CN113483660A - Method for evaluating radial circular run-out error of optical axis - Google Patents

Abstract

The invention belongs to the field of precision metering and computer application, and relates to a stable, quick and simple-form evaluation method for radial run-out errors of an optical axis. The method comprises the following steps: 1: acquiring an initial measuring point set of a measured section; 2: acquiring an initial measuring point set after prepositioning; 3: establishing a characteristic row vector set and a state element set; 4: establishing an analysis matrix and an analysis column vector; 5: performing rank analysis; 6: solving the analysis matrix and the analysis column vector; 7: updating the coordinates of the circle center and the state element set of the measured point of the measured section; 8: updating a reference section measuring point set; 9: establishing a characteristic row vector set and a state element set; 10: adding the measuring point sequence numbers of the measuring points of the reference segment into the key point set; 11: establishing an analysis matrix and an analysis column vector; 12: performing rank analysis; 13: solving the analysis matrix and the analysis column vector; 14: updating the measuring point set and the state element set; 15: and (5) judging the qualification.

Description

Method for evaluating radial circular run-out error of optical axis

Technical Field

The invention belongs to the field of precision measurement and computer application, and relates to a stable, quick and simple evaluation method for radial run-out errors of a shaft, which can be used for evaluating the radial run-out errors of shaft parts with self axes as reference axes and provides guidance for the improvement of the machining process of the shaft parts.

Background

The shaft parts are common typical parts in mechanical hardware fittings, dimensional errors and form and position errors (short for form errors and position errors) generated in the machining process directly influence the product quality, the assembly and the service life of the assembly, the part errors are calculated quickly and accurately, and the method has important significance.

The radial run-out is one of important indexes of geometric tolerance of shaft parts, and the national standards GBT 1182-2008, GBT 1958-2004 and ISO 1101:2012 (E) give definition and detection methods of radial run-out errors, but do not give methods for calculating radial run-out error values from specific measurement data. Moreover, the five types of currently popular assessment methods are difficult to directly assess the radial run-out error of the shaft.

First, a specialized geometric assessment method. And gradually searching radial circular run-out errors meeting the definition and/or judgment conditions of national standards and ISO standards according to the translation and deformation strategies of the circumscribed cylinder by utilizing the geometric properties of the cylinder. The method has high speed, but the form of the mathematical model is complex, and the method is not easy to popularize and use.

Second, convex hull or convex hull-like evaluation methods. And constructing a convex hull or a similar convex hull by using the properties of the convex hull, acquiring effective measurement data, reducing the scale of the data to be evaluated, and finally acquiring the radial circular run-out error meeting the definition and/or judgment conditions of the national standard and the ISO standard by using an enumeration method. This type of approach has significant advantages when dealing with medium scale station data. For larger data sizes, the data size can also be reduced by constructing convex hulls, but when such methods are used for direct evaluation, the efficiency is insufficient.

And in the third category, a linear or nonlinear target optimization function is constructed, optimization solution is carried out by adopting a general optimization method, and the optimization value of the target optimization function is used as the radial run-out error. The method is simple and easy to understand, and realizes a standard solution method in a plurality of software, so the method is easy to popularize. However, the efficiency of this type of method is generally not high, since no geometrical features for radial run-out error evaluation are added, and the situation of large data scale in the evaluation task is not considered.

The fourth category, artificial intelligence/biological intelligence algorithms. The advantage of this type of method over the third type of method is to analyze the "objective function with complex gradient or no apparent analytic expression" and to find the "global optimum". The method also realizes standard solutions in a plurality of software at present, so the method is easy to popularize. Although this type of method is popular at present, it is not suitable for the evaluation of radial run-out error. This is because the gradient of the objective function for radial run-out error assessment is the sum of a large number of simple analytical expressions, and the "local optimum" of the objective function is the "global optimum". Therefore, this type of method does not have significant advantages over the third type of method.

The fifth category, active set methods. The active set method is a method specially used for processing large-scale planning problems and is characterized in that the processing of 'invalid constraint' is reduced as much as possible in the optimization process. When the method is applied to radial circular runout error evaluation, the efficiency is equivalent to that of the first method, the algorithm maturity and the software integration are equivalent to that of the third method and the fourth method, and the method is a rapid and simple radial circular runout error evaluation method at present. However, this method is very sensitive to initial values and does not always perform the geometric assessment task stably.

In summary, the conventional geometric evaluation method cannot simultaneously consider stability, rapidness and simple form when applied to evaluation of radial run-out errors of a shaft.

Disclosure of Invention

The purpose of the invention is:

aiming at the problems and the defects in the prior art, the invention provides the stable and quick evaluation method of the radial circular runout error of the shaft part with the self axis as the reference axis, and the evaluation result can provide guidance for the improvement of the machining process.

The method is suitable for evaluating the radial circular run-out error of the shaft part which has higher requirements on the processing precision of the shaft such as a linear optical axis in a linear cylindrical guide rail and takes the self axis as the reference.

The invention can simultaneously realize the evaluation of the straightness error of the axis of the shaft part by taking the axis of the shaft part as the reference.

The invention takes a linear optical axis as an example, and verifies a fast, stable and simple evaluation method for the radial run-out error of a shaft.

