CN110597183A - An Efficient Compensation Method for Key Grinding Errors - Google Patents

Abstract

The invention discloses a high-efficiency compensation method for key errors in gear grinding. Firstly, based on the geometric error distribution of a form grinding machine tool and the actual kinematic chain of the machine tool, the actual forward kinematics model of gear grinding processing is constructed to reflect the position in the tool coordinate system under the influence of geometric errors. The functional relationship between the tool pose and the tool position data in the workpiece coordinate system; then, based on the actual inverse kinematics compensation principle, the analytical expression of the actual motion command of the motion axis after error compensation is derived, revealing the geometric error, the ideal tool position data The mapping law between the actual motion command and the actual motion command; finally, according to the principle of conjugate grinding, the geometric error-tooth surface error model is established, the actual tooth profile and tooth direction accuracy are calculated and evaluated, and the key error sources of the tooth profile deviation are identified. The actual inverse kinematics compensation method is simplified to realize efficient error compensation for tooth profile deviation reduction.

Description

An Efficient Compensation Method for Key Grinding Errors

technical field

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

Background technique

CNC form gear grinding machine is a special machine tool for gear finishing. The gear grinding accuracy is affected by multi-source errors, including machine tool geometry error, thermal error, force error, servo control error, etc. Among them, the geometric error is considered to be a quasi-static error, which does not change or changes slightly with time, and can be eliminated by compensation.

In order to compensate the geometric errors of multi-axis machine tools, experts and scholars have proposed a large number of methods, including differential operator decoupling methods, iterative regression calculation methods, differential error prediction methods, etc. The existing error compensation methods are inefficient, and mainly focus on solving the terminal error vector of the tool relative to the workpiece, and do not pay attention to the specific influence of the terminal error vector on the workpiece.

In addition, some scholars have proposed an actual inverse kinematics compensation method for the geometric error of ordinary five-axis CNC machine tools, which can realize geometric error compensation in the post-processing process, and the compensation efficiency of this method is relatively high. However, the form gear grinding machine is a special processing machine tool for gears. Although the structure of the machine tool is not special, due to the processing characteristics such as spiral grinding and conjugate contact, as well as the particularity of the gear accuracy evaluation index, there is currently no special tool for form grinding gear. Machine geometric error compensation method based on actual inverse kinematics.

In addition, the existing actual inverse kinematics method is to compensate for the influence of geometric errors on the tool position data. Since the actual grinding process is not considered, the final improvement of the single grinding accuracy evaluation index may not be completely ideal.

Contents of the invention

In view of this, the purpose of the present invention is to provide an efficient compensation method for the key error of gear grinding, which can deduce the analytical expression of the actual motion command of the motion axis after error compensation, and reveal the geometric error, the ideal tool position data and the actual motion command. Mapping law; and analyze the key error sources of gear accuracy evaluation basis such as tooth profile accuracy and tooth direction accuracy, so as to simplify the traditional actual inverse kinematics method and realize efficient compensation for tooth grinding errors such as tooth profile deviation and tooth direction deviation .

To achieve the above object, the present invention provides the following technical solutions:

The high-efficiency compensation method for gear grinding key errors provided by the present invention includes the following steps:

Step 1: Modeling the geometric error of the form grinding system;

(1) Geometric error analysis of the form grinding system: determine the full kinematic chain of the machine tool and the geometric error of the form grinding machine according to the basic structure of the form grinding machine tool; the full kinematic chain of the machine tool includes the workpiece from the RCS reference coordinate system to the WCS workpiece coordinate system Chain RCw and tool chain RXZAYt from RCS reference coordinate system to TCS tool coordinate system;

(2) Construct the actual forward kinematics model, as follows:

Model the actual forward kinematics of the workpiece chain:

Model the actual forward kinematics of the toolchain:

Establish the actual forward kinematics model of the whole kinematic chain of the gear grinding machine:

Step 2: Geometric error compensation method based on actual inverse kinematics

(1) Follow-up error compensation strategy based on the actual inverse kinematics compensation principle;

