CN107747931B - For the self-alignment reading head optimization placement method of angular encoder - Google Patents
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- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
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Abstract
The present invention relates to one kind to be used for the self-alignment reading head optimization placement method of angular encoder comprising following steps: determining the transmission function in angle error analysis using Fourier analysis method and analyzes its characteristic；Comprehensively consider the factors such as harmonics restraint, random noise and calibration accuracy and establishes optimization aim；Determine the constraint conditions such as the initial parameters such as reading head number, calibration harmonic wave sum and reading head installation minimum angles interval, and then solving optimization problem etc..It is proposed by the present invention to can be realized the high-precision to angular encoder for the self-alignment reading head optimization placement method of angular encoder, calibrate quickly, in real time, in place, and cost is lower, wherein two reading head the optimum layout schemes are especially suitable for conventional high precision measurement system, and three reading head optimization systems more can be used for the calibration occasion of superhigh precision.
Description
Technical field
The present invention relates to a kind of calibration methods of angle measurement unit in field of precision measurement, are used for more particularly to one kind
The self-alignment reading head optimization placement method of angular encoder.
Background technique
Angular encoder is the critical component in field of precision measurement, is mainly used in the high precision measuring device of angle value
In, such as machine tool rotary working table, precise rotating platform and angular measure National primary standard etc..Typically angular encoder mainly includes
Circular gratings code-disc and reading head.In practical work process, the measurement accuracy of angular encoder is affected by many factors, comprising:
The accuracy of grating pitch, signal subdivision quality, grating installation deformation, the installation of reading head and alignment error, grating relative to
The setting-up eccentricity of bearing, bearing bounce even temperature change etc..Therefore just needs determined by being calibrated to encoder into
And the measurement error caused by compensating by these factors, improving measurement accuracy.
A kind of traditional calibration method is by the angle-measuring equipment (reference encoder device or polygon etc.) of a higher precision
It is co-axially mounted with tested angular encoder, determines the error of tested angular encoder by continuously measuring the difference of the two.But
It is that this method real-time is poor, cumbersome time-consuming, simultaneously because having ignored the correlated error with reference to angle-measuring equipment, therefore its school
Quasi- precision is limited.Another traditional calibration method is known as sewage station, and cardinal principle is by by tested angular encoder
It is co-axially mounted with detection angle apparatus (such as polygon, multiteeth indexing table) is accompanied, the comparison measurement of repeatedly indexing is carried out, according to circle
All closure principles carry out error separate, finally may separate out the angle for eliminating the tested angular encoder for accompanying checking device error to influence
Spend error.Sewage station has very high calibration accuracy, but there is the problems such as operation is time-consuming, calibration points are limited.
In order to avoid using other angle-measuring equipments, while realizing real-time quick school of the angular encoder in its installation axle
Standard, self-calibration technique is also widely used in recent years.Retrieval discovery by existing technical literature, Patent No.
The United States Patent (USP) of US7840372B2 TDR method collimation angle encoder, but this method can only be used to calibrate air bearing or magnetic is floating
The negligible frictional resistance class main shaft such as bearing, and its algorithm complexity is not easy to operate；The United States Patent (USP) of Patent No. US7143518B2
Using M reading head around Circular gratings equiangularly spaced equally distributed mode, can calibrate to obtain in angular error except M and
Times all harmonic components of residue outer for several times of M, but in order to obtain higher calibration accuracy, required reading head generally is
Five or more, such method had not only had that reading head mounting and adjusting difficulty was big, but also increased corresponding cost.Also there are other
Document give it is several for the self-alignment reading head arrangement of angular encoder, but all exist using reading head number compared with
More, arrangement does not pass through the problems such as optimization design.
Summary of the invention
Technical problem to be solved by the invention is to provide one kind to optimize cloth for the self-alignment reading head of angular encoder
Method is set, the high-precision to angular encoder is can be realized, calibrates quickly, in real time, in place, and cost is lower, wherein two read
Several the optimum layout schemes are especially suitable for conventional high precision measurement system, and three reading head optimization systems more can be used for surpassing
High-precision calibration occasion.