The scheme adopted by the invention is as follows:

a method for evaluating radial run-out errors of a fast, stable and simple shaft comprises the following steps:

step 0: obtaining initial measuring point set of measured segmentQ _i，j *}: taking the cylindrical surface of a cylinder as a measured section and the axis of the cylinder as a reference section, and cutting the cylindrical surface at equal intervals along the axis directioniEach cross section being uniformly selected over the circumference of each cross-sectional circlejA measuring point, each section of which is perpendicular to the axis of the cylinder, thei×jInitial measuring point set for a segment to be measured consisting of measuring pointsQ _i，j }; wherein:

i=1, 2, 3, …, N；ithe serial number of the cross section is shown,Nis the total number of the sections;

j=1, 2, 3, …, M；jthe serial numbers of the measuring points on the single cross section,Mthe total number of the measuring points is;

Q _i，j *={x _i，j *，y _i，j *，z _i，j and the original space rectangular coordinate of the measured segment is marked.

After step 0, step 1 is performed.

Step 1: the cylinder is pre-positioned as follows: selecting coordinates of measured points of all measured segmentsx _i，j *_maxAndx _i，j *_mincalculating the average valuex _o* =（x _i，j *_max+ x _i，j *_min) Selecting coordinates of measured points of all measured sectionsy _i，j *_maxAndy _i，j *_mincalculating the average valuey _o*=（y _i，j *_max+y _i，j *_min) 2; obtaining a pre-positioned measured section measuring pointQ _i，j And use it in combinationConstitute a measured segment measuring point setQ _i，j }; obtaining the preset center coordinates of each cross-section circle after prepositioningP _i And using the initial measuring point set of reference segmentP _i }; wherein:

Q _i，j ={x _i，j ，y _i，j ，z _i，j and the space rectangular coordinate of the measured point of the measured section after pre-positioning, wherein:x _i，j = x _i，j *-x _o*，y _i，j = y _i，j *- y _o*，z _i，j = z _i，j and the cylinder axis being close to the coordinate systemzThe central planes of the two end surfaces of the measured cylinder are approximately parallel to the XOY plane of the coordinate system;

P _i *={x _i *，y _i *，z _i x is the spatial rectangular coordinate of the predetermined center of each cross-sectional circle after pre-positioning, wherein:x _i *=y _i *=0，z _i *= z _i，j *。

after step 1, step 1.1 is performed.

Step 1.1: in each section, a great face is collected according to the measured segment measuring pointQ _i，j Respectively establishing a characteristic line vector setW _i，j Great, set of boundary elementsb _i，j Great Chinese character and state element sett _i，j }; obtaining the center coordinates of each section circleP _i And using it to form a reference segment measuring point setP _i }; wherein:

W _i，j =（[x _i，j /t _i，j ，y _i，j /t _i，j ]) Is a feature row vector, all feature row vectorsW _i，j Is a set of characteristic line vectorsW _i，j }；

b _i，j =bIs a real number greater than 0, all boundary elementsb _i，j Is a set of boundary elementsb _i，j }；

P _i ={x _i ，y _i ，z _i The space rectangular coordinate of the circle center of each section is used, and initially,x _i = x _i *，y _i = y _i *，z _i = z _i *；

all state elements in the segment under testt _i，j The set of (a) is a state element set of a measured segment measuring pointt _i，j }。

After step 1.1 is finished, step 1.2 is performed.

Step 1.2: in each cross section, the cross section is taken ont _i，j Maximum value oft _maxCorresponding serial numberk ₁Is a key serial number, and willk ₁The key serial number sets respectively added to respective sectionskIn (c) }.

After step 1.2 is finished, step 1.3 is performed.

Step 1.3: according to the key sequence number set of each sectionkRespectively establishing analysis matrixesWAnd analyzing the column vectorsbWherein:

W= […，W _m ^T，…，W _n ^T，…]^Tis aKA matrix of rows and 2 columns of,Kis a critical sequence number setkThe number of the elements in the (C),m，nis a critical sequence number setkThe elements in (1);

b= […，b _m ，…，b _n ，…]^Tis aKA column vector of rows.

After step 1.3, step 1.4 is performed.

Step 1.4: for analysis matrixWAnd an augmented analysis matrixW，b]Rank analysis was performed.

Computingr _W =rank(W)，r _Wb =rank([W，b]) And comparer _W Andr _Wb there are two cases:

the first condition is as follows: if it is notr _W =r _Wb If the optimization is needed to be continued, step 1.5 is executed;

case two: if it is notr _W <r _Wb Attempting to derive a secondary analysis matrixWAnd analyzing the column vectorsbMiddle removed key serial number setkSome element ofkCorresponding rows, resulting in a reduced matrixW _k- And reducing the column vectorb _k- According toW _k- U _k- = b _k- Solved to obtainU _k- =U _k-0Then calculate b _k- =W _k U _k-0(ii) a If the key sequence number setkAll the elements in the Chinese character have been tried, and none of them is obtainedb _k- >bThen the optimization should be ended, jumping to step 1.7; if the critical sequence number set is triedkElements in (b) }kWhen it is obtainedb _k- >bThen the matrix will be reducedW _k- And reducing the column vectorb _k- Respectively as analysis matrixWAnd analyzing the column vectorsbWill elementkMovable key serial number setkAnd jumping to step 1.5; wherein:U _k- =[w _k-,1，w _k-,2]^T，U _k-0= [w _{k- ,01}，w _{k- ,02}]^T。

step 1.5: solving a system of linear equationsWU=bSolution of (2)U=U ₀WhereinU=[U ₁，U ₂]^T，U ₀=[U _,01，U _,02]^T；

After step 1.5, step 1.6 is performed.

