CN108267106B - A fast, stable and simple cylindricity error evaluation method - Google Patents

Abstract

The invention belongs to the field of precision measurement and computer application, and relates to a stable, fast and simple-form cylindricity error evaluation method. , boundary element set and state element set; Step 2: Take the measurement point corresponding to the minimum value of the state element set as the key point, and add its measurement point number to the key point set; Step 3: Establish an analysis matrix and analysis based on the key point set Column vector; Step 4: Perform rank analysis on the analysis matrix and the augmented analysis matrix to determine whether to continue the optimization, eliminate key points or terminate the program and obtain the optimal value; Step 5: Solve the analysis matrix and analyze the column vector to obtain the optimization direction ; Step 6: Solve new key points with the tracking problem, update the state set of measuring points, and enter the next cycle; Step 7, terminate the program and obtain the optimal value.

Description

A fast, stable and simple cylindricity error evaluation method

technical field

The invention belongs to the fields of precision measurement and computer application, and relates to a stable, fast and simple cylindricity error evaluation method, which can be used for the evaluation of the cylindricity error of parts with a structure of revolution, and improves the processing technology thereof. Provide guidance.

Background technique

Dimensional error, shape and position error (abbreviation for shape error and position error) directly affect product quality, assembly and its service life, and it is of great significance to calculate part error quickly and accurately. The definition and discrimination method of cylindricity error are given in the national standard and ISO standard, but the method of calculating the cylindricity error value from the measurement data is not given. At present, the evaluation method of cylindricity error is a research hotspot in academia, and it is mainly divided into the following five types of evaluation methods.

The first category, specialized geometric evaluation methods. Using the geometric properties of the cylinder, according to the translation and deformation strategy of the inscribed cylinder and/or the circumscribed cylinder, step by step to find the cylindricity error that meets the definition and/or discrimination conditions of the national standard and ISO standard. This kind of method is faster, but the form of the mathematical model is more complicated, and it is not easy to popularize and use.

The second category, convex hull or convex hull-like assessment methods. Use the properties of convex hulls to construct convex hulls or convex hulls, obtain effective measurement data, reduce the scale of data to be evaluated, and finally obtain cylindricity errors that meet the definition and/or discrimination conditions of national standards and ISO standards through enumeration methods. This kind of method has obvious advantages when dealing with medium-scale measuring point data. When the data scale is large, the data scale can still be reduced by constructing a convex hull. However, the efficiency of such methods for direct assessment has been insufficient.

The third category is to construct a linear or nonlinear objective optimization function, and use the common optimization method to optimize the solution, and the optimized value of the objective optimization function is used as the cylindricity error. This kind of method is simple and easy to understand, and standard solutions are implemented in many softwares, so it is easy to generalize. Because the geometric characteristics of cylindricity error evaluation are not added, and the large scale of data in the evaluation task is not considered, such methods are generally inefficient.

The fourth category is artificial intelligence/biological intelligence algorithms. The advantage of this type of method over the third type of method is to analyze "objective functions with complex gradient analytic expressions or no obvious analytic expressions" and to find "global optimal values". This type of method is also currently implemented in many software as standard solutions, so it is also easy to generalize. Although this kind of method is relatively popular at present, it is not suitable for evaluating the cylindricity error. This is because the gradient of the objective function evaluated by the cylindricity error 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 a clear advantage over the third type of method.

The fifth category is the effective set method. The efficient set method is a method specially dealing with large-scale planning problems, which is characterized by minimizing the processing of "invalid constraints" in the optimization process. When applied to cylindricity evaluation, the efficiency is comparable to the first type of method, and the algorithm maturity and software integration are comparable to those of the third and fourth types of methods. It is a relatively fast and simple cylindricity error evaluation method at present. However, this method is very sensitive to the initial value and does not always perform the cylindricity error assessment task stably.

To sum up, there is still a lack of a stable, fast and simple cylindricity error evaluation method.

SUMMARY OF THE INVENTION

The purpose of this invention is:

Aiming at the problems existing in the prior art, the present invention provides a stable, fast and simple cylindricity error evaluation method, which can be used for the evaluation of the cylindricity error of parts with a structure of revolution, and the processing technology thereof provide guidance for improvement.

