CN110597183A

CN110597183A - Efficient compensation method for gear grinding key errors

Info

Publication number: CN110597183A
Application number: CN201910752446.2A
Authority: CN
Inventors: 夏长久; 王时龙; 肖雨亮; 康玲; 马驰; 王四宝; 周杰; 黄筱调
Original assignee: Chongqing University
Current assignee: Chongqing University
Priority date: 2019-08-15
Filing date: 2019-08-15
Publication date: 2019-12-20
Anticipated expiration: 2039-08-15
Also published as: CN110597183B

Abstract

The invention discloses a high-efficiency compensation method for key errors of gear grinding, which comprises the steps of firstly, constructing an actual forward kinematics model of gear grinding processing based on geometric error distribution of a forming gear grinding machine tool and an actual kinematic chain of the machine tool, and reflecting the functional relation between tool position and pose in a tool coordinate system and tool position data in a workpiece coordinate system under the influence of geometric errors; then, based on the actual reverse kinematics compensation principle, deducing an analytical expression of the actual motion command of the motion axis after error compensation, and revealing a mapping rule between the geometric error, the ideal tool bit data and the actual motion command; and finally, according to a conjugate grinding principle, establishing a geometric error-tooth surface error model, calculating and evaluating actual tooth profile and tooth direction accuracy, identifying a key error source of tooth profile deviation, simplifying an actual reverse kinematic compensation method, and realizing efficient error compensation facing tooth profile deviation reduction.

Description

Efficient compensation method for gear grinding key errors

Technical Field

The invention relates to the technical field of numerical control machine tool error analysis and precision control, in particular to a gear grinding key error efficient compensation method.

Background

The numerical control forming gear grinding machine is a special machine tool for gear finish machining, and the gear grinding precision is cooperatively influenced by multi-source errors, including machine tool geometric errors, thermal errors, force errors, servo control errors and the like. Wherein, the geometric error is regarded as quasi-static error, does not change with time or changes slightly, can be compensated and eliminated.

In order to compensate for geometric errors of multi-axis machine tools, experts and scholars have proposed a number of methods including a differential operator decoupling method, an iterative regression calculation method, a differential error prediction method, and the like. The existing error compensation method is low in efficiency, mainly focuses on solving a terminal error vector of a tool relative to a workpiece, and does not pay attention to the specific influence of the terminal error vector on the machined workpiece.

In addition, a practical inverse kinematics compensation method aiming at geometric errors of a common five-axis numerical control machine tool is also provided by a learner, the geometric errors are compensated in a post-processing process, and the compensation efficiency of the method is high. However, although the machine tool structure of the forming gear grinding machine as a special gear processing machine tool is not special, due to the problems that the machining characteristics such as spiral grinding and conjugate contact and the like and the gear precision evaluation index has specificity and the like, a geometric error compensation method based on actual inverse kinematics for the forming gear grinding machine is not available at present.

In addition, the existing actual inverse kinematics method compensates for the influence of geometric errors on tool bit data, and because the actual gear grinding process is not considered, the final improvement of the evaluation index of the accuracy of a single gear grinding is not necessarily completely ideal.

Disclosure of Invention

In view of this, the present invention provides a method for efficiently compensating a gear grinding key error, which can derive an analytical expression of an actual motion command of a motion axis after error compensation, and reveal a mapping rule between a geometric error, ideal tool position data and the actual motion command; and key error sources of gear precision evaluation bases such as tooth profile precision, tooth direction precision and the like are analyzed, so that the traditional actual reverse kinematics method is simplified, and tooth grinding errors such as tooth profile deviation, tooth direction deviation and the like are efficiently compensated.

In order to achieve the purpose, the invention provides the following technical scheme:

the invention provides a high-efficiency compensation method for key errors of gear grinding, which comprises the following steps:

the method comprises the following steps: modeling geometric errors of a forming grinding system;

(1) geometric error analysis of a forming grinding system: determining the full kinematic chain of the machine tool and the geometric error of the forming gear grinding machine according to the basic structure of the forming gear grinding machine tool; the machine tool full motion chain comprises a workpiece chain RCw from an RCS reference coordinate system to a WCS workpiece coordinate system and a tool chain RXAYt from the RCS reference coordinate system to a TCS tool coordinate system;

(2) constructing an actual forward kinematics model as follows:

establishing an actual forward kinematic model of the workpiece chain:

establishing an actual forward kinematic model of the tool chain:

establishing an actual forward kinematics model of the full kinematic chain of the gear grinding machine tool:

step two: geometric error compensation method based on actual inverse kinematics

(1) Carrying out a subsequent error compensation strategy based on an actual reverse kinematics compensation principle;

(2) acquiring an analytic expression of the actual motion instruction of the rotating shaft: solving an actual motion instruction analytical expression of the rotating shaft by utilizing a mapping relation between ideal cutter shaft vector data and machine tool geometric errors;

(3) acquiring an analysis expression of the actual motion instruction of the linear axis: solving the actual motion instruction analytic expression of the linear axis by utilizing the mapping relation between the ideal cutter position data and the geometric error of the machine tool according to the solved actual motion instruction analytic expression of the rotating axis;

step three: key error identification and compensation model simplification

(1) Geometric error-tooth surface error model

(2) Respectively carrying out identification analysis on the tooth profile deviation and the spiral line deviation on key error sources;

(3) the key error efficient compensation method comprises the following steps: and according to the actual motion instruction analytic expression of the motion axis, obtaining the numerical control code after compensation of the corresponding key error source, and realizing efficient error compensation for tooth profile deviation reduction.

Further, the geometric errors in the first step include position-independent geometric errors PIGEs and position-dependent geometric errors PDGEs.

Further, the actual forward kinematics model is constructed according to the following steps:

calculating an ideal pose transformation matrix between adjacent component coordinate systems: i.e. by successive multiplication of the installation matrix T_pQNAnd motion matrix T_mQNObtaining;

calculating an actual pose transformation matrix: i.e. by successive multiplication of the installation matrix T_pQNMounting error matrix T_peQNMotion matrix T_mQNAnd a motion error matrix T_meQNObtaining;

the actual forward kinematics model of the workpiece chain is calculated according to the following formula:

wherein the content of the first and second substances,

an actual transformation matrix representing the WCS (workpiece coordinate system) to RCS (reference coordinate system);

T_RCan actual transformation matrix representing a C-axis coordinate system to RCS (reference coordinate system);

T_CWan actual transformation matrix representing the WCS (workpiece coordinate system) to the C-axis coordinate system;

T_pRCa mounting matrix representing a C-axis coordinate system to an RCS (reference coordinate system);

T_peRCrepresenting the C-axis coordinate system to RCS (reference coordinate system)A standard system) of mounting error matrices;

T_mRCa motion matrix representing a C-axis coordinate system to an RCS (reference coordinate system);

T_meRCa motion error matrix representing a C-axis coordinate system to an RCS (reference coordinate system);

T_pCWa mounting matrix representing WCS (workpiece coordinate system) to C-axis coordinate system;

T_peCWa mounting error matrix representing the WCS (workpiece coordinate system) to the C-axis coordinate system;

T_mCWa motion matrix representing the WCS (workpiece coordinate system) to the C-axis coordinate system;

T_meCWa motion error matrix representing the WCS (workpiece coordinate system) to the C-axis coordinate system;

the actual forward kinematics model of the tool chain is calculated according to the following formula:

wherein the content of the first and second substances,

an actual transformation matrix representing TCS (tool coordinate system) to RCS (reference coordinate system);

T_RXan actual transformation matrix representing an X-axis coordinate system to an RCS (reference coordinate system);