Step 1.6: in each section, calculate separatelyu _i，j = W _i，j U ₀Then calculateτ _i = (t _max-t _i，j )/(b-u _i，j ) (ii) a Respectively takeτ _i Minimum value of the part ofτ _minCorresponding serial numberk ₂Is a new key serial number and willk ₂Key serial number set added to each sectionkIn (c) }.

According to the cross sectionτ _minAndU ₀the center coordinates of each cross-sectional circleP _i Is updated toP _i +τ _min∙[U ₁，U ₂，0]^T；

In each section, the center coordinates of the section circle after updating are respectively used as the basisP _i And the coordinates of the measuring points on the sectionQ _i，j Updating state element sett _i，j }，t _maxIs updated tot _i，j Is measured.

And (5) finishing one-time optimization after the step 1.6 is finished, and performing the step 1.3.

Step 1.7: extracting the center coordinates of the section circle of each section which is finally updatedP _i And using it to update reference segment measuring point setP _i }。

After step 1.7, step 2 is performed.

Step 2: obtaining a last updated measurement point set of a reference segmentP _i According toP _i Establishing a characteristic line vector setA _i Great, set of boundary elementsb _i Great, state element sets _i }; extracting measured section measuring point set of step 1Q _i，j According toQ _i，j Re-establishing a set of state elementst’ _i，j }; wherein:

P _i ={x _i ，y _i ，z _i the space rectangular coordinate of the reference segment measuring point is obtained;

A _i =（[-x _i /s _i ，-y _i /s _i ，y _i z _i /s _i ，-x _i z _i /s _i ]) Is a feature row vector, all feature row vectorsA _i Is a characteristic row vectorA _i }；

b _i =bIs a real number greater than 0, all boundary elementsb _i Is a set of boundary elementsb _i }；

All state elements of the reference segments _i The set of (1) is a state element set of a reference segment measuring points _i }；

All state elements in the segment under testt’ _i，j The set of (a) is a state element set of a measured segment measuring pointt’ _i，j }。

After step 2, step 2.1 is performed.

Step 2.1: gets _i Maximum values _maxCorresponding serial numberl ₁Is a key serial number, and willl ₁Last page added to key serial number setlIn (c) }.

After step 2.1 is finished, step 2.2 is performed.

Step 2.2: according to the key sequence numberlEstablishment of an analysis matrixAAnd analyzing the column vectorsb', wherein:

A=[…，A _p ^T，…，A _q ^T，…]^Tis aLA matrix of rows and 4 columns,Lis a critical sequence number setlThe number of the elements in the (C),p，qis a critical sequence number setlThe elements in (1);

b’=[…，b _p ，…，b _q ，…]^Tis aLA column vector of rows.

After step 2.2 is finished, step 2.3 is performed.

Step 2.3: for analysis matrixAAnd the broadening matrix [ alpha ]A，b’]Performing rank analysis;

computingr _A = rank(A)，r _Ab’ = rank([A，b’]) And comparer _A Andr _Ab’there are two cases:

the first condition is as follows: if it is notr _A = r _Ab’If the optimization is needed to be continued, step 2.4 is executed;

case two: if it is notr _A <r _Ab Attempting to derive a secondary analysis matrixAAnd analyzing the column vectorsb' middle removing key serial number setlSome element oflCorresponding rows, resulting in a reduced matrixA _l- And reducing the column vectorb’ _l- According toA _l- Ψ _l- = b’ _l- Solved to obtainΨ _l- =Ψ _l-0Then calculate b’ _l- =A _l Ψ _l-0(ii) a If the key sequence number setlAll the elements in the Chinese character have been tried, and none of them is obtainedb’ _l- >bThen the optimization should be ended, and jump to step 3; if the critical sequence number set is triedlElements in (b) }lWhen it is obtainedb’ _l- >bThen the matrix will be reducedA _l- And reducing the column vectorb’ _l- Respectively as analysis matrixAAnd analyzing the column vectorsb', will elementlMovable key serial number setlAnd jumping to step 2.4; whereinΨ _l- =[v _l-,1，v _l-,2，v _l-,3，v _l-,4]^T，Ψ _l-0=[v _{l- ,01}，v _{l- ,02}，v _{l- ,03}，v _{l- ,04}]^T。

Step 2.4: solving a system of linear equationsAΨ=b' solution ofΨ=Ψ ₀WhereinΨ=[Ψ ₁，Ψ ₂，Ψ ₃，Ψ ₄]^T，Ψ ₀=[Ψ _0,1，Ψ _0,2，Ψ _0,3，Ψ _0,4]^T。

After step 2.4, step 2.5 is performed.

Step 2.5: computingv _i =A _i Ψ ₀Then calculateτ’ _i =(s _max-s _i )/(b-v _i ) (ii) a Getτ’ _i Minimum value of the part ofτ’_minCorresponding serial numberl ₂Is a new key serial number and willl ₂Last page added to key serial number setlIn (1) };

set the reference segment measuring pointP _i Is updated toP _i +τ’_min∙V _i Wherein:

；

according to the updatedP _i Updating state element sets _i }，s _maxIs updated tos _i Is measured.

