CN1927535A - Tracking method of on-line measuring roundness error and machine tool main axle error - Google Patents

Abstract

The invention relates to a method for online tracking and measuring the roundness error and the machine tool main spindle error. It measures the tracking roundness and the diameter making use of three displacement sensors in the online measuring device, and transforms the time-domain signal is collected when measuring into frequency domain signal then analyses the signal. It separates the roundness error coming from the eccentric rotations from system error online, so achieves online measure of the roundness error and system error and enhances the measurement accuracy.

Description

The method of tracking mode on-line measurement deviation from circular from and machine tool chief axis error

Technical field

The present invention relates to a kind of method that deviation from circular from and machine tool chief axis error are carried out on-line measurement, particularly a kind of off-centre operation is followed the tracks of the measuring method of deviation from circular from and machine tool chief axis error in the grinding.

Background technology

Deviation from circular from be meant circular pieces with the round degree of the planar interior surfaces shape of its axis normal, it belongs to macroscopical error in geometrical form, deviation from circular from directly affects the quality of fit of parts, running accuracy, friction, vibration, noise etc., thereby the high-precision rotary part has proposed more and more higher requirement to deviation from circular from.The tradition roundness fault separating method in use, it is motionless that sensor is installed in the fixed position, the rotation of detected workpiece wraparound commentaries on classics center, if therefore setting-up eccentricity is greater than transducer range, perhaps (workpiece centre does not overlap with the centre of gyration measured workpiece when pivot is done eccentric rotation, as the grinding crank-shaft link neck), just can not online deviation from circular from and machine tool system error separate measured workpiece.Along with people to product high accuracy, high efficiency pursuit, workpiece to be machined is implemented on-line measurement, and deviation from circular from can be separated with the machine tool system error, not only can improve certainty of measurement, and the data after separating can also be used for the compensation control of digital control processing, help improving the machining accuracy and the efficient of product.

Summary of the invention

The objective of the invention is to propose the method for a kind of tracking mode on-line measurement deviation from circular from and machine tool chief axis error, when the off-centre operation workpiece is followed the tracks of grinding, pursuit movement by three displacement transducers, obtain the redundancy on workpiece to be machined surface, and set up dynamic 3 circularity errors and separate equation, realize roundness error of workpiece and machine tool chief axis error online measuring.

For achieving the above object, the present invention adopts following technical proposals:

The method of a kind of tracking mode on-line measurement deviation from circular from and machine tool chief axis error, it is characterized in that transforming to frequency domain analysis according to the time-domain signal that three displacement transducers in tracking mode circularity and the diameter on-line measurement mechanism collect in tracking measurement, the online deviation from circular from that will make the eccentric workpiece that rotatablely moves separates with systematic error.

As shown in Figure 1, above-mentioned tracking mode circularity includes the measuring mechanism pedestal 1 of fixedlying connected with grinding carriage with diameter on-line measurement mechanism, and a tracking rocking arm 3 of swinging with emery wheel 2 coaxial lines is installed on the measuring mechanism pedestal 1.Follow the tracks of the outer end of rocking arm 3 three displacement transducer 4,5,6 contact workpieces 7 cylindricals realization measurements are installed, middle sensor 5 measured points should be positioned on the line of emery wheel 2 centers and grinding workpiece 7 central points.

The schematic diagram of the dynamic three point measurement methods of deviation from circular from as shown in Figure 2.Three sensor P are set respectively ₁, P ₂, P ₃, and allow three extension sensor lines intersect at the A point, promptly joining overlaps with the crank-shaft link neck center.When beginning to measure, sensor P ₁Be positioned at horizontal level, and with crank-shaft link neck center A, emery wheel center O ₁Be in sight alignment.If P ₁P ₂Between angle be  ₂, P ₂P ₃Between angle be  ₃, working space and the requirement of linear sensor correlation during in conjunction with on-line measurement, the setting angle of three displacement transducers adopts two kinds of schemes as shown in Table 1 among the present invention:

Table one sensor setting angle

The above-mentioned tracking mode on-line measurement deviation from circular from and the method for machine tool chief axis error, concrete steps are:

(1) respectively three displacement transducers 4,5,6 are gathered signal and carry out the back summation of name power, draw dynamic 3 circularity error measuring values, its formula is:

H(θ _n)＝r ₁(θ _n)+k ₂r ₂(θ _n+φ ₂)+k ₃r ₃(θ _n+φ ₂+φ ₃)

+R _Y(θ _n)[cosβ+k ₂cos(φ ₂+β)+k ₃cos(φ ₂+φ ₃+β)] (1)