The scheme adopted in the present invention is:

A fast, stable and simple cylindricity error evaluation method is realized by the following steps:

Step 1: Obtain the set of measuring points { p _i }, and establish the set of characteristic row vectors { A _𝛼 }, the set of boundary elements { b _𝛼 } and the set of state elements { t _𝛼 } according to { p _i }, where:

i =1, 2, 3, …, N ; 𝛼 =1, 2, 3, …, N , N+1, …,2 N ; i is the measurement point number, N is the total number of measurement points;

p _i ={ x _i , y _i , z _i } is the plane Cartesian coordinate of the measuring point i , and the axis of the measured cylinder is close to the z -axis of the coordinate system, and the center plane of the two bottom surfaces of the measured cylinder is close to the XOY of the coordinate system flat;

，所有的状态元素t _𝛼 的集合为状态元素集{t _𝛼 }； t _i = t _i+N =

, the set of all state elements t _𝛼 is the state element set { t _𝛼 };

A _i =- A _{i + N} =([ x _i / t _i , y _i / t _i , - y _i z _i / t _i , x _i z _i / t _i , 1]), is the eigenrow vector, all The set of feature row vectors A _𝛼 is the feature row vector set { A _𝛼 };

b _i = b _i+N = b , which is a real number greater than 0, and the set of all boundary elements b _𝛼 is the boundary element set { b _𝛼 }.

After step 1, go to step 2.

Step 2: Take the serial number l ₁ corresponding to the minimum value t _min,in of t _i as the key serial number, and add l ₁ to the key serial number set { l }; take the serial number l corresponding to the maximum value t _{i + N} t _{max,out 2} is the key sequence number, and l ₂ is added to the key sequence number set { l }.

After step 2, go to step 3.

Step 3: Establish an analysis matrix A and an analysis column vector b according to the key sequence number set { l }, where:

A =[…, A _j ^T , …, A _k ^T , …] ^T , is a matrix with L rows and 5 columns, L is the number of elements in the key sequence number set { l }, j , k are the key sequence number set { l element in };

b =[…, b , …] ^T , which is a column vector of L rows.

After step 3, go to step 4.

Step 4: Perform rank analysis on the analysis matrix A and the augmented analysis matrix [ A , b ].

Calculate r _A = rank( A ), r _Ab = rank([ A , b ]), and compare r _A and r _Ab , with only the following two cases:

Case 1: If r _A = r _Ab , then the optimization should be continued and skip to step 5;

Case 2: If r _A < r _Ab , then try to delete the row corresponding to a certain element l in the key sequence number set { l } from the analysis matrix A and the analysis column vector b to obtain the reduced matrix A _l- and the reduced column vector b _l- , find the solution of the linear equation A _l- v _l- = b _l- v _l- = v _l-0 , and then calculate b _l- = A _l v _l-0 ; if the key sequence number set { l } have tried all the elements of , and did not get any b _l- > b , then you should end the optimization and skip to step 7; if you get b _l- > when trying the element l in the key sequence number set { l } b , then, take the reduced matrix A _l- and the reduced column vector b _l- as the A matrix and the analytical column vector b respectively, move the element l out of the key sequence number set { l }, and skip to step 5; where v _l- = [ v _{l-, 1} , v _{l-, 2} , v _{l-, 3} , v _{l-, 4} , v _{l-, 5} ] ^T , v _l-0 =[ v _{l-0, 1} , v _{l-0 , 2} , v _{l-0, 3} , v _{l-0, 4} , v _{l-0, 5} ] ^T .

Step 5: Find the solution to the linear equation Av = b v = v ₀ , where v =[ v ₁ , v ₂ , v ₃ , v ₄ , v ₅ ] ^T , v ₀ =[ v _{0, 1} , v _{0, 2} , v _{0, 3} , v _{0, 4} , v _{0, 5} ] ^T .

After step 5, go to step 6.

