CN103791836A - Numerically controlled tool cutting edge measuring method based on laser scanning confocal technology - Google Patents

Abstract

The invention discloses a numerically controlled tool cutting edge measuring method based on a laser scanning confocal technology. The numerically controlled tool cutting edge measuring method is characterized by including the steps of preprocessing of a tool cutting edge, installation of a tool, acquisition of data of the tool cutting edge and postprocessing of the data. The preprocessing of the tool cutting edge includes the steps that the tool cutting edge is cleaned, and residual chips of the cutting edge after cutter sharpening are removed to guarantee measurement accuracy. The installation of the tool includes the steps that a gauge block with the appropriate angle is selected, and the tool is correctly installed on a clamp to guarantee that the cutting edge is located at the appropriate measurement position; the acquisition of the data of the tool cutting edge includes the steps that the tool cutting edge is positioned and focused through a laser scanning confocal microscope, and layering scanning is conducted to acquire three-dimensional images of the cutting edge. The postprocessing of the data includes the steps of data smooth denoising, data calibration and cutting edge transition circular arc fitting. Representation of the transition circular arc of the tool cutting edge is completed according to the cutting edge acquired by the laser confocal scanning microscope. The method has the advantage that measurement accuracy of the cutting edge can be effectively improved without damaging the structure of the tool.

Description

NC cutting tool cutting edge measuring method based on laser scanning co-focusing technology

Technical field

The present invention relates to a kind of measuring technique, especially a kind of contactless cutting edge roundness measuring technique, specifically a kind of NC cutting tool cutting edge measuring method based on laser scanning co-focusing technology.

Background technology

As everyone knows, cutter is after sharpening, and cutting edge can exist microdefect in various degree, and these microdefects finally can affect the cutting ability of cutter.Therefore, after tool sharpening, the general mode that adopts passivation is further processed cutting edge.When cutting edge passivation, the selection of tooth shape and cutting edge parameter is most important.At present, conventional tooth shape has negative chamfered edge, rounding and both mixing.Negative chamfered edge parameter and chamfering radius can produce material impact to cutting force, cutting temperature, cutter life and machined surface quality.

Pass judgment on the impact of cutting edge roundness on process, first will cutting edge be measured and be characterized.The accurate measurement of cutting edge roundness is more difficult, and existing measuring method mainly contains light cross-section method, sciagraphy, duplicating method and scanning electron microscopic observation method etc.Wherein light cross-section method adopts light-section microscope, measures loaded down with trivial details and not too accurate; Sciagraphy cuts shape to cutting edge and carries out projective amplification, then measures with osculating circle, can only measure larger radius value, and result is also not too accurate; Duplicating method is a kind of indirect measurement method, can be subject to the impact of the processes such as embedding sample, grinding, brings measuring error; The precision of scanning electron microscopic observation is higher, but must intercept cutting edge section, is a kind of damage type measuring method.

Confocal laser scanning microscope (LSCM) has overcome conventional optical microscope all shortcomings of imaging in addition of the structure of certain observed object depth scope, only to the structure imaging in focal plane, has avoided the impact of remainder, and precision is higher.When confocal laser scanning microscope imaging, object is divided into some optical faults, successively scanning imagery, has high depth resolution between layers, and image is clear.LSCM is a kind of desirable three-dimensional micro imaging system, can be used for the non-damage type precision measurement of cutting edge roundness.

Summary of the invention

The object of the invention is the problems such as the not high and economy of precision for existing cutting edge roundness measuring method is bad, invent a kind of cutting edge roundness measuring method based on laser scanning co-focusing technology, to improve blade measuring accuracy, for accurate sign and the follow-up cutting edge Study on mechanism of cutting edge roundness provide condition.