+R _X(θ _n)[sinβ+k ₂sin(φ ₂+β)+k ₃sin(φ ₂+φ ₃+β)]

K in the formula (1) ₂, k ₃Be respectively the weight coefficient of lower sensor 4,6;  ₂For angle being installed,  between lower sensor 6 and the middle sensor 5 ₃For between upper sensor 4 and the lower sensor 6 angle being installed;

α be measured workpiece 7 around the pivot O anglec of rotation, β is for measuring rocking arm 3 around emery wheel 2 center O ₁Pendulum angle, RSin α=(R _S+ R _W) Sin β, wherein R is a crank-shaft link neck center eccentricity, R _SBe emery wheel 2 radiuses, R _WBe the crank-shaft link neck radius; Three displacement transducers 4,5,6 and the workpiece 7 surfaces angle θ that relatively moves _n=alpha+beta, n is a surface of the work sampling number weekly in the formula;

(2) solve weight coefficient $k_{}$ ${}_2=-\frac{\sin ({\mathrm{\&phi;}}_2+{\mathrm{\&phi;}}_3)}{\sin {\mathrm{\&phi;}}_3},{\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">k</figure-callout>}}_3=\frac{\sin {\mathrm{\&phi;}}_2}{\sin {\mathrm{\&phi;}}_3},$ Obtain weight function G (k)=1+k after in the fundamental formular of dynamic 3 the circularity error measuring values of substitution ₂E ^{J2 π pk/N}+ k ₃E ^{J2 π (p+q) k/N}, in the formula, p, q are integers, and: $p=\frac{{\mathrm{\&phi;}}_2\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{\&pi;}},q=\frac{{\mathrm{\&phi;}}_3\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{\&pi;}},$ N-is the isogonism sampling number weekly, this moment n=0,1,2.....N-1;

(3) solve deviation from circular from sequence and machine tool chief axis rotation error sequence at last:

$r\left(n\right)=F^{-1}\left\{\frac{H\left(k\right)}{G\left(k\right)}\right\}$

$R_X\left(n\right)=\frac{\cos \mathrm{\&beta;}}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]-\frac{\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]$

$R_Y\left(n\right)=\frac{{\sin \mathrm{\&phi;}}_2-\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\cos \mathrm{\&beta;}\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]+\frac{1}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]$

R in the formula (n) represents the deviation from circular from sequence; R _X(n) and R _Y(n) represent spindle rotation error X in rectangular coordinate system, the discrete series of Y both direction projection.

Principle of the present invention is as follows:

With the A point is the center, with sensor P ₁The dead in line direction is a Y ' axle, sets up dynamic coordinate system X ' AY '.In the tracking measurement process, measured workpiece is measured rocking arm around the emery wheel center O in the time of spindle axis O rotation alpha angle ₁Swing β angle.Relation between α angle and the β angle can be used formula (1) expression, and wherein R represents crank-shaft link neck center eccentric throw, R _sThe expression grinding wheel radius, R _wExpression crank-shaft link neck radius:

Rsinα＝(R _s+R _w)sinβ (1)

Suppose that deviation from circular from is r _i(θ _n), and R _X(θ _n) and R _Y(θ _n) be respectively the projection components of machine tool chief axis Radial Motion Error on X and Y-axis, H _i(θ _n) represent the reading of i sensor.Wherein (i=1,2,3), n represents surface of the work sampling number weekly; θ _n=alpha+beta, representative are when n sampled point, and with respect to initial position, sensor is in measured workpiece surface move angle size.Set up following formula:

H ₁(θ _n)＝r ₁(θ _n)+R _Y(θ _n)·cosβ+R _X(θ _n)·sinβ (2)

H ₂(θ _n)＝r ₂(θ _n+φ ₂)+R _Y(θ _n)·cos(φ ₂+β)+R _X(θ _n)·sin(φ ₂+β) (3)

H ₃(θ _n)＝r ₃(θ _n+φ ₂+φ ₃)+R _Y(θ _n)·cos(φ ₂+φ ₃+β)+R _X(θ _n)·sin(φ ₃+φ ₃+β) (4)

If sensor P ₁, P ₂, P ₃Weight coefficient be respectively 1, k ₂, k ₃, and make three sensor output signals and H (θ _n)=H ₁(θ _n)+k ₂H ₂(θ _n)+k ₃H ₃(θ _n), so draw the fundamental formular of dynamic line-of-sight course roundness error measurement:

H(θ _n)＝r ₁(θ _n)+k ₂r ₂(θ _n+φ ₂)+k ₃r ₃(θ _n+φ ₂+φ ₃)

+R _Y(θ _n)[cosβ+k ₂cos(φ ₂+β)+k ₃cos(φ ₂+φ ₃+β)] (5)