The present invention is that above-mentioned technical problem is solved by following technical proposals: one kind being used for angular encoder self calibration
Reading head optimization placement method, which is characterized in that it is described be used for the self-alignment reading head optimization placement method of angular encoder
The following steps are included:
Step 1: it determines the transmission function in angle error analysis and analyzes its characteristic:
K-th of reading head HkRelative to first reading head H1Installation interval angle be αk(α1=0), then in Circular gratings
During code-disc rotates, reading head HkReading Hk(θ) had not only contained true rotation angle, θ, but also contained at the measuring point
Angle error ε (θ+αk), shown in following formula:
Hk(θ)=θ+ε (θ+αk)
Angle, θ can be eliminated by the relative difference of two reading head readings in actual test, obtaining only includes angle measurement
The relational expression of error, reading head HkWith HrThe following formula of relative difference shown in:
δkr(θ)=Hk(θ)-Hr(θ)=ε (θ+αk)-ε(θ+αr)
Again since angular encoder will repeat original error for each revolution, angle error has the period
Property, it can be indicated with fourier series, shown in following formula:
In formula, n indicates each order harmonic component, and N is the harmonic wave to be analyzed sum, and F (n) is Fourier coefficient；Based on circle
All closure principles, DC component F (0)=0；E is the nature truth of a matter, real number 2.718281；
It equally can also be to difference functions δkr(θ) carries out Fourier transform, if EkrIt (n) is δkrFourier leaf system corresponding to (θ)
It counts, then Ekr(n) and shown in the following formula of the relativeness of F (n):
W in formulakrIt (n) is transmission function, it can be seen that Wkr(n) interval angles only between reading head are related, because
This, measures difference functions δkrAfter (θ), its fourier coefficient E is obtained by Discrete Fourier Transformkr(n), and then transmitting is combined
Function Wkr(n) the Fourier coefficient F (n) that angle error can be obtained acquires angle finally by inverse discrete Fourier transformer inverse-discrete and compiles
Code device angle error function of ε (θ)；
Further for the encoder self-calibration system of reading head number M > 2, it is ok by any two of them reading head
F (n) is obtained, therefore a total ofA combination, the estimation precision of F (n) can further be promoted by comprehensively utilizing all combinations, as follows
Shown in formula:
Wherein, pkrIt (n) is weighting coefficient, shown in following formula:
In order to analyze the characteristic of transmission function, the mould for seeking transmission function is shown below:
Due to | exp (in αr) | ≡ 1, therefore for the harmonic component of specific order, two reading head transmission function moulds it is big
Angle between small and the two reading heads is related, further analysis can obtain 0≤| Wkr(n) |≤2, it in addition can also be with by formula
Find out, when | Wkr(n) | when=0, harmonics restraint can be led to the problem of, i.e. the n-th harmonic component of angle error is difficult to use Fourier
It is zero that method, which acquires and can only set the order harmonic component by force, and calibration result is caused to deviate its actual value；Furthermore in measurement process
Noise and random disturbances are inevitably present, and can give difference functions δ in this waykrEach rank fourier coefficient E corresponding to (θ)kr
(n) certain measurement error is brought, and for an equal amount of deviation, | Wkr(n) | value it is smaller, lead to the F finally released
(n) error is bigger；Only | Wkr(n) | value be greater than 1 when, can just play inhibiting effect to noise, comprehensively consider factors above,
The method of the present invention recommendation takes | Wkr(n) | threshold value be 1, as | Wkr(n) | the order harmonic component can be measured effectively when > 1；
Step 2: reasonable optimization problem is determined according to the characteristic of step 1:
For the system of reading head number M > 2, it is believed that the abundant item that n-th certain harmonic components can be detected effectively
Part is: as long as wherein there are two the corresponding transmission functions of reading head to meet | Wkr(n) | > 1, that is to say, that system detection is not
To the following formula of condition of n-th harmonic component:
In formula, | | W (n) | |∞It indicates by each element Wkr(n) Infinite Norm of the vector W (n) formed, setting need to detect
Harmonic wave sum be preceding n times, then optimization problem can be summarized as following formula:
s.t.α0≤α2≤α3–α2≤,···,≤αk–αk-1≤,···,≤αM–αM-1≤360°–αM
Wherein optimization aim means that the harmonic wave sum that cannot be effectively detected is minimum；The selection of constraint condition be for
The uniqueness solved, avoids the occurrence of more solutions to waste computing resource, α0Indicate the minimum angles interval between two reading heads,
It can be established according to factors such as the geometric dimensions of reading head；
Step 3: solving optimization problem:
Firstly the need of setting initial parameter when being solved for the optimization problem in step 2, comprising: reading head used
Number M, need the harmonic wave sum N detected, minimum angles interval α between reading head0And | Wkr(n) | threshold value, due to optimization
The objective function of problem is not convex function, globally optimal solution is not present, therefore can solve using heuritic approach, for example hereditary
Algorithm, differential evolution algorithm or particle swarm algorithm, when the dimension of problem is smaller, increment uses traversal also at an angle
It can be solved；
In fact in the industrial application of angular encoder, in angle error one as caused by Circular gratings code-disc setting-up eccentricity
Subharmonic error is occupied an leading position, and adds other several biggish low-order harmonics of amplitude outside, and higher hamonic wave is smaller due to amplitude
Substantially can ignore, present invention discover that being optimized in arrangement result using the reading head that above-mentioned optimization method obtains, two reading head systems
System is enough suitable for conventional high-precision calibration occasion, such as time of machine tool rotary working table, lathe C axis, mechanical bearing support
Transport the application of moving platform angularly encoder；And three reading head systems are enough the calibration occasion suitable for superhigh precision,
Such as use High-precision angle encoder and the ultra-precise revolving motion platform of air-bearing.