T_XZrepresenting an actual transformation matrix from a Z-axis coordinate system to an X-axis coordinate system;

T_ZArepresenting an actual transformation matrix from an A-axis coordinate system to a Z-axis coordinate system;

T_AYrepresenting an actual transformation matrix from a Y-axis coordinate system to an A-axis coordinate system;

T_YTan actual transformation matrix representing TCS (tool coordinate system) to Y-axis coordinate system;

T_pRXa mounting matrix representing an X-axis coordinate system to an RCS (reference coordinate system);

T_peRXrepresenting mounting errors of an X-axis coordinate system to an RCS (reference coordinate system)A matrix;

T_mRXa motion matrix representing an X-axis coordinate system to an RCS (reference coordinate system);

T_meRXa motion error matrix representing an X-axis coordinate system to an RCS (reference coordinate system);

T_pXZrepresenting an installation matrix from a Z-axis coordinate system to an X-axis coordinate system;

T_peXZrepresenting an installation error matrix from a Z-axis coordinate system to an X-axis coordinate system;

T_mXZrepresenting a motion matrix from a Z-axis coordinate system to an X-axis coordinate system;

T_meXZrepresenting a motion error matrix from a Z-axis coordinate system to an X-axis coordinate system;

T_pZArepresenting an installation matrix from an A-axis coordinate system to a Z-axis coordinate system;

T_peZArepresenting an installation error matrix from an A-axis coordinate system to a Z-axis coordinate system;

T_mZArepresenting a motion matrix from an A-axis coordinate system to a Z-axis coordinate system;

T_meZArepresenting a motion error matrix from an A-axis coordinate system to a Z-axis coordinate system;

T_pAYrepresenting an installation matrix from a Y-axis coordinate system to an A-axis coordinate system;

T_peAYrepresenting an installation error matrix from a Y-axis coordinate system to an A-axis coordinate system;

T_mAYrepresenting a motion matrix from a Y-axis coordinate system to an A-axis coordinate system;

T_meAYrepresenting a motion error matrix from a Y-axis coordinate system to an A-axis coordinate system;

T_pYTa mounting matrix representing TCS (tool coordinate System) to Y-axis coordinate System;

T_peYTa mounting error matrix representing TCS (tool coordinate system) to Y-axis coordinate system;

T_mYTa motion matrix representing TCS (tool coordinate system) to Y-axis coordinate system;

T_meYTa motion error matrix representing TCS (tool coordinate system) to Y-axis coordinate system;

wherein the content of the first and second substances,

b₁₁～b₃₄after the matrix operation is represented, the matrix is calculated,the value of each element;

calculating and establishing an actual forward kinematic model of the full kinematic chain of the gear grinding machine according to the following formula:

wherein the content of the first and second substances,

an actual transformation matrix representing TCS (tool coordinate system) to WCS (workpiece coordinate system);

calculating the tool position data under the influence of the geometric error according to the following formula:

wherein the content of the first and second substances,

Q′_w＝[i′,j′,k′]^Trepresenting actual arbor vector data;

P_w′＝[x′,y′,z′]^Trepresenting actual tip position data;

Q_tin presentation of TCSThe knife axis vector of (a);

P_tindicating the tip position in TCS;

i ', j ', k ' represents the x, y, z coordinates of the actual arbor vector data;

x ', y ', z ' represent the x, y, z coordinates of the actual nose position data.

Further, the subsequent error compensation strategy in the second step is performed as follows:

the ideal tool position data is guaranteed to be unchanged, the motion instruction of the motion axis is considered to be changed under the influence of geometric errors, and the actual motion instruction value after error compensation can be reversely solved according to the formula, so that the actual machining code is obtained, and error compensation is realized.

Further, the second step further comprises the following steps:

determining ideal cutter position data, wherein the ideal cutter position data comprises cutter position data and cutter axis vector data;

calculating the mapping relation between the cutter position data and the cutter axis vector data according to the following formula:

wherein the content of the first and second substances,

P_w＝[x,y,z]^Trepresenting ideal nose position data;

Q_w＝[i,j,k]^Trepresenting ideal arbor vector data;

P_t＝[0,0,0]^Tindicating the tip position in TCS;

Q_t＝[0,0,1]^Trepresenting the arbor vector in TCS.

Further, the analytic expression of the actual motion instruction of the rotating shaft in the step two is obtained according to the following mode:

solving an A-axis actual motion instruction and a C-axis actual motion instruction of the rotating shaft by utilizing a mapping relation between ideal cutter shaft vector data and machine tool geometric errors;

calculating an analytic expression of the A-axis actual motion command according to the following formula:

wherein the content of the first and second substances,

a represents an A-axis actual motion command;

arccos () represents an inverse cosine function;

ε_y(C) representing the y-direction angle error of the C-axis motion;

ε_x(C) representing the x-direction angle error of the C-axis movement;

ε_x(X) represents X-direction angular error of X-axis motion;

ε_x(Y) represents the x-direction angular error of the Y-axis motion;

ε_x(Z) represents the x-direction angular error of the Z-axis motion;

ε_x(A) representing the x-direction angle error of the A-axis movement;

S_YZrepresenting Y, Z the perpendicularity error between the axes;

α_CYrepresenting the angle error of the C axis installation in the x direction;

representing the tangent angle;

k represents an ideal cutter axis vector data z coordinate;

j represents the y coordinate of ideal cutter axis vector data;

i represents an ideal cutter axis vector data x coordinate;

or

Calculating the analytic expression of the C-axis actual motion command according to the following formula:

wherein the content of the first and second substances,

c represents an actual motion command of a C axis;

arcsin () represents an arcsine function;

ε_y(Y) represents the Y-direction angular error of the Y-axis motion;

ε_y(A) representing the angle error of the movement y direction of the A axis;

ε_y(Z) represents the Z axis motion y-direction angle error;

ε_y(X) represents the X-axis motion y-direction angular error;

ε_y(C) representing the y-direction angle error of the C-axis motion;

ε_z(C) representing the z-direction angle error of the C-axis motion;

ε_z(Z) represents Z-direction angular error of Z-axis motion;

ε_z(X) represents the z-direction angular error of the X-axis motion;

ε_x(C) representing the x-direction angle error of the C-axis movement;

β_CXrepresenting the angle error of the C axis installation in the y direction;

S_ZXrepresenting Z, X the perpendicularity error between the axes;

β_AZthe angle error of the axis A in the mounting y direction is shown;

γ_AYthe Z-direction angle error of the A-axis installation is shown;

or

Calculating an analytical expression of the actual motion instruction of the linear axis according to the following formula, wherein the analytical expression of the actual motion instruction of the linear axis comprises an analytical expression of an actual motion instruction of the Y axis, an analytical expression of an actual motion instruction of the Z axis and an analytical expression of an actual motion instruction of the X axis;

the analytical expression of the Y-axis actual motion instruction is as follows:

Y＝{-sinC(x+zε_y(C)-yε_z(C)+δ_x(C))-cosC(y+xε_z(C)-zε_x(C)+δ_y(C))

-(δ_z(A)+δ_z(Y))sinA+(δ_y(A)+δ_y(Y))cosA+δ_y(X)+δ_y(Z)+δ_Ay-zε_x(X)

+zα_CY-δ_Cy}/((ε_x(Z)+ε_x(A)+S_YZ)sinA-cosA)

wherein the content of the first and second substances,

y represents a Y-axis actual motion command;

x represents an ideal tool nose position data x coordinate;

y represents the ideal tool tip position data y coordinate;

z represents an ideal tool tip position data z coordinate;