(2) Obtain the analytical expression of the actual motion command of the rotary axis: use the mapping relationship between the ideal tool axis vector data and the geometric error of the machine tool to solve the analytical expression of the actual motion command of the rotary axis;

(3) Obtain the analytical expression of the actual motion command of the linear axis: According to the analytical expression of the actual motion command of the rotary axis, use the mapping relationship between the ideal tool position data and the geometric error of the machine tool to solve the analytical expression of the actual motion command of the linear axis ;

Step 3: Key error identification and compensation model simplification

(1) Geometric error-tooth surface error model

(2) Identify and analyze key error sources for tooth profile deviation and helix deviation;

(3) Efficient compensation method for key errors: According to the analytical expression of the actual motion command of the motion axis, the NC code corresponding to the key error source after compensation is obtained, and the efficient error compensation for tooth profile deviation reduction is realized.

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:

Calculate the ideal pose transformation matrix between the coordinate systems of adjacent parts: that is, it is obtained by sequentially multiplying the installation matrix T _pQN and the motion matrix T _mQN ;

Calculate the actual pose transformation matrix: it is obtained by sequentially multiplying the installation matrix T _pQN , the installation error matrix T _peQN , the motion matrix T _mQN and the motion error matrix T _meQN ;

Calculate the actual forward kinematic model of the workpiece chain according to the following formula:

in,

Indicates the actual transformation matrix from WCS (workpiece coordinate system) to RCS (reference coordinate system);

T _RC represents the actual transformation matrix from the C-axis coordinate system to the RCS (reference coordinate system);

T _CW represents the actual transformation matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

T _pRC represents the installation matrix from the C-axis coordinate system to the RCS (reference coordinate system);

T _peRC represents the installation error matrix from the C-axis coordinate system to the RCS (reference coordinate system);

T _mRC represents the motion matrix from the C-axis coordinate system to the RCS (reference coordinate system);

T _meRC represents the motion error matrix from the C-axis coordinate system to the RCS (reference coordinate system);

T _pCW represents the installation matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

T _peCW represents the installation error matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

T _mCW represents the motion matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

T _meCW represents the motion error matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

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

in,

Indicates the actual transformation matrix from TCS (tool coordinate system) to RCS (reference coordinate system);

T _RX represents the actual transformation matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _XZ represents the actual transformation matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _ZA represents the actual transformation matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _AY represents the actual transformation matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _YT represents the actual transformation matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _pRX represents the installation matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _peRX represents the installation error matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _mRX represents the motion matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _meRX represents the motion error matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _pXZ represents the installation matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _peXZ represents the installation error matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _mXZ represents the motion matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _meXZ represents the motion error matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _pZA represents the installation matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _peZA represents the installation error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _mZA represents the motion matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _meZA represents the motion error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _pAY represents the installation matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _peAY represents the installation error matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _mAY represents the motion matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _meAY represents the motion error matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _pYT represents the installation matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _peYT represents the installation error matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _mYT represents the motion matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _meYT represents the motion error matrix from TCS (tool coordinate system) to Y-axis coordinate system;

in,

b ₁₁ ~ b ₃₄ indicate that after matrix operation, the value of each element;

Calculate and establish the actual forward kinematics model of the whole kinematic chain of the gear grinding machine according to the following formula:

in,

Indicates the actual transformation matrix from TCS (tool coordinate system) to WCS (workpiece coordinate system);

Indicates the actual transformation matrix from WCS (workpiece coordinate system) to RCS (reference coordinate system);

Indicates the actual transformation matrix from TCS (tool coordinate system) to RCS (reference coordinate system);

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

in,

Q′ _w =[i′,j′,k′] ^T represents the actual tool axis vector data;

P _w '=[x',y',z'] ^T represents the actual tool nose position data;

Q _t represents the tool axis vector in TCS;

P _t represents the position of the tool tip in TCS;

i', j', k' represent the x, y, z coordinates of the actual tool axis vector data;

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

Further, the subsequent error compensation strategy in the step 2 is performed in the following manner:

With the goal of keeping the ideal tool position data unchanged, it is considered that the motion command of the motion axis will be changed due to the influence of geometric errors. The actual motion command value after error compensation can be reversely solved according to the above formula, so as to obtain the actual processing code and realize error compensation.