Preferably, described to obtain two reading head systems for the self-alignment reading head optimization placement method of angular encoder
Self calibration the optimum layout scheme: two reading heads are on five position of halving of circumference.
Preferably, described to obtain three reading head systems for the self-alignment reading head optimization placement method of angular encoder
The optimum layout scheme.
The positive effect of the present invention is that: it is proposed by the present invention for the self-alignment reading head optimization of angular encoder
Method for arranging can be realized the high-precision to angular encoder, calibrate quickly, in real time, in place, and cost is lower, wherein two readings
Head the optimum layout scheme is especially suitable for conventional high precision measurement system, and three reading head optimization systems more can be used for superelevation
The calibration occasion of precision.Characteristic of the present invention according to angle error transmission function in calibration process, establishes optimization object function, and
Constraint condition is obtained in conjunction with practical application, and then to optimization problem solving to obtain ideal preferred arrangement scheme.
Detailed description of the invention
Fig. 1 is more reading head arrangement schematic diagrams in angular encoder.
Fig. 2 is the flow chart of the method for the present invention.
Fig. 3 is the optimum layout scheme schematic diagram of two reading head systems.
The arrangement schematic diagram that Fig. 3 a is reading head angle when being 72 degree.
The arrangement schematic diagram that Fig. 3 b is reading head angle when being 144 degree.
Fig. 4 is the amplitude schematic diagram of two reading head system each harmonic transmission functions.
Fig. 5 is two reading head system calibration performance simulation schematic diagrames.
Fig. 6 is the optimum layout scheme schematic diagram of three reading head systems.
Fig. 6 a is the first the optimum layout scheme schematic diagram of three reading head systems.
Fig. 6 b is second of the optimum layout scheme schematic diagram of three reading head systems.
Fig. 6 c is the third the optimum layout scheme schematic diagram of three reading head systems.
Fig. 7 is the amplitude schematic diagram of three reading head system each harmonic transmission functions.
Fig. 8 is three reading head system calibration performance simulation schematic diagrames.
Specific embodiment
Present pre-ferred embodiments are provided with reference to the accompanying drawing, in order to explain the technical scheme of the invention in detail.