δ_x(C) representing x-direction linearity error of C-axis motion;

δ_y(C) representing the y-direction linearity error of the C-axis motion;

δ_z(A) represents the z-direction linearity error of the A-axis motion;

δ_z(Y) represents the Y axis motion z direction linearity error;

δ_y(A) represents the linear error of the movement y direction of the A axis;

δ_y(Y) represents the Y-direction linearity error of the Y-axis motion;

S_YZrepresenting Y, Z the perpendicularity error between the axes;

δ_y(X) represents X-axis motion y-direction linearity error;

δ_y(Z) represents the Z axis motion y-direction linearity error;

δ_Cyrepresenting the linear error of the C axis installation in the y direction;

δ_Aythe linear error of the axis A in the installation y direction is shown;

the analytic expression of the Z-axis actual motion instruction is as follows:

Z＝{x(-cosA(ε_y(C)+β_CXcosC+(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY)sinC)-sinA(sinC+ε_z(C)cosC))

+y(cosA(ε_x(C)+β_CXsinC-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY)cosC)-sinA(cosC-ε_z(C)sinC))

+z(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY-ε_x(C)cosC+ε_y(C)sinC)sinA)

-cosA((δ_y(A)+δ_y(Y))sinA+(δ_z(A)+δ_z(Y))cosA+δ_z(X)+δ_z(Z)+δ_Az-δ_z(C))

+sinA((δ_y(A)+δ_y(Y))cosA-(δ_z(A)+δ_z(Y))sinA+δ_y(X)+δ_y(Z)+δ_Ay-δ_Cy-δ_y(C)cosC-δ_x(C)sinC)}

/(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-ε_x(X))sinA)

wherein the content of the first and second substances,

z represents a Z-axis actual motion command;

δ_z(X) represents X-axis motion z-direction linearity error;

δ_z(Z) represents Z-direction linearity error of Z-axis motion;

δ_z(C) represents the z-direction linearity error of the C-axis motion;

δ_Azshowing the linear error of the axis A in the installation z direction;

δ_y(C) representing the y-direction linearity error of the C-axis motion;

δ_x(C) representing x-direction linearity error of C-axis motion;

the analytical expression of the X-axis actual motion instruction is as follows:

X＝x(cosC-ε_z(C)sinC)+y(-sinC-ε_z(C)cosC)+z(β_CX+ε_y(C)cosC+ε_x(C)sinC)

-Z(ε_y(X)+S_ZX)-Y((ε_y(X)+ε_y(Z)+S_ZX+β_AZ)sinA-(ε_z(X)+ε_z(Z)+γ_AY)cosA-S_YX-ε_z(A))

-(δ_x(X)+δ_x(Y)+δ_x(Z)+δ_x(A)-δ_Cx)+δ_x(C)cosC-δ_y(C)sinC

wherein the content of the first and second substances,

x represents an X-axis actual motion command;

δ_x(X) represents X-direction linearity error of X-axis motion;

δ_x(Y) represents the x-direction linearity error of the Y-axis motion;

δ_x(Z) represents the x-direction linearity error of the Z-axis motion;

δ_x(A) representing the x-direction linearity error of the A-axis motion;

δ_Cxrepresenting the linear error of the C axis installation in the x direction;

δ_x(C) representing x-direction linearity error of C-axis motion;

δ_y(C) representing the y-direction linearity error of the C-axis motion;

ε_z(C) representing the z-direction angle error of the C-axis motion;

ε_y(C) representing the y-direction angle error of the C-axis motion;

ε_z(A) representing the z-direction angle error of the A-axis movement;

S_YXrepresenting Y, X the perpendicularity error between the axes;

S_ZXrepresenting Z, X the perpendicularity error between the axes;

γ_AYthe Z-direction angle error of the A-axis installation is shown;

ε_z(Z) represents Z-direction angular error of Z-axis motion;

ε_z(X) represents the z-direction angular error of the X-axis motion;

β_AZthe angle error of the axis A in the mounting y direction is shown;

ε_y(Z) represents the Z axis motion y-direction angle error;

ε_y(X) represents the X-axis motion y-direction angleAn error;

ε_y(X) represents the y-direction angular error of the X-axis motion.

Further, the geometric error-tool space pose error model in the third step is established according to the following formula:

wherein the content of the first and second substances,

delta i represents the x coordinate error of the cutter shaft vector data;

delta j represents the coordinate error of the cutter shaft vector data y;

delta k represents the z coordinate error of the cutter shaft vector data;

the delta z represents the z coordinate error of the tool nose position data;

delta y represents the y coordinate error of the tool nose position data;

the delta x represents the x coordinate error of the tool nose position data;

Q′_wrepresenting actual arbor vector data;

P′_wrepresenting actual tip position data;

Q_wrepresenting ideal arbor vector data;

P_wrepresenting ideal nose position data;

an ideal transformation matrix representing TCS (tool coordinate system) to WCS (workpiece coordinate system);

Q_trepresenting the arbor vector in TCS;

P_tindicating the tip position in TCS.

Further, the geometric error-tooth surface pose error model is established by the following formula:

establishing a parameter equation of the axial profile of the grinding wheel:

wherein the content of the first and second substances,

r^waprepresenting the axial profile of the grinding wheel;

representing the x coordinate of the axial profile of the grinding wheel;

representing the y coordinate of the axial profile of the grinding wheel;

eta represents the axial profile parameter of the grinding wheel,

phi represents the rotation angle of the axial profile of the grinding wheel,

r^wtrepresenting the profile of the grinding wheel in the TCS, wherein the first superscript indicates the grinding wheel and the second superscript indicates the TCS;

establishing a grinding wheel curved surface dual-parameter equation in TCS:

wherein the content of the first and second substances,

representing the x coordinate of the profile of the grinding wheel in TCS;

representing the y coordinate of the profile of the grinding wheel in TCS;

representing the grinding wheel profile z coordinate in TCS;

r^wtrepresenting the grinding wheel profile in TCS;

the unit normal vector of the curved surface of the grinding wheel in the TCS is calculated according to the following formula:

n^wtrepresents the grinding wheel unit normal vector in TCS;

is represented by r^wtPartial derivation of eta;

is represented by r^wtPartial derivatives of phi;

calculating a curved surface parameter equation of the grinding wheel in the WCS according to the following formula:

wherein the content of the first and second substances,

r^wwirepresents the ideal profile of the grinding wheel in the WCS;

n^wwithe ideal unit normal vector of the grinding wheel in the WCS is shown;

r^wtrepresenting the grinding wheel profile in TCS;

n^wtrepresents the grinding wheel unit normal vector in TCS;

r^wwaindicating actual profile of grinding wheel in WCS；

n^wwaThe actual unit normal vector of the grinding wheel in the WCS is shown;

r^wtrepresenting the grinding wheel profile in TCS;

n^wtrepresents the grinding wheel unit normal vector in TCS;

the first superscript of r and n denotes the grinding wheel, the second superscript denotes the WCS, the third superscript denotes the ideal state i or the actual state a;

the grinding contact point is calculated according to the following formula:

f＝(k^gw×r^ww+pk^gw)·n^ww＝0

wherein the content of the first and second substances,

k^gwrepresenting the gear axis in the WCS;

r^wwa radial vector from the WCS origin to a point on the grinding wheel curve;

p represents a helix parameter;

n^wwa unit normal vector representing a point on the curved surface of the grinding wheel in the WCS;

the tooth surface numerical model is calculated according to the following formula:

wherein the content of the first and second substances,

r^gwrepresenting a tooth surface coordinate vector;

n^gwrepresents the tooth surface unit normal vector;

a coordinate vector representing a jth discrete point on the kth touch line;

a unit normal vector representing a jth discrete point on the kth contact line;

λ represents the number of contact lines constituting the tooth surface formed by grinding;

n represents the number of discrete contact points on the contact line;

establishing a tooth surface pose error model according to the following formula:

wherein the content of the first and second substances,

δ_TSindicating a tooth surface position error;

ε_TSrepresenting a tooth surface attitude error;

δ_x,δ_y,δ_zrepresenting errors in x, y, z directions of the tooth surface position;

ε_x,ε_y,ε_zrepresenting errors of the x, y and z directions of the tooth surface attitude;

r^gwarepresenting an actual tooth surface coordinate vector;

n^gwarepresenting the actual tooth surface unit normal vector;

r^gwirepresenting an ideal tooth surface coordinate vector;

n^gwirepresenting the ideal tooth surface unit normal vector.