Technical scheme of the present invention is:

A NC cutting tool cutting edge measuring method based on laser scanning co-focusing technology, is characterized in that it comprises that it comprises the obtaining and the aftertreatment of data of installation, cutting edge roundness data of pre-service, the cutter of cutting edge roundness;

(1) pre-service of described cutting edge roundness refers to cutting edge roundness is cleaned, and removes the residual chip of blade after sharpening, guarantees measuring accuracy;

(2) installation of described cutter refers to that cutter is correctly installed on fixture, guarantees that cutting edge and both sides front and rear knife face have part to be scanned;

(3) described obtaining of cutting edge roundness data comprises:

Step 1: cutting edge roundness is positioned and focused; First select low power objective, regulate X to Y-direction position, cutter is moved on to observation place, regulate focus, until can clearly see cutting edge image on display;

Step 2: wheel measuring head, need to select high power objective according to actual measurement range, adopt the mode of automatically measuring, automatically adjust the displacement of camera lens Z-direction by controller, cutting edge is carried out to demixing scan, obtain the three-dimensional height image of blade;

(4) aftertreatment of described data comprises the following steps:

Step 1: data smoothing denoising;

Step 2: data calibration; According to the altitude information gathering, in the middle of intercepting near horizontal profile calculate the mounted angle of cutter;

Step 3: intercept cutting edge Partial Height data after treatment, carry out coordinate alignment, adopt least square fitting cutting edge circular arc.

Described cutting edge roundness pre-service refers to and in ultrasonic vibration cleaning machine, adds acetone reagent, and cutter is put into wherein and cleaned and be no less than 10 minutes.

Described sectional fixture comprises angle gauge block, locating piece, pressing plate, back-moving spring and trip bolt; Described angle gauge block cross sectional shape is right-angled trapezium, and the bevel angle θ of described angle gauge block is determined by following formula:

θ=(90°+α ₀+γ ₀)/2

α in formula ₀for tool orthogonal rake, γ ₀for tool clearance;

Departure within the scope of ± 5 °, described back-moving spring in the time of mounting cutter not in the raw, after mounting cutter in compressive state.

Described microscope is laser scanning co-focusing microscope, and total magnification is up to 16000, and height Measurement Resolution is not less than 0.005um, and measurement light source is 658nm red semiconductor laser; Described low power objective refers to 10 times of object lens, and high power objective enlargement factor is 20,50 or 100 times of object lens.

Described data smoothing denoising adopts average weighted method, and smoothing formula is as follows:

$h_i^{\′}=\frac{f_{i-2}{h}_{i-2}+f_{i-1}{h}_{i-1}+f_i{h}_i+f_{i+1}{h}_{i+1}+f_{i+2}{h}_{i+2}}{f_{i-2}+f_{i-1}+f_i+f_{i+1}+f_{i+2}}$

In formula: h ' _irepresent the height value after i measurement point smoothly;

H _irepresent that i measurement point is without level and smooth height value;

F _irepresent i the weighting coefficient that measurement point height is level and smooth.

Described Cutting tool installation manner inclination alpha is calculated as follows,

$\mathrm{\α}=\frac{\underset{i=1}{\overset{5}{\mathrm{\Σ}}}\mathrm{arctg}|{\mathrm{\Δh}}_{a_i{b}_i}/{\mathrm{\Δl}}_{a_i{b}_i}|}{5}$

Altitude information calibration formula h'=h-l × tg α calculates;

In formula:

for a _iand b _ithe difference in height of 2;

for a _iand b _ithe horizontal range of 2;

H' is the height value after measurement point calibration;

H is the height value of measurement point without calibration;

L is the horizontal range of the initial measurement point of current measurement point distance.

