CN114636387B - Circular grating encoder double-reading-head asymmetric installation eccentric error compensation method - Google Patents
Circular grating encoder double-reading-head asymmetric installation eccentric error compensation method Download PDFInfo
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Abstract
The invention discloses a method for compensating an eccentric error of double-reading head asymmetric installation of a circular grating encoder, which is characterized in that a model function is used for performing linear approximation on a parameter to be estimated, the linear approximation is converted into a principle of linear least square problem, a circular grating eccentric simulation model is established to obtain an eccentric parameter, the calculation speed is high, and the result is accurate; by analyzing the angle measurement error compensation principle of the double-reading head under an ideal condition, establishing an average method error compensation ideal model of the double-reading head symmetrical installation, adding installation errors and random errors existing between the reading heads under an actual condition, and further improving the average method error compensation model to obtain an asymmetrical installation error compensation model of the double-reading head; and finally, substituting the eccentric parameters obtained by solving the circular grating eccentric simulation model into the double-reading-head asymmetric installation error compensation model, eliminating the influence of the installation error of the reading head on the angle measurement precision in actual measurement, and realizing the accurate compensation of the double-reading-head angle measurement eccentric error in the asymmetric installation of the circular grating encoder.
Description
Technical Field
The invention belongs to the technical field of precision measurement, and particularly relates to a method for compensating an asymmetric installation eccentric error of a double-reading head of a circular grating encoder.
Background
The circular grating encoder has the advantages of high resolution, small volume, convenience in installation, high response speed, simple processing circuit and the like, and is widely applied to the fields of aerospace, intelligent robots, high-grade numerical control machines, high-precision coordinate measuring machines and the like. With the development of science and technology, various instruments tend to be miniaturized and have higher and higher requirements on angle measurement accuracy, and small-size circular gratings perfectly meet the requirements on miniaturization and high accuracy. Meanwhile, the photoelectric pulse signal of the circular grating encoder has strong anti-interference capability, and the running and machining precision of the numerical control machine tool can be well guaranteed. With the continuous development of the technology, the requirement on the measurement accuracy of the circular grating encoder is higher and higher. The research report of international high-end circular grating sensor manufacturing and selling tap enterprises, heidenhain company of Germany clearly provides that the installation eccentricity error accounts for more than 80% of the angle measurement error, the installation eccentricity error has great influence on the angle measurement precision of a circular grating encoder, and the measurement for calculating the installation eccentricity error parameter and compensating the measurement error caused by the eccentricity error is very important. During installation, the circular grating inevitably generates eccentricity relative to a shaft system, and the installation eccentricity causes an encoder to generate an angle measurement error. Therefore, in order to ensure the system test accuracy, the measurement angle of the circular grating needs to be corrected.
At present, domestic researches on the influence of eccentric errors of circular grating encoder assembly on angle measurement precision are many, and various eccentric error compensation methods are provided, wherein the eccentric errors are eliminated simply and reliably by using double reading heads, and the cost is low. However, the study on the eccentricity error of the double-reading-head circular grating encoder generally considers that the double-reading heads which are radially arranged on the circumference of the grating are in an ideal state, namely the two reading heads are symmetrically arranged with an included angle of 180 degrees. However, in the actual installation process, no matter how the installation accuracy is improved, installation deviation exists between the two reading heads, and for the circular grating encoder with high angle measurement accuracy requirement, the influence of the installation deviation of the double reading heads on the angle measurement accuracy is not negligible.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for compensating the eccentric error of the asymmetric installation of the double reading heads of the circular grating encoder aiming at the defects in the prior art, based on the principle that sinusoidal signals detected by the reading heads at different positions at the same moment in the circular grating angle measurement process have phase difference, an error compensation model is constructed by combining the eccentric error compensation principle of the symmetric installation of the double reading heads in an ideal state, the eccentric parameter of the circular grating installation in the non-ideal state is obtained by solving a model based on the eccentric parameter of the circular grating based on an L-M algorithm, and the eccentric error compensation model of the angle measurement of the circular grating encoder is established on the basis of the eccentric parameter of the circular grating, so that the accurate compensation of the eccentric error of the angle measurement of the asymmetrically installed double reading heads of the circular grating encoder is realized.
The invention adopts the following technical scheme:
a method for compensating the eccentric error of the asymmetric installation of the double reading heads of a circular grating encoder comprises the following steps:
s1, solving the eccentricity and the eccentricity direction of the circular grating based on an L-M algorithm according to the synthesis of sinusoidal signals with different phases in the circumferential direction of the circular grating, and establishing a circular grating eccentricity parameter simulation model;
s2, establishing an error compensation ideal model of a mean value method with symmetrically installed double reading heads;
s3, adding a reading head installation error and a random error into the ideal error compensation model of the averaging method established in the step S2, and establishing a double-reading-head asymmetric installation error compensation model;
s4, measuring through a microscope to obtain the eccentricity of the circular grating, substituting the eccentricity serving as an initial value into the circular grating eccentricity parameter simulation model established in the step S1, and obtaining a real eccentricity through iteration;
and S5, substituting the real eccentricity obtained in the step S4 into the double-reading-head asymmetric installation error compensation model established in the step S3, and realizing accurate compensation of the angle measurement error caused by the installation eccentricity of the circular grating.