Method shown in the present invention is described in further detail with reference to the accompanying drawings, angular encoder is equipped with M reading head, this hair
How bright solution arranges this M reading head, so that between calibration curve obtained and the true angular error curve of encoder
Relative error it is minimum, i.e., calibration accuracy highest, the present invention are used for the self-alignment reading head optimization placement method of angular encoder
The following steps are included:
Step 1: it determines the transmission function in angle error analysis and analyzes its characteristic
K-th of reading head HkRelative to first reading head H1Installation interval angle be αk(α1=0), then in Circular gratings
During code-disc rotates, reading head HkReading Hk(θ) had not only contained true rotation angle, θ, but also contained at the measuring point
Angle error ε (θ+αk), shown in following formula (1):
Hk(θ)=θ+ε (θ+αk)……(1)
Angle, θ can be eliminated by the relative difference of two reading head readings in actual test, obtaining only includes angle measurement
The relational expression of error, reading head HkWith HrThe following formula of relative difference (2) shown in:
δkr(θ)=Hk(θ)-Hr(θ)=ε (θ+αk)-ε(θ+αr)……(2)
Again since angular encoder will repeat original error for each revolution, angle error has the period
Property, it can be indicated with fourier series, shown in following formula (3):
In formula, n indicates each order harmonic component, and N is the harmonic wave to be analyzed sum, and F (n) is Fourier coefficient；Based on circle
All closure principles, DC component F (0)=0；E is the nature truth of a matter, real number 2.718281；
It equally can also be to difference functions δkr(θ) carries out Fourier transform, if EkrIt (n) is δkrFourier leaf system corresponding to (θ)
It counts, then Ekr(n) and shown in the following formula of the relationship of F (n) (4) and (5):
W in formulakrIt (n) is transmission function, it can be seen that Wkr(n) interval angles only between reading head are related, because
This, measures difference functions δkrAfter (θ), its fourier coefficient E is obtained by Discrete Fourier Transformkr(n), and then transmitting is combined
Function Wkr(n) the Fourier coefficient F (n) that angle error can be obtained acquires angle finally by inverse discrete Fourier transformer inverse-discrete and compiles
Code device angle error function of ε (θ)；
Further for the encoder self-calibration system of reading head number M > 2, it is ok by any two of them reading head
F (n) is obtained, therefore a total ofA combination, the estimation precision of F (n) can further be promoted by comprehensively utilizing all combinations, as follows
Shown in formula (6):
Wherein, pkrIt (n) is weighting coefficient, shown in following formula (7):
In order to analyze the characteristic of transmission function, shown in the mould such as following formula (8) for seeking transmission function:
Due to | exp (in αr) | ≡ 1, therefore for the harmonic component of specific order, two reading head transmission function moulds it is big
Angle between small and the two reading heads is related, further analysis can obtain 0≤| Wkr(n) |≤2, it in addition can also be with by formula
Find out, when | Wkr(n) | when=0, harmonics restraint can be led to the problem of, i.e. the n-th harmonic component of angle error is difficult to use Fourier
It is zero that method, which acquires and can only set the order harmonic component by force, and calibration result is caused to deviate its actual value；Furthermore in measurement process
Noise and random disturbances are inevitably present, and can give difference functions δ in this waykrEach rank fourier coefficient E corresponding to (θ)kr
(n) certain measurement error is brought, and for an equal amount of deviation, | Wkr(n) | value it is smaller, lead to the F finally released
(n) error is bigger；Only | Wkr(n) | value be greater than 1 when, can just play inhibiting effect to noise, comprehensively consider factors above,
The method of the present invention recommendation takes | Wkr(n) | threshold value be 1, as | Wkr(n) | the order harmonic component can be measured effectively when > 1；
Step 2: reasonable optimization problem is determined according to the characteristic of step 1
For the system of reading head number M > 2, it is believed that the abundant item that n-th certain harmonic components can be detected effectively
Part is: as long as wherein there are two the corresponding transmission functions of reading head to meet | Wkr(n) | > 1, that is to say, that system detection is not
To the following formula of condition (9) of n-th harmonic component:
In formula, | | W (n) | |∞It indicates by each element Wkr(n) Infinite Norm of the vector W (n) formed, setting need to detect
Harmonic wave sum be preceding n times, then it is shown can be summarized as following formula (10) and (11) for optimization problem:
s.t.α0≤α2≤α3–α2≤,···,≤αk–αk-1≤,···,≤αM–αM-1≤360°–αM……(11)
Wherein optimization aim means that the harmonic wave sum that cannot be effectively detected is minimum；The selection of constraint condition be for
The uniqueness solved avoids the occurrence of more solutions to waste computing resource, and α 0 indicates the minimum angles interval between two reading heads,
It can be established according to factors such as the geometric dimensions of reading head；
Step 3: solving optimization problem
Firstly the need of setting initial parameter when being solved for the optimization problem in step 2, comprising: reading head used
Number M, need the harmonic wave sum N detected, minimum angles interval α between reading head0And | Wkr(n) | threshold value, due to optimization
The objective function of problem is not convex function, globally optimal solution is not present, therefore can solve using heuritic approach, for example hereditary
Algorithm, differential evolution algorithm or particle swarm algorithm, when the dimension of problem is smaller, increment uses traversal also at an angle
It can be solved；
In fact in the industrial application of angular encoder, in angle error one as caused by Circular gratings code-disc setting-up eccentricity
Subharmonic error is occupied an leading position, and adds other several biggish low-order harmonics of amplitude outside, and higher hamonic wave is smaller due to amplitude
Substantially can ignore, present invention discover that being optimized in arrangement result using the reading head that above-mentioned optimization method obtains, two reading head systems
System is enough suitable for conventional high-precision calibration occasion, such as time of machine tool rotary working table, lathe C axis, mechanical bearing support
Transport the application of moving platform angularly encoder；And three reading head systems are enough the calibration occasion suitable for superhigh precision,
Such as use High-precision angle encoder and the ultra-precise revolving motion platform of air-bearing；
The present invention provides two reading heads and three reading heads acquired according to optimization aim formula and constraint condition formula below
Optimum results, using the traversal of 0.1 ° of angle step, the initial parameter of selection are as follows: N=100, α0=10 °.
