阅读说明：本技术 一种基于数字化量规的任意方向上直线度评定方法 (Method for evaluating straightness in any direction based on digital gauge ) 是由 刘廷伟 黄美发 唐哲敏 于 2019-07-10 设计创作，主要内容包括：本发明属于精密计量与计算机应用领域,将实体量规以数学模型的方式来评定零件,具有涉及一种快速、稳定、形式简单,数字化的任意方向上直线度评定方法,步骤：1：获取测点集,并根据测点集建立边界元素集和状态元素集；2：根据测点集以及相应的法向量建立特征行向量集；3：加入一个关键点；4：建立分析矩阵和分析列向量；5：对分析矩阵及增广分析矩阵进行秩分析,以确定继续寻优、剔除关键点还是终止程序；6：计算寻优方向；7：以追及问题求解新的关键点,进行优化步骤计算,再按步骤1中的公式更新,进入下一次循环,或者按步骤4进入下一次循环；8：终止程序并得到最优值。(The invention belongs to the field of precision measurement and computer application, and relates to a method for evaluating the straightness of an entity gauge in any direction, which is rapid, stable, simple in form and digitalized, wherein the method comprises the following steps: 1: acquiring a measuring point set, and establishing a boundary element set and a state element set according to the measuring point set; 2: establishing a characteristic line vector set according to the measuring point set and the corresponding normal vector; 3: adding a key point; 4: establishing an analysis matrix and an analysis column vector; 5: performing rank analysis on the analysis matrix and the augmented analysis matrix to determine whether to continue optimizing, eliminating key points or terminating the program; 6: calculating an optimizing direction; 7: solving new key points by using the pursuit problem, calculating the optimization step, updating according to the formula in the step 1, and entering the next cycle, or entering the next cycle according to the step 4; 8: the procedure is terminated and the optimum value is obtained.)

1. A method for evaluating the straightness in any direction based on a digital gauge is realized by the following steps:

step 1: placing the part to be measured on a measuring platform, measuring in a coordinate system of a measuring machine and obtaining a measuring pointQ _i ^* ={X _i , Y _i ，Z _i Great, collect the measurementQ _i ^* And obtaining a z axis with the position of the measuring point as the minimum area line and close to the coordinate system through conventional coordinate transformation, wherein the central planes at two ends of the measuring point are close to the xoy plane of the coordinate system. Calculating according to a conventional fitting circle to obtain the center of each circle and obtain the coordinate value of a center measuring point passing through prepositioningQ _i ={x _i , y _i , z _i GreatQ _i }. According toMeasuring point setQ _i Establishing a boundary element setb _i Great Chinese character and state element sett _i }; wherein:i=1, 2, 3, …, N；ithe serial numbers of the measuring points are shown,Nthe total number of the measuring points is;

all state elementst _i Is a set of state elementst _i }；

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

After the step 1 is finished, performing a step 2;

step 2:Q _i ={x _i , y _i , z _i is the measurement pointiIs expressed in a space rectangular coordinate of (a), the envelope boundary is expressed with the Z-axis as an axis,t _maxForming a cylindrical surface for an initial radius, the collection of all measuring pointsQ _i The envelope boundary is positioned in the envelope boundary, and the envelope boundary contracts towards the Z axis at an assumed speed to enable the measuring point to adjust the position until the measuring point is in the minimum area;

measuring point setQ _i The method is regarded as a rigid body, the measuring point set is pushed to slightly translate and rotate when the envelope boundary shrinks, and the normal vector of each measuring point isN _i =[x _i , y _i ]^TAnd projecting the motion speed of the measuring point to the normal vector. According to each measuring pointQ _i And establishing characteristic row vectors by using corresponding normal vectorsA _i Namely: A _i =([-x _i , -y _i , y _i z _i , -x _i z _i ])/t _i all characteristic line vectorsA _i Is a set of characteristic line vectorsA _i }；

