CN100523722C - Double offset parameter circle contour measurement model and biased error separation method - Google Patents

Abstract

A double offset parameter circle contour measurement model and an offset error separation method belong to the technical field of surface shape measurement. The model simultaneously includes two offset error components: offset error of the detected component (e, Alpha) and offset error d of the measuring head of a sensor. The measurement model is that Rho i is equal to ecos(Theta i-Alpha)+((ro+Delta ri)<2>-(d+esin(Theta i - Alpha))<2>)<1/2>. The method of parameter optimization is adopted to realize accurate estimation and direct solving of offset error parameters and other parameters in the model, thus separating the offset parameters one by one from the measuring data and acquiring the actual circle contour of the detected piece. The measurement model provided by the invention can completely and accurately reflect the influence of the offset error parameters to the measurement of the circle contour and overcome the problems of principle defect and low parameter estimation accuracy existed in the generally used measurement model at the present time.

Description

A kind of biased error separation method of double offset parameter circle contour measurement model

Technical field

The invention belongs to the surface shape measuring technical field, particularly a kind of double offset parameter circle contour measurement model and biased error separation method.

Background technology

Deviation from circular from is an important technology parameter of control revolution type component quality.Along with developing rapidly of precision engineering technology and national defence sophisticated technology, the widespread use of a large amount of ultra-precise revolving body parts, as the deviation from circular from of the round standard-quartzy standard hemisphere appointment circular section of comparison and calibration usefulness between 5nm～50nm; Electrostatically sus pended gyroscope rotor circularity requires in 10nm, and these have all proposed high requirement to the ultra precise measurement of deviation from circular from.

Biased error is the important errors source that influences the circle contour ultra precise measurement, the geometric center that during at present both at home and abroad circle contour is measured the research of biased error is mainly concentrated on test specimen and the measurement centre of gyration do not overlap the eccentric error of generation, therefore in the circle contour measurement model, also only introduced offset parameter of eccentric error of test specimen, widely used measurement model is to be proposed and the Limacon model of proof by Britain Spragg, this model think the e that satisfies condition＜＜r _o(it is generally acknowledged 10 ^-3On the order of magnitude) time, its model tormulation is: r _i=ecos (θ _i-α)+r _o+ Δ r _i,, adopt least-square fitting approach can obtain eccentric error (e, approximate evaluation value α) according to this model.Yet, along with more and more higher, particularly in a lot of occasions, when the measuring accuracy of circularity is all required to reach several nanometer to the requirement of circle contour measuring accuracy, the defective exposed day by day of Limacon model is come out, as the principle defective because of considering that not the transducer probe assembly offset error produces in measurement model; In finding the solution the eccentric error process, measurement model is simplified and the truncation error and the parameter estimating error of generation; And when carrying out the circle contour ultra precise measurement, eccentric amount e must be adjusted in the very little scope, just can obtain more accurate measurement result, greatly increased the adjustment difficulty of measuring, reduced efficiency of measurement.More than these factors all make the Limacon model can't satisfy current accuracy requirement to the circle contour ultra precise measurement.

Current another reason that the super accurate measurement accuracy of measurement of circle contour is difficult to further to improve is in circle contour is measured, also exist an other biased error, this error is ignored by people for a long time always, the gauge head offset error that this error causes when being the sensor measurement line not by the measurement centre of gyration.A large amount of theoretical analysis and experiments of measuring show that when the circle contour measuring accuracy was reached nanometer scale, this error had become the important errors source that the restriction measuring accuracy further improves.

Taiwan's scholars Cha ' o-Kuang Chen has mentioned simply just that in the precision measurement of carrying out circularity the transducer probe assembly offset error can have influence on the accuracy of measurement, but do not propose how to solve accurately this error (Cha ' o-Kuang Chen.The study on the error separation and eccentricity self-compensationmethods for improving the precision of a roundness machine.Proceedings of the SecondInternational Symposium on Instrumentation Science and Technology, 2002, (1), 459-465).

