A Gas Film Modeling Method for an Air-Floating Precision Electromechanical System

Abstract

The present invention belongs to the technical field related to mechatronic control, and discloses a method for gas film modeling of a gas-floating precision mechatronic system. The method includes the following steps: S1 constructing a fractional-order component and connecting the fractional-order component with a spring to obtain a gas film model; S2 performing dynamic testing on the gas-floating motion platform to be processed to obtain the relationship between the excitation force and displacement / acceleration of the gas-floating motion platform to be processed in the frequency domain, and calculating the unknown parameters in the gas film model by using the relationship; S3 solving the equivalent stiffness and equivalent damping corresponding to the gas film model by using the unknown parameters to determine the motion equation and transfer function of the gas-floating motion platform, that is, obtaining the gas film equivalent model of the gas-floating precision mechatronic system. Through the present invention, the problem of precise and rapid modeling of the dynamic characteristics of the gas film in the control process of the gas-floating precision system is solved, and an effective and accurate theoretical model basis can be provided for the active suppression of the gas-floating motion platform.

Description

Technical Field

[0001] The present invention belongs to the technical field related to mechatronic control, and more specifically, relates to a method for gas film modeling of an air-bearing precision mechatronic system. Background Art

[0002] Ultra-precision manufacturing equipment represents the highest development level of precision manufacturing. Compared with contact bearings, air bearings have the advantages of high speed, high precision, low power consumption, and long service life. They can meet the requirements of ultra-precision equipment such as IC manufacturing and optical element processing for motion support. The dynamic characteristics of the bearing gas film are one of the most important characteristics of air bearings, and its characteristics affect the control accuracy of the precision mechatronic platform. However, in the prior art, traditional model methods have problems such as complex modeling and low accuracy.

[0003] The equivalent spring-damper-mass system composed of the gas lubrication film and the load mass is also an essentially nonlinear system. Obviously, the conclusion that the dynamic stiffness value of a linear system at low frequencies is equal to its static stiffness does not immediately apply to this nonlinear system of the gas film mass. For the gas film mass system, establishing a model that can more accurately equivalent the dynamic characteristics of the gas film has important practical significance. Summary of the Invention

[0004] In view of the above defects or improvement requirements of the prior art, the present invention provides a method for gas film modeling of an air-bearing precision mechatronic system, which solves the problem of precise and rapid modeling of the dynamic characteristics of the gas film in the control process of the air-bearing precision system.

[0005] To achieve the above object, according to one aspect of the present invention, a method for gas film modeling of an air-bearing precision mechatronic system is provided. The method includes the following steps:

[0006] S1 Construct a fractional-order component and connect the fractional-order component with a spring to obtain a gas film model;

[0007] S2 Perform dynamic testing on the air-bearing moving platform to be processed, obtain the relationship between the excitation force and displacement / acceleration of the air-bearing moving platform to be processed in the frequency domain, and calculate the unknown parameters in the gas film model using this relationship;

[0008] S3 Solve the equivalent stiffness and equivalent damping corresponding to the gas film model using the unknown parameters to determine the motion equation and transfer function of the air-bearing moving platform, that is, obtain the gas film equivalent model of the air-bearing precision mechatronic system.

[0009] Further preferably, in step S1, the fractional-order component is carried out according to the following relational expression:

[0010]

[0011] Among them, F(t) is the applied force, and η is the correlation coefficient. is the α-th derivative of the displacement x(t) with respect to time t.

[0012] Further preferably, in step S1, the air film model is:

[0013]

[0014] where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 and η2 are the correlation coefficients of the fractional dampers, α and β are the fractional orders, and a, b, c, and d are intermediate variables.

[0015] Further preferably, in step S1, the air film model is:

[0016]

[0017] where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 and η2 are the correlation coefficients of the fractional dampers, α and β are the fractional orders, and a, b, c, and d are intermediate variables.

[0018] Further preferably, in step S1, the air film model is:

[0019]

[0020]

[0021]

[0022] where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 and η2 are the correlation coefficients of the fractional dampers, α and β are the fractional orders, and a, b, c, and d are intermediate variables.