A tooth surface normal error model is established according to the following formula:

δn＝dot(δ_TS,n^gwi)＝dot(r^gwa-r^gwi,n^gwi)

wherein the content of the first and second substances,

δ n represents the tooth surface normal error;

n^gwiexpressing an ideal tooth surface unit normal vector;

dot () represents a dot product operation;

r^gwarepresenting an actual tooth surface coordinate vector;

n^gwirepresenting the ideal tooth surface unit normal vector.

Further, the identification and analysis of the key error source of the tooth profile deviation specifically comprises the following steps:

1) generating a random sampling matrix H based on a sampling sequence_N×2m；

Wherein N represents the number of samples of each error, and m represents the number of errors;

2) according to a random sampling matrix H_N×2mConstruction of the geometric error input matrix A_N×m、B_N×mAnd A_B ⁱ _N×m；

H is to be_N×2mThe first m columns of (A) are taken as a matrix A, the last m columns are taken as a matrix B, and m derivative matrices A are constructed_B ⁱ；

3) Taking each row of the input matrix as a group of geometric error parameters, and sharing N (m +2) groups;

and will be represented by A, B and A_B ⁱEach set of decomposed errors is respectively brought into the tooth profile deviation model F_αIn (f) and (g),

wherein G represents a set of geometric errors,

calculating N (m +2) times to obtain corresponding tooth profile deviations f (A), f (B) and f (A)_B ⁱ)；

Finally, calculating the sensitivity index of the geometric error element by using the Monte Carlo estimation formula, wherein the sensitivity index comprises a first-order sensitivity index S_iAnd global sensitivity index S_Ti：

Wherein the content of the first and second substances,

S_ia first order sensitivity index representing an ith error element;

S_Tia global sensitivity index representing an ith error element;

v represents the total variance of the error model output;

f(B)_jrepresenting the tooth profile deviation corresponding to the jth row input error of the matrix B;

representation matrixInputting the tooth profile deviation corresponding to the error in the j row;

f(A)_jrepresenting the tooth profile deviation corresponding to the j row input error of the matrix A;

n represents the number of samples per error.

The invention has the beneficial effects that:

the invention provides a high-efficiency compensation method for key errors of gear grinding, which comprises the steps of firstly, constructing an actual forward kinematics model of gear grinding processing based on geometric error distribution of a forming gear grinding machine tool and an actual kinematic chain of the machine tool, and reflecting the functional relation between tool position and pose in a tool coordinate system and tool position data in a workpiece coordinate system under the influence of geometric errors; then, based on the actual reverse kinematics compensation principle, deducing an analytical expression of the actual motion command of the motion axis after error compensation, and revealing a mapping rule between the geometric error, the ideal tool bit data and the actual motion command; and finally, according to a conjugate grinding principle, establishing a geometric error-tooth surface error model, calculating and evaluating actual tooth profile and tooth direction accuracy, identifying a key error source of tooth profile deviation, simplifying an actual reverse kinematic compensation method, and realizing efficient error compensation facing tooth profile deviation reduction.

Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.

Drawings

In order to make the object, technical scheme and beneficial effect of the invention more clear, the invention provides the following drawings for explanation:

figure 1 is a schematic view of a forming gear grinding machine.

FIG. 2 is a schematic diagram of a machine tool full kinematic chain and 41 geometric errors.

Fig. 3 is a flow chart of key error identification.

FIG. 4 is a graphical illustration of the error source sensitivity index for tooth profile deviations.

FIG. 5 is a flow chart of a method for efficiently compensating for key errors in gear grinding.

Detailed Description

The present invention is further described with reference to the following drawings and specific examples so that those skilled in the art can better understand the present invention and can practice the present invention, but the examples are not intended to limit the present invention.

As shown in fig. 5, the method for efficiently compensating a key error of gear grinding provided by this embodiment includes the following steps:

(1) form grinding system geometric error analysis

The basic structure of a tYAZXRCw type five-axis forming gear grinding machine shown in figure 1 comprises X, Y, Z three linear motion axes and two rotation axes, wherein R represents a machine base, t represents a cutter, namely a grinding wheel, and w represents a workpiece gear.

The machine tool full-motion chain is composed of a workpiece chain RCw from RCS (reference coordinate system) to WCS (workpiece coordinate system), and a tool chain RXZAYt from RCS to TCS (tool coordinate system).

The X axis is set as a basic axis without installation error, the Z axis is set as a secondary axis, and the origin of a machine tool coordinate system (reference coordinate system) is set as the intersection point of the central lines of the A and C rotating shafts when each moving shaft is at zero position. On the basis of configuring corresponding sub-coordinate systems for five motion axes, a machine base, a workpiece and a cutter, the position-independent geometric errors (PIGEs) and the position-dependent geometric errors (PDGEs) shown in the table 1 are considered, the actual full motion chain of the machine tool is shown in fig. 2, and fig. 2 shows the full motion chain of the machine tool and 41 geometric errors.

TABLE 1 elements of geometric errors of forming gear grinding machine

Wherein the content of the first and second substances,

δ_x(X),δ_y(X),δ_z(X) represents X, y, z direction linearity error of X axis motion;

δ_x(Y),δ_y(Y),δ_z(Y) represents x, Y, z-direction linearity error of Y-axis motion;

δ_x(Z),δ_y(Z),δ_z(Z) represents the x, y, Z direction linearity error of the Z axis motion;

δ_x(A),δ_y(A),δ_z(A) representing the linear error of the A axis movement in the x, y and z directions;

δ_x(C),δ_y(C),δ_z(C) representing the linear error of the X, Y and Z directions of the C-axis movement;

ε_x(X),ε_y(X),ε_z(X) represents X, y, z angular error of X-axis motion;

ε_x(Y),ε_y(Y),ε_z(Y) represents the angular error of the X, Y, Z direction of the Y axis movement;

ε_x(Z),ε_y(Z),ε_z(Z) represents the x, y, Z angular error of the Z motion;

ε_x(A),ε_y(A),ε_z(A) representing the angle error of the A axis movement in the x, y and z directions;

ε_x(C),ε_y(C),ε_z(C) representing the angle error of the X, Y and Z directions of the C-axis movement;

S_YX,S_YZrepresenting Y, X interaxial perpendicularity error and Y, Z interaxial perpendicularity error;

S_ZXrepresenting Z, X the perpendicularity error between the axes;

δ_Ay,δ_Az,β_AZ,γ_AYthe linear error of the axis A in the y and z directions and the angular error of the axis C in the y and z directions are shown;

δ_Cx,δ_Cy,α_CY,β_CXthe linear error of the C axis installation in the x and y directions and the angular error of the C axis installation in the x and y directions are shown;

(2) constructing a realistic forward kinematics model

If the ideal pose transformation matrix between the coordinate systems of adjacent components (e.g., N to Q coordinate systems) can be sequentially multiplied by the installation matrixT_pQNMotion matrix T_mQNThus obtaining the product.

