Temperature error compensation method for CNC machine tools based on fractional-order model synchronization

Abstract

The present invention discloses a method for compensating temperature errors of numerically controlled machines based on fractional-order model synchronization, comprising: a first step: establishing a fractional-order model of temperature errors of numerically controlled machines and identifying the parameters of the fractional-order model according to the temperature error and the deviation of the model output; a second step: designing an equivalent circuit according to the established fractional-order model of temperature errors; the output of the equivalent circuit is consistent with the temperature error of the numerically controlled machine, and error compensation is performed; the circuit includes a real-time temperature detection unit, which is the same temperature detection unit used for fractional-order modeling; the model equivalent circuit receives a temperature signal and generates an error signal according to the temperature signal. The present invention avoids complex calculations and enables real-time compensation of temperature errors, thereby improving the machining accuracy of numerically controlled machines. It avoids temperature error modeling based on complex mechanisms and can approximate the actual temperature error model of numerically controlled machines with arbitrary accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of high-precision CNC machine tool temperature error modeling and compensation, and in particular to a CNC machine tool temperature error compensation method based on fractional order model synchronization. Background Art

[0002] CNC machine tools are basic equipment in the field of processing and play an important role in the field of intelligent manufacturing. Various errors are inevitable in the operation and processing of machine tools, including temperature errors, geometric errors and errors caused by loads. Among them, temperature errors account for about 40% to 70% of the total errors of high-precision CNC machine tools. The existence of errors will inevitably affect the processing accuracy. The rapid development of industrial and agricultural production, scientific research, military industry and other fields has put forward higher requirements on the processing accuracy of mechanical parts. It can be seen that it is of great significance to study the temperature error modeling and error compensation mechanism of CNC machine tools and improve the processing accuracy of CNC machine tools.

[0003] The multivariate linear regression model based on least squares, BP neural network model, distributed lag model, grey theory model and support vector machine model have all been studied in the field of temperature error of CNC machine tools, but it is difficult to achieve ideal results. The causes of temperature effect errors of CNC machine tools are complex: ambient temperature changes, heating during cutting, uneven heat dissipation, etc. These factors lead to irregular distribution of temperature field, and it is difficult to establish an error mathematical model based on the temperature field distribution.

[0004] If modeling is done through physical mechanisms, the model will be extremely complex, parameter identification will be difficult, the amount of data calculation will be large, the calculation time will be long, and it will be difficult to meet the high-precision performance requirements of real-time processing of CNC machine tools. Summary of the invention

[0005] In view of the above-mentioned deficiencies of the prior art, the present invention provides a CNC machine tool temperature error compensation method based on fractional-order model synchronization, and a CNC machine tool temperature error compensation system with an equivalent compensation circuit designed according to the model, so as to solve the problems of modeling difficulty and error compensation lag in the prior art.

[0006] In order to solve the above technical problems, the technical solution of the present invention is as follows: a temperature error compensation method for CNC machine tools based on fractional order model synchronization, the steps are as follows:

[0007] Step 1: Establish a fractional-order model of temperature error for CNC machine tools and identify the parameters of the fractional-order model based on the temperature error and the model output deviation; including:

[0008] Establish a fractional-order model, collect temperature signals according to the temperature detection unit, and obtain the output signal of the fractional-order model; design the fractional-order model parameter identification rule by using the difference between the fractional-order model output signal and the temperature signal of the CNC machine tool;

[0009] When the difference between the fractional-order model output and the temperature error signal of the CNC machine tool meets a certain threshold condition, the parameter identification is stopped and the fractional-order model of the temperature error of the CNC machine tool is obtained;

[0010] Step 2: Design an equivalent circuit based on the established fractional-order model of temperature error; the output of the equivalent circuit is consistent with the temperature error of the CNC machine tool, and error compensation is performed; including:

[0011] The circuit includes a real-time temperature detection unit, which is the same temperature detection unit used in fractional order modeling; the model equivalent circuit receives a temperature signal and generates an error signal according to the temperature signal;

[0012] It also includes an error compensation module for the temperature error output signal of the CNC machine tool and the output signal of the model equivalent circuit; the model equivalent circuit is an analog circuit, and the temperature signal and the temperature effect error signal as the input and output signals of the model equivalent circuit are both analog signals; the temperature error output signal of the CNC machine tool is an analog signal; and the signal difference output by the temperature error compensation module is an analog signal.