Step 3 is carried out after step 2 is finished;

and step 3: gett _i Maximum valuet _maxCorresponding measuring pointQ _m Is a key point and the serial number of the measuring point ismJoined to a set of key pointsmIn (1) };

step 4 is carried out after step 3 is finished;

and 4, step 4: according to the set of key pointsmEstablishment of an analysis matrixAAnd analyzing the column vectorsbWherein:

A=[…, A _j ^T, …, A _k ^T, …]^Tis aLA matrix of rows and 4 columns,Lis a set of key pointsmThe number of the elements in the (C),j, kis a set of key pointsmThe elements in (1);

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

step 5 is carried out after step 4 is finished;

and 5: for analysis matrixAAnd an augmented analysis matrixA, b]Performing rank analysis;

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

the first condition is as follows: if the judgment criterion is:r _A =r _Ab then, the optimization should be continued, jumping to step 6;

case two: if the judgment criterion is:r _A < r _Ab then, an attempt is made to determine from the analysis matrixAAnd analyzing the column vectorsbMiddle deleted key point setmOne of the elementsmCorresponding rows, obtaining a reduced matrixA _m- And reducing the column vectorb _m- Calculating a matrixA _m- The rank of (c) is determined,r _{Am_} =rank(A _{m_} ) Extension matrix [ alpha ]A _{m_} , b _{m_} ]Rank ofr _{Am_bm_} =rank([A _{m_} , b _{m_} ]) Judgment ofr _{Am_} Andr _{Am_bm_} whether they are equal; in thatr _{Am_} Andr _{Am_bm_} if they are equal, then solve the linear equationA _m- v _m- = b _l- Solution of (2)v _m- =v _m-0 Then calculateb _m- =A _m v _m-0 (ii) a If the key point set is triedlElements in (b) }lWhen it is obtainedb _m- >bThen, the matrix will be reducedA _m- And reducing the column vectorb _m- Are respectively updated toAMatrix and analysis column vectorbWill elementmMoving out key point setmAnd jumps to step 6, where,v _m- =[v _m-,1, v _m-,2, v _m-,3, v _m-,4]^T，v _m-0 =[v _m-0,1, v _m-0,2, v _m-0,3, v _m-0,4]^T. Key point setmEach of themmThe elements have tried, eachmElement is inr _{Am_} Andr _{Am_bm_} in case of inequality, that is to say none is obtainedb _m- >bThen, the optimization should be ended, jumping to step 8;

step 6: solving linear equationsAv= bSolution of (2)v=v ₀ Wherein, in the step (A),v=[v ₁, v ₂, v ₃, v ₄]^T，v ₀ =[v _0,1, v _0,2, v _0,3, v _0,4]^T；

step 7 is carried out after step 6 is finished;

and 7: computingv _i =A _i v ₀ Then calculateτ _i =(t _max– t _i )÷(b - v _i ). Getτ _i Minimum value in the part of greater than zeroτ _minCorresponding measuring pointQ _m Is a new key point and the measured point is numberedmJoined to a set of key pointsmAll will bet _i Is updated tot _i –τ _min∙ v _i ，t _maxIs updated tot _i And updating the feature row vector set according to the formulas in step 1 and step 2A _i Great, boundary element setb _i Great Chinese character and state element sett _i }；

Finishing one-time optimization after the step 7 is finished, and performing the step 4;

and 8: computingt=2 t _max Is the error value of straightness in any direction.

2. The method of claim 1, wherein the linear degree in any direction is evaluated based on a digital gauge, wherein the linear degree is assumed to be linearb=1。

3. A method for assessing straightness in any direction based on a digitized gauge as claimed in claim 1, wherein for ease of numerical calculations the measurement set is pre-positioned by: firstly, moving according to the average value of the coordinates, or secondly, moving according to the extreme value of the coordinates, or thirdly, moving according to the root mean square minimum principle of the coordinates.

