CN104898564B - A kind of method for reducing three-shaft linkage profile errors - Google Patents

Abstract

A kind of method for reducing three-shaft linkage profile errors of the present invention belongs to complex curved surface parts precise high-efficiency manufacture field, is related to a kind of processing feed speed to plan again and is combined with profile errors precompensation, the method for reducing Digit Control Machine Tool three-shaft linkage profile errors.The method processes feed speed and preferable cutter location information in extracting numerical control code first, with each processing feed shaft plus acceleration limit, acceleration limiting as constraints, each program segment processing feed speed in numerical control program is planned again；And the method for using cumulative inborn parameter space cubic spline approximately expecting profile, estimate profile errors value；Each feed shaft profile errors compensation rate, the cutter location after being pre-compensated for are calculated using the method for Taylor series expansion；Said process iterative cycles, the machining code after being pre-compensated for realizes the high-quality and high-efficiency processing of complex curved surface parts.The present invention is simple and practical, can give full play to Digit Control Machine Tool performance, improves multi-shaft interlocked contour accuracy.

Description

A kind of method for reducing three-shaft linkage profile errors

Technical field

The invention belongs to complex curved surface parts precise high-efficiency manufacture field, more particularly to a kind of processing feed speed is planned again It is combined with a kind of multi-shaft interlocked profile errors precompensation, the method for reducing three-shaft linkage profile errors.

Background technology

Variable curvature curved surface part with complicated face structure is widely used in high-end equipment field.With Important Project The fast development in field and the continuous propulsion of high-end equipment batch product process, such crucial complex curved surface parts is not only to profile essence Degree has high requirement, and the demand to production efficiency is also increasingly lifted.To improve processing efficiency, current many high-grade, digitally controlled machine tools Larger main motion and processing feed motion speed can be provided.However, due to lathe servo feed system lagging characteristics and The factors such as feed shaft mechanical inertia, each operation axle can produce larger following error when being processed using feed speed high, and then induce Multi-shaft interlocked profile errors under continuous path operation cooked mode；Further, since deep camber knife rail when complex curved surface parts are processed Presence and machine tool motion performance limitation, processing feed shaft actual motion speed do not reach command speed value easily so that Exacerbate multi-shaft interlocked profile errors.When above-mentioned phenomenon can cause High-speed machining variable curvature complex parts, the performance of Digit Control Machine Tool Cannot be not fully exerted, the contour accuracy of part cannot meet requirement.Therefore, the precise high-efficiency of complex curved surface parts is processed As industry problem.

To solve this problem, the multi-shaft interlocked profile errors in feed speed processing high get the attention.Document “Feed speed scheduling method for parts with rapidly varied geometric feature Based on drive constraint of NC machine tool ", Zhenyuan Jia etc., International Journal of Machine Tools and Manufacture, 2014,87：73-88, the document is directed to and becomes with deep camber The Curve Machining profile errors problem of change, proposes a kind of subregion variable element control precision machining method, by set up profile errors with Relation between geometric properties, processing feed speed, classifying rationally is carried out to machining area, and distribute different processing to each subregion Feed speed, realizes variable curvature curve high accuracy and processes.However, the method is merely by planning processing feed speed reduction wheel Wide error, the influence to processing efficiency is larger.Document " Generalized Taylor series expansion for Free-form two-dimensional contour error compensation ", Feng Huo etc., International Journal of Machine Tools and Manufacture, 2012,53：91-99, the document is carried Go out a kind of popularization Taylor series expansion profile errors compensation method, missed by the profile for estimating free curve each moment in sampling period Difference, using the method calculation error offset of Taylor series expansion, for profile errors compensation, effectively increases 2-axis linkage certainly By Curve Machining contour accuracy, but it is not generalized in the middle of three-shaft linkage space curve.Similarly, document " profile of spatial curves Error real-time estimation is studied with compensation method ", Xiao Xiaoping etc., Sichuan University's journal (engineering science version), 2015,47 (1)：215- 222, the document predicts servo-drive system subsequent time output valve and estimates profile errors using transform, realizes space curve wheel Wide real-time error compensation.However, the method not only needs the accurate model of servo feed system, also need to change digital control system knot Structure, increases compensator, and this realizes difficulty for Highgrade integration Digit Control Machine Tool；Additionally, the method is mainly used in compensation watching Take parameter and lose a profile errors for inducing, it is impossible to reduce machine tooling feed shaft produced many under continuous path operational mode Axle linkage profile errors.

