Waveform Error Compensation Device and Control Method Based on Multi-Axis Nano-Ultrafast System

Abstract

The present invention provides a waviness error compensation device and a control method based on a multi-axis nano-ultrafast system. The compensation device includes four drive units of the same model, a flexible hinge, a vacuum adsorption platform, a connecting flange, and a fixed base. The drive unit includes a piezoelectric ceramic actuator and a displacement sensor. The piezoelectric ceramic actuator is located at the bottom of the flexible hinge. The vacuum adsorption platform is connected to the flexible hinge. The lower end of the piezoelectric ceramic actuator is connected to the fixed base for support. Four groups of flexible hinges are symmetrically distributed in parallel on the connecting flange. The connecting flange is located on the fixed base. The waviness error compensation device has three degrees of freedom. By applying a voltage to drive the piezoelectric ceramic actuator, the pose of the vacuum adsorption platform is adjusted. Through the solution of the present invention, the relative vibration between the tool tip and the workpiece can be compensated in real time, and the arc-shaped waviness error of fly-cutting can be suppressed, thereby significantly improving the quality and accuracy of the machined surface.

Description

Technical Field

[0001] The present invention belongs to the technical field of precision machining, and particularly relates to a waviness error compensation device and a control method based on a multi-axis nano-ultrafast system. Background Art

[0002] In the field of machining, especially in the fields of precision engineering and micro-nano technology, the requirements for the quality of the machined surface are getting higher and higher. Among them, waviness is one of the important quality indicators of the machined surface. During the machining process of a single-point diamond fly-cutting machine tool, the relative vibration between the tool tip and the workpiece is the main reason for the arc-shaped waviness error on the machined surface. This error will seriously affect the accuracy of the workpiece surface shape and limit the improvement of machining quality. During the actual cutting process, factors such as the dynamic characteristics of the machine tool and the spindle, the vibration caused by cutting, and external disturbances may all bring relative vibration between the tool tip and the workpiece. Summary of the Invention

[0003] To solve the above problems, the present invention proposes a waviness error compensation device and a control method based on a multi-axis nano-ultrafast system. Through a control algorithm composed of hysteresis compensation feedforward, active disturbance rejection control, and zero-phase feedforward, it can compensate the relative vibration between the tool tip and the workpiece in real time, thereby significantly improving the quality and accuracy of the machined surface.

[0004] On the one hand, the present invention proposes a waviness error compensation device based on a multi-axis nano-ultrafast system. The device includes four drive units of the same model, four groups of flexible hinges, a vacuum adsorption platform, four connecting flanges, and a fixed base. Each drive unit includes a piezoelectric ceramic actuator and a displacement sensor. The piezoelectric ceramic actuator is located at the bottom of the flexible hinge. The vacuum adsorption platform is connected to the flexible hinge. The vacuum adsorption platform uses four piezoelectric ceramic actuators as legs through the flexible hinge. The lower end of the piezoelectric ceramic actuator is connected to the fixed base for support. The four groups of flexible hinges are distributed in parallel and symmetrically on the connecting flange, and the connecting flange is located on the fixed base; four displacement sensors are arranged at four points on the edge of the vacuum adsorption platform in the same way as the piezoelectric ceramic actuators, in a cross distribution, to measure the pose height at the corresponding positions, so that the waviness error compensation device has three degrees of freedom. The three degrees of freedom are the rotational degrees of freedom RX and RY in two horizontal directions of the platform plane of the vacuum adsorption platform and the translational degree of freedom Z in the vertical direction. By applying a voltage to drive the piezoelectric ceramic actuator, the pose of the vacuum adsorption platform is adjusted.

[0005] Preferably, the stiffness of the piezoelectric ceramic actuator is 50 N / μm, the driving voltage is 100V, the stroke is 32μm, and the natural frequency is 40kHz.

[0006] Preferably, a capacitive sensor is selected as the displacement sensor in the present invention, and there is a direct linear relationship between its output voltage and the displacement.

[0007] Preferably, each flexible hinge is composed of two circular flexible hinges, and the production material of the flexible hinge is aluminum alloy.