4. A method for assessing the linearity of any direction based on a digital gauge as claimed in claim 1, wherein the following optimization is performed for obtaining a more accurate solution: in step 7, in order to control the amount of movement displacement of the measuring point set,assume a threshold valueqIf, ifτ _min∙ v _i Single order value ofτ _min∙v _i Or accumulated values sigma of several iterationsτ _min∙v _i Greater than a given thresholdqThen, willτ _min∙v _i Single value of (S) or accumulated value (sigma) of past iterationsτ _min∙ v _i Is equal to a threshold valueqCollecting measuring pointsQ _i Is updated toQ _i + τ _min∙vOrQ _i +∑τ _min∙v。

5. The method for assessing the linearity of any direction based on the digital gauge as claimed in claim 1, wherein the method can be used for detecting and assessing the qualification of the linearity of the axis of the stepped shaft of the winding machine equipment.

Technical Field

The invention belongs to the field of precision metering and computer application, and relates to a rapid, stable, simple-form and digital method for evaluating straightness in any direction, which can be used for detecting and evaluating the axial straightness qualification of a stepped shaft of a winch device and provides guidance for the improvement of a machining process of the stepped shaft.

Background

The shaft and hole parts are common geometric elements in mechanical parts, the precision of the shaft and hole parts has important influence on the quality, performance and service life of products, and the straightness is a main technical index of the shaft and hole parts. According to the definition and the judgment method of the straightness in any direction given in the national standard GB/T11336-2004 and ISO standard, the straightness error evaluation of the part in any direction needs to meet the minimum region condition to obtain a better evaluation verification result. In view of the widespread application of computer digitization technology and the rapid development of digital measurement in current social production, the way of combining the physical gauge with measurement data to process parts in a mathematical model is undoubtedly superior to the traditional physical gauge detection. The loss of the gauge caused by use is avoided, and the geometric parameters of the gauge can be changed according to the detection requirements of different parts due to the fact that the gauge exists in a virtual model mode. Only the limitation of qualification can be detected relative to a physical gauge, and a digital gauge can also give an exact error value in detection.

At present, the evaluation of straightness errors of a digital gauge used in a given plane is a research hotspot of academia, and two main problems exist in the realization of the evaluation. One of the methods is how to establish an evaluation method model to optimally match the spatial position of the measuring point with an ideal straight line, so that the measuring point is in a minimum area taking the ideal straight line as an axis. At present, scholars at home and abroad propose corresponding optimization evaluation methods, such as: and carrying out minimum region search by using algorithms such as a genetic algorithm, a particle swarm algorithm, a containment region iterative search algorithm, a longicorn algorithm and the like. The models of the methods are complex in form and random in the calculation process, the stability of the result is ensured by the corresponding methods, and when the number of the measuring points is large, the calculation efficiency is not high so that the production requirement cannot be met.

In summary, a method for evaluating the straightness in any direction, which is rapid, stable, simple in form and digital, is still lacking at present.

Disclosure of Invention

The purpose of the invention is:

aiming at the problems in the prior art, the invention provides a quick, stable, simple-form and digital method for evaluating the straightness in any direction, which can be used for detecting and evaluating the axial straightness qualification of the stepped shaft of the winch equipment and provides guidance for the improvement of the processing technology of the stepped shaft.

The scheme adopted by the invention is as follows:

a method for evaluating the straightness in any direction based on a digital gauge is realized by the following steps:

step 1: placing the part to be measured on a measuring platform, measuring in a coordinate system of a measuring machine and obtaining a measuring pointQ _i ^* ={X _i , Y _i ，Z _i Great, collect the measurementQ _i ^* And obtaining a z axis with the position of the measuring point as the minimum area line and close to the coordinate system through conventional coordinate transformation, wherein the central planes at two ends of the measuring point are close to the xoy plane of the coordinate system. Calculating according to a conventional fitting circle to obtain the center of each circle and obtain the coordinate value of a center measuring point passing through prepositioningQ _i ={x _i , y _i , z _i GreatQ _i }. According to a set of measuring pointsQ _i Establishing a boundary element setb _i Great Chinese character and state element sett _i }; wherein:i=1, 2, 3, …, N；ithe serial numbers of the measuring points are shown,Nthe total number of the measuring points is;