The content of the invention

It is contemplated that overcoming prior art defect, a kind of method for reducing three-shaft linkage profile errors is invented, will processed Feed speed is planned to be pre-compensated for profile errors and combined, effectively reduces when Digit Control Machine Tool processing feed shaft continuous path runs Three-shaft linkage profile errors.

The technical scheme is that a kind of method for reducing three-shaft linkage profile errors, its characteristic is that the method is first First to extract and process feed speed and preferable cutter location information in numerical control code, it is considered to processed under lathe continuous path operational mode into To the prediction characteristic of speed, with each processing feed shaft plus acceleration limit, acceleration limiting as constraints, in numerical control program Each program segment processing feed speed is planned again；Secondly, actual cutter location is estimated using stable state following error model, and is utilized The method that cumulative inborn parameter space cubic spline approximately expects profile, estimates profile errors value；Using Taylor series expansion Method calculates each feed shaft profile errors compensation rate, the cutter location after being pre-compensated for；Said process iterative cycles, you can obtain Cutter location after the planning post-processing feed speed of whole piece knife rail and compensation, so as to the machining code after being pre-compensated for, realizes multiple The high-quality and high-efficiency processing of miscellaneous curved surface part.Overall flow is comprised the following steps that referring to accompanying drawing 1：

1) processing feed speed is planned again

If i-th cutter location extracted in numerical control machining code is R_i(Rx_i,Ry_i,Rz_i), i-th program segment directive processing Feed speed is f_i, 1≤i≤n, n are cutter location sum, it is considered to before processing feed speed under continuous path operation cooked mode Look forward or upwards algorithm, at each cutter location preferable processing feed speed direction should with the desired profile trajectory tangential of the point, to avoid the need for The prior information of machining profile Mathematical Modeling, using vectorDirection as R_iLocate the near of profile traces tangential direction Seemingly, therefore, the preferable processing feed speed vector v of the point_iFor：

In formula,Make κ axles In, κ=x, y, z represent processing feed shaft X, Y, Z, then R_iThe preferable processing feed speed component v of place κ axles_{κ_i}For：

If κ axle plus acceleration limits areAcceleration limiting isJudge i-th program segment theory process timeIt is interior,WithConstraint under, can κ axle reality processings feed speed from v_{κ_i-1}Plus/minus speed arrives v_{κ_i}, If can not, program segment κ axles processing feed speed component is optimized, if the feed speed component after optimization isShould Meet theoretical process time after optimizationIt is interior andWithConstraint under, κ axle reality processings feed speed can be from v_{κ_i-1}Plus/minus speed is arrivedComputational methods be：

Using optimizing post-processing feed speed componentSynthesize processing feed speed f after calculation optimization_i ^p, finally, to keep away Exempt from speed big ups and downs, repeatedly smoothed feed speed is processed using B-spline curve method, obtain finally planning again Feed speed afterwards is f_i ^s；

2) profile errors are estimated

First, intended using cumulative inborn parameter space cubic spline curve between the adjacent cutter location of any two on knife rail Close and expect machining profile, obtain R_i-1And R_iBetween approximate expectation profile traces s_iIt is expressed as：

s_i=S (u_i-1,u_i,u) (4)

Wherein, parameter u ∈ [u_i-1,u_i], u_iComputational methods be：

Secondly, P is used_iI-th actual cutter location is represented, its coordinate is P_i(Px_i,Py_i,Pz_i), order A is natural number, is judged：

The minimum a values that inequality (6) will be met are designated as m, in matched curve s_mUpper utilization " dichotomy " is found apart from P_iRecently Point Q_i(Qx_i,Qy_i,Qz_i), then the profile errors vector estimate ε at the cutter location_i(ε_{x_i},ε_{y_i},ε_{z_i}) be：

3) profile errors precompensation

First, according to the processing feed speed f after planning again_i ^sAnd stable state following error model, estimation continuous path fortune Correspond to preferable cutter location R during row_iActual cutter location P_i(Px_i,Py_i,Pz_i), its coordinate representation is preferable cutter spacing point coordinates Function：

The expression of each function is：

In formula, Kx, Ky, Kz are respectively the servo gain of X, Y, Z feed shaft control system；

Secondly, the profile errors component ε of each processing feeding direction of principal axis is calculated_{x_i}、ε_{y_i}、ε_{z_i}；Finally utilize Taylor series exhibition Open method and calculate each feed shaft profile errors compensation rate, if profile errors compensation rate is Δ R_i=[Δ Rx_i ΔRy_i ΔRz_i]^T, then According to formula (8), actual cutter location after compensationFor：