Summary of the invention

Purpose of the present invention is exactly the problem that exists at above-mentioned prior art, a kind of double offset parameter circle contour measurement model and biased error separation method are proposed, comprise two biased error components of test specimen eccentric error and transducer probe assembly offset error in this measurement model simultaneously, can completely accurately reflect the influence that each biased error component is measured circle contour; The biased error separation method that proposes based on parameter optimization, can realize simultaneously to the accurate estimation of other parameter in transducer probe assembly offset error, test specimen eccentric error and the model with directly find the solution, and then in measurement data, isolate transducer probe assembly offset error and test specimen eccentric error component one by one, reach the purpose of further raising circle contour ultra precise measurement precision.

Above-mentioned purpose realizes by following technical scheme:

A kind of double offset parameter circle contour measurement model, comprise simultaneously in this model the test specimen eccentric error (e, α) and two biased error components of transducer probe assembly offset error d, its measurement model is:

ρ _i＝ecos(θ _i-α)+((r _o+Δr _i) ²-(d+esin(θ _i-α)) ²) ^1/2， i＝0，1，2，…，N-1

In the formula, ρ _iCertain is put to the distance of transient measurement center o ' on-test specimen circle contour; D-transducer probe assembly offset error; E-test specimen offset; α-eccentric angle; θ _i-angle position; r _o-least square radius of circle; Δ r _iCertain is put to the deviation of least square circle on the-test specimen circle contour; N-sampling number.

A kind of biased error separation method of double offset parameter circle contour measurement model, this method may further comprise the steps:

1) known comprise the test specimen eccentric error (e, α) and the measurement model of two biased error components of transducer probe assembly offset error d be:

ρ _i＝ecos(θ _i-α)+((r _o+Δr _i) ²-(d+esin(θ _i-α)) ²) ^1/2， i＝0，1，2，…，N-1

According to above-mentioned measurement model, have

Δr _i＝((ρ _i-ecos(θ _i-α)) ²+(d+esin(θ _i-α)) ²) ^1/2-r _o

i＝0，1，2，…，N-1

2) set up objective function (e, α, r0, d);

3) adopt parameter optimization method to objective function (e, α, r _o, d) directly find the solution, obtain in strict conformity with the double offset parameter circle contour measurement model transducer probe assembly offset error d, test specimen eccentric error (e, α) and associated arguments r _oAccurate estimated value

4) with above-mentioned estimated value

Respectively substitution circle contour expression formula and deviation from circular from expression formula, pointwise separate simultaneously transducer probe assembly offset error d and test specimen eccentric error (e, α);

$\left\{\begin{array}{c}\widehat{{\mathrm{\&rho;}}_{\mathrm{oi}}}={\{{[{\mathrm{\&rho;}}_i-\hat{e}\cos ({\mathrm{\&theta;}}_i-\widehat{\mathrm{\&alpha;}})]}^2+{[\hat{d}+\hat{e}\sin ({\mathrm{\&theta;}}_i-\widehat{\mathrm{\&alpha;}})]}^2\}}^{1/2}i=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N-1\\ \widehat{{\mathrm{\&Delta;r}}_i}=\widehat{{\mathrm{\&rho;}}_{\mathrm{oi}}}-\widehat{r_o}\end{array}\right.$

5) separate biased error through above-mentioned pointwise, promptly transducer probe assembly offset error parameter d and test specimen eccentric error parameter (e, α) after, can obtain the circle contour error information of " pure "

A kind of double offset parameter circle contour measurement model, this model can also be expressed as following form:

ρ _i＝acosθ _i+bsinθ _i+((r _o+Δr _i) ²-(d+asinθ _i-bcosθ _i) ²) ^1/2

i＝0，1，2，…，N-1

And have $\left\{\begin{array}{c}e={(a^2+b^2)}^{1/2}\\ \mathrm{\&alpha;}={\tan }^{-1}(b/a)\end{array}\right..$

The present invention has following characteristics and beneficial effect:

1, the double offset parameter circle contour measurement model of the present invention's proposition has comprised test specimen eccentric error (e simultaneously, α) with two biased error components of transducer probe assembly offset error d, can completely accurately reflect the influence that each biased error component is measured circle contour, further perfect measurement model, the principle defective of having avoided existing Limacon measurement model to exist, this is one of innovative point that is different from prior art;

2, the present invention utilizes parameters optimization method, can not be implemented in measurement model and parameter estimation procedure are carried out under the prerequisite of any simplification, finish to the accurate estimation of other parameter in two biased error components and the model with directly find the solution, significantly improved the accuracy of parameter estimation, can obtain the accurate circle contour of test specimen, solved the low problem of principle defective, estimated accuracy that existing method for parameter estimation causes because of model simplification, this be different from prior art innovative point two;

In addition, utilize the precise Estimation Method of measurement model of the present invention and model parameter, can from raw measurement data, isolate the biased error component accurately, therefore can suitably relax when measuring adjustment requirement to biased error, this has alleviated the labour intensity of survey crew to a certain extent, and has improved efficiency of measurement.

Description of drawings

Fig. 1 is a double offset parameter circle contour measurement model schematic diagram;

Fig. 2 measures the distortion amplitude curve for circle contour.

Among the figure: 1, the theoretical measuring position of transducer probe assembly; 2, transducer probe assembly actual measurement location; O ₁-measurement the centre of gyration; O ₂-least square circle center; O '-transient measurement center; ρ _iOn-test specimen the circle contour certain put to the transient measurement center O ' distance; r _iCertain point is to measuring centre of gyration O on the-test specimen circle contour ₁Polar radius; D-transducer probe assembly offset error; E-test specimen offset; α-eccentric angle; θ _i-angle position; r _o-least square radius of circle; Δ r _iCertain is put to the deviation of least square circle on the-test specimen circle contour; N-sampling number.

Embodiment

Below in conjunction with accompanying drawing embodiments of the invention are elaborated.

As shown in Figure 1, be the double offset parameter circle contour measurement model that the present invention proposes, remove in this model the test specimen eccentric error introduced in the traditional measurement model (e, α) outside, also introduced transducer probe assembly offset error d, promptly worked as the sensor measurement direction by apparatus measures centre of gyration o ₁, but measuring cross section bias internal d, and test specimen is measured cross section least square circle center o ₂With measurement centre of gyration o ₁Do not overlap, and the test specimen offset is when being e that what sensor measured is not with respect to measuring centre of gyration o ₁Polar radius r _i, but be nonlinear polar radius ρ with respect to transient measurement center o ' and enlargement ratio _i, corresponding measurement model can be write as:

ρ _i＝ecos(θ _i-α)+((r _o+Δr _i) ²-(d+esin(θ _i-α)) ²) ^1/2，i＝0，1，2，…，N-1 (1)

In the formula, ρ _iCertain is put to the distance of transient measurement center o ' on-test specimen circle contour; D-transducer probe assembly offset error; E-test specimen offset; α-eccentric angle; θ _i-angle position; r _o-least square radius of circle; Δ r _iCertain is put to the deviation of least square circle on the-test specimen circle contour; N-sampling number.

For the circle contour ultra precise measurement, it is key issue that the accurate estimation of above-mentioned model parameter is found the solution.Have according to formula (1)

Δr _i＝((ρ _i-ecos(θ _i-α)) ²+(d+esin(θ _i-α)) ²) ^1/2-r _o

(2)

i＝0，1，2，…，N-1

For finding the solution parameter (e, α, r _o, d), set up following objective function according to the principle of least square:

By formula (3) as can be known, objective function to be found the solution is that the unconstrained optimization of the complexity of nonlinear function is found the solution problem.For addressing this problem, this paper adopts optimized Algorithm directly to find the solution.Model parameter is being optimized in the solution procedure, for overcoming the multipole value problem that may occur, use the global optimizing algorithm, like this, can guarantee the local optimum ability of algorithm, can guarantee that again algorithm jumps out the local optimum point, and then in global scope, find optimum solution, be implemented in measurement model and Estimation of Parameters process are not done under the situation of any simplification, obtain the exact solution of each parameter.

