Li-Min Zhu

Modeling and Compensation of Asymmetric Hysteresis Nonlinearity for Piezoceramic Actuators With a Modified Prandtl–Ishlinskii Model

This paper presents a modified Prandtl-Ishlinskii (P-I) (MPI) model for the asymmetric hysteresis description and compensation of piezoelectric actuators. Considering the fact that the classical P-I (CPI) model is only efficient for the symmetric hysteresis description, the MPI model is proposed to describe the asymmetric hysteresis nonlinearity of piezoceramic actuators (PCAs). Different from the commonly used approach for the development of asymmetric P-I models by replacing the classical play operator with complex nonlinear operators, the proposed MPI model still utilizes the classical play operator as the elementary operator, while a generalized input function is introduced to replace the linear input function in the CPI model. By this way, the developed MPI model has a relative simple mathematic format with fewer parameters to characterize the asymmetric hysteresis behavior of PCAs. The benefit for the developed MPI model also lies in the fact that an analytic inverse model of the CPI model can be directly applied for the inverse compensation of the asymmetric hysteresis nonlinearity represented by the developed MPI model in real-time applications. To validate the developed MPI model and the inverse hysteresis compensator, simulation, and experimental results on a piezoceramic actuated platform are presented.

Motion Control of Piezoelectric Positioning Stages: Modeling, Controller Design, and Experimental Evaluation

In this paper, a general skeleton on modeling, controller design, and applications of the piezoelectric positioning stages is presented. Toward this framework, a general model is first proposed to characterize dynamic behaviors of the stage, including frequency response of the stage, voltage-charge hysteresis and nonlinear electric behavior. To illustrate the validity of the proposed general model, a dynamic backlash-like model is adopted as one of hysteresis models to describe the hysteresis effect, which is confirmed by experimental tests. Thus, the developed model provides a general frame for controller design. As an illustration to this aspect, a robust adaptive controller is developed based on a reduced dynamic model under both unknown hysteresis nonlinearities and parameter uncertainties. The proposed control law ensures the boundedness of the closed-loop signals and desired tracking precision. Finally, experimental tests with different motion trajectories are conducted to verify the proposed general model and the robust control law. Experimental results demonstrate the excellent tracking performance, which validates the feasibility and effectiveness of the proposed approach.

Modeling and Control of Piezo-Actuated Nanopositioning Stages: A Survey

Piezo-actuated stages have become more and more promising in nanopositioning applications due to the excellent advantages of the fast response time, large mechanical force, and extremely fine resolution. Modeling and control are critical to achieve objectives for high-precision motion. However, piezo-actuated stages themselves suffer from the inherent drawbacks produced by the inherent creep and hysteresis nonlinearities and vibration caused by the lightly damped resonant dynamics, which make modeling and control of such systems challenging. To address these challenges, various techniques have been reported in the literature. This paper surveys and discusses the progresses of different modeling and control approaches for piezo-actuated nanopositioning stages and highlights new opportunities for the extended studies.

Real-time inverse hysteresis compensation of piezoelectric actuators with a modified Prandtl-Ishlinskii model

This paper presents a novel real-time inverse hysteresis compensation method for piezoelectric actuators exhibiting asymmetric hysteresis effect. The proposed method directly utilizes a modified Prandtl-Ishlinskii hysteresis model to characterize the inverse hysteresis effect of piezoelectric actuators. The hysteresis model is then cascaded in the feedforward path for hysteresis cancellation. It avoids the complex and difficult mathematical procedure for constructing an inversion of the hysteresis model. For the purpose of validation, an experimental platform is established. To identify the model parameters, an adaptive particle swarm optimization algorithm is adopted. Based on the identified model parameters, a real-time feedforward controller is implemented for fast hysteresis compensation. Finally, tests are conducted with various kinds of trajectories. The experimental results show that the tracking errors caused by the hysteresis effect are reduced by about 90%, which clearly demonstrates the effectiveness of the proposed inverse compensation method with the modified Prandtl-Ishlinskii model.

Design and control of a decoupled two degree of freedom translational parallel micro-positioning stage

This paper presents a novel decoupled two degrees of freedom (2-DOF) translational parallel micro-positioning stage. The stage consists of a monolithic compliant mechanism driven by two piezoelectric actuators. The end-effector of the stage is connected to the base by four independent kinematic limbs. Two types of compound flexure module are serially connected to provide 2-DOF for each limb. The compound flexure modules and mirror symmetric distribution of the four limbs significantly reduce the input and output cross couplings and the parasitic motions. Based on the stiffness matrix method, static and dynamic models are constructed and optimal design is performed under certain constraints. The finite element analysis results are then given to validate the design model and a prototype of the XY stage is fabricated for performance tests. Open-loop tests show that maximum static and dynamic cross couplings between the two linear motions are below 0.5% and -45 dB, which are low enough to utilize the single-input-single-out control strategies. Finally, according to the identified dynamic model, an inversion-based feedforward controller in conjunction with a proportional-integral-derivative controller is applied to compensate for the nonlinearities and uncertainties. The experimental results show that good positioning and tracking performances are achieved, which verifies the effectiveness of the proposed mechanism and controller design. The resonant frequencies of the loaded stage at 2 kg and 5 kg are 105 Hz and 68 Hz, respectively. Therefore, the performance of the stage is reasonably good in term of a 200 N load capacity.