Set of measured segment measuring pointsQ _i，j Is updated toQ _i，j +τ’_min∙ V’ _i Wherein:

；

according to the updatedQ _i，j Updating state element sett’ _i，j }。

And finishing one-time optimization after the step 2.5 is finished, and performing the step 2.2.

And step 3: obtaining the final state element set of the measured segment measuring pointt ^’ _i，j Comparing the measured points of all the measured segmentst ^’ _i，j And judging whether the measured shaft meets the size requirement or not, wherein the maximum value and the minimum value are the minimum circumscribed cylindrical radius and the maximum inscribed cylindrical radius of the measured shaft.

Obtaining the final state element set of the reference segment measuring points _i Comparing state element sets _i Of all the points ins _i Value to obtains _maxThe straightness error of the axis of the measured shaft is 2s _maxAnd judging whether the axis of the measured shaft meets the linearity tolerance requirement.

In each section, comparing the measured points of each section to be measured on the sectiont’ _i，j Value, calculationt’= t’_max- t’_minThe radial circle run-out error of the section circle is obtained; compare allt' value, where the maximum is the radial run-out error of the cylinder; judging whether the radial circle run-out error of the measured shaft meets the radial circle run-out tolerance requirement; wherein:t’_maxfor measuring point on the cross-sectiont’ _i，j The maximum value of (a) is,t’_minfor measuring points on the cross-sectiont’ _i，j Is measured.

To get a more accurate solution, the following optimization can be done:

in step 2.5, ifτ ^’ _min ∙V _i Of single or several iterationsτ ^’ _min∙ V _i Greater than a given thresholdaThen, according to the updated reference segment measuring point setP _i Jump to step 2 to update feature line vector setA _i Great, set of boundary elementsb _i Great, state element sets _i }。

To facilitate numerical calculation, can makebTaking a specific value greater than 0, but not limited to 1.

The invention has the beneficial effects that:

1. the geometric characteristics of radial circular runout are fully considered, and the evaluation form is simplified, so that the method is easier to popularize than the first type of evaluation method. 2. The geometric characteristics of radial circular runout are fully considered, a better value is obtained through mature linear operation in each iteration, and the minimum radial circular runout error can be finally obtained, so that the algorithm is stable, and the problem of initial value sensitivity of the fifth method does not exist. 3. The fact that most of the measuring points are invalid measuring points in the radial circular runout error evaluation is implied, and the invalid measuring points are not added with iteration, so that the iteration times of the method are fewer and are equivalent to the first type evaluation method and the fifth type evaluation method.

4. When calculating optimizing direction, only considering key sequence number setlThe corresponding measuring point, therefore, each timeThe iterative operation amount is small and is equivalent to the fifth type evaluation method. 5. Because the iteration times are less and the operation amount of each iteration is less, the total operation speed is equivalent to the first type evaluation method and the fifth type evaluation method.

The invention provides a rapid, stable and simple-form radial circular run-out error evaluation method, which can be used for detecting and evaluating the radial circular run-out error of the shaft part with the axis of the shaft part as a reference axis, and provides guidance for the improvement of the machining process of the shaft part, so that the method has industrial possibility.

Drawings

FIG. 1 is a flow chart of the present invention.

FIG. 2 is a tolerance layout of parts in an exemplary embodiment.

Detailed Description

The following are specific embodiments of the present invention, and the embodiments of the present invention will be further described with reference to the drawings, but the present invention is not limited to these embodiments.

The radial run out error of one axis is evaluated and its tolerance design specification is shown in figure 2.

Step 0: obtaining initial measuring point set of measured segmentQ _i，j The following:

after step 0, step 1 is performed.

Step 1: obtaining a predetermined measured segment measuring point setQ _i，j The method comprises the following steps:

obtaining initial measuring point set of pre-positioned reference segmentP _i The following:

after step 1, step 1.1 is performed.

Step 1.1: taking the section 1 as an example, according to the measured segment measuring point set corresponding to each measuring point on the section 1Q _i，j Great atQ _，j1Construction of characteristic line vector setW _，j1The method comprises the following steps:

establishing a set of boundary elementsb _，j1The method comprises the following steps:

{b _，j1}=[1，1，1，1，1，1]^T；

establishing a set of state elementst _，j1The method comprises the following steps:

the center coordinates of the cross-sectional circle in the cross-section 1 are obtained as follows:

P ₁={0，0，0}。

after step 1.1 is finished, step 1.2 is performed.

Step 1.2: in section 1, taket _，j1Maximum value oft _maxCorresponding serial number 1 is a key serial number, and 1 is added to the key serial number set of the section 1kChinese, a large aperturek}={1}。

After step 1.2 is finished, step 1.3 is performed.

Step 1.3: according to the key sequence number set of section 1kEstablishment of an analysis matrixWAnd analyzing the column vectorsbWherein:

W= W ₁=[0.50001，0.86602]^T，

Wis a matrix with 1 row and 2 columns, and a key sequence number setkThe number of elements in the element is 1, and the element is 1;

b=1。

after step 1.3, step 1.4 is performed.

Step 1.4: for analysis matrixWAnd an augmented analysis matrixW，b]Rank analysis was performed.

Computingr _W =rank(W)=1，r _Wb =rank([W，b]) =1, and comparingr _W Andr _Wb . Because of the fact thatr _W =r _Wb So the optimization should continue, step 1.5 is performed.