Further, the step 2 also includes the following steps:

Determining ideal tool position data, the ideal tool position data includes tool position data and tool axis vector data;

Calculate the mapping relationship between the tool position data and the tool axis vector data according to the following formula:

in,

Indicates the actual transformation matrix from TCS (tool coordinate system) to WCS (workpiece coordinate system);

P _w ＝[x,y,z] ^T represents the ideal tool nose position data;

Q _w ＝[i, j, k] ^T represents the ideal tool axis vector data;

P _t ＝[0,0,0] ^T represents the tool nose position in TCS;

Q _t =[0,0,1] ^T represents the tool axis vector in TCS.

Further, the analytic expression of the actual motion command of the rotary axis in the second step is obtained in the following manner:

Use the mapping relationship between the ideal tool axis vector data and the geometric error of the machine tool to solve the actual motion command of the A axis and the actual motion command of the C axis of the rotary axis;

The analytical expression for calculating the actual motion command of the A-axis according to the following formula is:

in,

A represents the actual movement command of the A axis;

arccos() represents the arccosine function;

ε _y (C) represents the y-direction angle error of C-axis movement;

ε _x (C) represents the angular error of the C-axis motion in the x direction;

ε _x (X) represents the x-direction angle error of the X-axis movement;

ε _x (Y) represents the angular error of the Y-axis movement in the x direction;

ε _x (Z) represents the angular error of the Z-axis movement in the x direction;

ε _x (A) represents the angular error of the A-axis movement in the x direction;

S _YZ indicates the verticality error between Y and Z axes;

α _CY represents the x-direction angle error of C-axis installation;

Indicates the tangent angle;

k represents the z coordinate of the ideal tool axis vector data;

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

i represents the x coordinate of the ideal tool axis vector data;

or

The analytical expression for calculating the actual motion command of the C-axis according to the following formula is:

in,

C represents the actual movement command of the C axis;

arcsin() represents the arcsine function;

ε _y (Y) represents the y-direction angle error of the Y-axis movement;

ε _y (A) represents the y-direction angle error of A-axis movement;

ε _y (Z) represents the y-direction angle error of the Z-axis motion;

ε _y (X) represents the angular error in the y direction of the X-axis movement;

ε _y (C) represents the y-direction angle error of C-axis movement;

ε _z (C) represents the angular error in the z direction of the C-axis motion;

ε _z (Z) represents the z-direction angle error of Z-axis movement;

ε _z (X) represents the angular error of the X-axis motion in the z direction;

ε _x (C) represents the angular error of the C-axis motion in the x direction;

β _CX represents the y-direction angle error of C-axis installation;

S _ZX indicates the verticality error between the Z and X axes;

β _AZ represents the y-direction angle error of A-axis installation;

γ _AY indicates the z-direction angle error of A-axis installation;

or

The analytic expression of the actual motion command of the linear axis is calculated according to the following formula, and the analytic expression of the actual motion command of the linear axis includes the analytic expression of the actual motion command of the Y axis, the analytical expression of the actual motion command of the Z axis, and the expression of the actual motion command of the X axis. Analytical expression;

The analytical expression of the actual motion command of the Y-axis 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)

in,

Y represents the actual movement command of the Y axis;

x represents the x coordinate of ideal tool nose position data;

y represents the y coordinate of the ideal tool nose position data;

z represents the z coordinate of the ideal tool nose position data;

δ _x (C) represents the linear error of the C-axis movement in the x direction;

δ _y (C) represents the linear error of the C-axis movement in the y direction;

δ _z (A) represents the linear error of the A-axis movement in the z direction;

δ _z (Y) represents the linear error of the Y-axis motion in the z direction;

δ _y (A) represents the y-direction linear error of A-axis motion;