By above formula in point (Rx_i,Ry_i,Rz_i) place carries out first order Taylor series expansion and obtain：

To realize that profile errors are compensated, the actual cutter spacing point coordinates P κ of κ axles before and after compensation_i、Exist with profile errors vector κ axle components ε_{κ_i}Between should meet relation：

Formula (8) and formula (12) are substituted into formula (11), and omits higher-order shear deformation to obtain：

Therefore, at i-th cutter location, profile errors compensation rate Δ R_iFor：

4) said process iterative cycles, you can obtain profile errors compensation rate at each cutter location, and then refer to after being compensated Make cutter locationFor：

Using the processing feed speed f after planning again_i ^sAnd the cutter location after precompensationNumerical control adds after generation compensation Work code, for reality processing.

The beneficial effects of the invention are as follows：(1) the feeding speed under Digit Control Machine Tool continuous path operation cooked mode is taken into full account Degree prediction characteristic and feed shaft kinematics characteristic are planned have again to giving full play to machine tool capability to processing feed speed Significance；(2) without feed servo-system accurate model and machining profile mathematic(al) representation, only by machining code Realize that profile errors are pre-compensated for, highly versatile；(3) processing feed speed is planned again and is combined with cutter location precompensation method, it is right Reducing space profiles error has important directive significance.

Brief description of the drawings

Fig. 1 --- method overall flow figure；

Fig. 2 --- space astroid geometrical model figure；

Fig. 3 --- processing feed speed plan again, profile errors precompensation before profile errors value；Wherein, A axles represent knife Site sequence number, B axle represents profile errors absolute value, and unit is mm

Fig. 4 --- processing feed speed plan again, profile errors precompensation after profile errors value；Wherein, A axles represent knife Site sequence number, B axle represents profile errors absolute value, and unit is mm

Specific embodiment

Combination technology scheme describes specific embodiment of the invention in detail with accompanying drawing.

During using feed speed processed complex curved surface part high, presence and Digit Control Machine Tool dynamic due to variable curvature knife rail The factors such as characteristic, processing feed shaft mechanical inertia, it may appear that larger following error and reality processing feed speed easily reaches Less than the phenomenon of command speed value, so as to induce multi-shaft interlocked profile errors, it is unfavorable for the precision of variable curvature complex curved surface parts Highly-efficient processing.Accordingly, the multi-shaft interlocked profile errors in being processed for feed speed high, have invented a kind of reduction three-shaft linkage wheel The method of wide error.

By taking the space astroid knife rail that non-homogeneous B spline curve is represented as an example, calculate and imitate by MATLAB softwares Very, implementation process of the present invention is described in detail, overall flow is referring to accompanying drawing 1.

1) as shown in Figure 1, first, using NX/CAM softwares, with constant processing feed speed 50mm/s, space star is generated The initial numerical control machining code of shape line, the non-uniform rational B-spline parameter of space astroid is, exponent number：3, control point：[(0,0, 0), (24,12,10), (24,32,20), (48,16,10), (72,20,0), (54,0, -20), (72, -20, -10), (48, - 16,10), (24, -32, -10), (24, -12,5), (0,0,0)], weight factor：[1,1,1,1,0.7,1,0.7,1,1,1,1], section Point vector：[0,0,0,1/9,2/9,3/9,4/9,5/9,6/9,7/9,8/9,1,1,1], its geometrical model is referring to accompanying drawing 2；

2) cutter location R is extracted from the numerical control machining code for being generated_iAnd each program segment processing feed speed f_i, obtain Cutter location sum n=321；Take each processing feed shaft plus acceleration limit and acceleration limiting is respectivelyUsing step 1 in technical scheme) method, Consider processing feed speed prediction characteristic, be constraint with machine tooling feed shaft acceleration, acceleration limiting, calculating is planned again Each program segment processing feed speed f afterwards_i ^s, 1≤i≤321；

3) at i-th cutter location, calculated using formula (8) and formula (9) and correspond to theoretical cutter location R_iActual knife Site P_iCoordinate；Using cumulative inborn parameter space Cubic Spline Fitting ideal cutter location, machining profile is approximately expected；

4) according to technical scheme steps 2) method and formula (7) calculate P_iLocate profile errors vector ε_i(ε_{x_i},ε_{y_i}, ε_{z_i}), calculate dot profile error compensation amount Δ R using formula (14)_i, the value of each partial derivative is in formula：