[0023] Further preferably, in step S1, the air film model is

[0024]

[0025]

[0026]

[0027] where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 and η2 are the correlation coefficients of the fractional dampers, α and β are the fractional orders, and a, b, c, and d are intermediate variables.

[0028] Further preferably, in step S2, the dynamic test adopts the force hammer method or the shaker method.

[0029] Further preferably, in step S2, a double-layer algorithm combining a genetic algorithm and a particle swarm algorithm is used to calculate the unknown parameters in the gas film model.

[0030] Further preferably, the fitness function of the double-layer algorithm combining the genetic algorithm and the particle swarm algorithm is:

[0031]

[0032] where Z d (k) is the transfer function value measured experimentally, Z(k) is the fractional-order gas film model value, and n is the data length.

[0033] Further preferably, in step S2, the unknown parameters are the spring stiffness coefficients k1, k2, the fractional-order component correlation coefficients η1, η2, and the fractional-order orders α, β.

[0034] Generally speaking, compared with the prior art, the above technical solution conceived by the present invention has the following beneficial effects:

[0035] 1. By using the experimental air-floating response data, the present invention uses the GA-PSO algorithm to identify the parameters of the fractional-order model, and can prove the effectiveness of this method through experiments, realizing the design, implementation, and optimization of the fractional-order gas film equivalent model, and proving the effectiveness and simplicity of the gas film equivalent model of the present invention;

[0036] 2. The fractional-order components adopted in the present invention, where the fractional order is a mathematical generalization with a fractional order in calculus. Using the fractional-order model can accurately and equivalently fit the gas film characteristics, having the advantages of simple form and accurate model, providing a new idea for accurate dynamic modeling of the gas film;

[0037] 3. Compared with the models established by the finite element method and the finite difference method, the gas film equivalent model constructed by the present invention is simpler, has fewer parameters, is easier to calculate, and is more suitable for establishing a gas film model in the electromechanical control model; in addition, compared with the simple spring-damper equivalent model, the gas film model can significantly improve the equivalent accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 is the gas film modeling method for the air-floating precision electromechanical system constructed according to the preferred embodiment of the present invention;

[0039] Figure 2 is the schematic diagram of the FO-Zener gas film model constructed according to the preferred embodiment of the present invention;

[0040] Figure 3 is the schematic diagram of the FO-Maxwell gas film model constructed according to the preferred embodiment of the present invention;

[0041] Figure 4 It is a schematic diagram of the FO-3-parameter air film model constructed according to the preferred embodiment of the present invention;

[0042] Figure 5 It is a schematic diagram of the FO-Burgess air film model constructed according to the preferred embodiment of the present invention;

[0043] Figure 6 It is a schematic diagram of the GA-PSO algorithm flow constructed according to the preferred embodiment of the present invention;

[0044] Figure 7 It is a comparison chart of the performance of various models and the SD model constructed according to the preferred embodiment of the present invention. Among them, (a) is the real part of the transfer function H(f), (b) is the imaginary part of the transfer function H(f), (c) is the transfer function H(f), and (d) is the phase of the transfer function H(f);

[0045] Figure 8 It is a comparison chart of the experimental errors of various models and the SD model constructed according to the preferred embodiment of the present invention. Among them, (a) is the RMS real part error of the transfer function H(f), (b) is the RMS imaginary part error of the transfer function H(f), and (c) is the RMS error of the transfer function H(f). Detailed implementation manners

[0046] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0047] An air film modeling method for an air-floating precision electromechanical system, which is based on fractional-order modeling. Fractional-order modeling is a newly emerging method for analysis and modeling using fractional-order calculus theory. Its non-linear characteristics are very similar to the dynamic characteristics of the air film, making fractional-order modeling have the advantages of simplicity and accuracy; therefore, the present invention combines fractional-order calculus theory to propose a new type of air film equivalent model, providing a new method for accurate air film modeling, as Figure 1 shown, including the following steps:

[0048] S1 Establish fractional-order components and connect them in series and parallel with springs to obtain an air film model. The following 4 types of air film equivalent models can be established, where w = 2πf, f is the frequency (Hz), k1, k2 are the stiffness coefficients of the corresponding springs, η1, η2 are the fractional-order component related coefficients, and α, β are the fractional-order orders.