The actual pose transformation matrix can be sequentially multiplied by the installation matrix T_pQNMounting error matrix T_peQNMotion matrix T_mQNMotion error matrix T_meQNAnd (4) obtaining.

Then for workpiece chain (RCw), the actual transformation matrix of WCS with respect to RCS, i.e. the actual forward kinematics model of the workpiece chain, is:

similarly, an actual forward kinematics model of the tool chain can be established, i.e. an actual transformation matrix of TCS versus RCS:

wherein the content of the first and second substances,

the actual forward kinematic models of the tool chain and the tool chain are integrated, so that the actual forward kinematic model of the full kinematic chain of the gear grinding machine tool can be established, namely an actual transformation matrix of the TCS relative to the WCS:

therefore, the actual transformation matrix of TCS relative to WCS is essentially a matrix function with ideal motion commands of motion axes and geometric errors as arguments.

If it is matched with the knife edge position P in TCS_tSum arbor vector Q_tMultiplying to obtain the tool position data under the influence of the geometric error:

wherein, P'_w＝[x′,y′,z′]^TAnd Q'_w＝[i′,j′,k′]^TRespectively representing the actual tip position and the actual knife axis vector.

(1) Principle of actual inverse kinematics compensation

Step one, actually providing an actual tool position data explicit calculation method considering the influence of geometric errors, and solving the actual tool position data through known ideal motion instructions of the motion axis and the geometric errors measured and calibrated.

The subsequent error compensation strategy is typically: firstly, calculating an error vector between actual cutter position data and ideal cutter position data, then adding a compensation vector which is equal to the vector in size and opposite in direction on the basis of not changing an ideal machining code, and reversely solving a corresponding compensation machining code, thereby realizing error compensation and improving the machining precision.

In order to simplify the calculation flow of error compensation and improve the compensation efficiency, in this embodiment, the ideal tool position data is not changed, the motion instruction of the motion axis is changed due to the influence of the geometric error, and the actual motion instruction value after error compensation can be obtained by solving reversely according to the above formula, so as to obtain the actual processing code and implement error compensation.

Therefore, ideal tool position data including tool position data and tool axis vector data, i.e., the tool tip position P, is given in advance_w＝[x,y,z]^TSum arbor vector Q_w＝[i,j,k]^T. Meanwhile, the tool nose position and the cutter shaft vector in the TCS are respectively P_t＝[0,0,0]^TAnd Q_t＝[0,0,1]^TThen, the mapping relationship between the two can be expressed as:

that is to say that the first and second electrodes,

(2) rotating shaft actual motion instruction analytic expression derivation

Considering that the cutter shaft vector data is not influenced by the motion of a straight line shaft, the mapping relation between the ideal cutter shaft vector data and the geometric error of a machine tool is firstly utilized to separately solve the actual motion instructions A and C of the rotating shaft,

that is to say that the first and second electrodes,

wherein the content of the first and second substances,

if the C-axis motion matrix function is separately converted to the left side of the equation, simplification can be achieved

Wherein the content of the first and second substances,

thus, it is possible to obtain:

(T_mCWT_meCW)[i,j,k,0]^T＝[b₁₃-b₃₃β_CX,b₂₃+b₃₃α_CY,b₃₃+b₁₃β_CX-b₂₃α_CY,0]^T

since the left side of the equation is a matrix function related to the C-axis rotational motion only, it can be known that the z-direction data of the arbor vector is not affected by the C-axis rotational motion according to the rotation invariant theory.

Therefore, the calculation formula of the z-direction cutter axis vector in the formula can be extracted independently, and the second order and the higher order terms are omitted, so that the equation only related to the motion instruction A is obtained,

-ε_y(C)i+ε_x(C)j+k＝cosA-(ε_x(X)+ε_x(Y)+ε_x(Z)+ε_x(A)+S_YZ-α_CY)sinA

the analysis expression of the actual A-axis motion instruction is as follows according to the auxiliary angle formula and the trigonometric inverse function:

wherein the content of the first and second substances,

similarly, if the a-axis motion matrix function is separately transformed to the right of the equation, the simplification can be obtained:

wherein the content of the first and second substances,

an actual transformation matrix representing the TCS (tool coordinate system) to A-axis coordinate system;

since the right side of the equation is a matrix function only related to the rotational motion of the a axis, it can be known that the x-direction data of the arbor vector is not affected by the rotational motion of the a axis according to the rotation invariant theory. Therefore, the calculation formula of the x-direction cutter axis vector in the formula can be extracted independently, and the second order and the higher order terms are omitted, so that the equation only related to the motion command C is obtained,

[(i(ε_z(X)+ε_z(Z)+γ_AY-ε_z(C))-j+kε_x(C))]sinC+

[i+j(ε_z(X)+ε_z(Z)+γ_AY-ε_z(C))+kε_y(C)]cosC

＝ε_y(Y)+ε_y(A)-k(β_CX-ε_y(X)-ε_y(Z)-S_ZX-β_AZ)

according to the auxiliary angle formula and the inverse trigonometric function,

sin(C+φ)＝ε_y(Y)+ε_y(A)-k(β_CX-ε_y(X)-ε_y(Z)-S_ZX-β_AZ)

wherein the content of the first and second substances,

thus, the actual C-axis motion command analytic expression may be obtained as:

(3) linear axis actual motion instruction analytic expression derivation

Based on the obtained actual motion command of the rotating shaft, the mapping relation between the ideal tool position data and the geometric error of the machine tool is utilized to solve the actual motion command of the linear shaft:

that is to say that the first and second electrodes,

wherein the content of the first and second substances,

T_peRXa mounting error matrix representing an X-axis coordinate system to an RCS (reference coordinate system);

T_pAYto representAn installation matrix from a Y-axis coordinate system to an A-axis coordinate system;

T_meYTrepresenting the motion error matrix of TCS (tool coordinate system) to Y-axis coordinate system.

Based on the translation feature separation theory of block computation, it can be:

if it is provided with

Wherein the content of the first and second substances,

f₁₁～f₃₄represents T_meRXT_peXZAfter matrix operation, the value of each matrix element;

g₁₁～g₃₄represents T_meRXT_peXZT_meXZT_peZAT_mZAT_meZAT_peAYAfter matrix operation, the value of each matrix element;

h₁₁～h₃₄representing the value of each matrix element obtained by multiplying the matrix;

it is possible to obtain,

by expanding it into a polynomial equation system, we can obtain:

from the above equation, an analytical expression for the linear axis motion command (X, Y, Z) with errors can be derived.