Specifically, in step S1, the sinusoidal signal phase difference Δ ψ detected by two reading heads at different positions in the circumferential direction at the same time 12 Comprises the following steps:
where d is the grating pitch, 360 represents the number of angles in one period of the electrical fringe signal,
is the mounting angle between the two reading heads, and e is the eccentricity.
Further, the eccentricity e is specifically as follows:
where Δ ψ is the total phase difference read by the two read heads.
Specifically, in step S2, the ideal model for error compensation by the averaging method specifically includes:
wherein,
for the error compensation result of the averaging method, theta is the theoretical rotation angle without error, tau 1 Is the mounting position of the first reading head, τ 2 Is the mounting position of the second reading head, psi (theta) 1 ) Angle measurement error, delta (theta), caused by inconsistencies within the first readhead 1 ) For random errors due to other uncertainties of the first readhead, ψ (θ) 2 ) Angle error, delta (theta), caused by inconsistencies within the second read head 2 ) Random errors due to other uncertainties of the second read head.
Specifically, in step S3, the asymmetric installation error compensation model with two reading heads specifically includes:
wherein,
representing the measured value of the angle of rotation compensated by the new method,
for the compensated angle measurement error, e is the eccentricity, R is the code disc radius, omega is the eccentricity angle, theta 1 ，θ 2 The first reading head and the rotation angle measured by the first reading head during the rotation process.
Further, the angle measurement errors of the first reading head and the second reading head are respectively:
wherein, theta is the theoretical rotation angle without error.
Specifically, in step S4, solving the eccentricity and the eccentricity direction of the circular grating based on the L-M algorithm specifically includes:
setting initial values of eccentricity e, eccentricity angle omega and code disc radius R; calculating a Jacobian matrix J and a function value; calculating an error, and if the error is smaller than a set threshold value, terminating; otherwise, continuing; calculating the value of the Hessian matrix: h = J' × J; calculating a search step length d; calculating new parameters and function values; calculating a new error; if the error is smaller than the last error, the parameters are updated and the damping coefficient mu is reduced, if the error is larger, the damping coefficient is increased, the parameters are not updated, the Jacobian matrix J and the function value are calculated, and iteration is continued.
Further, x i Search step d of points i Comprises the following steps:
wherein, J i Is a Jacobian matrix, mu i Is damping coefficient, I is identity matrix, F (x) i ) Error theta for measuring angle of first reading head and second reading head 2 -θ 1 The functional expression of the formula.
Specifically, in step S5, the compensation model for the angle measurement error in the asymmetric installation of the dual-reading head specifically includes:
and the eccentricity direction calculated by the circular grating eccentricity parameter solving model are used as input variables, and the measured angle eccentricity error compensation value is used as an output variable, so that the accurate compensation of the measured angle error of the circular grating encoder caused by the installation eccentricity and the reading head installation deviation is realized.
In a second aspect, an embodiment of the present invention provides a system for compensating an eccentric error in asymmetric installation of dual-reading heads of a circular grating encoder, including:
the simulation module is used for solving the eccentricity and the eccentricity direction of the circular grating based on an L-M algorithm according to the synthesis of sinusoidal signals with different phases in the circumferential direction of the circular grating, and establishing a circular grating eccentricity parameter simulation model;
the ideal module is used for establishing an error compensation ideal model of an averaging method with symmetrically installed double reading heads;
the real module is used for adding a reading head installation error and a random error into the ideal error compensation model established by the ideal module by using the mean value method, and establishing a double-reading-head asymmetric installation error compensation model;
the iteration module is used for obtaining the eccentricity of the circular grating through microscope measurement, substituting the eccentricity as an initial value into the circular grating eccentricity parameter simulation model established by the simulation module, and obtaining the real eccentricity through iteration;
and the compensation module substitutes the real eccentricity obtained by the iteration module into the double-reading-head asymmetric installation error compensation model established by the reality module to realize accurate compensation of the angle measurement error caused by the installation eccentricity of the circular grating.
Compared with the prior art, the invention has at least the following beneficial effects:
the invention relates to a method for compensating an eccentric error of double-reading head asymmetric installation of a circular grating encoder, which comprises the steps of establishing a circular grating eccentric simulation model, establishing an average error compensation ideal model of the double-reading head symmetric installation, establishing a double-reading head asymmetric installation error compensation model and establishing an eccentric parameter simulation model based on an L-M algorithm, wherein the adjustment band of the circular grating is 30 microns, so that the difficulty of measuring the eccentric distance by using a microscope is high and inaccurate; under an ideal condition, no installation error exists between the two reading heads, an average method error compensation ideal model of the double-reading-head symmetrical installation is established on the basis, and the installation error between the reading heads is not considered under an actual condition, so that the average method error compensation model needs to be improved, and the installation error and the random error between the reading heads are added to obtain an asymmetric installation error compensation model of the double-reading-head; and substituting the eccentricity calculated by the circular grating eccentricity simulation model as an initial value into the double-reading-head asymmetric installation error compensation model to realize accurate compensation of the angle measurement error caused by the circular grating installation eccentricity.