δ _y (Y) represents the linear error of the Y-axis movement in the y direction;

S _YZ indicates the verticality error between Y and Z axes;

α _CY represents the x-direction angle error of C-axis installation;

δ _y (X) represents the linear error of the X-axis movement in the y direction;

δ _y (Z) represents the y-direction linear error of Z-axis movement;

δ _Cy represents the y-direction linear error of the C-axis installation;

δ _Ay represents the y-direction linear error of A-axis installation;

The analytical expression of the Z-axis actual motion command is:

Z＝{x(-cosA(ε _y (C)+β _CX cosC+(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 _CX sinC-(ε _x (X)+ε _x (Z)+ε _x (A)+S _YZ -α _CY )cosC)-sinA(cosC-ε _z ( C) sin C))

+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)

in,

Z represents the actual motion command of the Z axis;

δ _z (X) represents the linear error of the X-axis motion in the z direction;

δ _z (Z) represents the linear error of the Z-axis motion in the z direction;

δ _z (C) represents the linear error of the C-axis motion in the z direction;

δ _Az represents the linear error of the A-axis installation in the z direction;

δ _y (C) represents the linear error of the C-axis movement in the y direction;

δ _x (C) represents the linear error of the C-axis movement in the x direction;

The analytical expression of the actual motion command of the X-axis 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)

-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

in,

X represents the actual motion command of the X-axis;

δ _x (X) represents the linear error of the X-axis motion in the x direction;

δ _x (Y) represents the linear error of the Y-axis movement in the x direction;

δ _x (Z) represents the linear error of the Z-axis movement in the x direction;

δ _x (A) represents the linear error of the A-axis movement in the x direction;

δ _Cx represents the linear error of the C-axis installation in the x direction;

δ _x (C) represents the linear error of the C-axis movement in the x direction;

δ _y (C) represents the linear error of the C-axis movement in the y direction;

ε _z (C) represents the angular error in the z direction of the C-axis motion;

ε _y (C) represents the y-direction angle error of C-axis movement;

ε _z (A) represents the angular error of the A-axis movement in the z direction;

S _YX indicates the verticality error between the Y and X axes;

S _ZX indicates the verticality error between the Z and X axes;

γ _AY indicates the z-direction angle error of A-axis installation;

ε _z (Z) represents the z-direction angle error of Z-axis movement;

ε _z (X) represents the angular error of the X-axis motion in the z direction;

β _AZ represents the y-direction angle error of A-axis installation;

ε _y (Z) represents the y-direction angle error of the Z-axis movement;

ε _y (X) represents the angular error in the y direction of the X-axis motion;

ε _y (X) represents the angular error in the y direction of the X-axis motion.

Further, the geometric error-tool space pose error model in the step 3 is established according to the following formula:

in,

△i represents the x-coordinate error of the tool axis vector data;

△j represents the y coordinate error of the tool axis vector data;

△k represents the z coordinate error of the tool axis vector data;

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

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

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

Q′ _w represents the actual tool axis vector data;

P′ _w represents the actual tool nose position data;

Q _w represents the ideal tool axis vector data;

P _w represents the ideal tool nose position data;

Indicates the actual transformation matrix from TCS (tool coordinate system) to WCS (workpiece coordinate system);

Indicates the ideal transformation matrix from TCS (tool coordinate system) to WCS (workpiece coordinate system);

Q _t represents the tool axis vector in TCS;

P _t represents the tool tip position in TCS.

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

Establish the parameter equation of the axial profile of the grinding wheel:

in,

r ^wap represents the axial profile of the grinding wheel;

Indicates the x-coordinate of the axial profile of the grinding wheel;

Indicates the y-coordinate of the axial profile of the grinding wheel;

η represents the axial profile parameter of the grinding wheel,

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

r ^wt represents the sand profile in TCS, where the first superscript refers to the grinding wheel, and the second superscript represents TCS;

Establish the two-parameter equation of the grinding wheel surface in TCS:

in,

Indicates the x-coordinate of the sand profile in TCS;

Indicates the y-coordinate of the sand profile in TCS;