Biased error is separated and is comprised two aspect contents.At first, adopt parameter optimization method that the double offset parameter circle contour measurement model is directly found the solution, (e is α) with associated arguments r in strict conformity with the skew parameter d of double offset parameter circle contour measurement model, eccentric parameter in acquisition _oAccurate estimated value

Then, with above-mentioned estimated value difference substitution circle contour expression formula and deviation from circular from expression formula, pointwise separates offset error parameter and eccentric error parameter simultaneously, that is:

$\left\{\begin{array}{c}\widehat{{\mathrm{\&rho;}}_{\mathrm{oi}}}={\left\{{[{\mathrm{\&rho;}}_i-\hat{e}\cos ({\mathrm{\&theta;}}_i-\widehat{\mathrm{\&alpha;}})]}^2+{[\hat{d}+\hat{e}\sin ({\mathrm{\&theta;}}_i-\widehat{\mathrm{\&alpha;}})]}^2\right\}}^{1/2}i=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N-1\\ \widehat{{\mathrm{\&Delta;r}}_i}=\widehat{{\mathrm{\&rho;}}_{\mathrm{oi}}}-\widehat{r_o}\end{array}\right.---\left(4\right)$

Separate biased error through above-mentioned pointwise and (contain the transducer probe assembly offset error

With the test specimen eccentric error (

) after, can obtain the circle contour error information of " pure "

For the influence of verificating sensor gauge head offset error d to the circle contour ultra precise measurement, formula (1) is carried out power series expansion, have:

${\mathrm{\&rho;}}_i=e\cos ({\mathrm{\&theta;}}_i-\mathrm{\&alpha;})+(r_o+\mathrm{\&Delta;}{r}_i)-\frac{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="C200810136908E00111.png"\; state="\{\{state\}\}">d</figure-callout>}}^2}{2(r_o+\mathrm{\&Delta;}{r}_i)}-\frac{\mathrm{de}}{(r_o+\mathrm{\&Delta;}{r}_i)}\sin ({\mathrm{\&theta;}}_i-\mathrm{\&alpha;})-\frac{e^2}{2(r_o+\mathrm{\&Delta;}{r}_i)}{\sin }^2({\mathrm{\&theta;}}_i-\mathrm{\&alpha;})\&CenterDot;\&CenterDot;\&CenterDot;\left(5\right)$

At first analyze first and second, these two irrelevant with transducer probe assembly side-play amount d.Through observing, can find easily that these two is the canonical form of traditional Limacon measurement model.Its measurement model is:

r _i＝ecos(θ _i-α)+r _o+Δr _i (6)

Then formula (5) subtracts formula (6),

$\left|{\mathrm{\&delta;}}_{\mathrm{ri}}\right|=|{\mathrm{\&rho;}}_i-r_i|=|\frac{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="C200810136908E00111.png"\; state="\{\{state\}\}">d</figure-callout>}}^2}{2(r_o+\mathrm{\&Delta;}{r}_i)}+\frac{\mathrm{de}}{(r_o+\mathrm{\&Delta;}{r}_i)}\sin ({\mathrm{\&theta;}}_i-\mathrm{\&alpha;})+\&CenterDot;\&CenterDot;\&CenterDot;|---\left(7\right)$

But preceding two of the formula that high spot reviews d plays a major role (7), by observing as can be known, during the circle contour ultra precise measurement, Δ r _iRelative r _oBe worth very for a short time, it is generally acknowledged e/r _o＜10 ^-3So,

Item is almost constant, can think very little to the influence of circle contour amplitude, can ignore; The changes in amplitude amount that the high-order term of back causes also can be ignored.And