Further, as can be seen from the principle of circular grating angle measurement, monochromatic light emitted by the laser passes through the grid lines of the grating disk to generate interference fringes, and the photoelectric receiver can generate a sinusoidal signal in the oscilloscope after receiving light and dark fringe light signals generated by interference. When the circular grating has eccentricity, sinusoidal signals detected by reading heads at different positions in the circumferential direction at the same moment have phase difference. By utilizing the characteristic, the eccentric direction and the eccentric distance of the circular grating can be detected according to the synthesis of sinusoidal signals with different phases.
Furthermore, the eccentricity of the circular grating can be calculated by using the total phase difference delta psi read by the 2 reading heads when the circular grating rotates 180 degrees anticlockwise from the minimum value point to the maximum value point, namely, a circular grating eccentricity simulation model is established on the basis of the data of the two reading heads so as to solve the eccentricity, so that the measurement error caused by the method of measuring the eccentricity by using a microscope is avoided, and the result is more accurate. .
Furthermore, under the ideal condition, namely under the condition that no installation error exists between the two reading heads, the ideal model of the mean error compensation of the symmetrical installation of the double reading heads is established, the theoretical analysis of the angle measurement error of the double reading heads can be simply and clearly carried out, and the foundation is laid for establishing the error compensation model under the actual condition.
Furthermore, in actual conditions, a certain installation error exists between the two reading heads more or less, so that the ideal error compensation model for the symmetrical installation of the double reading heads by the averaging method cannot completely compensate the angle measurement error of the double reading heads, the installation error of the two reading heads can be considered by establishing the asymmetrical installation error compensation model of the double reading heads, the angle measurement error is compensated better, and the measurement precision is improved on the basis of ensuring the accuracy of the measurement result.
Furthermore, installation error solving formulas of the first reading head and the second reading head are respectively given, so that the two formulas are connected, namely, the data of the two reading heads are connected together, and a foundation is laid for solving the eccentricity through an eccentric simulation model.
Furthermore, the key of the L-M algorithm is that a model function is used for performing linear approximation on the parameter to be estimated in the field of the model function, derivative terms above the second order are omitted, and therefore the linear least square problem is converted, and the convergence speed is high. Therefore, the eccentricity and the eccentricity direction of the circular grating are solved based on the L-M algorithm, the calculation speed is high, and the calculation result is accurate.
Further, the Gauss-Newton algorithm requires that the Jacobian matrix must be column-full-rank, and thus for x i Search step d of points i The improvement is carried out to obtain the L-M algorithm, the limitation that the Jacobian matrix must be column full rank is broken through, and the flexibility of the algorithm is improved.
Furthermore, in practical situations, due to the influence of factors such as machining and manufacturing errors of parts, irregular installation and the like, the double-reading head is difficult to ensure complete radial installation in the installation process, so that installation errors between the two reading heads are inevitable, the installation errors of the double-reading head are taken into consideration by establishing an asymmetrical double-reading head installation angle measurement error compensation model, and the influence of the double-reading head installation errors on the circular grating angle measurement precision is further effectively eliminated.
In summary, the invention makes linear approximation to the parameter to be estimated in the field by using the model function, omits the derivative term above the second order, and thus converts the linear least square problem into the principle to establish the circular grating eccentric simulation model and further solve to obtain the accurate eccentric parameter, so that the calculation speed is high, and the calculation result is also accurate; by analyzing the angle measurement error compensation principle of the double-reading head under an ideal condition, establishing an average method error compensation ideal model of the double-reading head symmetrical installation, adding installation errors and random errors existing between the reading heads under an actual condition on the basis of the model, and further improving the average method error compensation model to obtain a double-reading head asymmetric installation error compensation model; and finally, substituting the eccentric parameters obtained by solving the circular grating eccentric simulation model into the double-reading-head asymmetric installation error compensation model, eliminating the influence of the installation error of the reading head on the angle measurement precision in actual measurement, and finally realizing the accurate compensation of the double-reading-head angle measurement eccentric error in the asymmetric installation of the circular grating encoder.
The technical solution of the present invention is further described in detail by the accompanying drawings and embodiments.
Drawings
FIG. 1 is a schematic view of eccentric installation of a circular grating encoder;
FIG. 2 is a schematic diagram of the eccentricity error of a circular grating encoder;
FIG. 3 is a schematic diagram of eccentric error measurement by a dual-reading head;
FIG. 4 is a graph of measurement error introduced by eccentricity;
FIG. 5 is a diagram of a measurement system;
FIG. 6 is a flow chart of eccentricity parameter solving based on L-M algorithm;
FIG. 7 is a graph of angular error before and after compensation of a circular grating encoder.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In the description of the present invention, it should be understood that the terms "comprises" and/or "comprising" indicate the presence of the stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.