[0013] Furthermore, the steps to establish the fractional-order model of temperature error of CNC machine tools are as follows:

[0014] Step 1.1: Design the fractional order model of temperature error of CNC machine tools as follows:

[0015]

[0016] In the formula, t represents time, T represents the ambient temperature of the CNC machine tool, and e T (t) represents temperature error, c represents fixed error, and k represents coefficient; represents Caputo fractional differential, and its definition expression is:

[0017]

[0018] In the formula, Γ(·) represents the gamma function, 0 represents the initial time; τ is the integral variable, 0<α<1 represents the differential order; the parameters c, k, α in the model are all unknown constants;

[0019] Step 1.2: Design the estimation model of the fractional-order model of temperature error of the CNC machine tool in step 1 as follows:

[0020]

[0021] in, is the temperature error of CNC machine tools e T (t) Estimated value; parameter are the estimated values ​​of the unknown parameters c, k, and α respectively;

[0022] Step 1.3: Define the difference between the temperature error and parameter error estimate and the true value as:

[0023]

[0024]

[0025] The following equation is obtained:

[0026]

[0027]

[0028] Step 1.4: Design parameter estimation value identification rules to make the estimated value consistent with the true value; the parameter identification rules are as follows:

[0029]

[0030] when This rule ensures that the following expression always holds:

[0031]

[0032] According to the fractional-order Lyapunov theory, E k (t) and E c (t) gradually approaches 0, realizing the identification of unknown parameters k, c;

[0033] Step 1.5: Divide the range (0,1) into five equal parts and take the middle value of each interval as the estimated value of the differential order, that is: Collect temperature module data, identify parameters according to the parameter identification rule of formula (6), and obtain parameters Five sets of parameters;

[0034] Step 1.6: Use the temperature data collected by the temperature sensor as the input of the fractional-order model, obtain the temperature error estimate according to the established model, subtract the temperature error estimate from the measured temperature error of the CNC machine tool, and obtain the temperature error estimate deviation; according to step 1.5, obtain five groups of temperature error estimate deviations

[0035] Step 1.7: Using the five groups of estimated value deviations obtained in step 1.6, calculate the mean square error of the five groups of estimated value deviations respectively;

[0036] Step 1.8, according to the minimum mean square error corresponding to And The neighborhood of is the new interval and then divided into five equal parts and the middle value is taken; repeat steps 1.6 to 1.7, and exit the loop when the mean square error is less than a certain threshold;

[0037] Model differential order is the current value of α, E k (t),E c The value (t) is the value of k and c in the model. Substituting it into formula (1) finally obtains the fractional order model of temperature error of CNC machine tools.

[0038] Furthermore, the steps of designing an equivalent circuit and performing synchronous compensation according to the fractional-order model of temperature error are as follows:

[0039] Step 2.1: The Laplace transform of the fractional-order temperature error model satisfies

[0040]

[0041] Among them, E T (s) and T(s) are e T (t) and the Laplace transform of T; s represents the Laplace operator, and α is the differential order of the fractional order model;

[0042] Step 2.2: Using the frequency domain equivalent relationship between fractional-order differentials and integer-order differentials, establish an equivalent model of the model in step 2.1; that is, obtain an equivalent model of the fractional-order model according to the differential order of step 2.1;

[0043] Step 2.3: According to the equivalent model of the fractional-order model obtained in step 2.2, establish the input-output relationship expression of temperature and temperature error, and decompose the expression into the subsystem multiplication form as the system function. The expression is as follows:

[0044]

