CN101493321B - Planarity assessment method for decreasing number of measuring points - Google Patents

Abstract

The invention relates to a planeness assessment method for reducing the number of measuring points, and has the following specific steps: the plane error values at a plurality of point positions on a plane are measured according to the existing planeness measuring method; the probability distribution of the measured plane errors is estimated by the maximum entropy method according to the limited measuring point values; according to the estimated probability distribution, the estimated value of the plane errors between the measuring points is generated at other positions of the measured plane, that is, the measuring points are interpolated; the planeness error assessment methods such as least envelope zone method, least square method or diagonal method are adopted to assesses the planeness error based on the measured value of the planeness error and the estimated value of the plane error at interpolation position; the planeness errors are respectively calculated according to the probability, statistical theory and a plurality of estimated values at interpolation position, and the mean value of the calculated results is taken as the final planeness error assessment value. On the basis of maximum entropy method, the invention uses limited measuring data sample to determine the probability distribution of the measured plane error.

Description

Reduce the planarity assessment method of measure dot number

Technical field

The present invention relates to a kind of method measuring flatness, especially a kind of measurement of flatness error and assessment method.

Background technology

Flatness is one of main project of form tolerance, and the measurement of flatness error has great significance in geometric measurement with evaluation.According to the regulation of form and position tolerance national standard, flatness error is meant the variation of measured surface to the ideal plane, and the orientation of ideal plane should meet minimal condition, and promptly its orientation should make measured surface that the maximum variation of ideal plane is minimum.In the evaluation of flatness error, the number of plane error measurement point has bigger influence to its evaluation result.Because the measured surface position is represented by several measurement points, and adopt when measuring sampled point, be subjected to the restriction of sampling number, tended to underestimate actual error amount error evaluation.On the other hand, be subjected to the restriction of Measuring Time, the flatness error measurement point can not be a lot.Therefore, how under the measure dot number condition of limited, the precision that improves the flatness error evaluation just becomes a major issue.

Summary of the invention

The present invention will solve under the measure dot number condition of limited, improves the technical matters of the precision of flatness error evaluation, and a kind of planarity assessment method that reduces measure dot number is provided.

For solving the problems of the technologies described above, technical scheme of the present invention is: a kind of planarity assessment method that reduces measure dot number, and concrete steps are:

1) according to existing method measuring flatness, the locational plane error value of some spots on the measurement plane;

2) according to these limited measurement point values,, estimate the probability distribution of tested plane error by maximum entropy method;

3) according to estimated probability distribution, on other position on tested plane, produce the estimated value of plane error between measurement point, promptly measurement point is carried out interpolation;

4) utilize the flatness error assessment method of minimum containment region method, least square method or diagonal method, by the estimated value of the measured value and the location of interpolation plane error of flatness error, evaluation flatness error;

5) estimated value repeatedly on the theoretical and location of interpolation according to Probability ﹠ Statistics is calculated flatness error respectively, and the mean value of getting these result of calculations is as final flatness error evaluation value.

Above-mentioned least square method flatness error assessment method is:

If the coordinate figure of any point is M on the tested plane _i(x _i, y _i, z _i), the equation of ideal plane is: z=Ax+By+C; By least square method, objective function is:

minS＝∑(z _i-z) ²＝∑(z _i-Ax _i-By _i-C) ² (1)

Determine the value of A, B, C by formula (1), promptly determine the position of ideal plane, ask the distance of each measuring point and ideal plane again, the flatness error that promptly gets each measuring point place is:

e _i＝z _i-(Ax _i+By _i+C) (2)

Use the probability density function that maximum entropy method calculates plane error, tested plane is with respect to the ideal plane error e _iThe information entropy of probability density p (e) be defined as:

$H=-\underset{R}{\∫}p\left(e\right)\ln p\left(e\right)\mathrm{de}---\left(3\right)$

In the formula (3), R is the integration space, and constraint condition is:

$\underset{R}{\∫}p\left(e\right)\mathrm{de}=1---\left(4\right)$

$\underset{R}{\∫}{e}^{i}p\left(e\right)\mathrm{de}=m_i,i=\mathrm{1,2},\·\·\·,m---\left(5\right)$

In the formula, m is the exponent number of used square, m _iIt is i rank moment of the orign.