Due to f₃₃＝1,f₂₃＝-ε_x(X), then (f)₂₃And- ②, omitting higher-order terms above the second order to obtain:

x(f₂₃a₃₁-a₂₁)+y(f₂₃a₃₂-a₂₂)+z(f₂₃a₃₃-a₂₃)+(f₂₃a₃₄-a₂₄)-f₂₃h₃₄+h₂₄＝Y(f₂₃g₃₂-g₂₂)

therefore, the analytical expression of the Y-axis actual motion command is:

Y＝{-sinC(x+zε_y(C)-yε_z(C)+δ_x(C))-cosC(y+xε_z(C)-zε_x(C)+δ_y(C))

-(δ_z(A)+δ_z(Y))sinA+(δ_y(A)+δ_y(Y))cosA+δ_y(X)+δ_y(Z)+δ_Ay-zε_x(X)

+zα_CY-δ_Cy}/((ε_x(Z)+ε_x(A)+S_YZ)sinA-cosA)

is composed of (c) g₂₂-②×g₃₂And omitting the higher-order terms of more than two orders, so as to obtain:

x(-cosA(ε_y(C)+β_CXcosC+(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY)sinC)-sinA(sinC+ε_z(C)cosC))

+z(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY-ε_x(C)cosC+ε_y(C)sinC)sinA)

+sinA((δ_y(A)+δ_y(Y))cosA-(δ_z(A)+δ_z(Y))sinA+δ_y(X)+δ_y(Z)+δ_Ay-δ_Cy-δ_y(C)cosC-δ_x(C)sinC)

＝Z(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-ε_x(X))sinA)

therefore, the analytic expression of the Z-axis actual motion command is:

+z(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY-ε_x(C)cosC+ε_y(C)sinC)sinA)

/(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-ε_x(X))sinA)

based on the obtained Y, Z-axis actual motion command, the following equation (i) is obtained:

X＝xa₁₁+ya₁₂+za₁₃+a₁₄-h₁₄-Zf₁₃-Yg₁₂

therefore, the analytical expression of the X-axis actual motion command is:

X＝x(cosC-ε_z(C)sinC)+y(-sinC-ε_z(C)cosC)+z(β_CX+ε_y(C)cosC+ε_x(C)sinC)

-(δ_x(X)+δ_x(Y)+δ_x(Z)+δ_x(A)-δ_Cx)+δ_x(C)cosC-δ_y(C)sinC

so far, the actual motion instructions and geometric errors of five motion axes of the numerical control forming gear grinding machine tool and the mapping rule of ideal cutter position data are clearly shown. On the premise that the ideal cutter bit data is given and the geometric error of the machine tool is effectively calibrated, the actual numerical control machining instruction with error compensation can be directly obtained according to the analytical expression, so that the error compensation is realized.

Step three: key error identification and compensation model simplification

(1) Geometric error-tooth surface error model

The numerical control forming gear grinding forward kinematics model describes the transformation relation between the position and the posture between a gear and a grinding wheel coordinate system. If the actual situation considering the geometric error is compared with the ideal state neglecting the error influence, and then the product is made with the homogeneous coordinate of the grinding wheel center position and the posture in the TCS, a geometric error-cutter space pose error model of the gear grinding machine tool can be established, namely the relative pose error between the grinding wheel and the gear blank in the WCS:

considering that the geometric error-cutter spatial error model ignores the influence of cutting motion interference between the cutter and the curved surface of the workpiece on the processing effect, a mapping model from the cutter spatial error to the tooth surface pose error needs to be further constructed based on a material removal mechanism in the actual grinding process, so that the geometric error-tooth surface pose error model which can more directly reflect the influence of the geometric error on the tooth surface pose error is established.

According to the conjugate grinding principle, the parameters of the axial profile of the grinding wheel are expressed by eta, and an axial profile parameter equation of the grinding wheel is established:

phi denotes the angle of rotation, r, of the axial profile of the grinding wheel^wtRepresenting a parametric equation for the profile of the grinding wheel, wherein the first superscript denotes the grinding wheelAnd the second superscript represents TCS, then a grinding wheel curved surface bi-parameter equation in TCS can be established:

wherein the content of the first and second substances,

from this, the unit normal vector of the curved surface of the grinding wheel in TCS is:

based on the transformation relation between the gear and the grinding wheel coordinate system established by the forward kinematics model, the curved surface parameter equation of the grinding wheel in the WCS under an ideal condition and an actual condition can be obtained as follows:

where the first superscript of r and n denotes the grinding wheel, the second superscript denotes the WCS, and the third superscript denotes the ideal state i or the actual state a.

When the grinding wheel is known, the conjugate grinding contact condition of the grinding wheel and the gear can be expressed as: radius vector r from WCS origin to one point on grinding wheel revolution surface^wwIf this is about the gear axis k^gwLinear velocity vector in spiral motion and normal n of the vector on curved surface of grinding wheel^wwPerpendicular, then this point is the grinding contact point.

f＝(k^gw×r^ww+pk^gw)·n^ww＝0

Respectively converting r in ideal and actual states^wwAnd n^wwBy taking into the above formula, the grinding track can be discretized and adoptedAnd (4) solving the numerical solution of the grinding contact line by a bisection method. If the tooth surface formed by grinding is composed of λ contact lines, each contact line being composed of n discrete contact points, the tooth surface numerical model can be expressed as:

the tooth surface under the actual state and the ideal state can be compared with each other to establish a tooth surface pose error model,

based on the geometric error-tooth surface pose error model, the tooth surface position error and the ideal tooth surface normal vector can be subjected to dot product operation, so that a tooth surface normal error model, namely a tooth surface normal error model is established

δn＝dot(δ_TS,n^gwi)＝(r^gwa-r^gwi,n^gwi)

As the grinding tooth surface is formed by the lambda contact lines containing n discrete points, if the normal error of the k contact line is rotationally transformed to the section of the gear end, a corresponding tooth profile deviation curve can be obtained, and the total tooth profile deviation (F) can be obtained by further analysis_α) Deviation of tooth profile shape (f)_fα) And tooth profile slope deviation (f)_Hα) And evaluating indexes; if the normal errors of all j-th points on the lambda contact lines are extracted independently, a corresponding spiral line deviation curve can be obtained, and the total deviation (F) of the spiral line can be obtained through further analysis_β) Deviation of helical line shape (f)_fβ) And the helix slope deviation (f)_Hβ) And the like.

(2) Key error source identification process

As shown in fig. 3, fig. 3 is a flow of identifying a key error, and according to the Sobol method, identification and analysis of a key error source can be performed for a tooth profile deviation and a spiral deviation, taking the key error source analysis of the tooth profile deviation as an example.

1) Generating a random sampling matrix H based on a sampling sequence_N×2mThe matrix being such that input errors are taken into accountThe probability distribution of the geometric terms. Where N represents the number of samples per error and m represents the number of errors.

2) According to a random sampling matrix H_N×2mConstruction of the geometric error input matrix A_N×m、B_N×mAnd A_B ⁱ _N×m. H is to be_N×2mThe first m columns of (A) are taken as a matrix A, the last m columns are taken as a matrix B, and m derivative matrices A are constructed_B ⁱExcept for the ith column, which is equal to B, the remaining columns are from A.

3) Each row of the input matrix is used as a group of geometric error parameters, and N (m +2) groups are provided. And will be represented by A, B and A_B ⁱEach set of decomposed errors is respectively brought into the tooth profile deviation model F_αIn f (G), G represents a geometric error set, N (m +2) times are calculated to obtain the corresponding tooth profile deviations f (a), f (b), f (a)_B ⁱ). Finally, calculating the sensitivity index of the geometric error element by using the Monte Carlo estimation formula, wherein the sensitivity index comprises a first-order sensitivity index S_iAnd global sensitivity index S_Ti。

And comparing the size of the global sensitivity index of the geometric error element of the tooth profile deviation, and judging a key error source of the total tooth profile deviation.

(3) Key error efficient compensation method

As shown in fig. 4, the basic diagram 4 is an error source sensitivity index of tooth profile deviation, and the error compensation model is simplified by neglecting other geometric error influences from the identified key error sources. For example, the geometric error source sensitivity index of tooth profile deviation. The abscissa of the graph represents the error sequence number, but the geometric error of the Y axis is not counted; the ordinate represents the error sensitivity index.