all state elementst _i Is a set of state elementst _i }；

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

After the step 1 is finished, performing a step 2;

step 2:Q _i ={x _i , y _i , z _i is the measurement pointiIs expressed in a space rectangular coordinate of (a), the envelope boundary is expressed with the Z-axis as an axis,t _maxForming a cylindrical surface for an initial radius, the collection of all measuring pointsQ _i The envelope boundary is positioned in the envelope boundary, and the envelope boundary contracts towards the Z axis at an assumed speed to enable the measuring point to adjust the position until the measuring point is in the minimum area;

measuring point setQ _i The method is regarded as a rigid body, the measuring point set is pushed to slightly translate and rotate when the envelope boundary shrinks, and the normal vector of each measuring point isN _i =[x _i , y _i ]^TAnd projecting the motion speed of the measuring point to the normal vector. According to each measuring pointQ _i And establishing characteristic row vectors by using corresponding normal vectorsA _i Namely: A _i =([-x _i , -y _i , y _i z _i , -x _i z _i ])/t _i all characteristic line vectorsA _i Is a set of characteristic line vectorsA _i }；

Step 3 is carried out after step 2 is finished;

and step 3: gett _i Maximum valuet _maxCorresponding measuring pointQ _m Is a key point and the serial number of the measuring point ismJoined to a set of key pointsmIn (1) };

step 4 is carried out after step 3 is finished;

and 4, step 4: according to the set of key pointsmEstablishment of an analysis matrixAAnd analyzing the column vectorsbWherein:

A=[…, A _j ^T, …, A _k ^T, …]^Tis aLA matrix of rows and 4 columns,Lis a set of key pointsmThe number of the elements in the (C),j, kis a set of key pointsmThe elements in (1);

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

step 5 is carried out after step 4 is finished;

and 5: for analysis matrixAAnd an augmented analysis matrixA, b]Performing rank analysis;

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

the first condition is as follows: if the judgment criterion is:r _A =r _Ab then, the optimization should be continued, jumping to step 6;

case two: if the judgment criterion is:r _A < r _Ab then, an attempt is made to determine from the analysis matrixAAnd analyzing the column vectorsbMiddle deleted key point setmOne of the elementsmCorresponding rows, obtaining a reduced matrixA _m- And reducing the column vectorb _m- Calculating a matrixA _m- The rank of (c) is determined,r _{Am_} =rank(A _{m_} ) Extension matrix [ alpha ]A _{m_} , b _{m_} ]Rank ofr _{Am_bm_} =rank([A _{m_} , b _{m_} ]) Judgment ofr _{Am_} Andr _{Am_bm_} whether they are equal; in thatr _{Am_} Andr _{Am_bm_} if they are equal, then solve the linear equationA _m- v _m- = b _l- Solution of (2)v _m- =v _m-0 Then calculateb _m- =A _m v _m-0 (ii) a If the key point set is triedlElements in (b) }lWhen it is obtainedb _m- >bThen, the matrix will be reducedA _m- And reducing the column vectorb _m- Are respectively updated toAMatrix and analysis column vectorbWill elementmMoving out key point setmAnd jumps to step 6, where,v _m- =[v _m-,1, v _m-,2, v _m-,3, v _m-,4]^T，v _m-0 =[v _m-0,1, v _m-0,2, v _m-0,3, v _m-0,4]^T. Key point setmEach of themmThe elements have tried, eachmElement is inr _{Am_} Andr _{Am_bm_} in case of inequality, that is to say none is obtainedb _m- >bThen, the optimization should be ended, jumping to step 8;

step 6: solving linear equationsAv= bSolution of (2)v=v ₀ Wherein, in the step (A),v=[v ₁, v ₂, v ₃, v ₄]^T，v ₀ =[v _0,1, v _0,2, v _0,3, v _0,4]^T；