Make information entropy reach maximal value by adjusting p (e), adopt method of Lagrange multipliers to find the solution this problem; If Lagrange multiplier is λ ₀, λ ₁..., λ _m, then have

$\stackrel{\‾}{H}=H\left(e\right)+({\mathrm{\λ}}_0+1)[\underset{R}{\∫}p\left(e\right)\mathrm{de}-1]+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_i[\underset{R}{\∫}{e}^{i}p\left(e\right)\mathrm{de}-m_i]---\left(6\right)$

Make dH/dp (e)=0, the analytical form of maximum entropy probability density function arranged:

$p\left(e\right)=\exp ({\mathrm{\λ}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_i{e}^i)---\left(7\right)$

Coordinate (x on the tested plane _k, y _k) locate, the estimated value computing method of flatness error are:

Produce an error amount e randomly according to maximum entropy probability density function p (e) _k, as (x _k, y _k) estimated value of plane error measured value of position.It is converted into the coordinate figure z of tested plane Z axle _k, according to formula (2)

z _k＝e _k+(Ax _k+By _k+C) (8)

Thereby obtain the estimated value of the flatness error measured value at assigned address interpolation point place.

Beneficial effect of the present invention:

The present invention utilizes limited measuring data sample based on maximum entropy method, determines the probability distribution of tested plane error.And estimate the not numerical value of measuring point according to the probability distribution of being tried to achieve, increase the number of measurement point with this, thereby improved the efficiency of measurement of flatness error effectively.

Description of drawings

Fig. 1 is a flatness error measurement point distribution plan.

Embodiment

The present invention is further illustrated below in conjunction with specific embodiment.

If the measuring point that a certain flatness error is measured distributes as shown in Figure 1, " zero " expression actual spot of measurement among Fig. 1,

Expression is according to maximum entropy method and the estimated measurement point value (is location of interpolation to call this position in the following text) of actual spot of measurement.Can carry out the evaluation of flatness error calculates according to these data.Because flatness error is a stochastic error,, also will be a kind of result of calculation at random therefore by with the flatness error that upper estimate was obtained.In order to obtain flatness error evaluation result accurately, need by maximum entropy method, according to limited actual measurement dot information, estimate the probability distribution of tested plane error, and probability distribution produces the measurement estimated value of each position thus.The mean value of getting repeatedly the flatness error evaluation then is as final flatness error evaluation result.

The basic step of summing up above flatness error evaluation process is:

(1) according to general method measuring flatness, the locational plane error value of some spots on the measurement plane;

(2) according to these limited measurement point values,, estimate the probability distribution of tested plane error by maximum entropy method;

(3) according to estimated probability distribution, on other position on tested plane, as shown in Figure 1

On the position, produce the estimated value of plane error between measurement point, promptly measurement point is carried out interpolation;

(4) utilize the flatness error assessment method, for example minimum containment region method, least square method or diagonal method etc., by the estimated value of the measured value and the location of interpolation plane error of flatness error, the evaluation flatness error;

(5) because the plane error on the location of interpolation for according to the error estimate that probability distribution produced, is a kind of stochastic error value, estimate that each time its value all can change.Therefore the flatness error of calculating by said method also is a kind of random value.According to the Probability ﹠ Statistics theory, in order to obtain flatness error evaluation result accurately, need calculate flatness error respectively, and the mean value of getting these result of calculations is as final flatness error evaluation value according to estimated value repeatedly on the location of interpolation.

For the computing method of flatness error, numerous books and document all have report, belong to knowledge.Gordian technique of the present invention has proposed to utilize maximum entropy method exactly, estimates the plane error measured value on the location of interpolation, and according to the Probability ﹠ Statistics theory, gets repeatedly the method for the mean value of flatness error result of calculation as final flatness error evaluation value.Below be example with least square method flatness error assessment method, narrate planarity assessment method proposed by the invention.

If the coordinate figure of any point is M on the tested plane _i(x _i, y _i, z _i), the equation of ideal plane is: z=Ax+By+C.By the basic thought of least square method, should make the quadratic sum minimum of measurement point to the coordinate figure on this plane by this ideal plane of measurement point match.So objective function is arranged:

minS＝∑(z _i-z) ²＝∑(z _i-Ax _i-By _i-C) ² (1)

According to least square method, can determine the value of A, B, C by formula (1), promptly determined the position of ideal plane, ask the distance of each measuring point and ideal plane again, get final product to such an extent that the flatness error at each measuring point place is:

e _i＝z _i-(Ax _i+By _i+C) (2)

Use the probability density function that maximum entropy method calculates plane error, tested plane is with respect to the ideal plane error e _iThe information entropy of probability density p (e) may be defined as:

$H=-\underset{R}{\∫}p\left(e\right)\ln p\left(e\right)\mathrm{de}---\left(3\right)$

In the formula (3), R is the integration space, and constraint condition is:

$\underset{R}{\∫}p\left(e\right)\mathrm{de}=1---\left(4\right)$

$\underset{R}{\∫}{e}^{i}p\left(e\right)\mathrm{de}=m_i,i=\mathrm{1,2},\·\·\·,m---\left(5\right)$

In the formula, m is the exponent number of used square, m _iIt is i rank moment of the orign.