While adopting least square method to carry out cutting edge roundness circular fitting, the solution of matching residual equation and matching circular parameters are given by the following formula:

$\begin{array}{c}\left[\begin{array}{c}{a}_1\\ a_2\\ {\mathrm{<figure-callout\; id="3"\; label="a"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">a</figure-callout>}}_3\end{array}\right]={\left[\begin{array}{ccc}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}& n\\ \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i{y}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i^2& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i\\ \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i^2& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i{y}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\end{array}\right]}^{-1}\left[\begin{array}{c}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(x_i^2+y_i^2)\\ -\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i(x_i^2+y_i^2)\\ -\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i(x_i^2+y_i^2)\end{array}\right]\\ \left\{\begin{array}{c}{x}_c=-a_1/2\\ y_c=-a_2/2\\ r=\sqrt{a_1^2+a_2^2-{4a}_3}/2\end{array}\right.\end{array}:$

In formula, each parameter represents respectively:

A ₁, a ₂and a ₃for the component of matching residual equation solution;

X _ifor the horizontal ordinate that after Data Post, i is ordered;

Y _ifor the ordinate that after Data Post, i is ordered;

X _cfor the center of circle horizontal ordinate of matching circular arc;

Y _cfor the center of circle ordinate of matching circular arc;

R is the radius of matching circular arc;

N is measurement point quantity.

Beneficial effect of the present invention:

The present invention utilizes confocal laser scanning microscope to carry out demixing scan measurement to cutting edge roundness, measurement data is carried out to subsequent treatment simultaneously, for direct precision measurement and the sign of cutting tool cutting edge micromechanism provide effective means, can effectively improve measuring accuracy, do not destroy cutter structure simultaneously.

Accompanying drawing explanation

Fig. 1 is the structural representation of measurement mechanism of the present invention.

Fig. 2 is the structural representation of sectional fixture of the present invention.

Fig. 3 is flow chart of data processing figure of the present invention.

Fig. 4 is the cutting edge height image schematic diagram obtaining in the embodiment of the present invention.

Fig. 5 is data calibration schematic diagram of the present invention.

Fig. 6 is the final fitting result schematic diagram of the cutting edge roundness of the embodiment of the present invention.

Embodiment

Below in conjunction with drawings and Examples, the present invention is further illustrated.

As shown in Fig. 1-6.

A NC cutting tool radius of edge measuring method based on laser scanning co-focusing technology, it comprise pre-service, the cutter of cutting edge roundness the obtaining of installation, cutting edge roundness data, data aftertreatment, as shown in Figure 3.Wherein:

(1) pre-service of cutting edge roundness refers to cutting edge roundness is cleaned, and removes the residual chip of blade after sharpening, guarantees measuring accuracy.When concrete enforcement, in ultrasonic vibration cleaning machine, add acetone reagent, cutter is put into and wherein cleaned 10 minutes.

(2) installation of cutter refers to the gauge block of selecting proper angle, and cutter is correctly installed on fixture as shown in Figure 2, and fixture is arranged on the measurement mechanism shown in Fig. 1, guarantees that cutting edge and both sides front and rear knife face have part to be scanned.When installation, be main locating surface take angle gauge block inclined-plane, locating piece auxiliary positioning, is fixed on cutter on the inclined-plane of angle gauge block and adjusts and make blade perpendicular to gauge block front-back.Fixture in Fig. 2 comprises angle gauge block 5, locating piece 6, pressing plate 8, back-moving spring 9 and trip bolt 10.Described angle gauge block 5 bevel angles are pressed formula (1) according to the anterior angle of cutter 7 and relief angle and are determined, error is controlled at ± 5 ° within the scope of.Described back-moving spring 9 in the time of mounting cutter not in the raw, after mounting cutter in compressive state.

θ=(90°+α ₀+γ ₀)/2 （1）

In formula: α ₀for tool orthogonal rake, γ ₀for tool clearance.

Cutter 7 is arranged on fixture 3, when installation, promote pressing plate 8 along trip bolt 10 axis directions, take tool base as main locating surface, cutter is placed on the inclined-plane of angle gauge block 5, locating piece 6 is close to in blade to be measured base simultaneously, make blade to be measured in measuring position, put down pressing plate 8, clamp-on tool under spring-force driven dual.Sectional fixture 3 is placed on rotating disc 2, and in desired location.