It is also to be understood that the terminology used in the description of the invention herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used in the specification of the present invention and the appended claims, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise.
It should be further understood that the term "and/or" as used in this specification and the appended claims refers to any and all possible combinations of one or more of the associated listed items, and including such combinations, e.g., a and/or B, may mean: a exists alone, A and B exist simultaneously, and B exists alone. In addition, the character "/" herein generally indicates that the former and latter related objects are in an "or" relationship.
It should be understood that although the terms first, second, third, etc. may be used to describe preset ranges, etc. in embodiments of the present invention, these preset ranges should not be limited to these terms. These terms are only used to distinguish preset ranges from each other. For example, the first preset range may also be referred to as a second preset range, and similarly, the second preset range may also be referred to as the first preset range, without departing from the scope of the embodiments of the present invention.
The word "if," as used herein, may be interpreted as "at … …" or "at … …" or "in response to a determination" or "in response to a detection," depending on the context. Similarly, the phrases "if determined" or "if detected (a stated condition or event)" may be interpreted as "when determined" or "in response to a determination" or "when detected (a stated condition or event)" or "in response to a detection (a stated condition or event)", depending on the context.
Various structural schematics according to the disclosed embodiments of the invention are shown in the drawings. The figures are not drawn to scale, wherein certain details are exaggerated and possibly omitted for clarity of presentation. The shapes of various regions, layers and their relative sizes and positional relationships shown in the drawings are merely exemplary, and deviations may occur in practice due to manufacturing tolerances or technical limitations, and a person skilled in the art may additionally design regions/layers having different shapes, sizes, relative positions, according to actual needs.
The invention provides a method for compensating eccentric error of double-reading head asymmetric installation of a circular grating encoder, which is characterized in that an error compensation model is constructed by utilizing the principle that sinusoidal signals detected by reading heads at different positions at the same moment in the angle measurement process of the circular grating and combining the eccentric error compensation principle of the double-reading head symmetric installation in an ideal state, the eccentric parameter of the circular grating is solved based on an L-M algorithm to obtain the eccentric parameter of the circular grating installation in the non-ideal state, and the angle measurement eccentric error compensation model of the circular grating encoder is established on the basis of the eccentric parameter of the circular grating, so that the accurate compensation of the eccentric error of the double-reading head asymmetric installation state of the circular grating encoder is realized.
Referring to fig. 1, the method for compensating the eccentric error of the asymmetric installation of the dual-reading head of the circular grating encoder of the present invention includes the following steps:
s1, based on a circular grating angle measurement error principle, when a circular grating has eccentricity, phase differences exist in sinusoidal signals detected by reading heads at different positions in the circumferential direction at the same moment, and a circular grating eccentricity parameter simulation model is established according to synthesis of sinusoidal signals with different phase differences;
based on the principle of circular grating angle measurement error, monochromatic light emitted by a laser generates interference fringes through grid lines of a grating disc, and a sine signal is generated in an oscilloscope after a photoelectric receiver receives the interference fringe light; when the circular grating is eccentric, sinusoidal signals detected by the reading heads at different positions in the circumferential direction at the same moment have phase differences; by utilizing the characteristic, the eccentric direction and the eccentric distance of the circular grating are detected according to the synthesis of sinusoidal signals with different phases, and then a circular grating eccentric detection model is established.
2 photoelectric reading heads are used for collecting fringe signals, and the first reading head and the second reading head are respectively arranged at A 1 、A 2 The positions are radially arranged by taking the rotation center OD as a center. The geometric center of the circular grating is eccentric relative to the rotating shaft, so that the phaseThe grating number of the first reading head and the grating number of the second reading head which pass through the first reading head in the same time are different, namely the lengths of the 2 photoelectric reading heads which pass through the optical circumference in the same time are different. Therefore, the sinusoidal signals detected by the first reading head and the second reading head have differences in frequency and phase.
When the rotating shaft rotates along a fixed direction, the difference of the lengths of the first reading head and the second reading head passing through the optical circumference at the same time is
From the theory of small angle approximation, it can be known that
And O is M E⊥A 1 A 2 According to the geometric similarity relationship shown in FIG. 2, the length difference
Expressed as:
wherein,
is eccentric direction and A 1 A 2 The included angle between the directions.
Difference in length
The number of angles corresponding to the period of the electrical fringe signals is the phase difference between the sinusoidal signals detected by the 2 reading heads, and the relative phase difference between the signals received by the first reading head and the second reading head is expressed as:
where d is the grating pitch and 360 represents the number of angles in one period of the electrical fringe signal. The phase difference between the first reading head and the second reading head changes along with the change of the rotation angle of the rotating shaft.