[0045] In the formula, h f(α) (s) represents f(α) different filter functions, i.e., decomposition subsystems. The value of f(α) is determined by the differential order α and the equivalent model; b(kT(s)+c / s) is the temperature input link function; b is a constant coefficient;

[0046] Step 2.4: Design the temperature error equivalent circuit of the CNC machine tool according to the system function obtained in step 2.3; design the temperature input circuit according to the temperature input link function; f(α) (s) Designing a filter circuit, connecting the temperature input circuit and the filter circuit in series, and obtaining a synchronous compensation circuit for temperature errors of CNC machine tools;

[0047] Step 2.5: Subtract the compensation circuit output from the CNC machine output to compensate for the temperature error of the CNC machine.

[0048] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0049] 1. The present invention utilizes a synchronous error compensation circuit to realize temperature error compensation of a CNC machine tool, which can not only realize real-time compensation but also improve the machining accuracy of the CNC machine tool.

[0050] 2. An equivalent fractional-order model of the temperature error of CNC machine tools is established, which not only avoids the temperature error modeling based on complex mechanisms, but also can approximate the actual temperature error model of CNC machine tools with arbitrary accuracy.

[0051] 3. The core of the present invention is to design an equivalent circuit using a fractional order model, and use the equivalent circuit to synchronize the error and compensate the temperature error output, which can avoid complex calculations and enable real-time compensation of temperature errors. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is the temperature error compensation scheme diagram of CNC machine tools;

[0053] Figure 2 is the equivalent circuit block diagram;

[0054] Figure 3 It is the temperature acquisition module circuit;

[0055] Figure 4 It is the h1(s) filter circuit;

[0056] Figure 5 It is the h2(s) filter circuit;

[0057] Figure 6 It is an h3(s) filter circuit. DETAILED DESCRIPTION

[0058] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0059] The temperature error compensation method for CNC machine tools based on fractional order model synchronization of the present invention is shown in the schematic diagram as follows: Figure 1 As shown, the specific steps are as follows:

[0060] Step 1: Establish a fractional-order model of temperature error for CNC machine tools and identify the parameters of the fractional-order model based on the temperature error and the model output deviation; including:

[0061] Establish a fractional-order model, collect temperature signals according to the temperature detection unit, and obtain the output signal of the fractional-order model; design the fractional-order model parameter identification rule by using the difference between the fractional-order model output signal and the temperature signal of the CNC machine tool;

[0062] When the difference between the fractional-order model output and the temperature error signal of the CNC machine tool meets a certain threshold condition, the parameter identification is stopped and the fractional-order model of the temperature error of the CNC machine tool is obtained.

[0063] Among them, the steps to establish the fractional-order model of temperature error of CNC machine tools are as follows:

[0064] Step 1.1: Design the fractional order model of temperature error of CNC machine tools as follows:

[0065]

[0066] In the formula, t represents time, T represents the ambient temperature of the CNC machine tool, and e T (t) represents temperature error, c represents fixed error, and k represents coefficient; represents Caputo fractional differential, and its definition expression is:

[0067]

[0068] In the formula, Γ(·) represents the gamma function, 0 represents the initial time; τ is the integral variable, 0<α<1 represents the differential order; the parameters c, k, α in the model are all unknown constants;

[0069] Step 1.2: Design the estimation model of the fractional-order model of temperature error of the CNC machine tool in step 1 as follows:

[0070]

[0071] in, is the temperature error of CNC machine tools e T (t) Estimated value; parameter are the estimated values ​​of the unknown parameters c, k, and α respectively;

[0072] Step 1.3: Define the difference between the temperature error and parameter error estimate and the true value as:

[0073]

[0074]

[0075] The following equation is obtained:

[0076]

[0077]

[0078] Step 1.4: Design parameter estimation value identification rules to make the estimated value consistent with the true value; the parameter identification rules are as follows:

[0079]

[0080] when This rule ensures that the following expression always holds:

[0081]

[0082] According to the fractional-order Lyapunov theory, E k (t) and E c (t) gradually approaches 0, realizing the identification of unknown parameters k, c;