Make information entropy reach maximal value by adjusting p (e), adopt method of Lagrange multipliers to find the solution this problem.If Lagrange multiplier is λ ₀, λ ₁..., λ _m, then have

$\stackrel{\‾}{H}=H\left(e\right)+({\mathrm{\λ}}_0+1)[\underset{R}{\∫}p\left(e\right)\mathrm{de}-1]+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_i[\underset{R}{\∫}{e}^{i}p\left(e\right)\mathrm{de}-m_i]---\left(6\right)$

Make dH/dp (e)=0, have

$p\left(e\right)=\exp ({\mathrm{\λ}}_0+\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_i{e}^i)---\left(7\right)$

Formula (7) is exactly the analytical form of maximum entropy probability density function.λ in the formula ₀, λ ₁..., λ _mCan try to achieve by the numerical computation method of being introduced on the pertinent literature, thereby obtain the analytical form of maximum entropy probability density function p (e).

Coordinate (x on the tested plane _k, y _k) locate, the estimated value computing method of flatness error are:

Produce an error amount e randomly according to maximum entropy probability density function p (e) _k, as (x _k, y _k) estimated value of plane error measured value of position.Because e _kBe the error amount of tested plane, so need it is converted into the coordinate figure z of tested plane Z axle with respect to the ideal plane _k, according to formula (2)

z _k＝e _k+(Ax _k+By _k+C) (8)

Thereby obtain the estimated value of the flatness error measured value at assigned address interpolation point place.

Estimated value according to measured point flatness error measured value and interpolation point flatness error measured value is calculated flatness error, and the accuracy of its evaluation result will be higher than the error evaluation value of only using measured point error measuring value gained.Therefore, adopt the inventive method under the situation of same measure dot number, can obtain the flatness error evaluation value of degree of precision, thereby can reduce the time that flatness detects effectively.

In information theory, the implication of maximum entropy is maximum uncertainty, and the big class problem that its solves is to make a strategic decision under the inadequate condition of priori or deduction etc.Its main meaning is: in the time of under the situation of only grasping partial information, will inferring to system state, should get meet constraint condition and entropy for maximum state as a kind of rational state.

Claims (2)

1. planarity assessment method that reduces measure dot number is characterized in that concrete steps are:

(1) according to existing method measuring flatness, the locational plane error value of some spots on the measurement plane;

(2) according to these limited measurement point values,, estimate the probability distribution of tested plane error by maximum entropy method;

(3) according to estimated probability distribution, on other position on tested plane, produce the estimated value of plane error between measurement point, promptly measurement point is carried out interpolation;

(4) utilize the flatness error assessment method of minimum containment region method, least square method or diagonal method, by the estimated value of the measured value and the location of interpolation plane error of flatness error, evaluation flatness error;

(5) estimated value repeatedly on the theoretical and location of interpolation according to Probability ﹠ Statistics is calculated flatness error respectively, and the mean value of getting these result of calculations is as final flatness error evaluation value.

2. the planarity assessment method of minimizing measure dot number according to claim 1 is characterized in that, described least square method flatness error assessment method is:

If the coordinate figure of any point is M on the tested plane _i(x _i, y _i, z _i), the equation of ideal plane is: z=Ax+By+C; By least square method, objective function is:

minS＝∑(z _i-z) ²＝∑(z _i-Ax _i-By _i-C) ²(1)

Determine the value of A, B, C by formula (1), promptly determine the position of ideal plane, ask the distance of each measuring point and ideal plane again, the flatness error that promptly gets each measuring point place is:

e _i＝z _i-(Ax _i+By _i+C)(2)

Use the probability density function that maximum entropy method calculates plane error, tested plane is with respect to the ideal plane error e _iThe information entropy of probability density p (e) be defined as:

In the formula (3), R is the integration space, and constraint condition is:

In the formula, m is the exponent number of used square, m _iBe i rank moment of the orign,

Make information entropy reach maximal value by adjusting p (e), adopt method of Lagrange multipliers to find the solution this problem; If Lagrange multiplier is λ ₀, λ ₁..., λ _m, then have

Make dH/dp (e)=0, the analytical form of maximum entropy probability density function arranged:

Coordinate (x on the tested plane _k, y _k) locate, the estimated value computing method of flatness error are:

Produce an error amount e randomly according to maximum entropy probability density function p (e) _k, as (x _k, y _k) estimated value of plane error measured value of position, it is converted into the coordinate figure z of tested plane Z axle _k, according to formula (2)

z _k＝e _k+(Ax _k+By _k+C)(8)

Thereby obtain the estimated value of the flatness error measured value at assigned address interpolation point place.

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