+R _X(θ _n)[sinβ+k ₂sin(φ ₂+β)+k ₃sin(φ ₂+φ ₃+β)]

In order to eliminate R _X(θ _n) and R _Y(θ _n) influence, can choose suitable k by finding the solution following equation ₂, k ₃Realize:

$\left\{\begin{array}{c}\cos \mathrm{\&beta;}+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">k</figure-callout>}}_2\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})+{\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">k</figure-callout>}}_3\cos ({\mathrm{\&phi;}}_2+{\mathrm{\&phi;}}_3+\mathrm{\&beta;})=0\\ \sin \mathrm{\&beta;}+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">k</figure-callout>}}_2\sin ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})+k_3\sin ({\mathrm{\&phi;}}_2+{\mathrm{\&phi;}}_3+\mathrm{\&beta;})=0\end{array}\right.---\left(6\right)$

Finding the solution as above, equation group can draw:

$k_2=-\frac{\sin ({\mathrm{\&phi;}}_2+{\mathrm{\&phi;}}_3)}{\sin {\mathrm{\&phi;}}_3},{\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">k</figure-callout>}}_3=\frac{\sin {\mathrm{\&phi;}}_2}{\sin {\mathrm{\&phi;}}_3}---\left(7\right)$

(7) formula substitution (5) formula is got:

H(θ _n)＝r ₁(θ _n)+k ₂r ₂(θ _n+φ ₂)+k ₃r ₃(θ _n+φ ₂+φ ₃) (8)

With formula (8) discretization:

H(n)＝r ₁(n)+k ₂r ₂(n+p)+k ₃r ₃(n+p+q) (9)

In the formula (9), p, q are integers, and: $p=\frac{{\mathrm{\&phi;}}_2\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{\&pi;}},q=\frac{{\mathrm{\&phi;}}_3\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{\&pi;}},$ N-is the isogonism sampling number weekly, this moment n=0,1,2.....N-1; Formula (9) both sides are done FFT and are got:

H(k)＝F{r(k)}G(k) (10)

F{r in the following formula (k) } represent the deviation from circular from sequence after discrete is carried out Fourier transform, G (k)=1+k ₂E ^{J2 π pk/N}+ k ₃E ^{J2 π (p+q) k/N}, k=0,1,2.....N-1 is called weight function.When the function G of holding power (k) is non-vanishing, have:

$F\left\{r\left(k\right)\right\}=\frac{H\left(k\right)}{G\left(k\right)}---\left(11\right)$

By inverse fast fourier transform, can obtain the discrete series of deviation from circular from again

$r\left(n\right)=F^{-1}\{\frac{H\left(k\right)}{G\left(k\right)}(n=\mathrm{0,1,2}.....N-1)---\left(12\right)$

In the formula $F^{-1}\left\{\frac{H\left(k\right)}{G\left(k\right)}\right\}$ It is right to represent $\frac{H\left(k\right)}{G\left(k\right)}$ Carry out inverse-Fourier transform.

The discrete form of deviation from circular from sequence r (n) the difference substitution formula (2) (3) (4) that solves in (12), can obtain spindle rotation error X in rectangular coordinate system, the discrete series R of Y both direction projection then _X(n) and R _Y(n):

$R_X\left(n\right)=\frac{\cos \mathrm{\&beta;}}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]-\frac{\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]---\left(13\right)$

$R_Y\left(n\right)=\frac{{\sin \mathrm{\&phi;}}_2-\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\cos \mathrm{\&beta;}\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]+\frac{1}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]---\left(14\right)$

Just can calculate the deviation from circular from and the machine tool chief axis rotation error of measured workpiece respectively by formula (12), (13) and (14), thereby reach the result that deviation from circular from is separated with systematic error.

The present invention compared with prior art, the present invention has following outstanding substantive distinguishing features and remarkable advantage: calculate simple, solved the rotatablely move deviation from circular from on-line measurement problem of workpiece of off-centre of doing, if measuring rocking arm maintains static, when being pendulum angle β identically vanishing, the deviation from circular from that just can be generalized to common axial workpiece is measured with the machine tool chief axis on-line error and is separated.

Description of drawings

Fig. 1 is a tracking measurement principle schematic of the present invention.

Fig. 2 is the dynamic line-of-sight course principle schematic of deviation from circular from of the present invention.

The specific embodiment

Details are as follows in conjunction with the accompanying drawings for a preferred embodiment of the present invention:

The method of this tracking mode on-line measurement deviation from circular from and machine tool chief axis error is to transform to frequency domain analysis according to the time-domain signal that three displacement transducers in tracking mode circularity and the straight line on-line measurement mechanism collect in tracking measurement, and the online deviation from circular from that will make the eccentric workpiece that rotatablely moves separates with systematic error.