[0083] Step 1.5: Divide the range (0,1) into five equal parts and take the middle value of each interval as the estimated value of the differential order, that is: Collect temperature module data, identify parameters according to the parameter identification rule of formula (6), and obtain parameters Five sets of parameters;

[0084] Step 1.6: Use the temperature data collected by the temperature sensor as the input of the fractional-order model, obtain the temperature error estimate based on the established model, subtract the temperature error estimate from the measured temperature error of the CNC machine tool to obtain the temperature error estimate deviation; according to step 1.5, obtain five groups of temperature error estimate deviations

[0085] Step 1.7: Using the five groups of estimated value deviations obtained in step 1.6, calculate the mean square error of the five groups of estimated value deviations respectively;

[0086] Step 1.8, according to the minimum mean square error corresponding to And The neighborhood of is the new interval and then divided into five equal parts and the middle value is taken; for example, the minimum mean square error corresponds to Then divide the interval (0.3, 0.7) into five equal parts and take the middle value; repeat steps 1.6 to 1.7, and exit the loop when the mean square error is less than a certain threshold;

[0087] Model differential order is the current value of α, E k (t),E c The value (t) is the value of k and c in the model. Substituting it into formula (1) finally obtains the fractional order model of temperature error of CNC machine tools.

[0088] Step 2: Design an equivalent circuit based on the established fractional-order model of temperature error; the output of the equivalent circuit is consistent with the temperature error of the CNC machine tool, and error compensation is performed; including:

[0089] The circuit includes a real-time temperature detection unit, which is the same temperature detection unit used in fractional order modeling; the model equivalent circuit receives a temperature signal and generates an error signal according to the temperature signal;

[0090] It also includes an error compensation module for the temperature error output signal of the CNC machine tool and the output signal of the model equivalent circuit; the model equivalent circuit is an analog circuit, and the temperature signal and the temperature effect error signal as the input and output signals of the model equivalent circuit are both analog signals; the temperature error output signal of the CNC machine tool is an analog signal; and the signal difference output by the temperature error compensation module is an analog signal.

[0091] Specifically, the steps of designing an equivalent circuit and performing synchronous compensation according to the fractional-order model of temperature error are as follows:

[0092] Step 2.1: The fractional-order differential Laplace transform has the following properties: The fractional-order Laplace transform of the temperature error satisfies

[0093]

[0094] Among them, E T (s) and T(s) are e T (t) and the Laplace transform of T; s represents the Laplace operator, and α is the differential order of the fractional order model;

[0095] Step 2.2: Using the frequency domain equivalent relationship between fractional-order differentials and integer-order differentials, establish an equivalent model of the model in step 2.1; that is, obtain the equivalent model of the fractional-order model according to the differential order of step 2.1. See Table 1;

[0096] Table 1 Frequency domain equivalent relationship between fractional-order model and integer-order model

[0097]

[0098] If α = 0.2, the following equivalent model can be obtained:

[0099]

[0100] Step 2.3: According to the equivalent model of the fractional-order model obtained in step 2.2, establish the input-output relationship expression of temperature and temperature error, and decompose the expression into the subsystem multiplication form as the system function. The expression is as follows:

[0101]

[0102] In the formula, h f(α) (s) represents f(α) different filter functions, that is, the decomposition subsystem, and the value of f(α) is determined by the differential order α and the equivalent model; b(kT(s)+c / s) is the temperature input link function; b is a constant coefficient.

[0103] Take α=0.2 as an example:

[0104] Establish the input-output relationship between temperature and temperature error, the expression is as follows:

[0105]

[0106] Decomposed into subsystem multiplication form, the expression is as follows:

[0107]

[0108] The following four subsystems are obtained:

[0109] Three filters h1(s), h2(s), h3(s),

[0110]

[0111] And the temperature input link:

[0112]

[0113] Step 2.4: Design the temperature error equivalent circuit of the CNC machine tool according to the system function obtained in step 2.3; design the temperature input circuit according to the temperature input link function; f(α) (s) Design a filter circuit, connect the temperature input circuit and the filter circuit in series, and obtain a synchronous compensation circuit for temperature errors of CNC machine tools.