(3) obtain (being DATA REASONING) of cutting edge roundness data as shown in Figure 4, it comprises:

Step 1: cutting edge roundness location and focusing: regulate rotating disc 2 to guarantee gauge block bottom surface minor face and microscope column parallel sided.The X that regulates confocal laser scanning microscope 1 to Y-direction position knob, cutter is moved on to position under object lens.Select 10 times of object lens, focus, until can clearly see cutting edge image on display.

Step 2: measurement range is set and automatically measured: wheel measuring head, need to select high power objective according to actual measurement range, object lens Z-direction height value when focusing and recording setting measurement scope up-and-down boundary place blur-free imaging, is made as respectively scanning upper limit position and the lower position of laser confocal scanning microscope 1 by it.Adopt the mode of automatically measuring, automatically adjust the displacement of camera lens Z-direction by controller, from top to down cutting edge is carried out to demixing scan, obtain the three-dimensional height image of blade.

Fig. 4 has shown the cutting edge height image obtaining in one embodiment of the invention.

(4) aftertreatment of data, comprises the following steps:

Data Post refers to that the cutting edge data to obtaining process to retain valid data, obtains measurement result.It can be divided into following step:

Step 1: data smoothing denoising: the some cloud level degrees of data of obtaining is carried out to smoothing denoising sound, level and smooth ± 2 average weighted methods that adopt, corresponding weighting coefficient is as shown in table 1, and computing formula is as the formula (2).Each pixel altitude information after level and smooth can calculate by the listed formula of table 2.

$h_i^{\&prime;}=\frac{f_{i-2}{h}_{i-2}+f_{i-1}{h}_{i-1}+f_i{h}_i+f_{i+1}{h}_{i+1}+f_{i+2}{h}_{i+2}}{f_{i-2}+f_{i-1}+f_i+f_{i+1}+f_{i+2}}---\left(2\right)$

Table 1 data smoothing weighting coefficient

Table 2 each point altitude information computing formula

Point present position	Each point altitude information h iComputing formula
		The first point	(17h 1+12h 2-3h 3)/26

Second point	(12h 1+17h 2+12h 3-3h 4)/38
		All the other point	(-3h i-2+12h i-1+17h i+12h i+1-3h i+2)/35
Point second from the bottom	(-3h i-2+12h i-1+17h i+12h i+1)/38
		Point last	(-3h i-2+12h i-1+17h i)/26

Step 2: data calibration: for eliminating the cutting edge altitude information error producing in Cutting tool installation manner process, altitude information after level and smooth is calibrated, when concrete enforcement, intercept as shown in Figure 5 near the horizontal profile centre recording, choose 10 totally 5 pairs of altitude informations, calculate mounted angle according to formula (3).Table 3 is selected point height and mounted angle in one embodiment of the invention.

$\mathrm{\&alpha;}=\frac{\underset{i=1}{\overset{5}{\mathrm{\&Sigma;}}}\mathrm{arctg}|{\mathrm{\&Delta;h}}_{a_i{b}_i}/{\mathrm{\&Delta;l}}_{a_i{b}_i}|}{5}---\left(3\right)$

Can calibrate by formula (4) the each point altitude information measuring,

h'=h-l×tgα （4）

The altitude information after calibration deducts the current measurement point distance horizontal range l of initial measurement point and the product of inclination angle tangent by the altitude information h before calibrating.Wherein initial measurement point is tilted to the left or is tilted to the right according to the horizontal profile of choosing, and chooses respectively the rightest and the most left measurement point.

Table 3 is for the selected point numerical value of data calibration

Step 3: matching cutting edge transition arc.The cutting edge roundness Partial Height data that intercept are carried out to coordinate alignment, when concrete enforcement, a selected plane right hand rectangular coordinate system, each point is processed rear height number as ordinate y numerical value, each point is carried out to setting coordinate from left to right simultaneously, origin coordinates is 0, and spacing is two measurement point spacing, determines successively each point horizontal ordinate x numerical value.