Item exists eccentric amount e and side-play amount d simultaneously, and d has amplification to e.Generally, d is 10 of e ²～10 ⁴The order of magnitude, even bigger, visible d not only has linear amplification to eccentric amount e when have side-play amount d, and different with the action direction of item once, so measured circle contour distortion amplitude will produce very large variation, has a strong impact on measurement result to the end.Especially more remarkable for the circle contour measurement influence of minor diameter test specimen, carry out the emulation experiment checking, get offset and be respectively e=0.1 μ m, 0.5 μ m, 1.0 μ m, 2.0 μ m; Least square radius of circle r ₀=10000 μ m (10mm); Eccentric angle α=30 °; Transducer probe assembly side-play amount d=100 μ m, 500 μ m, 800 μ m, 1000 μ m; The circle contour sampling number is N=1024 weekly.Consider ultraprecise test specimen Δ r _i≤ 0.1 μ m, then $\frac{1}{r_o+\mathrm{\&Delta;}{r}_i}\&ap;\frac{1}{r_o}.$ Order $f(d,e,\mathrm{\&alpha;},{\mathrm{\&theta;}}_i)=\frac{\mathrm{de}\sin ({\mathrm{\&theta;}}_i-\mathrm{\&alpha;})}{r_o}(i=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;,1023),$ Its distortion changes in amplitude curve as shown in Figure 2.

It is all common when the roundness measurement of minor diameter test specimen that this analyzes employed numerical value.By curve shown in Figure 2 as can be known, when the circularity of minor diameter test specimen is measured, because the amplification of d makes

The numerical value of distortion amplitude is very big, so can cause very large measuring error.In the scope that may control usually, the error component that causes has reached 0.02 μ m～0.2 μ m magnitude.Under a lot of situations, the measuring error that causes because of the transducer probe assembly skew can account for 30%～50% of roundness measurement value, even bigger.

Correctness for checking double offset parameter circle contour measurement model and biased error separation method, carry out emulation experiment, take place one group and comprise transducer probe assembly offset error component d and test specimen eccentric error component (e simultaneously, data α), and adopt traditional Limacon measurement model and double offset parameter circle contour measurement model to find the solution the biasing parameter respectively, the emulated data method for generation is as follows:

If contain simultaneously gauge head side-play amount d and eccentric error (e, data generation function α) is:

ρ _i＝ecos(θ _i-α)+((r _o+Δr _i) ²-(d+esin(θ _i-α)) ²) ^1/2

＝acosθ _i+bsinθ _i+((r _o+Δr _i) ²-(d+asinθ _i-bcosθ _i) ²) ^1/2

i＝0，1，2，…，N-1

In the formula:

$\left\{\begin{array}{c}e={(a^2+b^2)}^{1/2}\\ \mathrm{\&alpha;}={\tan }^{-1}(b/a)\end{array}\right.$

Get N=1024, and make Δ r _i=0, get two groups of data and verify, first group be test specimen eccentric error and transducer probe assembly offset error hour: make r _o=10000 μ m, a=0.1 μ m, b=0.1 μ m, d=100 μ m at first utilizes the Limacon model to carry out the estimation of eccentric error, and estimation formulas is: $\widehat{r_o}=\frac{1}{N}\underset{i=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{r}_i,$ $\hat{a}=\frac{2}{N}\underset{i=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{r}_i\cos {\mathrm{\&theta;}}_i,$ $\hat{b}=\frac{2}{N}\underset{i=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{r}_i\sin {\mathrm{\&theta;}}_i,$ Calculate estimator $\widehat{r_o}=9999.4999\mathrm{\&mu;m},$ $\hat{a}=0.1010\mathrm{\&mu;m},$ $\hat{b}=0.0989\mathrm{\&mu;m};$ Utilize parameter optimization method of the present invention, and adopt the Levenberg-Marquardt global optimization approach, model parameter is accurately found the solution, obtain estimating to separate: $\widehat{r_o}=999.9999,$ $\hat{a}=0.1000\mathrm{\&mu;m},$ $\hat{b}=0.099\mathrm{\&mu;m};$ $\hat{d}=99.9995\mathrm{\&mu;m};$