If 0.05 is taken as a sensitive threshold, preliminarily judging that the key error serial number of the tooth profile deviation is as follows: 4. 10, 17, 27, 28, 29. 32, 33, i.e. the geometric error term epsilon_x(X)、ε_x(Z)、ε_x(A)、ε_x(C)、ε_y(C)、ε_z(C)、α_CY、β_CX。

Therefore, the actual motion command analysis expressions of the rotary axis and the linear axis in the steps (2) and (3) can be simplified as follows:

wherein the content of the first and second substances,

Y＝{-sinC(x+zε_y(C)-yε_z(C))-cosC(y+xε_z(C)-zε_x(C))-zε_x(X)+zα_CY}

/((ε_x(Z)+ε_x(A))sinA-cosA)

Z＝{x(-cosA(ε_y(C)+β_CXcosC+(ε_x(X)+ε_x(Z)+ε_x(A)-α_CY)sinC)-sinA(sinC+ε_z(C)cosC))

+y(cosA(ε_x(C)+β_CXsinC-(ε_x(X)+ε_x(Z)+ε_x(A)-α_CY)cosC)-sinA(cosC-ε_z(C)sinC))

+z(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)-α_CY-ε_x(C)cosC+ε_y(C)sinC)sinA)}

/(cosA-(ε_x(Z)+ε_x(A))sinA)

X＝x(cosC-ε_z(C)sinC)+y(-sinC-ε_z(C)cosC)+z(β_CX+ε_y(C)cosC+ε_x(C)sinC)

according to the simplified analysis expression of the actual motion instruction of the motion axis, the numerical control code after corresponding key error compensation can be obtained to replace the original code to run, and the efficient error compensation aiming at tooth profile deviation reduction can be realized.

The above-mentioned embodiments are merely preferred embodiments for fully illustrating the present invention, and the scope of the present invention is not limited thereto. The equivalent substitution or change made by the technical personnel in the technical field on the basis of the invention is all within the protection scope of the invention. The protection scope of the invention is subject to the claims.

Claims

1. A gear grinding key error efficient compensation method is characterized by comprising the following steps: the method comprises the following steps:

(2) constructing an actual forward kinematics model as follows:

establishing an actual forward kinematic model of the workpiece chain:

establishing an actual forward kinematic model of the tool chain:

step three: key error identification and compensation model simplification

(1) Geometric error-tooth surface error model

2. The method of claim 1, wherein: the geometric errors in the first step comprise position-independent geometric errors PIGEs and position-dependent geometric errors PDGEs.

3. The method of claim 1, wherein: the actual forward kinematics model is constructed according to the following steps:

wherein the content of the first and second substances,

an actual transformation matrix representing the WCS workpiece coordinate system to the RCS reference coordinate system;

T_RCrepresenting an actual transformation matrix from a C-axis coordinate system to an RCS reference coordinate system;

T_CWrepresenting an actual transformation matrix from the WCS workpiece coordinate system to the C-axis coordinate system;

T_pRCrepresenting an installation matrix from a C-axis coordinate system to an RCS reference coordinate system;

T_peRCrepresenting an installation error matrix from a C-axis coordinate system to an RCS reference coordinate system;

T_mRCrepresenting a motion matrix from a C-axis coordinate system to an RCS reference coordinate system;

T_meRCrepresenting a motion error matrix from a C-axis coordinate system to an RCS reference coordinate system;

T_pCWan installation matrix representing a WCS workpiece coordinate system to a C-axis coordinate system;

T_peCWrepresenting an installation error matrix from a WCS workpiece coordinate system to a C-axis coordinate system;

T_mCWrepresenting a motion matrix from a WCS workpiece coordinate system to a C-axis coordinate system;

T_meCWrepresenting a motion error matrix from a WCS workpiece coordinate system to a C-axis coordinate system;

wherein the content of the first and second substances,

representing an actual transformation matrix from the TCS tool coordinate system to the RCS reference coordinate system;

T_RXrepresenting an actual transformation matrix from an X-axis coordinate system to an RCS reference coordinate system;

T_ZAto show the axis AA practical transformation matrix from a standard system to a Z-axis coordinate system;

T_YTrepresenting an actual transformation matrix from a TCS cutter coordinate system to a Y-axis coordinate system;

T_pRXan installation matrix representing an X-axis coordinate system to an RCS reference coordinate system;

T_peRXrepresenting an installation error matrix from an X-axis coordinate system to an RCS reference coordinate system;

T_mRXrepresenting a motion matrix from an X-axis coordinate system to an RCS reference coordinate system;

T_meRXrepresenting a motion error matrix from an X-axis coordinate system to an RCS reference coordinate system;

T_pYTrepresenting the TCS tool coordinate system toAn installation matrix of a Y-axis coordinate system;

T_peYTrepresenting an installation error matrix from a TCS cutter coordinate system to a Y-axis coordinate system;

T_mYTrepresenting a motion matrix from a TCS cutter coordinate system to a Y-axis coordinate system;

T_meYTrepresenting a motion error matrix from a TCS cutter coordinate system to a Y-axis coordinate system;

wherein the content of the first and second substances,

b₁₁～b₃₄after representing matrix operationsThe value of each element;

wherein the content of the first and second substances,

representing an actual transformation matrix from a TCS tool coordinate system to a WCS workpiece coordinate system;

wherein the content of the first and second substances,

Q′_w＝[i′,j′,k′]^Trepresenting actual arbor vector data;

P′_w＝[x′,y′,z′]^Trepresenting actual tip position data;

Q_tindicating a knife in TCSAn axis vector;

P_tindicating the tip position in TCS;

4. The method of claim 1, wherein: the subsequent error compensation strategy in the second step is carried out according to the following mode:

5. The method of claim 1, wherein: the second step further comprises the following steps:

wherein the content of the first and second substances,

P_w＝[x,y,z]^Trepresenting ideal nose position data;

Q_w＝[i,j,k]^Trepresenting ideal arbor vector data;

P_t＝[0,0,0]^Tindicating the tip position in TCS;

Q_t＝[0,0,1]^Trepresenting the arbor vector in TCS.

6. The method of claim 1, wherein: and the rotating shaft actual motion instruction analytic expression in the step two is obtained according to the following mode:

wherein the content of the first and second substances,

a represents an A-axis actual motion command;

arccos () represents an inverse cosine function;

ε_y(C) representing the y-direction angle error of the C-axis motion;

ε_x(C) representing the x-direction angle error of the C-axis movement;

ε_x(X) represents X-direction angular error of X-axis motion;

ε_x(Y) represents the x-direction angular error of the Y-axis motion;

ε_x(Z) represents the x-direction angular error of the Z-axis motion;

ε_x(A) representing the x-direction angle error of the A-axis movement;

S_YZrepresenting Y, Z the perpendicularity error between the axes;

representing the tangent angle;

k represents an ideal cutter axis vector data z coordinate;

j represents the y coordinate of ideal cutter axis vector data;

i represents an ideal cutter axis vector data x coordinate;

or

wherein the content of the first and second substances,

c represents an actual motion command of a C axis;

arcsin () represents an arcsine function;

ε_y(Y) represents the Y-direction angular error of the Y-axis motion;

ε_y(A) representing the angle error of the movement y direction of the A axis;

ε_y(Z) represents the Z axis motion y-direction angle error;

ε_y(X) represents the X-axis motion y-direction angular error;

ε_z(C) representing the z-direction angle error of the C-axis motion;

ε_z(Z) represents Z-direction angular error of Z-axis motion;

ε_z(X) represents the z-direction angular error of the X-axis motion;