To completing the data of coordinate alignment, adopt least square method to carry out matching.The general equation of given first cutting edge fitting circle is as follows:

x ²+y ²+a ₁x+a ₂y+a ₃=0 （5）

In the situation that waiting precision redundant observation, obtain matching residual equation and be:

$\left[\begin{array}{c}{v}_1\\ {\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\\ .\\ .\\ .\\ v_n\end{array}\right]=\left[\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">y</figure-callout>}}_2& 1\\ & ...& \\ x_n& y_n& 1\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]-\left[\begin{array}{c}-(x_1^2+y_1^2)\\ -({\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">x</figure-callout>}}_2^2+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">y</figure-callout>}}_2^2)\\ ...\\ -({\mathrm{<figure-callout\; id="2"\; label="x\; n"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">x</figure-callout>}}_n^{}+y_n^2)\end{array}\right]---\left(6\right)$

This equation least square solution can obtain by its normal equation, is specially:

$\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]={\left[\begin{array}{ccc}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i& n\\ \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i{y}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i^2& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i\\ \underset{i=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="n\; x\; i"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}{x}_i^{}{}_2^{}& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i{y}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\end{array}\right]}^{-1}\left[\begin{array}{c}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}({\mathrm{<figure-callout\; id="2"\; label="x\; i"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">x</figure-callout>}}_i^{}+y_i^2)\\ -\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i(x_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">i</figure-callout>}}^2+y_i^2)\\ -\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i(x_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA0000463208030000011.png"\; state="\{\{state\}\}">i</figure-callout>}}^2+y_i^2)\end{array}\right]---\left(7\right)$

After solving method equation, the center of circle of fitting circle and radius can be tried to achieve by following formula

$\left\{\begin{array}{c}{x}_c=-a_1/2\\ y_c=-a_2/2\\ r=\sqrt{a_1^2+a_2^2-{4a}_3}/2\end{array}\right.---\left(8\right)$

Table 4 is rear blade measurement point data of coordinate alignment in one embodiment of the invention.

Blade measurement point data after the alignment of table 4 coordinate

Can carry out the matching of least square circular arc to measurement data according to formula (7) and (8), adopt MATLAB software to realize when enforcement, specific implementation source code is as follows:

Fig. 6 utilizes the final fitting result schematic diagram of the inventive method to cutting edge.

The part that the present invention does not relate to all prior art that maybe can adopt same as the prior art is realized.

Claims (7)

1. the NC cutting tool cutting edge measuring method based on laser scanning co-focusing technology, is characterized in that it comprises the obtaining and the aftertreatment of data of installation, cutting edge roundness data of pre-service, the cutter of cutting edge roundness;

(1) pre-service of described cutting edge roundness refers to cutting edge roundness is cleaned, and removes the residual chip of blade after sharpening, guarantees measuring accuracy;

(2) installation of described cutter refers to that cutter is correctly installed on fixture, guarantees that cutting edge and both sides front and rear knife face have part to be scanned;

(3) described obtaining of cutting edge roundness data comprises:

Step 1: cutting edge roundness is positioned and focused; First select low power objective, regulate X to Y-direction position, cutter is moved on to observation place, regulate focus, until can clearly see cutting edge image on display;

Step 2: wheel measuring head, need to select high power objective according to actual measurement range, adopt the mode of automatically measuring, automatically adjust the displacement of camera lens Z-direction by controller, cutting edge is carried out to demixing scan, obtain the three-dimensional height image of blade;

(4) aftertreatment of described data comprises the following steps:

Step 1: data smoothing denoising;

Step 2: data calibration; According to the altitude information gathering, in the middle of intercepting near horizontal profile calculate the mounted angle of cutter;

Step 3: intercept cutting edge Partial Height data after treatment, carry out coordinate alignment, adopt least square fitting cutting edge circular arc.