Cutter location after compensating is calculated according to formula (15)Coordinate

5) judge whether i is equal to n, if, using cutter location after compensationInstead of mending Repay preceding cutter location R_i(Rx_i,Ry_i,Rz_i), substitute into formula (8) and calculate actual cutter location P after compensation_i ^comCoordinateI=i+1 is made, using actual cutter location after compensationInstead of Actual cutter location P before compensation_i-1(Px_i-1,Py_i-1,Pz_i-1), return to step 3), calculate next actual cutter location；Said process Iterative cycles, when i=n, that is, obtain at whole piece each cutter location of knife rail cutter spacing point coordinates after profile errors compensation rate and compensation1≤i≤321；

6) according to the processing feed speed f after planning again_i ^s, cutter location behind 1≤i≤321, and compensation1≤i≤ 321, numerical control machining code after generation compensation, for reality processing.Compensation before and compensation after profile errors simulation result referring to Accompanying drawing 3 and accompanying drawing 4, as a result show, profile errors maximum is reduced to 0.025mm from 0.053mm, using three axle of the invention Linkage space profiles error reduction method, can effectively reduce multi-shaft interlocked knife rail profile errors, improve machining profile precision.

It is actual when the present invention is for feed speed lower feeding axle continuous path high operation processing variable curvature complex curved surface parts Processing feed speed does not reach command speed value and the larger problem of knife rail profile errors easily, has invented processing feed speed Plan again and be combined with profile errors precompensation, the method for reducing three-shaft linkage profile errors, to numerical control machine tool capability Play and the precise high-efficiency processing of high-performance complex curved surface parts has Important Project Practical significance.

Claims (1)

1. a kind of method for reducing three-shaft linkage profile errors, its characteristic is, the method is processed in extracting numerical control code first Feed speed and preferable cutter location information, are processed feed speed and are added with each processing feed shaft under lathe continuous path operational mode Speed limit, acceleration limiting are constraints, and each program segment processing feed speed in numerical control program is planned again；Its It is secondary, actual cutter location is estimated using stable state following error model, and approximately expect using cumulative inborn parameter space cubic spline The method of profile, estimates profile errors value；Each feed shaft profile errors compensation rate is calculated using the method for Taylor series expansion, is obtained Cutter location after to precompensation；Said process iterative cycles, after obtaining planning post-processing feed speed and the compensation of whole piece knife rail Cutter location；So as to the machining code after being pre-compensated for, the high-quality and high-efficiency processing of complex curved surface parts is realized；Specific steps are such as Under：

1) processing feed speed is planned again

If in numerical control machining code, i-th cutter location of extraction is R_i(Rx_i,Ry_i,Rz_i), i-th program segment directive processing feeding Speed is f_i, 1≤i≤n, n are cutter location sum, feed speed are processed under continuous path operation cooked mode, in each cutter location The preferable processing feed speed direction in place and the desired profile trajectory tangential of the point, using vectorDirection as R_iPlace's wheel The approximate direction in wide orbit tangent direction, therefore, the preferable processing feed speed vector v of the point_iFor：

In formula,In making κ axles, κ =x, y, z, represent processing feed shaft X, Y, Z, then R_iThe preferable processing feed speed component v of place κ axles_{κ_i}For：

If κ axle plus acceleration limits areAcceleration limiting isJudge i-th program segment theory process timeIt is interior,WithConstraint under, can κ axle reality processings feed speed from v_{κ_i-1}Plus/minus speed arrives v_{κ_i}, If can not, program segment κ axles processing feed speed component is optimized, if the feed speed component after optimization isShould Meet theoretical process time after optimizationIt is interior andWithConstraint under, κ axle reality processings feed speed can be from v_{κ_i-1}Plus/minus speed is arrived Computational methods be：

Using optimizing post-processing feed speed componentSynthesize processing feed speed f after calculation optimization_i ^p, finally, to avoid speed Big ups and downs, are repeatedly smoothed using B-spline curve method by feed speed is processed, entering after obtaining finally planning again It is f to speed_i ^s；

2) profile errors are estimated

First, the cumulative inborn parameter space cubic spline interpolation phase is utilized between the adjacent cutter location of any two on knife rail Machining profile is hoped, R is obtained_i-1And R_iBetween approximate expectation profile traces s_iIt is expressed as：

s_i=S (u_i-1,u_i,u) (4)