Step 6: Calculate v _𝛼 = A _𝛼 v ₀ , then calculate τ _i =( t _i – t _min,in )÷( b - v _i ), τ _{i + N} =( t _max,out – t _i+N ) ÷( b - v _i+N ). Take the sequence number l ₃ corresponding to the minimum value τ _min in the part of τ _𝛼 greater than zero as the new key sequence number, and add l ₃ to the key sequence number set { l }.

update all ti to ti + τ _min ∙( v _i - v _{0, 5} ), update all _{ti + N} to ti + N - τ min _{∙ (} v _i + _{N + v} 0 _{, 5} ), t _min,in is updated to the minimum value of t _i , and t _max,out is updated to the maximum value of t _{i + N.}

After step 6 is completed, an optimization is completed, and step 3 is performed.

Step 7: Calculate t = t _max,out - t _min,in is the desired cylindricity error.

In order to conveniently obtain the set of measuring points { p _i } in step 1, the general measurement data { p _i ^* } can be processed by the following but not limited to the following methods to obtain the z -axis whose axis is close to the coordinate system and the two cylinders under test. The measuring point set { p _i } of the bottom center plane close to the XOY plane of the coordinate system: 1. Move according to the average value of the coordinates; 2. Move according to the extreme value of the coordinates; 3. Move according to the minimum principle of the root mean square of the coordinates.

In order to get a more accurate solution, the following optimizations can be performed:

In step 6, if the single value of τ _min ∙ v _i or the accumulated value of several iterations ∑ τ _min ∙ v _i is greater than the given threshold q , then the set of measuring points { p _i } is updated to p _i + τ _min ∙ v or p _i +∑ τ _min ∙ v , and update the feature row vector set { A _i }, the boundary element set { b _i }, and the state element set { t _i } according to the formula in step 1.

In order to facilitate numerical calculation, b can be set to take a specific value greater than 0, which can be but not limited to 1.

The beneficial effects of the present invention are:

1. Fully consider the geometric characteristics of cylindricity error and simplify the evaluation form. Therefore, it is easier to popularize than the first type of evaluation method. 2. Fully considering the geometric characteristics of cylindricity error, each iteration obtains a better value through mature linear operations, and finally obtains the smallest cylindricity error. Therefore, this algorithm is relatively stable, and there is no fifth type of method. initial value-sensitive problem. 3. The fact that "most of the measuring points are invalid measuring points" in the evaluation of implicit cylindricity error, these invalid measuring points will not be added to the iteration, therefore, the number of iterations of the present invention is less, which is different from the first type of evaluation method and the third type of evaluation method. The five evaluation methods are equivalent. 4. When calculating the optimization direction, only the measurement points corresponding to the key sequence number set { l } are considered. Therefore, the calculation amount of each iteration is small, which is equivalent to the fifth type of evaluation method. 5. Because the number of iterations is small and the calculation amount of each iteration is small, the total calculation speed is comparable to that of the first type of evaluation method and the fifth type of evaluation method.

The invention provides a cylindricity error evaluation method, which is stable, fast and simple in form, can be used for the evaluation of the cylindricity error of parts with a rotary body structure, and provides guidance for the improvement of its processing technology, so it has the advantages of industrial possibility.

Description of drawings

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

Detailed ways

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

Evaluate the cylindricity error of the set of measuring points { p _i }.

Step 1: Obtain the set of measuring points { p _i } as follows:

<i>i</i> <i>x</i>_<i>i</i> <i>y</i>_<i>i</i> z_<i>i</i> 1 9.5285 3.1018 -44.9417 2 5.8950 -8.0895 -39.9115 3 -5.8697 -8.0894 -34.9254 4 -9.5075 3.0930 -29.9394 5 0.0051 10.0065 -24.9598 6 9.5187 3.0979 -19.9390 7 5.8812 -8.0864 -14.9905 8 -5.8714 -8.0748 -9.9766 9 -9.4958 3.1040 -4.9176 10 0.0166 10.0059 0.0309 11 9.5210 3.0967 5.0832 12 5.8941 -8.0790 10.0263 13 -5.8642 -8.0855 15.0456 14 -9.5029 3.1009 20.0992 15 0.0151 10.0196 25.0235 16 9.5211 3.0912 30.0757 17 5.8899 -8.0730 35.0988 18 -5.8593 -8.0820 40.0000 19 -9.4997 3.0943 45.0219 20 0.0065 10.0019 50.0748

The set of state elements { t _𝛼 } is established as follows:

<i>𝛼</i> <i>𝛼</i> <i>t</i>_<i>𝛼</i> 1 twenty one 10.0207 2 twenty two 10.0095 3 twenty three 9.9946 4 twenty four 9.9980 5 25 10.0065 6 26 10.0101 7 27 9.9989 8 28 9.9838 9 29 9.9902 10 30 10.0059 11 31 10.0120 12 32 10.0006 13 33 9.9882 14 34 9.9960 15 35 10.0196 16 36 10.0104 17 37 9.9933 18 38 9.9825 19 39 9.9910 20 40 10.0019

Establish feature row vector set { A _𝛼 }, when 𝛼 = i , { A _i } is as follows:

When 𝛼 = i +20, A _{i+ 20} =- A _i .

The set of boundary elements { b _𝛼 } is established as follows:

{ b _𝛼 }=[1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] ^T .

After step 1, go to step 2.

Step 2: Take the serial number 18 corresponding to the minimum value of t _i t _min,in as the key serial number, and add 18 to the key serial number set { l }; take the serial number 21 corresponding to the maximum value of t _{i +20} t _max,out as the key serial number sequence number, and add 21 to the key sequence number set { l }, { l }={18,21};.

After step 2, go to step 3.

Step 3: Establish an analysis matrix A and an analysis column vector b according to the key sequence number set { l }, where:

A = is a matrix with 2 rows and 5 columns, the number of elements in the key sequence number set { l }={18,21} is 2, and the elements are 18, 21;

b =[1,1] ^T , a column vector with 2 rows.

After step 3, go to step 4.

Step 4: Perform rank analysis on the analysis matrix A and the augmented analysis matrix [ A , b ].

Calculate r _A = rank( A ) = 2, r _Ab = rank([ A , b ]) = 2, and compare r _A and r _Ab . Since r _A = r _Ab , the optimization should continue and skip to step 5.

Step 5: Find the solution of the linear equation Av = b v = v ₀ =[0.0000 , 0.0000 , 0.0626 , 0.0438 ,0.0000] ^T .

After step 5, go to step 6.

Step 6: When v _𝛼 = A _𝛼 v ₀ , 𝛼 = i , the result is as follows:

<i>i</i> <i>v</i>_<i>i</i> 1 -1.0000 2 -3.0491 3 -0.8721 4 1.8266 5 1.5625 6 -0.4438 7 -1.1453 8 -0.2484 9 0.3003 10 -0.0019 11 0.1132 12 0.7660 13 0.3759 14 -1.2271 15 -1.5654 16 0.6709 17 2.6814 18 1.0000 19 -2.7476 20 -3.1344

When 𝛼 = i +20, v _{i+ 20} =- v _i .

Then calculate τ _i =( t _i - t _min,in )÷( b - v _i ), τ _{i +10} =( t _max,out - t _{i +10} )÷( b - v _{i +10} ). Taking the part of τ _𝛼 greater than zero results as follows:

<i>𝛼</i> <i>τ</i>_<i>𝛼</i> 1 0.0191 2 0.0067 3 0.0065 6 0.0192 7 0.0077 8 0.0010 9 0.0111 10 0.0234 11 0.0333 12 0.0773 13 0.0092 14 0.0061 15 0.0145 16 0.0848 19 0.0023 20 0.0047 twenty three 0.2034 twenty four 0.0080 25 0.0055 26 0.0189 28 0.0491 29 0.0234 30 0.0148 31 0.0078 32 0.0114 33 0.0236 36 0.0061 37 0.0074 38 0.0191

The sequence number 8 corresponding to the minimum value τ _min is a new key sequence number, and 8 is added to the key sequence number set { l }. At this point, { l }={18,21,8}.

Update all t _i to t _i + τ _min ∙( v _i - v _{0, 5} ) = t _i + τ _min ∙( v _i -0), update all t _{i +20} to t _{i +20} - τ _min ∙( v _{i +20} + v _{0, 5} )= t _{i +20} - τ _min ∙( v _{i +20} +0), t _min,in is updated to the minimum value of t _i , and t _max,out is updated to t _{i +20} max.

After step 6 is completed, an optimization is completed, and step 3 is performed.

By analogy, after the sixth optimization, the key sequence number set { l } ={18,19,35,20,22,23}.