Second group is test specimen eccentric error and transducer probe assembly offset error when big: r _o=10000 μ m, a=0.5 μ m, b=0.8 μ m, d=1000 μ m at first utilizes the Limacon model to carry out the estimation of eccentric error, obtains estimator after the calculating $\widehat{r_o}=9949.8743\mathrm{\&mu;m},$ $\hat{a}=0.5804\mathrm{\&mu;m},$ $\hat{b}=0.7497\mathrm{\&mu;m},$ Utilize method of the present invention, obtain estimating to separate: $\widehat{r_o}=10000.0036\mathrm{\&mu;m},$ $\hat{a}=0.4999\mathrm{\&mu;m},$ $\hat{b}=0.8\mathrm{\&mu;m},$ $\hat{d}=1000.0037\mathrm{\&mu;m}.$

Be not difficult to find out that from The simulation experiment result along with the increase gradually of transducer probe assembly offset error d and test specimen test specimen eccentric amount e, the estimated bias that Limacon model and method for parameter estimation thereof produce is increasing; And adopt double offset parameter circle contour error separating model of the present invention to reach the optimization of biasing parameter is found the solution, the biasing Estimation of Parameters value that obtains is not subjected to the influence of biased error size substantially, the value of finding the solution and theoretical value deviation are very little, and solving precision is significantly higher than the Limacon method.With the accurate estimated value of biasing parameter that adopts the inventive method to obtain In the substitution formula (4), can obtain the real cross section of test specimen circle contour, and then realize the ultra precise measurement of circle contour.

Claims (1)

1, a kind of biased error separation method of double offset parameter circle contour measurement model, this method may further comprise the steps:

1) propose to comprise the test specimen eccentric error (e, α) and the measurement model of two biased error components of transducer probe assembly offset error d be:

ρ _i＝ecos(θ _i-α)+((r _o+Δr _i) ²-(d+esin(θ _i-α)) ²) ^1/2，i＝0，1，2，…，N-1

In the formula, ρ _iCertain is put to the distance of transient measurement center o ' on-test specimen circle contour; D-transducer probe assembly offset error; E-test specimen offset; α-eccentric angle; θ _i-angle position; r _o-least square radius of circle; Δ r _iCertain is put to the deviation of least square circle on the-test specimen circle contour; N-sampling number;

It is characterized in that having according to above-mentioned measurement model

Δr _i＝((ρ _i-ecos(θ _i-α)) ²+(d+esin(θ _i-α)) ²) ^1/2-r _o

i＝0，1，2，…，N-1

2) set up objective function

3) adopt parameter optimization method to objective function Directly find the solution, obtain in strict conformity with the double offset parameter circle contour measurement model transducer probe assembly offset error d, test specimen eccentric error (e, α) and associated arguments r _oAccurate estimated value

4) with above-mentioned estimated value

Respectively substitution circle contour expression formula and deviation from circular from expression formula, pointwise separate simultaneously transducer probe assembly offset error d and test specimen eccentric error (e, α);

$\left\{\begin{array}{c}\widehat{{\mathrm{\&rho;}}_{\mathrm{oi}}}={\{{[{\mathrm{\&rho;}}_i-\hat{e}\cos ({\mathrm{\&theta;}}_i-\widehat{\mathrm{\&alpha;}}))}^2+{(\hat{d}+\hat{e}\sin ({\mathrm{\&theta;}}_i-\widehat{\mathrm{\&alpha;}})]}^2\}}^{1/2}i=\mathrm{0,1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N-1\\ \widehat{{\mathrm{\&Delta;r}}_i}=\widehat{{\mathrm{\&rho;}}_{\mathrm{oi}}}-\widehat{r_o}\end{array}\right.$

5) separate biased error through above-mentioned pointwise, promptly transducer probe assembly offset error parameter d and test specimen eccentric error parameter (e, α) after, can obtain the circle contour error information of " pure "

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