Wherein, parameter u ∈ [u_i-1,u_i], u_iComputational methods be：

Secondly, P is used_iI-th actual cutter location is represented, its coordinate is P_i(Px_i,Py_i,Pz_i), ordera It is natural number, judges：

${\&dtri;}_i\left(R_{i-a}\right)\&CenterDot;{\&dtri;}_i\left(R_{i-a-1}\right)<0---\left(6\right)$

The minimum a values that inequality (6) will be met are designated as m, in matched curve s_mUpper utilization " dichotomy " is found apart from P_iNearest point Q_i(Qx_i,Qy_i,Qz_i), then the profile errors vector estimate ε at the cutter location_i(ε_{x_i},ε_{y_i},ε_{z_i}) be：

$\left\{\begin{array}{c}{\mathrm{\&epsiv;}}_{x\_i}={\mathrm{Qx}}_i-{\mathrm{Px}}_i\\ {\mathrm{\&epsiv;}}_{y\_i}={\mathrm{Qy}}_i-{\mathrm{Py}}_i\\ {\mathrm{\&epsiv;}}_{z\_i}={\mathrm{Qz}}_i-{\mathrm{Pz}}_i\end{array}\right.---\left(7\right)$

3) profile errors precompensation

First, according to the processing feed speed f after planning again_i ^sAnd stable state following error model, when estimation continuous path runs Corresponding to preferable cutter location R_iActual cutter location P_i(Px_i,Py_i,Pz_i), its coordinate representation is the function of preferable cutter spacing point coordinates：

$\left\{\begin{array}{c}{\mathrm{Px}}_i={\mathrm{fx}}_i({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)\\ {\mathrm{Py}}_i={\mathrm{fy}}_i({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)\\ {\mathrm{Pz}}_i={\mathrm{fz}}_i({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)\end{array}\right.---\left(8\right)$

The expression of each function is：

$\left\{\begin{array}{c}{\mathrm{fx}}_i(x,y,z)=x-\frac{f_i^s(x-{\mathrm{Px}}_{i-1})}{Kx\sqrt{{(x-{\mathrm{Px}}_{i-1})}^2+{(y-{\mathrm{Py}}_{i-1})}^2+{(z-{\mathrm{Pz}}_{i-1})}^2}}\\ {\mathrm{fy}}_i(x,y,z)=y-\frac{f_i^s(y-{\mathrm{Py}}_{i-1})}{Ky\sqrt{{(x-{\mathrm{Px}}_{i-1})}^2+{(y-{\mathrm{Py}}_{i-1})}^2+{(z-{\mathrm{Pz}}_{i-1})}^2}}\\ {\mathrm{fz}}_i(x,y,z)=z-\frac{f_i^s(z-{\mathrm{Pz}}_{i-1})}{Kx\sqrt{{(x-{\mathrm{Px}}_{i-1})}^2+{(y-{\mathrm{Py}}_{i-1})}^2+{(z-{\mathrm{Pz}}_{i-1})}^2}}\end{array}\right.---\left(9\right)$

In formula, Kx, Ky, Kz are respectively the servo gain of X, Y, Z feed shaft control system；

Secondly, the profile errors component ε of each processing feeding direction of principal axis is calculated_{x_i}、ε_{y_i}、ε_{z_i}；Finally utilize Taylor series expansion method Each feed shaft profile errors compensation rate is calculated, if profile errors compensation rate is Δ R_i=[Δ Rx_i ΔRy_i ΔRz_i]^T, then basis Formula (8), actual cutter location after compensationFor：

$\left\{\begin{array}{c}{\mathrm{Px}}_i^{com}={\mathrm{fx}}_i({\mathrm{Rx}}_i+{\mathrm{\&Delta;Rx}}_i,{\mathrm{Ry}}_i+{\mathrm{\&Delta;Ry}}_i,{\mathrm{Rz}}_i+{\mathrm{\&Delta;Rz}}_i)\\ {\mathrm{Py}}_i^{com}={\mathrm{fy}}_i({\mathrm{Rx}}_i+{\mathrm{\&Delta;Rx}}_i,{\mathrm{Ry}}_i+{\mathrm{\&Delta;Ry}}_i,{\mathrm{Rz}}_i+{\mathrm{\&Delta;Rz}}_i)\\ {\mathrm{Pz}}_i^{com}={\mathrm{fz}}_i({\mathrm{Rx}}_i+{\mathrm{\&Delta;Rx}}_i,{\mathrm{Ry}}_i+{\mathrm{\&Delta;Ry}}_i,{\mathrm{Rz}}_i+{\mathrm{\&Delta;Rz}}_i)\end{array}\right.---\left(10\right)$