At this time, proceed to step 3: establish an analysis matrix A and an analysis column vector b according to the key sequence number set { l } ={18,19,35,20,22,23}, where:

b =[1,1,1,1,1,1] ^T .

After step 3, go to step 4.

Step 4: Perform rank analysis on the analysis matrix A and the augmented analysis matrix [ A , b ].

Calculate r _A = rank( A ) = 5, r _Ab = rank([ A , b ]) = 6, r _A < r _Ab . First, try to delete the row corresponding to the first element 18 in the key sequence number set { l }={18,19,35,20,22,23} from the analysis matrix A and the analysis column vector b , and get the reduced matrix A _{18 -} :

and b _{18 -} column vector b _{18 -} =[1,1,1,1,1] ^T .

Find the solution of the linear equation A _{18 -} v _{18 -} = b _{18 -} v _{18 -} = v _{18 -0} =[-1.6091 , 0.2620 , -0.0798 , -0.0358 , -3.2544] ^T , then calculate b _{18 -} = A ₁₈ v _{18 -0} = -4.2658 <1= b ; b _{19 -} = 10.3093 >1= b .

Take A ₁₉ -matrix and b ₁₉ -matrix as A matrix and b matrix respectively, move element 19 out of the key sequence number set { l }, so that { l }={18, 35, 20, 22, 23}, and skip to step 5.

By analogy, after completing the seventh optimization, the key sequence number set { l } ={18, 35, 20, 22, 23, 17}.

At this time, proceed to step 3 first: establish an analysis matrix A and an analysis column vector b according to the key sequence number set { l } ={18, 35, 20, 22, 23, 17}.

After step 3, go to step 4.

Step 4: Perform rank analysis on the analysis matrix A and the augmented analysis matrix [ A , b ].

Calculate r _A = rank( A ) = 5, r _Ab = rank([ A , b ]) = 6, r _A < r _Ab .

Corresponding to { l } ={18, 35, 20, 22, 23, 17}, respectively obtain b _{18 -} = -12.6721 <1= b , b _{35 -} =-1.8263<1= b , b _{20 -} = -1.8264 <1= b , b _{22 -} = -12.7279 <1= b , b _{23 -} =-12.6335 <1= b , b _{17 -} = -12.6883<1= b . Skip to step 7.

Step 7: Calculate t = t _{max, out} - t _{min, in} = 0.0334 is the desired cylindricity error.

In the above description, the present invention has been described with reference to specific embodiments, but it should be understood by those skilled in the art that various modifications and changes can be made within the spirit and field of the invention without departing from the scope of the claims.

Claims (5)

1. a fast, stable and simple cylindricity error assessment method, is characterized in that, is made up of the following steps: Step 1: Obtain the set of measuring points { p _i }, and establish the set of characteristic row vectors { A _𝛼 }, the set of boundary elements { b _𝛼 } and the set of state elements { t _𝛼 } according to { p _i }, where: i =1, 2, 3, …, N ; 𝛼 =1, 2, 3, …, N , N+1, …,2 N ; i is the measurement point number, N is the total number of measurement points; p _i ={ x _i , y _i , z _i } is the plane Cartesian coordinate of the measuring point i , and the axis of the measured cylinder is close to the z -axis of the coordinate system, and the center plane of the two bottom surfaces of the measured cylinder is close to the XOY of the coordinate system flat; t _i = t _i+N =