[0008] On the other hand, the present invention also proposes a control method for a waviness error compensation device based on a multi-axis nano-ultrafast system. Regarding the vacuum adsorption platform as a rigid body, a displacement sensor detects the position height of the vacuum adsorption platform, and the actual pose of the vacuum adsorption platform is obtained by means of plane fitting calculation, which specifically includes the following steps:

[0009] Step S1, input the height of the corresponding measurement point into each driving unit. At this time, the displacement influence of the three piezoelectric ceramic drivers outside this driving unit on this driving unit is regarded as a disturbance, and the piezoelectric ceramic driver in this driving unit is used for compensation control to achieve target decoupling, so as to more conveniently control a single driving unit;

[0010] Step S2, convert the position information into an actual voltage input signal through a driving power supply, and drive the piezoelectric ceramic driver to push the flexible hinge upward to adjust the pose of the vacuum adsorption platform.

[0011] Preferably, the control method of a single driving unit in step S1 includes the following steps:

[0012] Step S11, model the piezoelectric drive unit system. This piezoelectric drive unit system includes the strong hysteresis nonlinear characteristics exhibited by the piezoelectric ceramic driver and the linear characteristics exhibited by the flexible hinge mechanism. Model the two characteristics respectively and synthesize them into the dynamic characteristics of the overall system to obtain a complete dynamic characteristic description model of the piezoelectric drive unit system for subsequent control method design;

[0013] Step S12, according to the description model established in step S11, model the hysteresis inverse model to calculate the corresponding input control voltage according to the target hysteresis intermediate quantity, eliminate the influence of the hysteresis characteristic, and linearize the piezoelectric drive unit system;

[0014] Step S13, compensate for the error caused by linearization through the active disturbance rejection control algorithm. This algorithm observes the disturbance of the system by using a linear form of the Luenberger state observer, compensates the disturbance to the input end, and simultaneously achieves the purpose of eliminating the internal disturbance and external disturbance of the system;

[0015] Step S14, under the condition of extended state observer compensation, the system characteristics approach the linear system model. In order to make the observer tracking accuracy higher, it is necessary to reduce the order of the system characteristics to a second-order system:

[0016] ;

[0017] where s is the complex variable of the transfer function of the multi-axis nano-ultrafast system, , , are the constants of the second-order system, obtained by simulation or experiment calibration. Based on the characteristics of the reduced-order second-order system, second-order state feedback control is performed on the original system. In order to make the system response characteristics approach an identical linear relationship in a wider frequency range, that is, both the amplitude response and the phase lag tend to 0. According to the response characteristics of the second-order system, the natural frequency of the closed-loop system needs to be increased, and the damping ratio needs to be reduced to near critical damping;

[0018] Step S15, for the system after closed-loop feedback regulation, its phase response still has a lag. Therefore, a zero-phase tracking feedforward compensation term is added before the closed-loop system to ensure that there is no phase lag in the system output theoretically, and generally improve the dynamic tracking performance and control accuracy of the system.

[0019] Preferably, in the step S11, the two characteristics are respectively modeled as the MPI model and the ARX model. Among them, the MPI model is the magnetic particle imaging model, which can better characterize the strong hysteresis nonlinear characteristics, and the ARX model is the autoregressive exogenous model, which can summarize the characteristics of the linear part of the system.

[0020] The beneficial technical effects of the present invention are:

[0021] A waviness error compensation device and control method based on a multi-axis nano-ultrafast system provided by the present invention, in addition to improving the characteristics of the machine tool itself, based on the multi-axis nano-ultrafast system driven by piezoelectric ceramics and flexure hinges, through precise control by piezoelectric ceramic actuators, it can be used as a micro-feed and micro-adjustment device for the workpiece to be machined. Through the control algorithm composed of hysteresis compensation feedforward, active disturbance rejection control and zero-phase feedforward, it can compensate the relative vibration between the tool tip and the workpiece in real time, suppress the arc-shaped waviness error of fly-cutting, and thus significantly improve the quality and accuracy of the machined surface. Compared with other technical means, the present invention reduces the requirements for machine tool component configuration and operation costs; at the same time, the feedforward linearization control method based on the hysteresis nonlinear model and the active disturbance rejection control algorithm with zero-phase feedforward proposed by the present invention achieve closed-loop control, which can greatly improve the tracking control accuracy of the system and effectively suppress the PV value, that is, the peak-to-valley value, of the arc-shaped surface shape during machining. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0023] Figure 1 It is a schematic structural diagram of a waviness error compensation device based on a multi-axis nano-ultrafast system provided by an embodiment of the present invention;