step 7 is carried out after step 6 is finished;

and 7: computingv _i =A _i v ₀ Then calculateτ _i =(t _max– t _i )÷(b - v _i ). Getτ _i Minimum value in the part of greater than zeroτ _minCorresponding measuring pointQ _m Is a new key point and the measured point is numberedmJoined to a set of key pointsmAll will bet _i Is updated tot _i –τ _min∙ v _i ，t _maxIs updated tot _i And updating the feature row vector set according to the formulas in step 1 and step 2A _i Great, boundary element setb _i Great Chinese character and state element sett _i }；

Finishing one-time optimization after the step 7 is finished, and performing the step 4;

and 8: computingt=2 t _max The straightness error value in any direction is obtained;

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

To facilitate numerical calculations, the measurement set is pre-positioned as: firstly, moving according to the average value of the coordinates, or secondly, moving according to the extreme value of the coordinates, or thirdly, moving according to the root mean square minimum principle of the coordinates.

To get a more accurate solution, the following optimization can be done: in step 7, a threshold value is assumed for controlling the amount of movement of the measurement point setqIf, ifτ _min∙ v _i Single order value ofτ _min∙v _i Or accumulated values sigma of several iterationsτ _min∙v _i Greater than a given thresholdqThen, willτ _min∙ v _i Single value of (S) or accumulated value (sigma) of past iterationsτ _min∙ v _i Is equal to a threshold valueqCollecting measuring pointsQ _i Is updated toQ _i + τ _min∙vOrQ _i +∑τ _min∙v。

A method for quickly and stably evaluating the linearity in any direction in a simple and digitalized manner features that the measuring points are used for measuringQ _i ^* ={X _i , Y _i ，Z _i Obtained from the step shaft of the winch equipment.

The invention has the beneficial effects that:

1. the geometric characteristics of straightness in any direction are fully considered, the evaluation form is simplified, and the evaluation method is easier to popularize. 2. The geometric characteristics of straightness in any direction are fully considered, and a better value is obtained through mature linear operation in each iterationAnd finally the minimum straightness error is obtained, the motion displacement of the measuring point set is controlled, and a threshold value is assumedqTherefore, the algorithm is stable. 3. The fact that most measuring points are invalid measuring points in the straightness evaluation in any direction is implicit, the invalid measuring points are not added into iteration, and therefore the iteration number is small. And only the keypoint set is considered for the last page in the calculation of the optimizing directionmThe corresponding measuring points are calculated, so that the calculation amount of each iteration is small, and the algorithm is quick;

the invention provides a quick, stable, simple-form and digital method for evaluating the straightness in any direction, which can be used for detecting and evaluating the straightness errors of the axes of shaft and hole parts and provides guidance for the improvement of the processing technology of the shaft and hole parts, thereby having 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 aspects of the present invention will be further described with reference to the drawings, but the present invention is not limited to these embodiments;

evaluation test setQ _i Straightness error in any direction.

Step 1: calculating to obtain measuring point setQ _i The method comprises the following steps:

establishing a set of state elementst _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 1 1 1 1 1 1]^T。

after step 1, step 2 is performed.

Step 2: establishing a feature line vector setA _i The method comprises the following steps:

after step 2, step 3 is performed.

And step 3: gett ₁₆ =0.0531 maximum valuet _maxCorresponding measuring pointQ ₁₆Is a key point, and adds the measuring point serial number 16 to a key point setmIn is made-m}={16}。

After step 3, step 4 is performed.

And 4, step 4: according to the set of key pointsmEstablishment of an analysis matrixAAnd analyzing the column vectorsbWherein:

A=[ A ₁₆ ^T]^T=[-0.68572 -0.72787 116.4585 -109.71516]is a matrix with 1 row and 4 columns;

b=[1]^Tand is a column vector of 1 row.

After step 4, step 5 is performed.