, the set of all state elements t _𝛼 is the state element set { t _𝛼 }; A _i =- A _{i + N} =([ x _i / t _i , y _i / t _i , - y _i z _i / t _i , x _i z _i / t _i , 1]), is the eigenrow vector, all The set of feature row vectors A _𝛼 is the feature row vector set { A _𝛼 }; b _i = b _i+N = b , is a real number greater than 0, and the set of all boundary elements b _𝛼 is the boundary element set { b _𝛼 }; After step 1, go to step 2; Step 2: Take the serial number l ₁ corresponding to the minimum value t _min,in of t _i as the key serial number, and add l ₁ to the key serial number set { l }; take the serial number l corresponding to the maximum value t _{i + N} t _{max,out 2} is the key sequence number, and l ₂ is added to the key sequence number set { l }; After step 2, go to step 3; Step 3: Establish an analysis matrix A and an analysis column vector b according to the key sequence number set { l }, where: A =[…, A _j ^T , …, A _k ^T , …] ^T , is a matrix with L rows and 5 columns, L is the number of elements in the key sequence number set { l }, j , k are the key sequence number set { l element in }; b =[…, b , …] ^T , is a column vector of L rows; After step 3, go to step 4; Step 4: Perform rank analysis on the analysis matrix A and the augmented analysis matrix [ A , b ]; Calculate r _A = rank( A ), r _Ab = rank([ A , b ]), and compare r _A and r _Ab , with only the following two cases: Case 1: If r _A = r _Ab , then the optimization should be continued and skip to step 5; Case 2: If r _A < r _Ab , then try to delete the row corresponding to a certain element l in the key sequence number set { l } from the analysis matrix A and the analysis column vector b to obtain the reduced matrix A _l- and the reduced column vector b _l- , find the solution of the linear equation A _l- v _l- = b _l- v _l- = v _l-0 , and then calculate b _l- = A _l v _l-0 ; if the key sequence number set { l } have tried all the elements of , and did not get any b _l- > b , then you should end the optimization and skip to step 7; if you get b _l- > when trying the element l in the key sequence number set { l } b , then, take the reduced matrix A _l- and the reduced column vector b _l- as the A matrix and the analytical column vector b respectively, move the element l out of the key sequence number set { l }, and skip to step 5; where v _l- = [ v _{l-, 1} , v _{l-, 2} , v _{l-, 3} , v _{l-, 4} , v _{l-, 5} ] ^T , v _l-0 =[ v _{l-0, 1} , v _{l-0 , 2} , v _{l-0, 3} , v _{l-0, 4} , v _{l-0, 5} ] ^T ; Step 5: Find the solution to the linear equation Av = b v = v ₀ , where v =[ v ₁ , v ₂ , v ₃ , v ₄ , v ₅ ] ^T , v ₀ =[ v _{0, 1} , v _{0, 2} , v _{0, 3} , v _{0, 4} , v _{0, 5} ] ^T ; After step 5, go to step 6; Step 6: Calculatev _𝛼 =A _𝛼 v ₀ , and then calculateτ _i =(t _i –t _min,in)÷(b-v _i ),τ _{i + N} =(t _max,out –t _i+N )÷(b- v _i+N );Pickτ _𝛼 the smallest value in the portion greater than zeroτ _mincorresponding serial numberl ₃is the new key sequence number and willl ₃add to key sequence number set {l}middle; update all ti to t _i + τ _min ∙( v _i - v _{0, 5} ), update all _{ti + N} to t i _{+ N} - τ _min ∙( v _i+N + v _{0, 5} ), t _min,in is updated to the minimum value of t _i , and t _max,out is updated to the maximum value of t _{i + N} ; After step 6, complete an optimization, and go to step 3; Step 7: Calculate t = t _max,out - t _min,in is the desired cylindricity error. 2. a kind of fast, stable and simple cylindricity error evaluation method as claimed in claim 1, is characterized in that, general measurement data { p _i ^* } is transformed through conventional coordinate, obtains the z -axis of the axis close to the coordinate system, the The measuring point set { p _i } where the center planes of the two bottom surfaces of the measuring cylinder are close to the XOY plane of the coordinate system. 3. a kind of fast, stable and simple cylindricity error evaluation method as claimed in claim 2 is characterized in that, described conventional coordinate transformation, is one, moves according to the average value of coordinates, or two, according to the extreme value of coordinates Move, or three, move according to the minimum principle of the root mean square of the coordinates. 4. A fast, stable and simple cylindricity error assessment method as claimed in claim 1, characterized in that, in step 6, if the single value of τ _min ∙ v _i or the accumulated value of several iterations ∑ τ _min ∙ v _i is greater than the given threshold q , then, update the measuring point set { p _i } to p _i + τ _min ∙ v or p _i +∑ τ _min ∙ v , and update the feature row vector according to the formula in step 1 set { A _i }, set of boundary elements { b _i } and set of state elements { t _i }. 5. A fast, stable and simple cylindricity error evaluation method as claimed in claim 1, characterized in that, b =1.

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