[0024] Figure 2 It is a structural block diagram of an inverse hysteresis feedforward linearization method provided by an embodiment of the present invention;

[0025] Figure 3 It is an equivalent system block diagram after inverse hysteresis feedforward linearization provided by an embodiment of the present invention, where (a) represents an equivalent system block diagram including each error component, and (b) represents an equivalent system block diagram including lumped errors;

[0026] Figure 4 It is a control block diagram of introducing an extended state observer for disturbance observation compensation provided by an embodiment of the present invention;

[0027] Figure 5 It is a system control block diagram of state feedback regulation provided by an embodiment of the present invention.

[0028] Explanation of reference numerals: 1. Driving unit; 2. Flexible hinge; 3. Vacuum adsorption platform; 4. Connecting flange; 5. Fixed base; 1-1. Piezoelectric ceramic actuator; 1-2. Displacement sensor. Detailed implementation manners

[0029] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0030] Such as Figure 1As shown in the figure, a waviness error compensation device based on a multi-axis nano-ultrafast system provided by the present invention includes four drive units 1 of the same model, four groups of flexible hinges 2, a vacuum adsorption platform 3, four connecting flanges 4, and a fixed base 5. Each drive unit 1 includes a piezoelectric ceramic actuator 1-1 and a displacement sensor 1-2. The piezoelectric ceramic actuator 1-1 is located at the bottom of the flexible hinge 2. The vacuum adsorption platform 3 is connected to the flexible hinge 2. The vacuum adsorption platform 3 uses four piezoelectric ceramic actuators 1-1 as legs through the flexible hinge 2. The lower end of the piezoelectric ceramic actuator 1-1 is connected to the fixed base 5 for support. The four groups of flexible hinges 2 are symmetrically distributed in parallel on the connecting flange 4. The connecting flange 4 is located on the fixed base 5. The four displacement sensors 1-2 are the same as the piezoelectric ceramic actuators 1-1 and are arranged at four points on the edge of the vacuum adsorption platform 3 in a cross distribution to measure the pose height at the corresponding positions, so that the waviness error compensation device has three degrees of freedom, namely the rotational degrees of freedom RX and RY in two horizontal directions of the platform plane of the vacuum adsorption platform 3 and the translational degree of freedom Z in the vertical direction. By applying a voltage to drive the piezoelectric ceramic actuator 1-1, the pose of the vacuum adsorption platform 3 is adjusted.

[0031] The stiffness of the piezoelectric ceramic actuator 1-1 is 50 N / μm, the driving voltage is 100 V, the stroke is 32 μm, and the natural frequency is 40 kHz.

[0032] The displacement sensor 1-2 is a capacitive sensor, and the specific model of the probe is C8-2.0-2.0. The suitable distance range between the probe of this sensor and the measurement target is 75-125 μm. There is a direct linear relationship between its output voltage and the displacement: when the output voltage is 10 V, it means the measurement object is 75 μm away from the probe; when the output voltage is -10 V, it corresponds to a distance of 125 μm. The measurement range of this sensor is 50 μm, and it has a measurement accuracy of up to 0.003% of the full scale.

[0033] Each flexible hinge 2 is composed of two circular flexible hinges. The production material of the flexible hinge 2 is aluminum alloy 7075-T6. The four groups of flexible hinges are arranged in a parallel symmetric structure, and each moving chain uses a flexible moving pair.

[0034] The present invention also provides a method for controlling the above-mentioned waviness error compensation device based on a multi-axis nano-ultrafast system. Regarding the vacuum adsorption platform 3 as a rigid body, the displacement sensor 1-2 detects the position height of the vacuum adsorption platform, and the actual pose of the vacuum adsorption platform is obtained by means of plane fitting calculation, which specifically includes the following steps:

[0035] Step S1: Input the height of the corresponding measurement point into each driving unit 1. At this time, the displacement influence of the three piezoelectric ceramic actuators 1-1 outside this driving unit 1 on this driving unit 1 is regarded as a disturbance, and is compensated and controlled by the piezoelectric ceramic actuator 1-1 in this driving unit 1 to achieve target decoupling, so as to more conveniently control a single driving unit 1.