2. method according to claim 1, is characterized in that described cutting edge roundness pre-service refers to and in ultrasonic vibration cleaning machine, adds acetone reagent, and cutter is put into wherein and cleaned and be no less than 10 minutes.

3. method according to claim 1, is characterized in that described sectional fixture comprises angle gauge block, locating piece, pressing plate, back-moving spring and trip bolt; Described angle gauge block cross sectional shape is right-angled trapezium, and the bevel angle θ of described angle gauge block is determined by following formula:

θ=(90°+α ₀+γ ₀)/2

α in formula ₀for tool orthogonal rake, γ ₀for tool clearance;

Departure within the scope of ± 5 °, described back-moving spring in the time of mounting cutter not in the raw, after mounting cutter in compressive state.

4. method according to claim 1, is characterized in that described microscope is laser scanning co-focusing microscope, and total magnification is up to 16000, and height Measurement Resolution is not less than 0.005um, and measurement light source is 658nm red semiconductor laser; Described low power objective refers to 10 times of object lens, and high power objective enlargement factor is 20,50 or 100 times of object lens.

5. method according to claim 1, is characterized in that described data smoothing denoising adopts average weighted method, and smoothing formula is as follows:

$h_i^{\&prime;}=\frac{f_{i-2}{h}_{i-2}+f_{i-1}{h}_{i-1}+f_i{h}_i+f_{i+1}{h}_{i+1}+f_{i+2}{h}_{i+2}}{f_{i-2}+f_{i-1}+f_i+f_{i+1}+f_{i+2}}$

In formula: h ' _irepresent the height value after i measurement point smoothly;

H _irepresent that i measurement point is without level and smooth height value;

F _irepresent i the weighting coefficient that measurement point height is level and smooth.

6. method according to claim 1, the Cutting tool installation manner inclination alpha described in it is characterized in that is calculated as follows,

$\mathrm{\&alpha;}=\frac{\underset{i=1}{\overset{5}{\mathrm{\&Sigma;}}}\mathrm{arctg}|{\mathrm{\&Delta;h}}_{a_i{b}_i}/{\mathrm{\&Delta;l}}_{a_i{b}_i}|}{5}$

Altitude information calibration formula h'=h-l × tg α calculates;

In formula: for a _iand b _ithe difference in height of 2;

for a _iand b _ithe horizontal range of 2;

H' is the height value after measurement point calibration;

H is the height value of measurement point without calibration;

L is the horizontal range of the initial measurement point of current measurement point distance.

7. method according to claim 1, while it is characterized in that adopting least square method to carry out cutting edge roundness circular fitting, the solution of matching residual equation and matching circular parameters are given by the following formula:

$\begin{array}{c}\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]={\left[\begin{array}{ccc}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}& n\\ \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i{y}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i^2& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i\\ \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i^2& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i{y}_i& \underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\end{array}\right]}^{-1}\left[\begin{array}{c}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(x_i^2+y_i^2)\\ -\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_i(x_i^2+y_i^2)\\ -\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i(x_i^2+y_i^2)\end{array}\right]\\ \left\{\begin{array}{c}{x}_c=-a_1/2\\ y_c=-a_2/2\\ r=\sqrt{a_1^2+a_2^2-{4a}_3}/2\end{array}\right.\end{array}:$

In formula, each parameter represents respectively:

A ₁, a ₂and a ₃for the component of matching residual equation solution;

X _ifor the horizontal ordinate that after Data Post, i is ordered;

Y _ifor the ordinate that after Data Post, i is ordered;

X _cfor the center of circle horizontal ordinate of matching circular arc;

Y _cfor the center of circle ordinate of matching circular arc;

R is the radius of matching circular arc;

N is measurement point quantity.

Images