In the process of one rotation of the measured shaft, the difference of the eccentric distance between the first reading head and the second reading head reaches the extreme point 2 times, as shown in fig. 2, corresponding to the eccentric direction and a 1 A 2 The direction is orthogonal for 2 points read by the receiver 2. Circular grating from minimum point P min Rotated 180 degrees counterclockwise to reach a maximum point P max The total phase difference read by the 2 reading heads is delta psi; and further converting the arrangement formula to obtain a calculation formula of the eccentricity e.
When the second reading head reads an extreme point, the recording is started, and after the circular grating rotates 90 degrees, the eccentric direction will be equal to A 1 A 2 The directions are consistent. Therefore, after the phase extreme point is captured in the oscilloscope, the eccentric direction of the circular grating can be determined according to the orthogonal relation.
S2, correcting and compensating angle measurement errors caused by installation eccentricity of the circular grating by adopting a double-reading-head symmetrical installation method, effectively compensating and correcting the angle measurement errors, and establishing an average error compensation ideal model of double-reading-head symmetrical installation based on the principle;
the method for correcting and compensating the angle measurement error caused by the installation eccentricity of the circular grating by adopting the double-reading head method can effectively compensate and correct the angle measurement error, and under the ideal installation condition, the establishment of the double-reading head symmetrical error compensation model can effectively eliminate the installation eccentricity of the circular grating and the radial motion deviation of a shaft system.
For a double-reading head circular grating angle measurement system, the method of averaging is a conventional error correction method. The ideal model for error compensation by the averaging method is as follows:
averaging is simple, but for two readsThe requirement on the installation precision of a plurality of heads is high; when the reading head has no installation error, wherein 1 ＝0°，τ 2 ＝180°。
At this time, the above formula may be transformed into:
the angle measurement error caused by the installation eccentricity of the circular grating and the radial runout of the shafting is eliminated. However, when there is a mounting error in the reading head, the averaging method cannot completely eliminate the influence of the two errors. Through the analysis of the error source and the influence mechanism in the circular grating angle measuring system, the application of the double-reading-head averaging method in the environment with larger installation errors of the reading head has certain limitation.
S3, adding a reading head installation error and a random error into the ideal error compensation model of the averaging method established in the step S2, and establishing a double-reading-head asymmetric installation error compensation model;
in the actual installation process, the double reading heads cannot reach an ideal state, namely, installation deviation exists between the two reading heads, so that the adverse effect of installation eccentricity on angle measurement precision cannot be effectively eliminated.
The method for correcting the eccentric error of the circular grating with the non-diameter-alignment double-reading head can reduce the adverse effect of the diameter-alignment installation error on the measurement result; on the other hand, the limitation that the reading head must be installed in a diameter direction is broken through, and the degree of freedom of the instrument and equipment in structural design is improved.
Under precision measurement conditions, the magnitudes of random and systematic errors are often close; after the system error is corrected correctly, the amplitude of the random error may exceed the amplitude of the system error, and the random error component is added into the double-reading-head symmetrical installation error compensation model, so that the influence of the random error on the compensation result can be effectively eliminated.
Please refer to fig. 3,O 1 Is the rotation center of a shaft system, O is the geometric center of a circular grating disc, and C, D is the installation positions of two reading heads;OO 1 the eccentricity of the circular grating disk is denoted by e; less than O 1 OD is the eccentric angle of the circular grating disc and is represented by omega; less than O 1 OC is represented by eta; for convenience of presentation, the shaft and circular grating disk are fixed in FIG. 3, and the reading head is wound around the rotation center O 1 Rotating counterclockwise, and rotating the angle theta from the point C, D to the point A and the point B respectively; theta 1 ，θ 2 The rotation angles measured by the two reading heads in the rotation process; mounting angle CO of two reading heads 1 For D
Representing; when the double reading head is strictly installed in a diameter-checking way, namely, the reading head is not installed in a position error,
the angle measurement error of the first reading head can be obtained by the fact that the triangle outer angle is equal to the sum of two non-adjacent inner angles:
and obtaining the angle measurement error of the second reading head in the same way:
from FIG. 3, it can be seen that
sinη[sinα+cos(ω+γ+φ)sin(ω-θ 1 )]+cosηsin(ω+γ+φ)sin(ω-θ 1 )＝0
Let m = sin α + cos (ω + γ + φ) · sin (ω - θ) 1 ),n＝sin(ω+γ+φ)·sin(ω-θ 1 ) And then:
m·sinη+n·cosη＝0
obtaining the following formula according to the auxiliary angle:
the relation between the measured values of the two reading heads and the eccentricity parameter can be obtained by the following formula:
wherein, only the reading of two reading heads and the installation eccentricity parameter of the circular grating are included. Two of the readheads read θ 1 ，θ 2 The acquisition may be performed by a data acquisition system. Rotating the shaft system for one circle to obtain a series of theta 1 ，θ 2 The circular grating installation eccentric parameters can be fitted by using a least square method, so that the calibration of the circular grating installation eccentric parameters is realized
The asymmetric installation error compensation model of the double-reading head obtained by the analysis is as follows:
wherein,
representing the measured value of the rotation angle compensated by the new method, and the compensated angle measurement error can be expressed as
In practical application, the measured shaft drives the circular grating to rotate for a circle to obtain the installation eccentricity parameter of the circular grating and the installation position error parameter of the two reading heads, and then the readings theta of the two reading heads are read 1 ，θ 2 And substituting to obtain a compensated angle measurement result, wherein the adverse effects of circular grating installation eccentricity and reading head diameter installation error are eliminated.