[0036] In order to be able to transform the precise control of the end pose of the positioning platform into the high-precision disturbance rejection control of a single driving unit 1 in this embodiment, it is necessary to construct a decoupling control framework for the redundant degrees of freedom of the multi-axis nano-ultrafast system. This framework requires a feedforward linearization control method based on a hysteresis nonlinear model and an active disturbance rejection control algorithm with zero-phase feedforward, so as to achieve the high-dynamic decoupling closed-loop control of this motion compensation system.

[0037] The control method of the single driving unit 1 in the step S1 includes the following steps:

[0038] Step S11: First, model the piezoelectric driving unit system. This piezoelectric driving unit system includes the strong hysteresis nonlinear characteristics exhibited by the piezoelectric ceramic actuator 1-1 and the linear characteristics exhibited by the flexible hinge mechanism. Therefore, it is necessary to model the two characteristics separately and synthesize them into the dynamic characteristics of the overall system to obtain a complete dynamic characteristic description model of the piezoelectric driving unit system, so as to facilitate the subsequent design of the control method.

[0039] In step S11, the two characteristics are respectively modeled as an MPI model and an ARX model. Among them, the MPI model is the magnetic particle imaging model, which can better characterize the strong hysteresis nonlinear characteristics, and the ARX model is the autoregressive exogenous model, which can summarize the linear part characteristics of the system.

[0040] Step S12: Model the hysteresis inverse model according to the description model established in step S11, so as to calculate the corresponding input control voltage according to the target hysteresis intermediate quantity, eliminate the influence of the hysteresis characteristics, and linearize the piezoelectric driving unit system.

[0041] As Figure 2 shown is the process of modeling the hysteresis inverse model. represents the ideal input of the system. represents the voltage input of the piezoelectric driving unit. represents the hysteresis output intermediate quantity. H [ u ] represents the hysteresis characteristics of the system. H n − 1 [ u ] then represents the inverse model of the hysteresis characteristics. Represents the linear characteristics of the system. The inverse hysteresis model is modeled according to the established description model to calculate the corresponding input control voltage based on the target hysteresis intermediate quantity, eliminate the influence of hysteresis characteristics, and linearize the piezoelectric drive unit system. Since the system after feedforward linearization still has some non-linear errors due to the difference between the theoretical model and the actual system, and because the control input is discretized, which will bring integration errors, these two errors are equivalent to the non-linear errors of inverse hysteresis compensation, denoted as Considering the system with modeling errors can be equivalent to Figure 3 As shown in (a), since there must be certain errors in the modeling and identification of the hysteresis model, there is where is the bounded system uncertainty. After equivalent of each part of the error, there is Δ w = Δ ⋅ w f + ( 1 + Δ ) ⋅ e [ w f ] where e [ w f ] represents the non-linear modeling error. After equivalent, it is as shown in Figure 3 (b). Figure 3 In (a) represents the equivalent system block diagram including each error component, and (b) represents the equivalent system block diagram including the lumped error.

[0042] Step S13, through the active disturbance rejection control algorithm, can better compensate for the errors brought by linearization. This algorithm observes the disturbance of the system through a linear form of the Luenberger state observer, compensates the disturbance to the input end, and simultaneously achieves the purpose of eliminating the internal and external disturbances of the system.

[0043] For the internal disturbance which has been obtained above, and in actual working conditions, external disturbances will also be encountered. represents the external disturbance related to the system characteristics, such as factors like input voltage fluctuation, machine tool vibration, etc., which can be equivalent to the input end. represents the external disturbance directly from the outside and unrelated to the system characteristics, such as coupled axis movement, cutting force excitation, etc., which can be equivalent to the output end. Under the total disturbance assumption of the active disturbance rejection algorithm, there is the total disturbance observation value where is the inverse of the linear characteristics obtained from the model. The schematic diagram of the control framework introducing the extended state observer for disturbance observation and compensation is as shown in Figure 4 . Due to the influence of the disturbance, where is the ideal input of the system after being compensated by the observer, is the output of the linearized system, is the displacement output of the system after being disturbed.

