CN114647878A - Geometric error parametric modeling method based on Chebyshev polynomial - Google Patents

Geometric error parametric modeling method based on Chebyshev polynomial Download PDF

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CN114647878A
CN114647878A CN202210346816.4A CN202210346816A CN114647878A CN 114647878 A CN114647878 A CN 114647878A CN 202210346816 A CN202210346816 A CN 202210346816A CN 114647878 A CN114647878 A CN 114647878A
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范晋伟
叶倩
郭树欣
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Beijing University of Technology
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Abstract

The invention discloses a geometric error parametric modeling method based on a Chebyshev polynomial, which is characterized by determining a translation shaft and a rotating shaft of a numerical control precision cylindrical grinder, analyzing geometric error elements of the shafts and obtaining 24 errors related to positions; measuring and identifying 24 basic geometric errors by using a laser interferometer and a nine-line method to obtain 24 motion error numerical values; carrying out variable conversion on the numerical control instruction according to the property of a variable definition domain in the Chebyshev polynomial; constructing Chebyshev polynomial basis functions of different times; and (4) combining the normalized numerical control instruction and the basis function of the known coefficient to build a geometric error parameterization model. The invention can reflect the error field distribution of the research object in different instruction states based on the parameterized model obtained by the error numerical value and the Chebyshev polynomial property, solves the problems of insufficient data processing space and longer time of numerical iterative operation in the compensation technology, and is convenient, rapid, intuitive and innovative.