[0049] Establish a fractional-order component:

[0050]

[0051] Wherein, F(t) is the applied force, and η is the correlation coefficient, is the α-order derivative of the displacement x(t) with respect to time t.

[0052] The result of the Laplace transform of this component is:

[0053]

[0054] Euler transform

[0055]

[0056] The air film model can be one of the following four types:

[0057] (1) As shown in Figure 2 the FO-Zener air film model

[0058]

[0059] Its equivalent stiffness and damping are:

[0060]

[0061]

[0062] In the formula:

[0063]

[0064]

[0065]

[0066]

[0067] (2) As shown in Figure 3 the FO-Maxwell air film model

[0068]

[0069] Its equivalent stiffness and damping are:

[0070]

[0071]

[0072] When n = 1, θ = α, and when n = 2, θ = β

[0073]

[0074]

[0075]

[0076]

[0077] (3) As Figure 4 shown, the FO-3 parameter air film model

[0078]

[0079] wherein:

[0080]

[0081] Its equivalent stiffness and damping are:

[0082]

[0083]

[0084] (4) As Figure 5 shown, the FO-burgess air film model

[0085]

[0086] wherein:

[0087]

[0088]

[0089] Its equivalent stiffness and damping are:

[0090]

[0091]

[0092] S2 Adopt the force hammer method or the exciter method to conduct dynamic tests on the air floating system, obtain the relationship between the excitation force and displacement / acceleration of the system in the frequency domain, optimize the parameters in the air film equivalent model, and thus obtain the parameter correlation coefficients k1, k2, η1, η2, and the fractional order orders α and β.

[0093] S3 Use the optimized parameters above to solve the equivalent stiffness and equivalent damping corresponding to each air film model, and thus determine the motion equation and transfer function of the air floating motion platform, that is, obtain the air film equivalent model of the air floating precision electromechanical system.

[0094] The motion equation of the air film - mass system (air - floating motion platform system) can be expressed as:

[0095]

[0096] Where c(w) and k(w) are the expressions of the equivalent stiffness and equivalent damping obtained from the established air - film model respectively. The transfer function of the air - film - mass system can be obtained from the equation as:

[0097]

[0098] The dynamic characteristics of the air film (A(w) and F(w)) are obtained from the fast Fourier transform of the impact force and the mover acceleration measured by the force hammer method / vibrator method.

[0099] As a further preference, in step S2, a double - layer algorithm (GA - PSO) combining the genetic algorithm and the particle swarm algorithm is used to optimize the parameters in the air - film equivalent model, and the air - film equivalent model of the air - floating precision electromechanical system is obtained.

[0100] As a further preference, as Figure 6 shown, the root - mean - square tracking error of the modulus is used as the fitness function of the optimization algorithm. With the goal of minimizing the fitness function, the model parameters are iteratively optimized by the GA - PSO algorithm, and its calculation formula is as follows:

[0101]

[0102] Where, Z d (k) is the value of the transfer function measured experimentally, Z(k) is the value of the air - film model, and n is the data length.

[0103] As a further preference, the fractional - order operator calculation formula is as follows:

[0104]

[0105] Where, Γ is the gamma function, where n - 1 ≤ α < n (n is an integer), α is the order of the fractional derivative, and f (n) (τ) is the n - th derivative of f(τ).

[0106] The present invention will be further described below with specific embodiments.