[0114] Taking α=0.2 as an example, the equivalent circuit diagram of the temperature error of CNC machine tools is as follows: Figure 2 According to the temperature input link function, the temperature input circuit is designed, that is, the temperature acquisition module circuit, as shown in Figure 3 As shown; where V dd =1V,R 01 =1 / kΩ,R 02 =79.432Ω,R 03 =1 / cΩ,R 04 =1kΩ, k, c comes from the model identification parameters; the temperature sensor output voltage T is output through the operational amplifier (T / R 01 )R 02 =-79.432kT, another V dd =1V After the operational amplifier, the output is (1 / R 03 )R 02 =-79.432c, the two signals are added to get the output Uo0 =-(79.432kT+79.432c).

[0115] Design the filter circuit according to h1(s), such as Figure 4 As shown; take R 101 =R 103 =316.255kΩ,C 101 =C 102 =100uF,R 105=1kΩ,R 102 =582Ω,R 104 =1kΩ; According to the RC filter circuit and the follower principle, the Laplace transform UI01(s) of the input voltage UI01 and the Laplace transform Uo01(s) of the output voltage Uo01 satisfy the following relationship:

[0116]

[0117] Design the filter circuit according to h2(s), such as Figure 5 As shown; take R 201 =R 203 =17.778kΩ,C 201 =C 202 =100uF,R 205 =1kΩ,R 202 =562.4Ω,R 204 =1kΩ; the Laplace transform UI02(s) of the input voltage UI02 and the Laplace transform Uo02(s) of the output voltage Uo02 satisfy the following relationship:

[0118]

[0119] Design the filter circuit according to h3(s), such as Figure 6 As shown; take R 301 =R 303 =56.24Ω,C 301 =C 302 =100uF,R 305 =1kΩ,R 302 =10kΩ,R 304 =1kΩ; the Laplace transform UI03(s) of the input voltage UI03 and the Laplace transform Uo03(s) of the output voltage Uo03 satisfy the following relationship:

[0120]

[0121] Connect the above circuits in series to obtain the equivalent circuit of synchronous compensation of temperature error of CNC machine tools. Short-circuit the output Uo0 of the temperature acquisition module with the input voltage UI01 of the h1(s) module circuit; short-circuit the output Uo01 of the h1(s) module with the input voltage UI02 of the h2(s) module circuit; short-circuit the output Uo02 of the h2(s) module with the input voltage UI03 of the h3(s) module circuit. The output of module h3(s) can be obtained.

[0122]

[0123] It can be seen that the output of the compensation circuit is consistent with the temperature error of the CNC machine tool.

[0124] Step 2.5: Subtract the output of the compensation circuit from the output of the CNC machine tool to effectively compensate for the temperature error of the CNC machine tool.