[0044] Step S14, under the condition of observing and compensating the expansion state, the system characteristics approach the linear system model. To make the observer tracking accuracy higher, it is necessary to reduce the system characteristics to a second-order system:

[0045] ;

[0046] where s is the complex variable of the transfer function of the multi-axis nano-ultrafast system, 、 、 are the constants of the second-order system, obtained by simulation or experiment calibration. Based on the reduced-order system characteristics, second-order state feedback control is performed on the original system. To make the system response characteristics approach an identical linear relationship in a wider frequency range, that is, both the amplitude response and the phase lag tend to 0. According to the response characteristics of the second-order system, the natural frequency of the closed-loop system needs to be increased, and the damping ratio needs to be reduced to near the critical damping. Preferably, the natural frequency of the target system can be set to , and the damping ratio is the critical damping . Under the condition of ensuring that the static amplitude gain of the system remains unchanged, the second-order system is transformed into an ideal closed-loop system:

[0047] .

[0048] where, 、 、 are the constants of the ideal closed-loop system. To perform feedback control, it is necessary to introduce the state feedback coefficient Figure 4 and the input gain coefficient on the basis of to ensure that the static gain of the system remains unchanged. The system control block diagram after closed-loop state adjustment is shown in Figure 5.

[0049] Step S15, for the system after closed-loop feedback adjustment, there is still a certain lag in its phase response. Therefore, zero-phase tracking control is added on the basis of the active disturbance rejection control algorithm. Adding a zero-phase tracking feedforward compensation term before the closed-loop system can ensure that there is no phase lag in the system output theoretically, and generally improve the dynamic tracking performance and control accuracy of the system.

[0050] Step S2, the position information is converted into an actual voltage input signal through the drive power supply, and the piezoelectric ceramic actuator 1-1 is driven to push the flexible hinge 2 upward to realize the pose adjustment of the vacuum adsorption platform 3.

[0051] The present invention verifies the dynamic characteristics of the multi-axis nano-ultrafast system through simulation modal analysis and experimental modal analysis to ensure that the system design meets the requirements of high-performance dynamic response for subsequent research on the control system.

[0052] It is required that the control system be accurate enough to reflect the machining state in real time and make corresponding pose adjustments in a timely and accurate manner, so as to reduce the waviness error. Therefore, in the present invention, the degrees of freedom of the platform are decoupled, and the pose control of the positioning platform is transformed into the individual compensation control of each driving unit 1 by the piezoelectric ceramic actuator 1-1 to cancel the coupling effect and simplify the control process. The proposed control algorithms are the feedforward linearization control method based on the hysteresis nonlinear model and the active disturbance rejection control algorithm with zero-phase feedforward. Considering the strong hysteresis nonlinear characteristics of the piezoelectric ceramic actuator 1-1 and the linear characteristics such as flexible hinges, the nonlinear characteristics are modeled by the model of MPI + high-order term linear system in series, and the linear characteristics are modeled by the ARX model, obtaining a complete dynamic characteristic description model of the driving unit electromechanical system. Based on this model, the piezoelectric drive unit system is linearized to eliminate the influence of nonlinear characteristics, and then the error caused by feedforward linearization and external disturbance is further compensated by using the active disturbance rejection control algorithm with zero-phase feedforward.