Step 1.5: solving a system of linear equationsWU=bSolution of (2)U=U ₀=[0.50001，0.86602]^T。

After step 1.5, step 1.6 is performed.

Step 1.6: in section 1, calculateu _，j1= W _，j1 U ₀The results are as follows:

then calculateτ _i = (t _max-t _，j1)/(b-u _，j1) Get itτ _i The results for the portion of the series greater than zero are as follows:

minimum value thereofτ _minThe corresponding serial number 3 is a new key serial number, and 3 is added into the key serial number set of the section 1kAt this time, a retaining openingk}={1，3}。

According to section 1τ _minAndU ₀coordinates of the center of a circle in the cross section 1P ₁Is updated toP ₁+τ _min∙[U ₁，U ₂，0]^TThe results are as follows:

P ₁={0.00128，0.00221，0}。

according to the updated center coordinates of the cross-section circleP ₁And the coordinates of the measuring points on the sectionQ _，j1Updating state element sett _，j1}，t _maxIs updated tot _i，j The results are as follows:

and (5) finishing one-time optimization after the step 1.6 is finished, and performing the step 1.3.

By analogy, after the second optimization, the key sequence number setk}={1，3，5}，P ₁={-0.00112，0.00636}。

At this time, step 1.3 is performed first: according to the key sequence number set of section 1k} = {1, 3, 5} establishing analysis matrixWAnd analyzing the column vectorsbWherein:

b=[1，1，1]^T。

after step 1.3, step 1.4 is performed.

Step 1.4: for analysis matrixWAnd an augmented analysis matrixW，b]Rank analysis was performed.

Computingr _W =rank(W)=2，r _Wb =rank([W，b])=3，r _W < r _Wb . First, an attempt is made to analyze the matrix fromWAnd analyzing the column vectorsbMiddle deleted key serial number setkRow corresponding to the first element 1 in = {1, 3, 5}, resulting in a reduced matrixW _-1：

Andb _1-column vector:

b _1-=[1，1]^T。

according toW _1- U _-1= b _1-Solved to obtainU _-1= U _-10=[-1.00066，-1.73405]^TThen calculate to obtainb _1-=W ₁ U _-10=

-2.00206<1=b。

Similarly, respectively obtainb _3-= -1.99859<1=b，b _5-= -0.00021<1=bNone is obtainedb _k- >bSo the seek should be ended, jumping to step 1.7.

Step 1.7: extracting the center coordinates of the finally updated section circle of the section 1P ₁={-0.00112，0.00636}。

By parity of reasoning, the center coordinates of the finally updated section circles of the other sections are obtainedP _i And using it to update reference segment measuring point setP _i The method comprises the following steps:

after step 1.7, step 2 is performed.

Step 2: extracting the measuring point set of the reference segment finally updated in the above stepP _i According toP _i Establishing state element sets _i The method comprises the following steps:

establishing a feature line vector setA _i The method comprises the following steps:

establishing a set of boundary elementsb _i The method comprises the following steps:

{b _i }=[1，1，1，1，1，1，1，1，1，1]^T。

extracting measured section measuring point set of step 1Q _i，j According toQ _i，j Re-establishing a set of state elementst’ _i，j The method comprises the following steps:

after step 2, step 2.1 is performed.

Step 2.1: gets _i Maximum values _maxThe corresponding serial number 10 is the key serial number, and 10 is added into the key serial number setlChinese, a large aperturel}=10。

After step 2.1 is finished, step 2.2 is performed.

Step 2.2: according to the key sequence numberlEstablishment of an analysis matrixAAnd analyzing the column vectorsb', wherein:

A= A ₁₀=[0.99576，0.09203，-0.82831，8.96180]^T

Ais a matrix with 1 row and 4 columns, and a key sequence number setlThe number of elements in the =10 is 1, and the element is 10;

b’=1。

after step 2.2 is finished, step 2.3 is performed.

Step 2.3: for analysis matrixAAnd the broadening matrix [ alpha ]A，b’]Rank analysis was performed.

Computingr _A = rank(A)=1，r _Ab’= rank([A，b’]) =1, comparisonr _A Andr _Ab’. Because of the fact thatr _A = r _Ab’So the optimization should continue, step 2.4 is performed.

Step 2.4: solving a system of linear equationsAΨ= b' solution ofΨ=Ψ ₀=[0.01214，0.00112，-0.01010，0.10929]^T。

After step 2.4, step 2.5 is performed.

Step 2.5: computingv _i =A _i Ψ ₀The results are as follows:

then calculateτ’ _i =(s _max-s _i )/(b-v _i ) Get itτ’ _i The results for the portion of the series greater than zero are as follows:

minimum value thereofτ’_minThe corresponding serial number 5 is a new key serial number, and 5 is added into the key serial number setlIn (c) }. At this time. {l}={10，5}。

Set the reference segment measuring pointP _i Is updated toP _i +τ’_min∙V _i Wherein:

obtaining updatedP _i The method comprises the following steps:

according to the updatedP _i Updating state element sets _i }，s _maxIs updated tos _i The results are as follows:

set of measured segment measuring pointsQ _i，j Is updated toQ _i，j +τ’_min∙ V’ _i Wherein:

obtaining updatedQ _i，j The method comprises the following steps:

according to the updatedQ _i，j Updating state element sett’ _i，j Results are as follows:

and finishing one-time optimization after the step 2.5 is finished, and performing the step 2.2.