Description

Geometric error parametric modeling method based on Chebyshev polynomial
Technical Field
The application relates to the field of machine tool geometric error parametric modeling, in particular to a geometric error parametric modeling method based on Chebyshev polynomials.
Background
And carrying out error compensation technical research on the numerical control machine tool, wherein geometric error element modeling mainly aims at motion errors. The motion characteristics of the moving axis and the rotating axis can restrict the motion error numerical value distribution, the measured motion error numerical value is discrete, the discrete data needs to be processed, the quasi-static characteristic of the motion error numerical value distribution determines that the error value is in a nonlinear characteristic, and an error element parameterized model is obtained, so that the machine tool comprehensive geometric error modeling and the error compensation are facilitated. Chebyshev Polynomials (Chebyshev Polynomials) are special functions in computational mathematics, originating from the expansion of cosine and sine functions of multiple angles. The method has wide application in the approximation theories of injection continuous function approximation problem, impedance transformation, digital signal processing, global navigation orbit determination, meteorology and the like. And obtaining a parameterized model of the basic geometric error term of each axis by adopting a nonlinear curve fitting mode, and using the parameterized model for error compensation of the machine tool. In order to meet the requirements, a geometric error parameterization modeling method based on Chebyshev polynomials is provided.
Disclosure of Invention
The method aims at the problems that the final effect of the fitting method is not ideal, the parameterized modeling precision is low and the expected modeling effect cannot be obtained due to too few measuring points or too high fitting times in actual measurement and the like. The invention provides a geometric error parameterization modeling method based on Chebyshev polynomials. According to the method, a special function of a Chebyshev polynomial is taken as a theoretical basis, a numerical control precise cylindrical grinding machine of a certain company is taken as an example, and firstly, according to the known 24 motion error values and the Chebyshev polynomial properties of the machine tool, the processing instruction of the numerical control precise cylindrical grinding machine is converted into the parameter quantity of the polynomial; then calculating coefficients of Chebyshev polynomial basis functions of different orders; and substituting the relation between the numerical control grinder processing instruction and the polynomial parameter into the basis function of the last step, and unfolding to obtain a corresponding motion error parameterized model. This parametric modeling process is operated via MATLAB R2018 a. The invention obtains the motion error parameterization model of each motion axis of the numerical control machine tool by taking the mathematical function of the Chebyshev polynomial as a theoretical basis, simplifies the nonlinearity and the complexity of the motion error changing along with the motion instruction in the past, can intuitively obtain the distribution field of each error element, further brings the model into a compensation model, greatly simplifies the error compensation technology, and has the advantages of simple and intuitive calculation and flexible and innovative application.
The technical scheme adopted by the invention for solving the technical problem is as follows: a geometric error parameterization modeling method based on Chebyshev polynomials comprises the following steps:
the method comprises the following steps: the method comprises the steps of determining a translation axis and a rotation axis of a research object (a numerical control precision cylindrical grinder of a certain company), analyzing geometric error elements of each axis, and obtaining 24 errors (motion errors) related to positions, wherein the motion errors change along with the change of machine tool instructions and show nonlinear correlation.
Step two: and measuring and identifying the 24 basic geometric errors by using a laser interferometer and a nine-line method to obtain 24 motion error numerical values.
Step three: and (3) carrying out variable conversion (linear normalization) on the numerical control instruction according to the property of the variable definition domain in the Chebyshev polynomial.
Step four: constructing Chebyshev polynomial basis functions of different degrees.
Step five: and combining the 24 basic geometric error values and the polynomial basis functions of different orders to obtain coefficients of the polynomial basis functions.
Step six: and (4) combining the normalized numerical control instruction and the basis function of the known coefficient to build a geometric error parameterization model.
The invention has the beneficial effects that: according to the method, a special function Chebyshev polynomial is taken as a theoretical basis, and geometric errors generated by an X, Z translational axis and a B, C rotary axis of a researched object (a numerical control precise cylindrical grinding machine) in space motion are analyzed to obtain 24 motion errors; and then measuring and identifying basic geometric error terms by using a high-precision error detection instrument laser interferometer and a more common nine-line method to obtain various error numerical values. Obtaining polynomial basis functions of different orders and a form after x, z, b and c variable conversion according to the recurrence relation and the variable range of the first class of Chebyshev polynomials; and combining the plurality of basis functions and the variables to obtain the basis functions with different times and known coefficients, and further building a geometric error parameterization model. This parametric modeling process may be operated via MATLAB R2018 a. The invention can reflect the error field distribution of the research object in different instruction states based on the parameterized model obtained by the error numerical value and the Chebyshev polynomial property, solves the problems of insufficient data processing space and longer time of numerical iterative operation in the compensation technology, and is convenient, rapid, intuitive and innovative.
The present invention will be described in detail with reference to the following embodiments.
Detailed Description
The example used a numerically controlled precision cylindrical grinder from a company. The rotation angles of the spindle C and the turntable B are C and B respectively, and the moving distances of the guide rail on the sliding seat Z and the guide rail on the sliding seat X are Z and X respectively.
The geometric error parametric modeling method based on the Chebyshev polynomial comprises the following steps:
the method comprises the following steps: the method comprises the steps of determining a translational axis and a rotating axis of a research object (a numerical control precision cylindrical grinding machine of a certain company), wherein each axis has six degrees of freedom according to a rigid body kinematics theory, correspondingly, errors in the directions of the six degrees of freedom and errors on a coordinate system of each axis are generated along with the machining process of the grinding machine due to the reasons of assembly, manufacturing and the like, and 24 geometric errors (motion errors) related to positions are obtained, and the motion errors change along with the change of machine tool commands, show nonlinear correlation and have a discrete characteristic. The motion errors, also called position-dependent errors, vary with the translation and rotation of the motion axes, and we refer to them here as elementary geometric error terms, for example the translation axis Z, with six elementary geometric error terms δx(z)、δy(z)、δz(z)、εx(z)、εy(z)、εz(Z) the six basic errors are non-linear functions related to Z (amount of movement of the translation axis Z/machine code command in the Z direction).