[0053] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A waviness error compensation device based on a multi-axis nano-ultrafast system, characterized in that: The device comprises four drive units (1) of the same model, four sets of flexible hinges (2), a vacuum adsorption platform (3), four connecting flanges (4) and a fixed base (5), each drive unit (1) comprises a piezoelectric ceramic drive (1-1) and a displacement sensor (1-2), the piezoelectric ceramic drive (1-1) is located at the bottom of the flexible hinge (2), the vacuum adsorption platform (3) is connected to the flexible hinge (2), the vacuum adsorption platform (3) is supported by the four piezoelectric ceramic drives (1-1) through the flexible hinge (2), the lower end of the piezoelectric ceramic drive (1-1) is connected to the fixed base (5) for support, and the four sets of The flexible hinges (2) are symmetrically distributed in parallel on the connecting flange (4), and the connecting flange (4) is located on the fixed base (5); four displacement sensors (1-2) and the piezoelectric ceramic driver (1-1) are arranged at four points on the edge of the vacuum adsorption platform (3) in a cross distribution, and are used to measure the posture height of the corresponding position, so that the corrugation error compensation device has three degrees of freedom, which are the rotational degrees of freedom RX and RY in two horizontal directions of the platform plane of the vacuum adsorption platform (3) and the movement degree of freedom Z in the vertical direction. The piezoelectric ceramic driver (1-1) is driven by applying voltage to adjust the posture of the vacuum adsorption platform (3); The vacuum adsorption platform (3) is regarded as a rigid body, and the displacement sensor (1-2) detects the position height of the vacuum adsorption platform, and the actual position and posture of the vacuum adsorption platform is obtained by plane fitting calculation, which specifically includes the following steps: Step S1, the height of the corresponding measuring point is input into each drive unit (1), and the displacement influence of the three piezoelectric ceramic drivers outside the drive unit is regarded as disturbance, and the piezoelectric ceramic driver in the drive unit performs compensation control to achieve target decoupling, so as to more conveniently control the single drive unit (1). The control of the single drive unit (1) specifically includes: Step S11, modeling a piezoelectric drive unit system, the piezoelectric drive unit system including the strong hysteresis nonlinear characteristics exhibited by the piezoelectric ceramic driver (1-1) and the linear characteristics exhibited by the flexible hinge mechanism, modeling the two characteristics separately and integrating them into the dynamic characteristics of the overall system, thereby obtaining a complete piezoelectric drive unit system dynamic characteristics description model; Step S12, building a hysteresis inverse model according to the description model established in step S11, so as to calculate the corresponding input control voltage according to the target hysteresis intermediate amount, eliminate the influence of hysteresis characteristics, and linearize the piezoelectric drive unit system; Step S13, compensating the error caused by linearization by using an anti-disturbance control algorithm, which observes the disturbance of the system by using a linear form of a Lomberg state observer, and compensates the disturbance to the input end, thereby achieving the purpose of eliminating internal and external disturbances of the system; Step S14: Under the condition of extended state observation compensation, the system characteristics approach the linear system model. In order to make the observer tracking more accurate, the system characteristics need to be reduced to a second-order system: Among them, s is the complex variable of the transfer function of the multi-axis nano-ultrafast system, a1, b1, b2 are constants of the second-order system, obtained by simulation or experimental calibration, and the original system is subjected to second-order state feedback control based on the reduced-order second-order system characteristics. In order to make the amplitude response and phase lag of the system approach 0, according to the response characteristics of the second-order system, the natural frequency of the closed-loop system needs to be increased, and the damping ratio needs to be reduced to near the critical damping; Step S15, after the closed-loop feedback adjustment, the phase response of the system still has a lag, so a zero-phase tracking feedforward compensation term is added before the closed-loop system to ensure that the system output has no phase lag in theory; Step S2, converting the position information into an actual voltage input signal through a driving power supply, driving the piezoelectric ceramic driver (1-1) to push the flexible hinge (2) to move upward, thereby achieving position adjustment of the vacuum adsorption platform (3).

2. The waviness error compensation device based on a multi-axis nano-ultrafast system according to claim 1, characterized in that: The piezoelectric ceramic driver (1-1) has a stiffness of 50 N / μm, a driving voltage of 100 V, a stroke of 32 μm and a natural frequency of 40 kHz.

3. The waviness error compensation device based on a multi-axis nano-ultrafast system according to claim 1 is characterized in that: The displacement sensor (1-2) is a capacitive sensor, and there is a direct linear relationship between its output voltage and displacement.

4. The waviness error compensation device based on a multi-axis nano-ultrafast system according to claim 1, characterized in that: Each flexible hinge (2) is composed of two circular flexible hinges, and the flexible hinge (2) is made of aluminum alloy.

5. The waviness error compensation device based on a multi-axis nano-ultrafast system according to claim 1, characterized in that: In step S11, the two characteristics are modeled as an MPI model and an ARX model respectively, wherein the MPI model is a magnetic particle imaging model, which can characterize the strong hysteresis nonlinear characteristics, and the ARX model is an organic regression model, which can summarize the linear part characteristics of the system.

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