[0125] The above is a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the technical principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. Temperature error compensation method for CNC machine tools based on fractional order model synchronization, Features: Here are the steps: Step 1: Establish a fractional-order model of temperature error for CNC machine tools and identify the parameters of the fractional-order model based on the temperature error and the model output deviation; including: Establish a fractional-order model, collect temperature signals according to the temperature detection unit, and obtain the output signal of the fractional-order model; design the fractional-order model parameter identification rule by using the difference between the fractional-order model output signal and the temperature signal of the CNC machine tool; When the difference between the fractional-order model output and the temperature error signal of the CNC machine tool meets a certain threshold condition, the parameter identification is stopped and the fractional-order model of the temperature error of the CNC machine tool is obtained; Step 1.1: Design the fractional order model of temperature error of CNC machine tools as follows: In the formula, t represents time, T represents the ambient temperature of the CNC machine tool, and e T (t) represents temperature error, c represents fixed error, and k represents coefficient; represents Caputo fractional differential, and its definition expression is: In the formula, Γ(·) represents the gamma function, 0 represents the initial time; τ is the integral variable, 0<α<1 represents the differential order; the parameters c, k, α in the model are all unknown constants; Step 1.2: Design the estimation model of the fractional-order model of temperature error of the CNC machine tool in step 1 as follows: in, is the temperature error of CNC machine tools e T (t) Estimated value; parameter are the estimated values ​​of the unknown parameters c, k, and α respectively; Step 1.3: Define the difference between the temperature error and parameter error estimate and the true value as: The following equation is obtained: Step 1.4: Design parameter estimation value identification rules to make the estimated value consistent with the true value; the parameter identification rules are as follows: when This rule ensures that the following expression always holds: According to the fractional-order Lyapunov theory, E k (t) and E c (t) gradually approaches 0, realizing the identification of unknown parameters k, c; Step 1.5: Divide the range (0,1) into five equal parts and take the middle value of each interval as the estimated value of the differential order, that is: Collect temperature module data, identify parameters according to the parameter identification rule of formula (6), and obtain parameters Five sets of parameters; Step 1.6: Use the temperature data collected by the temperature sensor as the input of the fractional-order model, obtain the temperature error estimate according to the established model, subtract the temperature error estimate from the measured temperature error of the CNC machine tool, and obtain the temperature error estimate deviation; according to step 1.5, obtain five groups of temperature error estimate deviations Step 1.7: Using the five groups of estimated value deviations obtained in step 1.6, calculate the mean square error of the five groups of estimated value deviations respectively; Step 1.8, according to the minimum mean square error corresponding to And The neighborhood of is the new interval and then divided into five equal parts and the middle value is taken; repeat steps 1.6 to 1.7, and exit the loop when the mean square error is less than a certain threshold; Model differential order is the current value of α, E k (t),E c (t) value is the value of k and c in the model, and it is substituted into formula (1) to finally obtain the fractional order model of temperature error of CNC machine tools; Step 2: Design an equivalent circuit based on the established fractional-order model of temperature error; the output of the equivalent circuit is consistent with the temperature error of the CNC machine tool, and error compensation is performed; including: The circuit includes a real-time temperature detection unit, which is the same temperature detection unit used in fractional order modeling; the model equivalent circuit receives a temperature signal and generates an error signal according to the temperature signal; It also includes an error compensation module for the temperature error output signal of the CNC machine tool and the output signal of the model equivalent circuit; the model equivalent circuit is an analog circuit, and the temperature signal and the temperature effect error signal as the input and output signals of the model equivalent circuit are both analog signals; the temperature error output signal of the CNC machine tool is an analog signal; the signal difference output by the temperature error compensation module is an analog signal; Step 2.1: The Laplace transform of the fractional-order temperature error model satisfies Among them, E T (s) and T(s) are e T (t) and the Laplace transform of T; s represents the Laplace operator, and α is the differential order of the fractional order model; Step 2.2: Using the frequency domain equivalent relationship between fractional-order differentials and integer-order differentials, establish an equivalent model of the model in step 2.1; that is, obtain an equivalent model of the fractional-order model according to the differential order of step 2.1; Step 2.3: According to the equivalent model of the fractional-order model obtained in step 2.2, establish the input-output relationship expression of temperature and temperature error, and decompose the expression into the subsystem multiplication form as the system function. The expression is as follows: In the formula, h f(α) (s) represents f(α) different filter functions, i.e., decomposition subsystems. The value of f(α) is determined by the differential order α and the equivalent model; b(kT(s)+c / s) is the temperature input link function; b is a constant coefficient; Step 2.4: Design the temperature error equivalent circuit of the CNC machine tool according to the system function obtained in step 2.3; design the temperature input circuit according to the temperature input link function; f(α) (s) Designing a filter circuit, connecting the temperature input circuit and the filter circuit in series, and obtaining a synchronous compensation circuit for temperature errors of CNC machine tools; Step 2.5: Subtract the compensation circuit output from the CNC machine output to compensate for the temperature error of the CNC machine.

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