Step two: and measuring partial geometric errors of the mark points in the motion range of each axis by using a laser interferometer, and performing expansion identification on the residual basic geometric errors according to a nine-line method to obtain 24 motion error numerical values.
Step three: the 4 motion axis motion amounts are subjected to variable conversion (linear normalization) according to the property that the chebyshev variable t range is [ -1,1 ]:
Figure BDA0003576787890000031
x∈[0,365]；z∈[0,1500]；；b∈[0,360]；c∈[0,360]
step four: constructing Chebyshev polynomial basis functions of different times as follows:
T0(t)＝1
T1(t)＝cosθ＝t
T2(t)＝cos2θ＝2t2-1
T3(t)＝cos3θ＝4t3-3t
T4(t)＝cos4θ＝8t4-8t2+1
T5(t)＝cos5θ＝16t5-20t3+5t
Figure BDA0003576787890000032
Tn+1(t)＝cos(n+1)θ＝2tTn(t)-Tn-1(t)
Tn(x) Is an algebraic polynomial of degree n.
Tn(x) The highest power term x ofnHas a coefficient of 2n-1。
Step five: and combining the 24 items of basic geometric error data in the second step and the polynomial basis functions of different orders in the fourth step to obtain coefficients of the polynomial basis functions.
Taking the first-order chebyshev polynomial model as an example, it can be expressed as:
Figure BDA0003576787890000041
after inversion, the regression coefficient of the first-order basis function is:
Figure BDA0003576787890000042
n is the number of data of a certain geometric error term
tjIs a Chebyshev polynomial variable after the variable conversion of each axis motion quantity
a0、a1Is a coefficient of a polynomial of a basis function
δjIs the j error data of a certain geometric error term
The regression coefficients of the quadratic basis functions are:
Figure BDA0003576787890000043
the regression coefficients of the m-th basis function are:
Figure BDA0003576787890000044
the principle of motion error data parameterization modeling is that the error values are approximated by different times of Chebyshev interpolation method, and when the times are higher, the Chebyshev curve is closer to the error values. But instead of making the order infinite, a polynomial is found that most accurately approximates the value in all orders.
Step six: and (4) combining the normalized numerical control instruction and the basis function of the known coefficient to build a geometric error parameterization model.
Due to the strong data processing capability of MATLAB numerical analysis software, a parametric modeling of Chebyshev polynomials for motion errors will be achieved using MATLAB numerical analysis tools.
The positioning error and the roll angle error of the translational axis X in the X direction are respectively as follows:
δx(x)＝a0T0(t)+a1T1(t)+a2T2(t)
＝23.66-11.83*(0.005479*x-1)*(0.01096*x-2)-0.06479*x
εx(x)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)+a5T5(t)
＝0.001444x-0.3004*(0.005479x-1)*(0.01096x-2)+5.335*(0.005479x-1)2-6.14*(0.005479x-1)3-5.335*(0.005479x-1)4+4.912*(0.005479x-1)5-0.6301
the straightness error and the pitch error of the translational axis X in the Y direction are respectively as follows:
δy(x)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝3.058*(0.005479x-1.0)(0.01096x-2.0)-0.006673x+14.76*(0.005479x-1)2-14.76*(0.005479x-1)4-3.686
εy(x)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)+a5T5(t)
＝0.009598x+0.445*(0.005479x-1)(0.01096x-2)+2.147*(0.005479x-1)2-7.882*(0.005479x-1)3-2.147*(0.005479x-1)4+6.306*(0.005479x-1)5-2.465
the straightness error and the deflection error of the translational axis X in the Z direction are respectively as follows:
δz(x)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)+a5T5(t)
＝0.001101*x+0.6285*(0.005479x-1)(0.01096*x-2)+6.299*(0.005479x-1)2-1.792*(0.005479x-1)3-6.299*(0.005479x-1)4+1.434*(0.005479x-1)5-1.617
εz(x)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)+a5T5(t)
＝16.78*(0.005479x-1)2-0.2403*(0.005479x-1)*(0.01096x-2)-0.00885x-3.582*(0.005479x-1)3-16.78*(0.005479x-1)4+2.866*(0.005479x-1)5-0.242
the straightness error and the roll angle error of the translational axis Z in the X direction are respectively as follows:
δx(z)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)
＝0.1646-0.0823*(0.001333z-1)(0.002667z-2)-0.0001097*z
εx(z)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝2.011*(0.001333z-1)2-0.0009795*z-2.011*(0.001333z-1)4-0.4784*(0.001333z-1)(0.002667z-2)+0.9616
the straightness error and the pitch error of the translational axis Z in the Y direction are respectively as follows:
δy(z)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)
＝0.2422-0.1211*(0.001333z-1)(0.002667z-2)-0.0001615*z
εy(z)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)
＝0.7628-0.3814*(0.001333z-1)(0.002667z-2)-0.0005085*z
the straightness error and the deflection error of the translational axis Z in the Z direction are respectively as follows:
δz(z)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)
＝0.5565-0.2781*(0.001333z-1)(0.002667z-2.0)-0.0003712*z
εz(z)＝a0T0(t)+a1T1(t)+a2T2(t)
＝1.85-0.925*(0.001333*z-1)(0.002667z-2)-0.001233z
the axial runout error and the angular positioning error of the rotating shaft B in the X direction are respectively as follows:
δx(b)＝a0T0(t)+a1T1(t)+a2T2(t)
＝0.1981-0.099*(0.005556b-1)(0.01111b-2)-0.0005506*b
εx(b)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝1.743*(0.005556b-1)2-0.0006994*b-1.743*(0.005556b-1)4+0.0887*(0.005556b-1)*(0.01111b-2)-0.1807
the radial run-out error and the angle error of the rotating shaft B in the Y direction are respectively as follows:
δy(b)＝a0T0(t)+a1T1(t)+a2T2(t)
＝0.3888-0.1955*(0.005556b-1)(0.01111b-2)-0.001074*b
εy(b)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝2.842*(0.005556b-1)2-0.0004878*b-2.842*(0.005556b-1)4+0.2679*(0.005556b-1)*(0.01111b-2)-0.5353
the radial run-out error and the angle error of the rotating shaft B in the Z direction are respectively as follows: deltaz(b)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝4.178*(0.005556b-1)2-0.00188*b-4.178*(0.005556b-1)4+0.1761*(0.005556b-1)*(0.01111b-2)-0.36
εz(b)＝a0T0(t)+a1T1(t)+a2T2(t)
＝1.217-0.6085*(0.005556b-1)*(0.01111b-2)-0.003381*b
The axial runout error and the angular positioning error of the rotating shaft C in the X direction are respectively as follows:
δx(c)＝a0T0(t)+a1T1(t)+a2T2(t)
＝0.2063-0.1032*(0.005556c-1)*(0.01111c-2)-0.0005728*c
εx(c)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)+a5T5(t)
＝2.779*(0.005556c-1)2-2.32*(0.005556c-1)3-2.779*(0.005556c-1)4-0.666+1.856*(0.005556c-1)5+0.0899*(0.005556c-1)*(0.01111c-2)+0.001271*c
the radial run-out error and the angular error of the rotating shaft C in the Y direction are respectively as follows:
δy(c)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)
＝0.6068-0.3034*(0.005556c-1)*(0.01111c-2)-0.001686*c
εy(c)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝1.727*(0.005556c-1)2-0.0002728*c-1.727*(0.005556c-1)4+0.1665*(0.005556c-1)*(0.01111c-2)-0.3333
the radial run-out error and the angular error of the rotating shaft C in the Z direction are respectively
δz(c)＝a0T0(t)+a1T1(t)+a2T2(t)
＝0.1549-0.0775*(0.005556c-1)*(0.01111c-2)-0.00043*c
εz(c)＝a0T0(t)+a1T1(t)+a2T2(t)+a3T3(t)+a4T4(t)
＝1.678*(0.005556c-1)2-5*c-1.678*(0.005556c-1)4+0.2033*(0.005556c-1)(0.01111c-2)-0.4049
The first three terms of the 24-term geometric error elements represent linear displacement errors in the direction of the coordinate axis i, the last three terms represent angular displacement errors in the direction of the coordinate axis i, and i represents each of the motion axes X, Z, B, C.
According to the embodiment, the basic geometric error parameterization model of the numerical control precision cylindrical grinding machine is quick, simple and convenient to build, requires a short time for MNATLAB calculation, can visually obtain the error value at any position in the motion range of the machine tool, and is conveniently used in a subsequent compensation method to replace a complex software iteration compensation method.