Referring to Fig. 1, above-mentioned tracking mode circularity includes the measuring mechanism pedestal 1 of fixedlying connected with grinding carriage with diameter on-line measurement mechanism, a tracking rocking arm 3 of swinging with emery wheel 2 coaxial lines is installed on the measuring mechanism pedestal 1, follow the tracks of the outer end of rocking arm 3 three displacement transducer 4,5,6 contact workpieces 7 cylindricals realization measurements are installed, the point that a middle sensor 5 is surveyed should be positioned on the line of emery wheel 2 centers and grinding workpiece 7 central points.

Referring to Fig. 2, above-mentioned three sensors, 4,5, the 6 extended lines A that intersects at a point, promptly joining overlaps with the crank-shaft link neck center of workpiece 7; Below established angle  between sensor 6 and the intermediate sensor 5 ₂Be 90 degree, and the established angle  between a top sensor 4 and the following sensor 6 ₃Be 181.05 degree, perhaps  ₂Be 60.82 the degree and  ₃Be 240.12 degree.

The method of this tracking mode on-line measurement deviation from circular from and machine tool chief axis error, concrete steps are:

(4) respectively three displacement transducers 4,5,6 are gathered signal and carry out the back summation of name power, draw dynamic 3 circularity error measuring values, its formula is:

H(θ _n)＝r ₁(θ _n)+k ₂r ₂(θ _n+φ ₂)+k ₃r ₃(θ _n+φ ₂+φ ₃)

+R _Y(θ _n)[cosβ+k ₂cos(φ ₂+β)+k ₃cos(φ ₃+φ ₃+β}] (1)

+R _X(θ _n)[sinβ+k ₂sin(φ ₂+β)+k ₃sin(φ ₂+φ ₃+β)]

K in the formula (1) ₂, k ₃Be respectively the weight coefficient of lower sensor 4,6;  ₂For angle being installed,  between lower sensor 6 and the middle sensor 5 ₃For between upper sensor 4 and the lower sensor 6 angle being installed;

α be measured workpiece 7 around the pivot O anglec of rotation, β is for measuring rocking arm 3 around emery wheel (2) center O ₁Pendulum angle, RSin α=(R _S+ R _W) Sin β, wherein R is a crank-shaft link neck center eccentricity, R _S Be emery wheel 2 radiuses, R _WBe the crank-shaft link neck radius; Three displacement transducers 4,5,6 and the workpiece 7 surfaces angle θ n=alpha+beta that relatively moves, n is a surface of the work sampling number weekly in the formula;

(5) solve weight coefficient $k_{}$ ${}_2=-\frac{\sin ({\mathrm{\&phi;}}_2+{\mathrm{\&phi;}}_3)}{\sin {\mathrm{\&phi;}}_3},{\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">k</figure-callout>}}_3=\frac{\sin {\mathrm{\&phi;}}_2}{\sin {\mathrm{\&phi;}}_3},$ Obtain weight function G (k)=1+k after in the fundamental formular of dynamic 3 the circularity error measuring values of substitution ₂E ^{J2 π pk/N}+ k ₃E ^{J2 π (p+q) k/N}, in the formula, p, q are integers, and: $p=\frac{{\mathrm{\&phi;}}_2\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{\&pi;}},q=\frac{{\mathrm{\&phi;}}_3\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20061002923900101.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{\&pi;}},$ N-is the isogonism sampling number weekly, this moment n=0,1,2.....N-1;

(6) solve deviation from circular from sequence and machine tool chief axis rotation error sequence at last:

$r\left(n\right)=F^{-1}\left\{\frac{H\left(k\right)}{G\left(k\right)}\right\}$

$R_X\left(n\right)=\frac{\cos \mathrm{\&beta;}}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]-\frac{\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]$

$R_Y=\frac{\sin {\mathrm{\&phi;}}_2-\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\cos \mathrm{\&beta;}\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]+\frac{1}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_n)]$

R in the formula (n) represents the deviation from circular from sequence; R _X(n) and R _Y(n) represent spindle rotation error X in rectangular coordinate system, the discrete series of Y both direction projection.

Claims (4)

1. the method for tracking mode on-line measurement deviation from circular from and machine tool chief axis error, it is characterized in that transforming to frequency domain analysis according to the time-domain signal that three displacement transducers in tracking mode circularity and the diameter on-line measurement mechanism collect in tracking measurement, the online deviation from circular from that will make the eccentric workpiece that rotatablely moves separates with systematic error.