S4, measuring by a microscope to obtain the eccentricity of the circular grating, substituting the eccentricity serving as an initial value into the circular grating eccentricity parameter simulation model established in the step S1, and obtaining the real eccentricity through iteration;
the circular grating eccentric parameter solving model specifically comprises the following steps:
based on a Gauss-Newton algorithm, the search step length is improved, the limitation that a Jacobian matrix in the Gauss-Newton algorithm must be column full rank is eliminated, and then a circular grating eccentricity parameter solving model based on an L-M algorithm is obtained.
The study used the L-M algorithm to solve for the eccentricity parameters of the circular grating.
Will be provided with
Is marked as F (x) i ) =0, wherein x i ＝[e i ，R i ，ω i ]. The eccentricity error parameter identification problem is then expressed as:
the gradient and the Hessian matrix of the objective function f are respectively:
wherein,
is a matrix of the Jacobian ratio,
using Newton iteration method to obtain:
ignoring S (x) therein, we get:
x i+1 ＝x i +d i
wherein,
is f (x) at x i The search step size of the point. The above equation is the Gauss-Newton algorithm, but this algorithm requires that the jacobian matrix must be column full rank.
For search step length d k The improvement is as follows:
wherein I is an identity matrix, mu i Is the damping coefficient. The equation obtained at this time is the L-M algorithm.
Please refer to fig. 6,L-M algorithm for solving the eccentricity parameter, the specific steps are as follows:
1) Setting initial values of eccentricity e, eccentricity angle omega and code disc radius R;
2) Calculating a Jacobian matrix J and a function value;
3) And calculating an error, and if the error is less than a set threshold value, terminating. Otherwise, continuing;
4) Calculating the value of the Hessian matrix, wherein H = J' × J;
5) Calculating a search step length d;
6) Calculating new parameters and function values;
7) Calculating a new error;
8) If the error is smaller than the last time, updating the parameters and reducing the damping coefficient mu, if the error is larger, increasing the damping coefficient, not updating the parameters, and returning to the step 2) to continue iteration.
The L-M algorithm can be implemented using the nlinfit function in MATLAB software. The solution of the eccentricity error compensation model needs to set initial values of 3 parameters, namely eccentricity e, eccentricity angle omega and code disc radius R. The initial values of the eccentricity e and the eccentricity angle omega are set by random numbers, and the radius of the code disc is set according to the size of the actually selected circular grating.
And S5, substituting the real eccentricity obtained in the step S4 into the double-reading-head asymmetric installation error compensation model established in the step S3, and realizing accurate compensation of the angle measurement error caused by the installation eccentricity of the circular grating.
The double-reading head asymmetric installation angle measurement error compensation model specifically comprises the following steps:
and the eccentricity direction calculated by the circular grating eccentricity parameter solving model are used as input variables, and the measured angle eccentricity error compensation value is used as an output variable, so that the accurate compensation of the measured angle error of the circular grating encoder caused by the installation eccentricity and the reading head installation deviation is realized.
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 5, the constant-speed driving device drives the platform to rotate through the coupler, the circular grating is mounted on the rotating platform and rotates with the platform, and the first reading head and the second reading head are mounted above the grid line of the circular grating to measure the rotating angle of the circular grating.
Referring to fig. 4, a curve shows the influence of the circular grating installation eccentricity on the angle measurement accuracy when a single reading head measures an angle, and the influence of the circular grating installation eccentricity on the angle measurement accuracy can be seen from the graph to be very obvious, so that the angle measurement error caused by the eccentricity is corrected by adopting a measurement method of symmetrically installing double reading heads.
Referring to fig. 7, two curves are measurement errors introduced by circular grating installation eccentricity before and after compensation, respectively, and it can be seen that after compensation is performed by using a double-reading-head asymmetric installation error compensation model, the angle measurement accuracy of the circular grating system is significantly improved.