Two reading head system self-calibration the optimum layout schemes that a kind of optimization method obtains: two reading heads are in
On five position of halving of circumference, i.e., 72 ° or 144 ° are divided between the two angle.
A kind of the optimum layout scheme of the optimization method three obtained reading head system: a total of 3 kinds, reading head it
Between angle interval combination be respectively (27 °, 36 °, 63 °), (36 °, 117 °, 153 °) and (81 °, 108 °, 171 °).
In conclusion the angular encoder self-alignment reading head optimization placement method proposed by the present invention that is used for is at less
Number reading heads on the basis of can obtain very high calibration accuracy, therefore can greatly reduce installation and debugging difficulty and correlation at
This, has good practical value.The solution of the present invention can be used in the high-precision of angular encoder, quick, real-time, school in place
Standard, and cost is lower, wherein two reading head preferred arrangement schemes are highly suitable for conventional high precision measurement system, and three read
Head optimization system more can be used for the calibration occasion of superhigh precision.
Particular embodiments described above, the technical issues of to solution of the invention, technical scheme and beneficial effects carry out
It is further described, it should be understood that the above is only a specific embodiment of the present invention, is not limited to
The present invention, all within the spirits and principles of the present invention, any modification, equivalent substitution, improvement and etc. done should be included in this
Within the protection scope of invention.
Claims (3)
1. one kind is used for the self-alignment reading head optimization placement method of angular encoder, which is characterized in that described to be compiled for angle
Code the self-alignment reading head optimization placement method of device the following steps are included:
Step 1: it determines the transmission function in angle error analysis and analyzes its characteristic:
K-th of reading head HkRelative to first reading head H1Installation interval angle be αk, wherein α1=0, then in Circular gratings code
During disc spins, reading head HkReading Hk(θ) had not only contained true rotation angle, θ, but also contained reading head HkMeasuring point
Angle error ε (θ+the α at placek), shown in following formula:
Hk(θ)=θ+ε (θ+αk)
Angle, θ is eliminated by the relative difference of two reading head readings in actual test, obtains the pass for including angle error
It is formula, reading head HkWith HrThe following formula of relative difference shown in:
δkr(θ)=Hk(θ)-Hr(θ)=ε (θ+αk)-ε(θ+αr)
Again since angular encoder will repeat original error for each revolution, angle error has periodically, uses
Fourier series indicates, shown in following formula:
In formula, n indicates each order harmonic component, and N is the harmonic wave to be analyzed sum, and F (n) is Fourier coefficient；It is sealed based on circumference
Close principle, DC component F (0)=0；E is the nature truth of a matter, real number 2.718281；
To difference functions δkr(θ) carries out Fourier transform, if EkrIt (n) is δkrFourier coefficient corresponding to (θ), then Ekr(n) with
Shown in the following formula of the relativeness of F (n):
W in formulakrIt (n) is transmission function, Wkr(n) interval angles only between reading head are related, it is therefore possible for the detected that difference functions
δkrAfter (θ), its fourier coefficient E is obtained by Discrete Fourier Transformkr(n), and then transmission function W is combinedkr(n) it is surveyed
The Fourier coefficient F (n) of angle error acquires angular encoder angle error function of ε finally by inverse discrete Fourier transformer inverse-discrete
(θ)；
For the encoder self-calibration system of reading head number M > 2, F (n) can be accessed by any two of them reading head,
Therefore a total ofA combination comprehensively utilizes all estimation precisions for combining and being able to ascend F (n), shown in following formula:
Wherein, pkrIt (n) is weighting coefficient, shown in following formula:
In order to analyze the characteristic of transmission function, the mould for seeking transmission function is shown below:
Due to | exp (in αr) | ≡ 1, therefore for the harmonic component of specific order, the size of two reading head transmission function moulds only with
Angle between the two reading heads is related, and analysis can obtain 0≤| Wkr(n) |≤2, take | Wkr(n) | threshold value be 1, as | Wkr(n)
| the order harmonic component can be measured effectively when > 1；
Step 2: reasonable optimization problem is determined according to the characteristic of step 1:
For the system of reading head number M > 2, the adequate condition that n-th certain harmonic components can be detected effectively is: as long as its
In there are two the corresponding transmission functions of reading head to meet | Wkr(n) | > 1, i.e. item of the system detection less than n-th harmonic component
The following formula of part:
In formula, | | W (n) | |∞It indicates by each element Wkr(n) Infinite Norm of the vector W (n) formed, setting need to detect humorous
Wave sum is preceding n times, then optimization problem is summarized as following formula:
s.t.α0≤α2≤α3–α2≤,···,≤αk–αk-1≤,···,≤αM–αM-1≤360°–αM
Wherein optimization aim refers to that the harmonic wave sum that cannot be effectively detected is minimum；The selection of constraint condition is for solution
Uniqueness, α0It indicates the minimum angles interval between two reading heads, is established according at least to the geometric dimension factor of reading head；
Step 3: solving optimization problem:
When being solved for the optimization problem in step 2, it is necessary first to set initial parameter, comprising: of reading head used
Minimum angles interval α between number M, the harmonic wave sum N for needing to detect, reading head0And | Wkr(n) | threshold value, using heuristic calculation
Method or traversal solve.
2. being used for the self-alignment reading head optimization placement method of angular encoder as described in claim 1, which is characterized in that institute
It states and obtains two reading head system self-calibration the optimum layout sides for the self-alignment reading head optimization placement method of angular encoder
Case: two reading heads are on five position of halving of circumference.
3. being used for the self-alignment reading head optimization placement method of angular encoder as described in claim 1, which is characterized in that adopt
With the traversal of 0.1 ° of angle step, the initial parameter of selection are as follows: N=100, α 0=10 °, it is described to be used for angular encoder self-correcting
Quasi- reading head optimization placement method obtains the optimum layout scheme of three reading head systems: the angle interval group between reading head
Close is respectively (27 °, 36 °, 63 °), (36 °, 117 °, 153 °) and (81 °, 108 °, 171 °).
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| CN109990823A (en) * | 2019-04-29 | 2019-07-09 | 吉林大学 | For detecting the detection device and detection method of photoelectric encoder precision |
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| JP3826207B2 (en) * | 2004-08-31 | 2006-09-27 | 独立行政法人産業技術総合研究所 | Angle detector with self-calibration function |
| JP4824415B2 (en) * | 2006-01-27 | 2011-11-30 | 株式会社 ソキア・トプコン | Rotary encoder |
| JP4984269B2 (en) * | 2009-11-09 | 2012-07-25 | 独立行政法人産業技術総合研究所 | Angle detector with complex self-calibration function |
| CN102022975B (en) * | 2010-11-22 | 2012-09-05 | 重庆理工大学 | Error self-calibration method of angle displacement sensor system |
| CN103162614B (en) * | 2013-03-28 | 2015-08-19 | 重庆理工大学 | The online self-calibrating method of a kind of angular displacement sensor |
| CN103217139B (en) * | 2013-03-28 | 2015-08-05 | 重庆理工大学 | The online self-calibrating method of a kind of angular displacement sensor based on double testing head |
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| CN106500631B (en) * | 2016-10-21 | 2018-11-23 | 北京信息科技大学 | Circular gratings eccentric error parameter identification and compensation method |
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| CN110296668A (en) * | 2019-04-24 | 2019-10-01 | 合肥工业大学 | A kind of circular raster sensor angle error modification method based on BP neural network |
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