2. the method for tracking mode on-line measurement deviation from circular from according to claim 1 and machine tool chief axis error, it is characterized in that described tracking mode circularity and diameter on-line measurement mechanism include the measuring mechanism pedestal (1) of fixedlying connected with grinding carriage, measuring mechanism pedestal (1) is gone up a tracking rocking arm (3) of swinging with emery wheel (2) coaxial line is installed, follow the tracks of the outer end of rocking arm (3) three displacement transducers (4 are installed, 5,6) contact workpiece (7) cylindrical is realized measuring, and the point that a middle sensor (5) is surveyed should be positioned on the line of emery wheel (2) center and grinding workpiece (7) central point.

3. the method for tracking mode on-line measurement deviation from circular from according to claim 2 and machine tool chief axis error, it is characterized in that described three sensors (4,5,6) extended line intersects at a point (A), promptly joining overlaps with the crank-shaft link neck center of workpiece (7); Below established angle  between a sensor (6) and the intermediate sensor (5) ₂Be 90 degree, and the established angle  between a top sensor (4) and the following sensor (6) ₃Be 181.05 degree, perhaps  ₂Be 60.82 the degree and  ₃Be 240.12 degree.

4. the method for tracking mode on-line measurement deviation from circular from according to claim 1 and machine tool chief axis error is characterized in that concrete steps are:

A) respectively three displacement transducers (4,5,6) are gathered signal and carry out the back summation of name power, draw dynamic 3 circularity error measuring values, its formula is:

H(θ _n)＝r ₁(θ _n)+k ₂r ₂(θ _n+φ ₂)+k ₃r ₃(θ _n+φ ₂+φ ₃)

+R _Y(θ _n)[cosβ+k ₂cos(φ ₂+β)+k ₃cos(φ ₂+φ ₃+β)] (1)

+R _X(θ _n)[sinβ+k ₂sin(φ ₂+β)+k ₃sin(φ ₂+φ ₃+β)]

K in the formula (1) ₂, k ₃Be respectively the weight coefficient of lower sensor (4,6);  ₂For angle being installed,  between lower sensor (6) and the middle sensor (5) ₃For between upper sensor (4) and the lower sensor (6) angle being installed; α be measured workpiece (7) around pivot (O) anglec of rotation, β is for measuring rocking arm (3) around emery wheel (2) center (O ₁) pendulum angle, RSin α=(R _S+ R _W) Sin β, wherein R is a crank-shaft link neck center eccentricity, R _SBe emery wheel (2) radius, R _WBe the crank-shaft link neck radius; Three displacement transducers (4,5,6) and workpiece (7) the surface angle θ that relatively moves _n=alpha+beta, n is a surface of the work sampling number weekly in the formula;

B) solve weight coefficient $k_2=-\frac{\sin ({\mathrm{\&phi;}}_2+{\mathrm{\&phi;}}_3)}{\sin {\mathrm{\&phi;}}_3},k_3=\frac{\sin {\mathrm{\&phi;}}_2}{\sin {\mathrm{\&phi;}}_3},$ Obtain weight function G (k)=1+k after in the fundamental formular of dynamic 3 the circularity error measuring values of substitution ₂E ^{J2 π pk/N}+ k ₃E ^{J2 π (p+q) k/N}, in the formula, p, q are integers, and: $p=\frac{{\mathrm{\&phi;}}_{2}N}{2\mathrm{\&pi;}},q=\frac{{\mathrm{\&phi;}}_{3}N}{2\mathrm{\&pi;}},$ N-is the isogonism sampling number weekly, this moment n=0,1,2.....N-1;

C) solve deviation from circular from sequence and machine tool chief axis rotation error sequence at last:

$r\left(n\right)=F^{-1}\left\{\frac{H\left(k\right)}{G\left(k\right)}\right\}$

$R_X\left(n\right)=\frac{\cos \mathrm{\&beta;}}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]-\frac{\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]$

$R_Y\left(n\right)=\frac{\sin {\mathrm{\&phi;}}_2-\cos ({\mathrm{\&phi;}}_2+\mathrm{\&beta;})}{\cos \mathrm{\&beta;}\sin {\mathrm{\&phi;}}_2}[H_1\left(n\right)-r_1\left(n\right)]+\frac{1}{\sin {\mathrm{\&phi;}}_2}[H_2\left(n\right)-r_2(n+{\mathrm{\&phi;}}_2)]$

R in the formula (n) represents the deviation from circular from sequence; R _X(n) and R _Y(n) represent spindle rotation error X in rectangular coordinate system, the discrete series of Y both direction projection.

Images