By analogy, after the 4 th optimization, the key sequence number is collectedl}={10，5，8，6，2}。

At this time, step 2.2 is performed first: according to the key sequence numberl} = {10, 5, 8, 6, 2} building analysis matricesAAnd analyzing the column vectorsb', wherein:

b’=[1，1，1，1，1]^T。

step 2.3: for analysis matrixAAnd the broadening matrix [ alpha ]A，b’]Rank analysis was performed.

Computingr _A = rank(A)=4，r _Ab’= rank([A，b’])=5， r _A < r _Ab’. First, an attempt is made to analyze the matrix fromAAnd analyzing the column vectorsb' middle removing key serial number setlRow corresponding to the first element 10 in = {10, 5, 8, 6, 2}, resulting in a reduced matrixA _-10：

And reducing the column vectorb’ _-10：

b’ _-10=[1，1，1，1]^T。

According toA _-10 Ψ _-10= b’ _-10Solved to obtainΨ _-10=Ψ _10-0=[ 19.44577，5.93810，0.66183，-4.50369]^TThen calculate to obtainb’₁₀= A ₁₀ Ψ _10-0= -20.99967<1=b。

Similarly, respectively obtainb’₅= -3.07654<1=b，b’₈= -2.91341<1=b，b’₆= -2.58009<1=b，b’₂= 0.26099<1=bNone is obtainedb’ _l- >bSo the seek should be ended, jumping to step 3.

And step 3: obtaining the final state element set of the measured segment measuring pointt’ _i，j The method comprises the following steps:

comparing measured points of all measured sectionst ^’ _i，j The maximum value 25.02715 is the minimum circumscribed cylindrical radius of the measured shaft, and the minimum value 24.95604 is the maximum inscribed cylindrical radius of the measured shaft, so that the minimum circumscribed cylindrical diameter of the measured shaft and the maximum inscribed cylindrical diameter of the measured shaft are 50.05430 and 49.91208 respectively, and the size requirements are not met from 49.961 to 50.

Obtaining the final state element set of the reference segment measuring points _i The method comprises the following steps:

comparing a set of state elementss _i Of all the points ins _i Value to obtains _max=0.01399，2s _max=0.02798<0.04, the axis of the measured shaft meets the linearity tolerance requirement.

On each section, comparing the measured points of each section to be measured on the sectiont’ _i，j Value, calculationt’ = t’_max- t’_minI.e. the cross-section circleThe radial run-out error of (a) is as follows:

compare allt' value, the maximum value 0.05618 of which is the radial run out error of the measured shaft, 0.05618>0.03, the measured shaft does not meet the radial run-out tolerance requirement.

In the above description, the present invention has been described by way of specific embodiments, but those skilled in the art will appreciate that various modifications and variations can be made within the spirit and scope of the invention as hereinafter claimed.

Claims (5)

1. A method for evaluating radial run-out error of an optical axis is characterized by comprising the following steps:

step 0: obtaining initial measuring point set of measured segmentQ _i，j *}: taking the cylindrical surface of a cylinder as a measured section and the axis of the cylinder as a reference section, and cutting the cylindrical surface at equal intervals along the axis directioniEach cross section being uniformly selected over the circumference of each cross-sectional circlejA measuring point, each section of which is perpendicular to the axis of the cylinder, thei×jInitial measuring point set for a segment to be measured consisting of measuring pointsQ _i，j }; wherein:

i=1, 2, 3, …, N；ithe serial number of the cross section is shown,Nis the total number of the sections;

j=1, 2, 3, …, M；jthe serial numbers of the measuring points on the single cross section,Mthe total number of the measuring points is;

Q _i，j *={x _i，j *，y _i，j *，z _i，j the coordinate is the initial space rectangular coordinate of the measured point of the measured segment;

after the step 0 is finished, performing a step 1;

step 1: the cylinder is pre-positioned as follows: selecting coordinates of measured points of all measured segmentsx _i，j *_maxAndx _i，j *_mincalculating the average valuex _o* =（x _i，j *_max+ x _i，j *_min) Selecting coordinates of measured points of all measured sectionsy _i，j *_maxAndy _i，j *_mincalculating the average valuey _o*=（y _i，j *_max+ y _i，j *_min) 2; obtaining a pre-positioned measured section measuring pointQ _i，j And using it to form a measuring point set of the segment to be measuredQ _i，j }; obtaining the preset center coordinates of each cross-section circle after prepositioningP _i And using the initial measuring point set of reference segmentP _i }; wherein:

Q _i，j ={x _i，j ，y _i，j ，z _i，j and the space rectangular coordinate of the measured point of the measured section after pre-positioning, wherein:x _i，j = x _i，j *- x _o*，y _i，j = y _i，j *- y _o*，z _i，j = z _i，j and the cylinder axis being close to the coordinate systemzThe central planes of the two end surfaces of the measured cylinder are approximately parallel to the XOY plane of the coordinate system;

P _i *={x _i *，y _i *，z _i x is the spatial rectangular coordinate of the predetermined center of each cross-sectional circle after pre-positioning, wherein:x _i *= y _i *=0，z _i *= z _i，j *；

step 1.1 is carried out after step 1 is finished;

step 1.1: in each section, a great face is collected according to the measured segment measuring pointQ _i，j Are constructed respectivelyVertical characteristic line vector setW _i，j Great, set of boundary elementsb _i，j Great Chinese character and state element sett _i，j }; obtaining the center coordinates of each section circleP _i And using it to form a reference segment measuring point setP _i }; wherein:

W _i，j =（[x _i，j /t _i，j ，y _i，j /t _i，j ]) Is a feature row vector, all feature row vectorsW _i，j Is a set of characteristic line vectorsW _i，j }；

b _i，j =bIs a real number greater than 0, all boundary elementsb _i，j Is a set of boundary elementsb _i，j }；

P _i ={x _i ，y _i ，z _i The space rectangular coordinate of the circle center of each section is used, and initially,x _i = x _i *，y _i = y _i *，z _i = z _i *；

all state elements in the segment under testt _i，j The set of (a) is a state element set of a measured segment measuring pointt _i，j }；

Step 1.2 is carried out after step 1.1 is finished;

step 1.2: in each cross section, the cross section is taken ont _i，j Maximum value oft _maxCorresponding serial numberk ₁Is a key serial number, and willk ₁Key to adding to respective cross-sectionSerial number collectionkIn (1) };

step 1.3 is carried out after step 1.2 is finished;

step 1.3: according to the key sequence number set of each sectionkRespectively establishing analysis matrixesWAnd analyzing the column vectorsbWherein:

W= […，W _m ^T，…，W _n ^T，…]^Tis aKA matrix of rows and 2 columns of,Kis a critical sequence number setkThe number of the elements in the (C),m，nis a critical sequence number setkThe elements in (1);

b= […，b _m ，…，b _n ，…]^Tis aKA column vector of rows;

step 1.4 is carried out after step 1.3 is finished;

step 1.4: for analysis matrixWAnd an augmented analysis matrixW，b]Performing rank analysis;

computingr _W =rank(W)，r _Wb =rank([W，b]) And comparer _W Andr _Wb there are two cases:

the first condition is as follows: if it is notr _W =r _Wb If the optimization is needed to be continued, step 1.5 is executed;

case two: if it is notr _W <r _Wb Attempting to derive a secondary analysis matrixWAnd analyzing the column vectorsbMiddle removed key serial number setkSome element ofkCorresponding rows, resulting in a reduced matrixW _k- And reducing the column vectorb _k- According toW _k- U _k- = b _k- Solved to obtainU _k- =U _k-0Then calculateb _k- =W _k U _k-0(ii) a If the key sequence number setkAll the elements in the Chinese character have been tried, and none of them is obtainedb _k- >bThen the optimization should be ended, jumping to step 1.7; if the critical sequence number set is triedkElements in (b) }kWhen it is obtainedb _k- >bThen the matrix will be reducedW _k- And reducing the column vectorb _k- Respectively as analysis matrixWAnd analyzing the column vectorsbWill elementkMovable key serial number setkAnd jumping to step 1.5; wherein:U _k- =[w _k-,1，w _k-,2]^T，U _k-0= [w _{k- ,01}，w _{k- ,02}]^T;

step 1.5: solving a system of linear equationsWU=bSolution of (2)U=U ₀WhereinU=[U ₁，U ₂]^T，U ₀=[U _,01，U _,02]^T；

After step 1.5 is finished, step 1.6 is carried out;

step 1.6: in each section, calculate separatelyu _i，j = W _i，j U ₀Then calculateτ _i = (t _max-t _i，j )/(b-u _i，j ) (ii) a Respectively takeτ _i Minimum value of the part ofτ _minCorresponding serial numberk ₂Is a new key serial number and willk ₂Key serial number set added to each sectionkIn (1) };

according to the cross sectionτ _minAndU ₀the center coordinates of each cross-sectional circleP _i Is updated toP _i +τ _min∙[U ₁，U ₂，0]^T；

In each section, the center coordinates of the section circle after updating are respectively used as the basisP _i And the coordinates of the measuring points on the sectionQ _i，j Updating state element sett _i，j }，t _maxIs updated tot _i，j Maximum value of (d);

finishing one-time optimization after the step 1.6 is finished, and performing the step 1.3;

step 1.7: extracting the center coordinates of the section circle of each section which is finally updatedP _i And using it to update reference segment measuring point setP _i }；

Step 2 is carried out after step 1.7 is finished;

step 2: obtaining a last updated measurement point set of a reference segmentP _i According toP _i Establishing a characteristic line vector setA _i Great, set of boundary elementsb _i Great, state element sets _i }; extracting measured section measuring point set of step 1Q _i，j According toQ _i，j Re-establishing a set of state elementst’ _i，j }; wherein:

P _i ={x _i ，y _i ，z _i the space rectangular coordinate of the reference segment measuring point is obtained;

A _i =（[-x _i /s _i ，-y _i /s _i ，y _i z _i /s _i ，-x _i z _i /s _i ]) Is a feature row vector, all feature row vectorsA _i Is a characteristic row vectorA _i }；

b _i =bIs a real number greater than 0, all boundary elementsb _i Is a set of boundary elementsb _i }；

All state elements of the reference segments _i The set of (1) is a state element set of a reference segment measuring points _i }；

All state elements in the segment under testt’ _i，j The set of (a) is a state element set of a measured segment measuring pointt’ _i，j }；