Claims (1)

1. A geometric error parameterization modeling method based on Chebyshev polynomials is built, and is characterized by comprising the following steps:
the method comprises the following steps: determining a translation axis and a rotation axis of the numerical control precision cylindrical grinding machine, wherein each axis has six degrees of freedom according to the rigid body kinematics theory, and correspondingly, errors in the directions of the six degrees of freedom and errors on coordinate systems of each axis are generated along with the machining process of the grinding machine due to the reasons of assembly, manufacture and the like, so as to obtain 24 geometric errors related to positions, namely motion errors;
step two: measuring partial geometric errors of the mark points in the motion range of each axis by using a laser interferometer, and performing expansion identification on the residual basic geometric errors according to a nine-line method to obtain 24 motion error numerical values;
step three: according to the property that the Chebyshev variable range is [ -1,1], carrying out variable conversion, namely linear normalization on the motion quantity of the 4 motion axes;
step four: constructing Chebyshev polynomial basis functions of different times;
step five: combining the 24 items of basic geometric error data in the second step and the polynomial basis functions of different orders in the fourth step to obtain coefficients of the polynomial basis functions;
step six: and (4) combining the normalized numerical control instruction and the basis function of the known coefficient to build a geometric error parameterization model.
CN202210346816.4A 2022-03-31 2022-03-31 Geometric error parametric modeling method based on Chebyshev polynomial Pending CN114647878A (en)

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