S_ZXrepresenting Z, X the perpendicularity error between the axes;

β_AZthe angle error of the axis A in the mounting y direction is shown;

γ_AYthe Z-direction angle error of the A-axis installation is shown;

or

Y＝{-sinC(x+zε_y(C)-yε_z(C)+δ_x(C))-cosC(y+xε_z(C)-zε_x(C)+δ_y(C))

-(δ_z(A)+δ_z(Y))sinA+(δ_y(A)+δ_y(Y))cosA+δ_y(X)+δ_y(Z)+δ_Ay-zε_x(X)

+zα_CY-δ_Cy}/((ε_x(Z)+ε_x(A)+S_YZ)sinA-cosA)

wherein the content of the first and second substances,

y represents a Y-axis actual motion command;

x represents an ideal tool nose position data x coordinate;

y represents the ideal tool tip position data y coordinate;

z represents an ideal tool tip position data z coordinate;

δ_x(C) representing x-direction linearity error of C-axis motion;

δ_y(C) representing the y-direction linearity error of the C-axis motion;

δ_z(A) represents the z-direction linearity error of the A-axis motion;

δ_z(Y) represents the Y axis motion z direction linearity error;

δ_y(A) represents the linear error of the movement y direction of the A axis;

δ_y(Y) represents the Y-direction linearity error of the Y-axis motion;

δ_y(X) represents X-axis motion y-direction linearity error;

δ_y(Z) represents the Z axis motion y-direction linearity error;

δ_Aythe linear error of the axis A in the installation y direction is shown;

the analytic expression of the Z-axis actual motion instruction is as follows:

+z(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-α_CY-ε_x(C)cosC+ε_y(C)sinC)sinA)

+sinA((δ_y(A)+δ_y(Y))cosA-(δ_z(A)+δ_z(Y))sinA+δ_y(X)+δ_y(Z)+δ_Ay-δ_Cy-δ_y(C)cosC-δ_x(C)sinC)}/(cosA-(ε_x(X)+ε_x(Z)+ε_x(A)+S_YZ-ε_x(X))sinA)

wherein the content of the first and second substances,

z represents a Z-axis actual motion command;

δ_z(X) represents X-axis motion z-direction linearity error;

δ_z(Z) represents Z-direction linearity error of Z-axis motion;

δ_z(C) represents the z-direction linearity error of the C-axis motion;

δ_Azshowing the linear error of the axis A in the installation z direction;

δ_x(C) representing x-direction linearity error of C-axis motion;

X＝x(cosC-ε_z(C)sinC)+y(-sinC-ε_z(C)cosC)+z(β_CX+ε_y(C)cosC+ε_x(C)sinC)

-(δ_x(X)+δ_x(Y)+δ_x(Z)+δ_x(A)-δ_Cx)+δ_x(C)cosC-δ_y(C)sinC

wherein the content of the first and second substances,

x represents an X-axis actual motion command;

δ_x(X) represents X-direction linearity error of X-axis motion;

δ_x(Y) represents the x-direction linearity error of the Y-axis motion;

δ_x(Z) represents the x-direction linearity error of the Z-axis motion;

δ_x(A) representing the x-direction linearity error of the A-axis motion;

δ_x(C) representing x-direction linearity error of C-axis motion;

ε_z(A) representing the z-direction angle error of the A-axis movement;

S_YXindicating Y, X the interaxial perpendicularity error.

7. The method of claim 1, wherein: the geometric error-cutter space pose error model in the third step is established according to the following formula:

wherein the content of the first and second substances,

delta i represents the x coordinate error of the cutter shaft vector data;

delta j represents the coordinate error of the cutter shaft vector data y;

delta k represents the z coordinate error of the cutter shaft vector data;

the delta z represents the z coordinate error of the tool nose position data;

delta y represents the y coordinate error of the tool nose position data;

the delta x represents the x coordinate error of the tool nose position data;

Q′_wrepresenting actual arbor vector data;

P′_wrepresenting actual tip position data;

an ideal transformation matrix representing TCS (tool coordinate system) to WCS (workpiece coordinate system).

8. The method of claim 1, wherein: the geometric error-tooth surface pose error model is established by the following formula:

establishing a parameter equation of the axial profile of the grinding wheel:

wherein the content of the first and second substances,

r^waprepresenting the axial profile of the grinding wheel;

representing the x coordinate of the axial profile of the grinding wheel;

representing the y coordinate of the axial profile of the grinding wheel;

eta represents the axial profile parameter of the grinding wheel;

phi represents the rotation angle of the axial profile of the grinding wheel;

establishing a grinding wheel curved surface dual-parameter equation in TCS:

wherein the content of the first and second substances,

representing the x coordinate of the profile of the grinding wheel in TCS;

representing the y coordinate of the profile of the grinding wheel in TCS;

representing the grinding wheel profile z coordinate in TCS;

r^wtrepresenting the grinding wheel profile in TCS;

n^wtrepresents the grinding wheel unit normal vector in TCS;

is represented by r^wtPartial derivation of eta;

is represented by r^wtPartial derivatives of phi;

wherein the content of the first and second substances,

r^wwirepresents the ideal profile of the grinding wheel in the WCS;

n^wwithe ideal unit normal vector of the grinding wheel in the WCS is shown;

an ideal transformation matrix representing the TCS tool coordinate system to the WCS workpiece coordinate system;

r^wtrepresenting the grinding wheel profile in TCS;

n^wtrepresents the grinding wheel unit normal vector in TCS;

r^wwarepresenting the actual profile of the grinding wheel in the WCS;

n^wwathe actual unit normal vector of the grinding wheel in the WCS is shown;

r^wtrepresenting the grinding wheel profile in TCS;

n^wtrepresents the grinding wheel unit normal vector in TCS;

the grinding contact point is calculated according to the following formula:

f＝(k^gw×r^ww+pk^gw)·n^ww＝0

wherein the content of the first and second substances,

k^gwrepresenting the gear axis in the WCS;

r^wwa radial vector from the WCS origin to a point on the grinding wheel curve;

p represents a helix parameter;

wherein the content of the first and second substances,

r^gwrepresenting a tooth surface coordinate vector;

n^gwrepresents the tooth surface unit normal vector;

a coordinate vector representing a jth discrete point on the kth touch line;

a unit normal vector representing a jth discrete point on the kth contact line;

n represents the number of discrete contact points on the contact line;

wherein the content of the first and second substances,

δ_TSindicating a tooth surface position error;

ε_TSrepresenting a tooth surface attitude error;

r^gwarepresenting an actual tooth surface coordinate vector;

n^gwarepresenting the actual tooth surface unit normal vector;

r^gwirepresenting an ideal tooth surface coordinate vector;

δn＝dot(δ_TS,n^gwi)＝dot(r^gwa-r^gwi,n^gwi)

wherein the content of the first and second substances,

δ n represents the tooth surface normal error;

dot () represents a dot product operation;

r^gwarepresenting an actual tooth surface coordinate vector;

n^gwirepresenting the ideal tooth surface unit normal vector.

9. The method of claim 1, wherein: the identification and analysis of the key error source of the tooth profile deviation comprises the following specific steps:

1) generating a random sampling matrix H based on a sampling sequence_N×2m；

2) according to a random sampling matrix H_N×2mConstruction of the geometric error input matrix A_N×m、B_N×mAnd

wherein G represents a set of geometric errors, F_αRepresenting tooth profile deviation;

Wherein the content of the first and second substances,

S_ia first order sensitivity index representing an ith error element;

S_Tia global sensitivity index representing an ith error element;

v represents the total variance of the error model output;

n represents the number of samples per error.