonlinearities including hysteresis and creep. To maximize the control bandwidth of system under the proposed dual-loop scheme, an optimization method is thus proposed for simultaneously tuning the parameters of the inner and the outer loop controllers. Finally, to validate the proposed dual-loop control scheme, comparative experiments are carried out on a piezoactuated nanopositioning stage. Results demonstrate that the proposed control scheme improves the bandwidth of the system from 497 Hz (with PI control) and 1543 Hz (with a commonly used positive acceleration, velocity, and position damping control and a PI controller) to 6546 Hz, which is 664 Hz larger than the first resonant frequency of the original system, validating the effectiveness of the proposed dual-loop scheme on high-bandwidth control. Note to Practitioners—The demand of high-bandwidth and high-accuracy piezoactuated nanopositioning stages increases rapidly. However, the lightly damped resonance of the mechanism and the intrinsic nonlinearities of the piezoelectric actuator limit the tracking performance of the stage. A dual-loop control structure is adopted in this work to improve the tracking performance of the nanopositioning stage. Different from most of the vibration control methods proposed in the literature which aimed only at improving the damping ratio, a state-feedback-based modal controller is designed in the inner-loop for improving both the damping ratio and the stiffness of the system. This task is realized by re-placing the resonant poles of the system to the optimized location. The outer-loop controller adopts the high-gain PI control for eliminating the tracking errors. More importantly, in order to realize the high-bandwidth and high-accuracy control, a numerical optimization method is proposed for simultaneously tuning the parameters of the controllers in inner and outer loops. The controller design is simple, and it can be applied to other systems with second or higher order in which the first resonant mode dominates the system dynamics.

Odd-harmonic repetitive control (ORC) has been successfully applied to improve the triangular trajectory tracking performance of nanopositioners. However, the conventional ORC tends to amplify the tracking errors at frequencies other than the odd-harmonic components, mainly the even harmonics of the fundamentals of the intended triangular trajectory to be tracked. Due to the influence from the hysteresis nonlinearity of the piezoelectric actuator, this would result in significant tracking errors. To overcome this limitation, this article proposes an enhanced odd-harmonic repetitive control (EORC) using the spectrum-selection filtering scheme to improve the loop-shaping property of the ORC. This effectively eliminates the problem of amplifying the tracking errors while preserving the advantages of the conventional ORC, such as fast convergence speed and low computation cost. The EORC is combined with a proportional–integral tracking controller to improve the tracking performance. The controller design, stability analysis, and performance evaluation are presented. The experimental results demonstrating the effectiveness of the proposed EORC-based control scheme are presented showing the excellent tracking of triangular trajectories with fundamental frequencies up to 1000 Hz. Moreover, a reduction in rms tracking errors by up to 52% is achieved.Note to Practitioners—The high-speed atomic force microscopy (AFM) plays an increasingly vital role in observing and manipulating objects at the nanoscale. In the raster scanning of AFMs, the most challenging issue is the triangular trajectory tracking of nanopositioning stages with high precision. Although the odd-harmonic repetitive control (ORC) has been successfully applied to improve the tracking performance, the conventional ORC would amplify the tracking errors distributed at the frequencies other than the odd harmonics, especially those at the even harmonics, resulting from the inherent complicated hysteresis nonlinearity. This problem of amplifying the tracking errors would lead to larger tracking errors, deteriorating the performance of AFM imaging. To address this issue, this article proposes an enhanced ORC using the spectrum-selection filtering scheme to improve the loop-shaping property of the ORC, so as to eliminate the problem of amplifying the tracking errors while preserving the advantages of the conventional ORC, such as fast convergence speed and low computation cost. The experimental results show that, with this simple modification, the positioning errors are reduced greatly, all-the-while preserving the benefits of the conventional ORC scheme– fast convergence speed and low computation cost. In terms of tracking accuracy and the simple structure, this development can be easily implemented to other systems that challenge from the tracking accuracy under ORC scheme.