Step 2.1 is carried out after step 2 is finished;

step 2.1: gets _i Maximum values _maxCorresponding serial numberl ₁Is a key serial number, and willl ₁Last page added to key serial number setlIn (1) };

step 2.2 is carried out after step 2.1 is finished;

step 2.2: according to the key sequence numberlEstablishment of an analysis matrixAAnd analyzing the column vectorsb', wherein:

A=[…，A _p ^T，…，A _q ^T，…]^Tis aLA matrix of rows and 4 columns,Lis a critical sequence number setlThe number of the elements in the (C),p，qis a critical sequence number setlThe elements in (1);

b’=[…，b _p ，…，b _q ，…]^Tis aLA column vector of rows;

step 2.3 is carried out after step 2.2 is finished;

step 2.3: for analysis matrixAAnd the broadening matrix [ alpha ]A，b’]Performing rank analysis;

computingr _A = rank(A)，r _Ab’ = rank([A，b’]) And comparer _A Andr _Ab’is divided intoThe following two cases:

the first condition is as follows: if it is notr _A = r _Ab’If the optimization is needed to be continued, step 2.4 is executed;

case two: if it is notr _A <r _Ab Attempting to derive a secondary analysis matrixAAnd analyzing the column vectorsb' middle removing key serial number setlSome element oflCorresponding rows, resulting in a reduced matrixA _l- And reducing the column vectorb’ _l- According toA _l- Ψ _l- = b’ _l- Solved to obtainΨ _l- =Ψ _l-0Then calculate b’ _l- =A _l Ψ _l-0(ii) a If the key sequence number setlAll the elements in the Chinese character have been tried, and none of them is obtainedb’ _l- >bThen the optimization should be ended, and jump to step 3; if the critical sequence number set is triedlElements in (b) }lWhen it is obtainedb’ _l- >bThen the matrix will be reducedA _l- And reducing the column vectorb’ _l- Respectively as analysis matrixAAnd analyzing the column vectorsb', will elementlMovable key serial number setlAnd jumping to step 2.4; whereinΨ _l- =[v _l-,1，v _l-,2，v _l-,3，v _l-,4]^T，Ψ _l-0=[v _{l- ,01}，v _{l- ,02}，v _{l- ,03}，v _{l- ,04}]^T；

Step 2.4: solving a system of linear equationsAΨ=b' solution ofΨ=Ψ ₀WhereinΨ=[Ψ ₁，Ψ ₂，Ψ ₃，Ψ ₄]^T，Ψ ₀=[Ψ _0,1，Ψ _0,2，Ψ _0,3，Ψ _0,4]^T；

Step 2.5 is carried out after step 2.4 is finished;

step 2.5: computingv _i =A _i Ψ ₀Then calculateτ’ _i =(s _max-s _i )/(b-v _i ) (ii) a Getτ’ _i Minimum value of the part ofτ’_minCorresponding serial numberl ₂Is a new key serial number and willl ₂Last page added to key serial number setlIn (1) };

set the reference segment measuring pointP _i Is updated toP _i +τ’_min∙V _i Wherein:

；

according to the updatedP _i Updating state element sets _i }，s _maxIs updated tos _i Maximum value of (d);

set of measured segment measuring pointsQ _i，j Is updated toQ _i，j +τ’_min∙ V’ _i Wherein:

；

according to the updatedQ _i，j Updating state element sett’ _i，j }；

Finishing one-time optimization after the step 2.5 is finished, and performing the step 2.2;

and step 3: obtaining the final state element set of the measured segment measuring pointt ^’ _i，j Comparing the measured points of all the measured segmentst ^’ _i，j The maximum value and the minimum value are the minimum circumscribed cylinder radius and the maximum inscribed cylinder radius of the measured shaft, and whether the measured shaft meets the size requirement is judged;

obtaining the final state element set of the reference segment measuring points _i Comparing state element sets _i Of all the points ins _i Value to obtains _maxThe straightness error of the axis of the measured shaft is 2s _maxJudging whether the axis of the measured shaft meets the requirement of straightness tolerance;

in each section, comparing the measured points of each section to be measured on the sectiont’ _i，j Value, calculationt’= t’_max- t’_minThe radial circle run-out error of the section circle is obtained; compare allt' value, where the maximum is the radial run-out error of the cylinder; judging whether the radial circle run-out error of the measured shaft meets the radial circle run-out tolerance requirement; wherein:t’_maxfor measuring point on the cross-sectiont’ _i，j The maximum value of (a) is,t’_minfor measuring points on the cross-sectiont’ _i，j Is measured.

2. A method of assessing radial run out error of an optical axis as claimed in claim 2, wherein said coordinate transformation is shifted by an average of the coordinates.

3. A method of assessing optical axis run-out errors as claimed in claim 1, wherein in step 2.5, if soτ ^’ _min ∙V _i Of single or several iterationsτ ^’ _min∙ V _i Greater than a given thresholdaThen, according to the updated reference segment measuring point setP _i Jump to step 2 to update feature line vector setA _i Great, set of boundary elementsb _i State elementCollection checks _i }。

4. A method of assessing radial run-out errors of an optical axis as claimed in claim 1,b =1。

5. the method is suitable for evaluating the radial circular run-out error of the shaft part which has higher requirements on the processing precision of the shaft such as a linear optical axis in a linear cylindrical guide rail and takes the self axis as the reference, and can simultaneously evaluate the straightness error of the axis of the shaft part which takes the self axis as the reference.

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