And (3) comprehensively analyzing a model result, setting the eccentricity e =10 μm, the radius R =26mm of the circular grating disc, the eccentricity angle omega =60 degrees and the installation error of the two reading heads to be 2 degrees, substituting the preset parameters into the model, fitting the simulation data by using a least square method to obtain the eccentricity e =10.070 μm, the eccentricity angle omega =59.465 867 degrees and the installation error of the two reading heads to be 2.000030 degrees, and basically consistent with the input parameters, thereby proving the accuracy of the eccentricity parameter fitting algorithm. Therefore, the circular grating carried on the air floating rotary table can be measured. Two reading heads A and B are arranged in the radial direction, and a circular grating encoder of RCDM20-108 series of Ranisha company is used, and the diameter of the circular grating is 108mm. And rotating the circular grating, observing the value of the reading head A, recording the values of the reading heads A and B at intervals of 15 degrees until 24 groups of data are obtained after one rotation, and solving the eccentric parameters of the circular grating by using an L-M algorithm according to experimental data. Solving needs to set initial values of 3 parameters of eccentricity ratio a, eccentricity angle w and included angle alpha of the two reading heads, because the two reading heads are installed in an approximate diameter-aligning mode, the initial value of the included angle alpha of the two reading heads is set to be 180 degrees. Initial values of eccentricity a and an eccentricity angle w are set by random numbers, and finally, eccentric parameters of the circular grating are solved by 69 times of iteration, wherein the eccentric distance e =1.8 μm, and an included angle alpha =179.250 degrees of the two reading heads. Substituting the solved parameters into the following formula
Through model analysis, the eccentricity error is generated due to the fact that the geometric center of the circular grating and the rotation center of the measured shaft are not coincident in the installation process, and is a regular system error, and the error value of the eccentricity error changes periodically along with the rotation angle of the rotating shaft. The larger the eccentricity is, the larger the angle measurement error is; under the same eccentric quantity, the larger the diameter of the grating ring is, the smaller the angle measurement error is; the angle measurement eccentric error changes periodically along with the rotation of the grating circular ring, and the error changes for a period when the shafting rotates for a circle, and the error shows as a first harmonic in an error frequency spectrum. The double-reading-head symmetric error compensation model can effectively eliminate the installation eccentricity of the circular grating and the radial motion deviation of a shaft system, so that the circular grating installation eccentricity compensation model is theoretically constructed. In consideration of the tiny installation deviation between the two reading heads in the actual installation process, the method for correcting the eccentric error of the circular grating with the non-diameter-aligning double reading heads can reduce the adverse effect of the diameter-aligning installation error on the measurement result on one hand; on the other hand, the limitation that the reading head must be installed in a diameter direction is broken through, and the degree of freedom of the instrument and equipment in structural design is improved. And based on a circular grating eccentricity parameter solving model, the eccentricity and the eccentricity direction calculated by the model are used as input variables, and the measured angle eccentricity error compensation value is used as an output variable, so that the accurate compensation of the measured angle error of the circular grating encoder caused by the installation eccentricity and the reading head installation deviation is realized. According to the experimental result, the average error of the angle measurement of the circular grating encoder before compensation is 6.81 ', the average error of the angle measurement of the circular grating encoder after compensation is 2.28 ', the angle measurement error caused by eccentricity can be effectively compensated by adopting a circular grating eccentricity error correction method of installing double reading heads in a non-diameter-aligning mode, and the angle measurement precision of the encoder is improved by 4.53 '. The pitch error peak-to-valley value before compensation was 10.52 ", the pitch error peak-to-valley value after compensation was 2.83", and the precision was improved by 7.69 ". In conclusion, the circular grating eccentricity error correction model with the non-diameter-alignment-mounted double-reading head can effectively eliminate the influence of the mounting eccentricity of the code wheel on the accuracy of circular grating angle measurement, and has important significance for improving the accuracy of circular grating encoder angle measurement.
In summary, according to the method for compensating the eccentric error of the double-reading head asymmetric installation of the circular grating encoder, the double-reading head symmetric error compensation model can effectively eliminate the installation eccentricity of the circular grating and the radial motion deviation of the shafting, so that the circular grating installation eccentric error compensation model is theoretically constructed. In consideration of the tiny installation deviation between the two reading heads in the actual installation process, the method for correcting the eccentric error of the circular grating with the non-diameter-aligning double reading heads can reduce the adverse effect of the diameter-aligning installation error on the measurement result on one hand; on the other hand, the limitation that the reading head must be installed in a diameter direction is broken through, and the degree of freedom of the instrument and equipment in structural design is improved. And based on a circular grating eccentricity parameter solving model, the eccentricity and the eccentricity direction calculated by the model are used as input variables, and the measured angle eccentricity error compensation value is used as an output variable, so that the accurate compensation of the measured angle error of the circular grating encoder caused by the installation eccentricity and the reading head installation deviation is realized. According to the experimental result, the average error of the angle measurement of the circular grating encoder before compensation is 6.81 ', the average error of the angle measurement of the circular grating encoder after compensation is 2.28 ', the angle measurement error caused by eccentricity can be effectively compensated by adopting the circular grating eccentricity error correction method of installing double reading heads in a non-diameter-alignment manner, and the angle measurement precision of the encoder is improved by 4.53 '. The pitch error peak-to-valley value before compensation was 10.52 ", the pitch error peak-to-valley value after compensation was 2.83", and the precision was improved by 7.69 ". In conclusion, the circular grating eccentricity error correction model with the non-radial installation of the double-reading head can effectively eliminate the influence of the installation eccentricity of the code disc on the accuracy of the circular grating angle measurement, and has important significance for improving the angle measurement accuracy of the circular grating encoder.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.