And 5: for analysis matrixAAnd an augmented analysis matrixA, b]Rank analysis was performed. Computingr _A =rank(A) =1，r _Ab =rank（[A, b]) =1, and comparingr _A Andr _Ab . Because of the fact thatr _A =r _Ab Then, the optimization should be continued, jumping to step 6;

step 6: solving linear equationsAv= bSolution of (2)v=v ₀ Wherein, in the step (A),v ₀ =[0，0，0.0045，-0.0043]^T。

after step 6, step 7 is performed.

And 7: byv _i =A _i v ₀ Calculating the measured point andrelative speed of its corresponding point on the gaugeb - v _i The following are:

then calculateτ _i =(t _max – t _i )÷(b - v _i ). Getτ _i Minimum value in the part of greater than zeroτ _minThe corresponding measuring point 6 is a new key point, and the measuring point serial number 6 is added into the key point setmIn is made-m} = {6,16 }; and updating the boundary element set according to the formula in step 1b _i Great Chinese character and state element sett _i }；

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 1 1 1 1 1 1]^T；

establishing a set of state elementst _i The method comprises the following steps:

updating characteristic line vector set according to formula in step 2A _i As follows:

and (4) jumping to the step 4 after the step 2 is finished.

And 4, step 4: according to the set of key pointsmEstablishment of an analysis matrixAAnd analyzing the column vectorsbWherein:

is a matrix with 2 rows and 4 columns;

b=[1 1]^Tand is a 2-row column vector.

After step 4, step 5 is performed.

And 5: for analysis matrixAAnd an augmented analysis matrixA, b]Rank analysis was performed. Computingr _A =rank(A) =2，r _Ab =rank（[A, b]) =2, and comparingr _A Andr _Ab . Because of the fact thatr _A =r _Ab Then, the optimization should be continued, jumping to step 6;

step 6: solving linear equationsAv= bSolution of (2)v=v ₀ Wherein, in the step (A),v ₀ =[0.0001，0.0002，0.0153，-0.0071]^T。

after step 6, step 7 is performed.

And 7: byv _i =A _i v ₀ Calculating the relative speed of the measuring point and the corresponding point on the gaugeb - v _i The following are:

then calculateτ _i =(t _max – t _i )÷(b - v _i ). Getτ _i Minimum value in the part of greater than zeroτ _minThe corresponding measuring point 6 is a new key point, and the measuring point serial number 6 is added into the key point setmIn is made-m} = {6,16 }; and updating the boundary element set according to the formulas in step 1 and step 2b _i Great Chinese character and state element sett _i Great, a feature line vector setA _i }；

And so on, iterating for the 16 th time to make the lastm} = {1,10,11,12,16 }; and updating the boundary element set according to the formula in step 1b _i Great Chinese character and state element sett _i }：

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 1 1 1 1 1 1]^T；

establishing a set of state elementst _i The method comprises the following steps:

jumping to the step 4 after the step 1 is finished;

and 4, step 4: according to the set of key pointsmEstablishment of an analysis matrixAAnd analyzing the column vectorsbWherein:

is a matrix with 2 rows and 4 columns;

b=[1 1]^Tand is a 2-row column vector.

After step 4, step 5 is performed.

And 5: for analysis matrixAAnd an augmented analysis matrixA, b]Rank analysis was performed. Computingr _A =rank(A) =2，r _Ab =rank（[A, b]) =2, and comparingr _A Andr _Ab . Because of the fact thatr _A =r _Ab Then, the optimization should be continued, jumping to step 6;

and (4) jumping to the step 4 after the step 2 is finished.

And 4, step 4: according to the set of key pointsmEstablishment of an analysis matrixAAnd analyzing the column vectorsbWherein:

is a matrix with 5 rows and 4 columns;

b=[1 1 1 1 1]^Tis 5 linesA column vector.

After step 4, step 5 is performed.