High-speed tracking control of piezoelectric actuators using an ellipse-based hysteresis model

In this paper, an ellipse-based mathematic model is developed to characterize the rate-dependent hysteresis in piezoelectric actuators. Based on the proposed model, an expanded input space is constructed to describe the multivalued hysteresis function H[u](t) by a multiple input single output (MISO) mapping Gamma:R(2)-->R. Subsequently, the inverse MISO mapping Gamma(-1)(H[u](t),H[u](t);u(t)) is proposed for real-time hysteresis compensation. In controller design, a hybrid control strategy combining a model-based feedforward controller and a proportional integral differential (PID) feedback loop is used for high-accuracy and high-speed tracking control of piezoelectric actuators. The real-time feedforward controller is developed to cancel the rate-dependent hysteresis based on the inverse hysteresis model, while the PID controller is used to compensate for the creep, modeling errors, and parameter uncertainties. Finally, experiments with and without hysteresis compensation are conducted and the experimental results are compared. The experimental results show that the hysteresis compensation in the feedforward path can reduce the hysteresis-caused error by up to 88% and the tracking performance of the hybrid controller is greatly improved in high-speed tracking control applications, e.g., the root-mean-square tracking error is reduced to only 0.34% of the displacement range under the input frequency of 100 Hz.

A Distance Function Based Approach for Localization and Profile Error Evaluation of Complex Surface

This paper presents a unified framework for best-fitting of complex rigid surface to measured 3-D coordinate data by adjusting its location (position/orientation). For a point expressed in the machine reference frame and a nominal surface represented in its own model frame, a signed point-to-surface distance function is defined, and its properties are investigated, especially, its increment with respect to the differential motion of the surface, up to the second order, is derived. On this basis, localization and profile error evaluation of complex surface are formulated as a nonlinear least-squares problem and nonlinear constrained optimization problem respectively, and sequential approximation algorithms are developed to solve them. The two algorithms have the advantages of implementational simplicity, computational efficiency and robustness. Also strategies for estimating initial solution and compensating probe radius are presented. Examples confirm the validity of the proposed approach.

Global optimization of tool path for five-axis flank milling with a conical cutter

In this paper, optimum positioning of the conical cutter for five-axis flank milling of slender surfaces is addressed from the perspective of approximating the tool envelope surface to the data points on the design surface following the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation. Based on the observation that a conical surface can be treated as a canal surface, i.e. envelope surface of one-parameter family of spheres, the swept envelope of a conical cutter is represented as a sphere-swept surface. Then, an approach is presented to efficiently compute the signed distance between a point in space and the swept surface without constructing the swept surface itself. The first order differential increment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory surface is derived. By using the distance function, tool path optimizations for semi-finish and finish millings are formulated as two constrained optimization problems in a unified framework, and a sequential approximation algorithm along with a hierarchical algorithmic structure is developed for the optimization. Numerical examples are given to confirm the validity and efficiency of the proposed approach. Comparing with the existing approaches, the present one improves the machining accuracy greatly. The rationale developed applies to general rotary cutters.

Design, analysis and testing of a parallel-kinematic high-bandwidth XY nanopositioning stage.

This paper presents the design, analysis, and testing of a parallel-kinematic high-bandwidth XY nanopositioning stage driven by piezoelectric stack actuators. The stage is designed with two kinematic chains. In each kinematic chain, the end-effector of the stage is connected to the base by two symmetrically distributed flexure modules, respectively. Each flexure module comprises a fixed-fixed beam and a parallelogram flexure serving as two orthogonal prismatic joints. With the purpose to achieve high resonance frequencies of the stage, a novel center-thickened beam which has large stiffness is proposed to act as the fixed-fixed beam. The center-thickened beam also contributes to reducing cross-coupling and restricting parasitic motion. To decouple the motion in two axes totally, a symmetric configuration is adopted for the parallelogram flexures. Based on the analytical models established in static and dynamic analysis, the dimensions of the stage are optimized in order to maximize the first resonance frequency. Then finite element analysis is utilized to validate the design and a prototype of the stage is fabricated for performance tests. According to the results of static and dynamic tests, the resonance frequencies of the developed stage are over 13.6 kHz and the workspace is 11.2 μm × 11.6 μm with the cross-coupling between two axes less than 0.52%. It is clearly demonstrated that the developed stage has high resonance frequencies, a relatively large travel range, and nearly decoupled performance between two axes. For high-speed tracking performance tests, an inversion-based feedforward controller is implemented for the stage to compensate for the positioning errors caused by mechanical vibration. The experimental results show that good tracking performance at high speed is achieved, which validates the effectiveness of the developed stage.

Analytical Expression of the Swept Surface of a Rotary Cutter Using the Envelope Theory of Sphere Congruence

Based on the observation that many surfaces of revolution can be treated as a canal surface, i.e., the envelope surface of a one-parameter family of spheres, the analytical expressions of the envelopes of the swept volumes generated by the commonly used rotary cutters undergoing general spatial motions are derived by using the envelope theory of sphere congruence. For the toroidal cutter, two methods for determining the effective patch of the envelope surface are proposed. With the present model, it is shown that the swept surfaces of a torus and a cylinder can be easily constructed without complicated calculations, and that the minimum distance (between the swept surface and a simple surface) and the signed distance (between the swept surface and a point in space) can be easily computed without constructing the swept surface itself. An example of global tool path optimization for flank milling of ruled surface with a conical tool, which requires to approximate the tool envelope surface to the point cloud on the design surface, is given to confirm the validity of the proposed approach.