Claims (9)
1. A method for compensating an eccentric error of double-reading head asymmetric installation of a circular grating encoder is characterized by comprising the following steps:
s1, solving the eccentricity and the eccentricity direction of the circular grating based on an L-M algorithm according to the synthesis of sinusoidal signals with different phases in the circumferential direction of the circular grating, and establishing a circular grating eccentricity parameter simulation model;
s2, establishing an error compensation ideal model of a mean value method with symmetrically installed double reading heads;
s3, adding a reading head installation error and a random error into the ideal error compensation model of the averaging method established in the step S2, and establishing a double-reading-head asymmetric installation error compensation model;
s4, measuring by a microscope to obtain the eccentricity of the circular grating, substituting the eccentricity serving as an initial value into the circular grating eccentricity parameter simulation model established in the step S1, and obtaining the real eccentricity through iteration;
and S5, substituting the real eccentricity obtained in the step S4 into the double-reading-head asymmetric installation error compensation model established in the step S3, and realizing accurate compensation of the angle measurement error caused by the installation eccentricity of the circular grating.
2. The method for compensating the eccentric error of the asymmetric installation of the double reading heads of the circular grating encoder as claimed in claim 1, wherein in step S1, the phase difference delta psi of the sinusoidal signals detected by the two reading heads at the same time at different positions in the circumferential direction 12 Comprises the following steps:
where d is the grating pitch, 360 represents the number of angles in one period of the electrical fringe signal,
is the mounting angle between the two reading heads, and e is the eccentricity.
3. The method for compensating the asymmetric installation eccentricity error of the double-reading head of the circular grating encoder as claimed in claim 2, wherein the eccentricity e is specifically as follows:
where Δ ψ is the total phase difference read by the two read heads.
4. The method for compensating the asymmetric installation eccentricity error of the double-reading head of the circular grating encoder as claimed in claim 1, wherein in the step S2, the ideal model for compensating the error by the averaging method specifically comprises:
wherein,
for the error compensation result of the averaging method, theta is the theoretical rotation angle without error, tau 1 Is the mounting position of the first reading head, τ 2 Is the mounting position of the second reading head, psi (theta) 1 ) Angle measurement error, delta (theta), caused by inconsistencies within the first readhead 1 ) For random errors due to other uncertainties of the first readhead, ψ (θ) 2 ) Angle error, delta (theta), caused by inconsistencies within the second read head 2 ) Random errors due to other uncertainties of the second read head.
5. The method for compensating the asymmetric installation eccentricity error of the double-reading head of the circular grating encoder according to claim 1, wherein in the step S3, the asymmetric installation error compensation model of the double-reading head specifically comprises:
wherein,
representing the measured value of the angle of rotation compensated by the new method,
for compensated angle measurement error, e is eccentricity, and R is code wheel halfDiameter, ω being the eccentric angle, θ 1 ，θ 2 Is the rotation angle of the first reading head and the first reading head measured in the rotation process, and eta is ° O 1 OC。
6. The method for compensating the asymmetric installation eccentricity error of the double reading heads of the circular grating encoder as claimed in claim 5, wherein the angle measurement errors of the first reading head and the second reading head are respectively as follows:
wherein, theta is the theoretical rotation angle without error.
7. The method for compensating the eccentric error of the asymmetric installation of the double reading heads of the circular grating encoder as claimed in claim 1, wherein in the step S4, the solution of the eccentricity and the eccentricity direction of the circular grating based on the L-M algorithm is specifically as follows:
setting an initial value of an eccentricity e, an eccentricity angle omega and a code disc radius R; calculating a Jacobian matrix J and a function value; calculating an error, and if the error is smaller than a set threshold value, terminating; otherwise, continuing; calculating the value of the Hessian matrix: h = J '× J, J' being the derivation of the jacobian matrix J; calculating a search step length d; calculating new parameters and function values; calculating a new error; if the error is smaller than the last error, the parameters are updated and the damping coefficient mu is reduced, if the error is larger, the damping coefficient is increased, the parameters are not updated, the Jacobian matrix J and the function value are calculated, and iteration is continued.
8. The method of claim 7, wherein x is x i Search step d of points i Comprises the following steps:
wherein, J i Is a Jacobian matrix, mu i For damping coefficient, I is an identity matrix, F (x) i ) Error theta for measuring angle of first reading head and second reading head 2 -θ 1 The functional expression of the expression.
9. The method for compensating the asymmetric installation eccentricity error of the double-reading head of the circular grating encoder according to claim 1, wherein in the step S5, the asymmetric installation angle measurement error compensation model of the double-reading head specifically comprises:
and the eccentricity direction calculated by the circular grating eccentricity parameter solving model are used as input variables, and the measured angle eccentricity error compensation value is used as an output variable, so that the accurate compensation of the measured angle error of the circular grating encoder caused by the installation eccentricity and the reading head installation deviation is realized.
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