And 5: for analysis matrixAAnd an augmented analysis matrixA, b]Rank analysis was performed. Computingr _A =rank(A) =4，r _Ab =rank（[A, b]) =5, and comparingr _A Andr _Ab . Because of the fact thatr _A <r _Ab Attempting to derive a secondary analysis matrixAAnd analyzing the column vectorsbMiddle deleted key point setmA row corresponding to one element 1,10,11,12 or 16 in (i);

firstly, deleting the row corresponding to the 1 st element in the key point to obtain a reduced matrixA ₁And reducing the column vectorb ₁Calculating a matrixA ₁The rank of (c) is determined, r _A1=rank(A ₁) =4, augmentation matrix [, ]A ₁, b ₁]Rank ofr _A1b1=rank([A ₁, b ₁]) =4 becauser _A1=r _A1b1 ，Then solve the linear equationA ₁ v ₁= b ₁Solution of (2)v ₁=v ₁₀Then calculateb ₁=A ₁ v ₂₀= -10.095 becauseb ₁<bSo key point 1 is not a virtual contact;

then, deleting the row corresponding to the 10 th element in the key point to obtain a reduced matrixA ₁₀And reducing the column vectorb ₁₀Calculating a matrixA ₁₀The rank of (c) is determined, r _A10=rank(A ₁₀) =4, augmentation matrix [, ]A _10, b ₁₀]Rank ofr _A10b10=rank([A ₁₀, b ₁₀]) =4 becauser _A10=r _A10b10 ，Then solve the linear equationA ₁₀ v ₄= b ₁₀Solution of (2)v ₁₀=v ₁₀₀Then calculateb ₁₀=A ₁₀ v ₁₀₀= -1.5445 becauseb ₁₀<bSo the key point 10 is not a virtual contact;

then, deleting the row corresponding to the 11 th element in the key point to obtain a reduced matrixA ₁₁And reducing the column vectorb ₁₁Calculating a matrixA ₁₁The rank of (c) is determined, r _A11=rank(A ₁₁) =4, augmentation matrix [, ]A _11, b ₁₁]Rank r of_A11b11=rank([A ₁₁, b ₁₁]) =4 becauser _A11=r _A11b11 ，Then solve the linear equationA ₁₁ v ₁₁= b ₁₁Solution of (2)v ₁₁=v ₁₁₀Then calculateb ₁₁=A ₁₁ v ₁₁₀= -24.6985 becauseb ₁₁<bSo the key point 11 is not a virtual contact point;

then, deleting the row corresponding to the 12 th element in the key point to obtain a reduced matrixA ₁₂And reducing the column vectorb ₁₂Calculating a matrixA ₁₂The rank of (c) is determined, r _A12=rank(A ₁₂) =4, augmentation matrix [, ]A _12, b ₁₂]Rank r of_A12b12=rank([A ₁₂, b ₁₂]) =4 becauser _A12=r _A12b12 ，Then solve the linear equationA ₁₂ v ₁₂= b ₁₂Solution of (2)v ₁₂=v ₁₂₀Then calculateb ₁₂=A ₁₂ v ₁₂₀= -1.3294 becauseb ₁₂<bSo the keypoint 112 is not a virtual contact;

then, deleting the row corresponding to the 16 th element in the key point to obtain a reduced matrixA ₁₆And reducing the column vectorb ₁₆Calculating a matrixA ₁₆The rank of (c) is determined, r _A16=rank(A ₁₆) =4, augmentation matrix [, ]A _16, b ₁₆]Rank r of_A16b16=rank([A ₁₆, b ₁₆]) =4 becauser _A16=r _A16b16 ，Then solve the linear equationA ₁₆ v ₁₆= b ₁₆Solution of (2)v ₁₆=v ₁₆₀Then calculateb ₁₆=A ₁₆ v ₁₆₀= -19.5520 becauseb ₁₆<bSo the keypoint 16 is not a virtual contact and the optimization is finished.

And jumping to the step 8 after the step 5 is finished.

And 8: computingt=2*0.0457=0.0914 is the straightness error value in any direction;

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 in the spirit and scope of the invention without departing from the scope of the claims;

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.