[0107] S1 Establish a fractional - order component and connect it in series and parallel with a spring to obtain the following air - film model (taking the FO - Zener air - film model as an example), and its transfer function is:

[0108]

[0109] Its equivalent stiffness and damping are:

[0110]

[0111]

[0112] In the formula:

[0113]

[0114]

[0115]

[0116]

[0117] S2 Establish a single-degree-of-freedom vibration system, and use the force hammer method or the shaker method to conduct dynamic tests on the air-floating system to obtain the relationship between the system force and displacement / acceleration in the frequency domain. Taking the acceleration sensor as an example, the theoretical expression of the frequency domain experiment is:

[0118]

[0119] S3 Substitute the established model equivalent stiffness expression into the experimental theoretical expression, and use the double-layer algorithm (GA-PSO) combining the genetic algorithm and the particle swarm algorithm to optimize the parameters in the air film equivalent model to obtain the air film equivalent model of the air-floating precision electromechanical system. The schematic diagram of the algorithm flow is as Figure 6 shown.

[0120] Introduce the root mean square error of the modulus as the fitness function to optimize the parameters.

[0121]

[0122] The search results are as Figure 7 and Figure 8 shown. The result of the fractional-order air film model established by the present invention can reduce the error by more than 80% at most compared with the spring-damping model.

[0123] In summary, a method for establishing a fractional-order air film model and its parameter identification is applied to the simulation modeling of the air film due to its characteristic of accurately describing nonlinear phenomena; based on the application of the fractional-order calculus theory to describe the nonlinear link of the air film, the present invention proposes a fractional-order air film model, which has the advantages of simple parameters and high modeling accuracy.

[0124] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for gas film modeling of an air-floating precision electromechanical system, characterized in that, The method includes the following steps: S1 Construct a fractional-order component and connect the fractional-order component with a spring to obtain an air film model; S2 Conduct a dynamic test on the air-bearing motion platform to be processed to obtain the relationship between the exciting force and displacement / acceleration of the air-bearing motion platform to be processed in the frequency domain, and calculate the unknown parameters in the air film model by using this relationship; S3 Solve the equivalent stiffness and equivalent damping corresponding to the air film model by using the unknown parameters to determine the motion equation and transfer function of the air-bearing motion platform, that is, obtain the air film equivalent model of the air-bearing precision electromechanical system; In step S1, the fractional-order component is as follows: where F(t) is the applied force and η is the correlation coefficient, is the α-th derivative of the displacement x(t) with respect to time t; In step S1, the air film model is: where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 is the fractional damper correlation coefficient, and α is the fractional order.

2. The air film modeling method of an air-floating precision electromechanical system according to claim 1, characterized in that, In step S1, the air film model can also be: where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 and η2 are the fractional damper correlation coefficients, and α and β are the fractional orders.

3. A method for gas film modeling of an air-floating precision electromechanical system according to claim 1, characterized in that, In step S1, the air film model can also be: where w = 2πf, f is the frequency, k1 is the stiffness coefficient of the corresponding spring, η1 and η2 are the fractional damper correlation coefficients, α and β are the fractional orders, and a, b, c, and d are intermediate variables.

4. The air film modeling method of an air floating precision electromechanical system according to claim 1, characterized in that, In step S1, the air film model can also be: where w = 2πf, f is the frequency, k1 and k2 are the stiffness coefficients of the corresponding springs, η1 and η2 are the fractional damper correlation coefficients, α and β are the fractional orders, and a, b, c, and d are intermediate variables.

5. A method for gas film modeling of an air-floating precision electromechanical system according to claim 1 or 2, characterized in that, In step S2, the dynamic test adopts the force hammer method or the shaker method.

6. A method for gas film modeling of an air-floating precision electromechanical system according to claim 1 or 2, characterized in that, In step S2, the calculation of the unknown parameters in the air film model adopts a two-layer algorithm combining the genetic algorithm and the particle swarm algorithm.

7. A method for gas film modeling of an air-floating precision electromechanical system according to claim 6, characterized in that, The fitness function of the two-layer algorithm combining the genetic algorithm and the particle swarm algorithm is: Among them, Z d (k) is the value of the transfer function measured experimentally, Z(k) is the value of the fractional-order gas film model, and n is the data length.

8. A method for gas film modeling of an air-floating precision electromechanical system according to claim 1 or 2, characterized in that In step S2, the unknown parameters are the stiffness coefficients k1, k2 of the spring, the fractional-order component correlation coefficients η1, η2, and the fractional-order orders α, β.

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