By above formula in point (Rx_i,Ry_i,Rz_i) place carries out first order Taylor series expansion and obtain：

$\left\{\begin{array}{c}{\mathrm{Px}}_i^{com}\&ap;{\mathrm{fx}}_i({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)+\frac{\&part;{\mathrm{fx}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Rx}}_i+\frac{\&part;{\mathrm{fx}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Ry}}_i+\frac{\&part;{\mathrm{fx}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Rz}}_i\\ {\mathrm{Py}}_i^{com}\&ap;{\mathrm{fy}}_i({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)+\frac{\&part;{\mathrm{fy}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Rx}}_i+\frac{\&part;{\mathrm{fy}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Ry}}_i+\frac{\&part;{\mathrm{fy}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Rz}}_i\\ {\mathrm{Pz}}_i^{com}\&ap;{\mathrm{fz}}_i({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)+\frac{\&part;{\mathrm{fz}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Rx}}_i+\frac{\&part;{\mathrm{fz}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Ry}}_i+\frac{\&part;{\mathrm{fz}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}{\mathrm{\&Delta;Rz}}_i\end{array}\right.---\left(11\right)$

To realize that profile errors are compensated, the actual cutter spacing point coordinates P κ of κ axles before and after compensation_i、With profile errors vector in κ axles Component ε_{κ_i}Between should meet relation：

${\mathrm{P\&kappa;}}_i^{com}-{\mathrm{P\&kappa;}}_i={\mathrm{\&epsiv;}}_{\mathrm{\&kappa;}\_i}---\left(12\right)$

Formula (8) and formula (12) are substituted into formula (11), and omits higher-order shear deformation to obtain：

$\left[\begin{array}{c}{\mathrm{\&epsiv;}}_{x\_i}\\ {\mathrm{\&epsiv;}}_{y\_i}\\ {\mathrm{\&epsiv;}}_{z\_i}\end{array}\right]=\left[\begin{array}{ccc}\frac{\&part;{\mathrm{fx}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fx}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fx}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}\\ \frac{\&part;{\mathrm{fy}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fy}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fy}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}\\ \frac{\&part;{\mathrm{fz}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fz}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fz}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{\mathrm{\&Delta;Rx}}_i\\ {\mathrm{\&Delta;Ry}}_i\\ {\mathrm{\&Delta;Rz}}_i\end{array}\right]---\left(13\right)$

Therefore, at i-th cutter location, profile errors compensation rate Δ R_iFor：

${\mathrm{\&Delta;R}}_i=\left[\begin{array}{c}{\mathrm{\&Delta;Rx}}_i\\ {\mathrm{\&Delta;Ry}}_i\\ {\mathrm{\&Delta;Rz}}_i\end{array}\right]={\left[\begin{array}{ccc}\frac{\&part;{\mathrm{fx}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fx}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fx}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}\\ \frac{\&part;{\mathrm{fy}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fy}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fy}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}\\ \frac{\&part;{\mathrm{fz}}_i}{\&part;x}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fz}}_i}{\&part;y}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}& \frac{\&part;{\mathrm{fz}}_i}{\&part;z}{|}_{({\mathrm{Rx}}_i,{\mathrm{Ry}}_i,{\mathrm{Rz}}_i)}\end{array}\right]}^{-1}\&CenterDot;\left[\begin{array}{c}{\mathrm{\&epsiv;}}_{x\_i}\\ {\mathrm{\&epsiv;}}_{y\_i}\\ {\mathrm{\&epsiv;}}_{z\_i}\end{array}\right]---\left(14\right)$

4) said process iterative cycles, you can obtain profile errors compensation rate at each cutter location, and then knife is instructed after being compensated SiteFor：

$\left\{\begin{array}{c}{\mathrm{Rx}}_i^{com}={\mathrm{Rx}}_i+{\mathrm{\&Delta;Rx}}_i\\ {\mathrm{Ry}}_i^{com}={\mathrm{Ry}}_i+{\mathrm{\&Delta;Ry}}_i\\ {\mathrm{Rz}}_i^{com}={\mathrm{Rz}}_i+{\mathrm{\&Delta;Rz}}_i\end{array}\right.---\left(15\right)$

Using the processing feed speed f after planning again_i ^sAnd the cutter location after precompensationDigital control processing generation after generation compensation Code, for reality processing.