Indicates the z coordinate of the sand profile in TCS;

r ^wt represents the sand profile in TCS;

Calculate the unit normal vector of the grinding wheel surface in TCS according to the following formula:

n ^wt represents the unit normal vector of the grinding wheel in TCS;

Represents the partial derivative of r ^wt to η;

Indicates the partial derivative of r ^wt to φ;

Calculate the grinding wheel surface parameter equation in WCS according to the following formula:

in,

r ^wwi represents the ideal profile of the grinding wheel in WCS;

n ^wwi represents the ideal unit normal vector of the grinding wheel in WCS;

Indicates the ideal transformation matrix from TCS (tool coordinate system) to WCS (workpiece coordinate system);

r ^wt represents the sand profile in TCS;

n ^wt represents the unit normal vector of the grinding wheel in TCS;

r ^wwa represents the actual profile of the grinding wheel in WCS;

n ^wwa represents the actual unit normal vector of the grinding wheel in WCS;

Indicates the actual transformation matrix from TCS (tool coordinate system) to WCS (workpiece coordinate system);

r ^wt represents the sand profile in TCS;

n ^wt represents the unit normal vector of the grinding wheel in TCS;

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

Calculate the grinding contact point according to the following formula:

f＝(k ^gw ×r ^ww +pk ^gw )·n ^ww ＝0

in,

k ^gw represents the gear axis in WCS;

r ^ww represents the radial vector from the origin of WCS to a point on the surface of the grinding wheel;

p represents the spiral parameter;

n ^ww represents the unit normal vector of a point on the surface of the grinding wheel in WCS;

Calculate the tooth surface numerical model according to the following formula:

in,

r ^gw represents the tooth surface coordinate vector;

n ^gw represents the unit normal vector of the tooth surface;

Represents the coordinate vector of the jth discrete point on the kth contact line;

Indicates the unit normal vector of the jth discrete point on the kth contact line;

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

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

The tooth surface pose error model is established according to the following formula:

in,

δ _TS represents the tooth surface position error;

ε _TS represents the attitude error of the tooth surface;

δ _x , δ _y , δ _z represent the errors in the x, y, and z directions of the tooth surface position;

ε _x , ε _y , ε _z represent the tooth surface attitude x, y, z direction error;

r ^gwa represents the actual tooth surface coordinate vector;

n ^gwa represents the unit normal vector of the actual tooth surface;

r ^gwi represents the ideal tooth surface coordinate vector;

n ^gwi represents the unit normal vector of the ideal tooth surface.

The 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 )

in,

δn represents the normal error of the tooth surface;

n ^gwi represents the unit normal vector of the ideal tooth surface;

dot() means dot product operation;

r ^gwa represents the actual tooth surface coordinate vector;

n ^gwi represents the unit normal vector of the ideal tooth surface.

Further, the specific steps of identifying and analyzing the key error sources of the tooth profile deviation are as follows:

1) Generate a random sampling matrix H _N×2m based on the sampling sequence;

Among them, N represents the number of samples for each error, and m represents the number of errors;

2) Construct geometric error input matrices A _N×m , B _N×m and A _B ⁱ _N×m according to the random sampling matrix H _N×2m ;

Take the first m columns of H _N×2m as matrix A, and the last m columns as matrix B, and construct m derived matrices A _B ⁱ ;

3) Each row of the input matrix is used as a set of geometric error parameters, and there are N (m+2) sets in total;

And bring each group of errors disassembled by A, B and A _B ⁱ into the tooth profile deviation model F _α =f(G),

Among them, G represents the set of geometric errors,

Calculate N(m+2) times to obtain the corresponding tooth profile deviation f(A), f(B), f(A _B ⁱ );

Finally, the sensitivity index of geometric error elements is calculated by using the Monte Carlo estimation formula, including the first-order sensitivity index S _i and the global sensitivity index S _Ti :

in,

S _i represents the first-order sensitivity index of the i-th error element;

S _Ti represents the global sensitivity index of the i-th error element;

V represents the total variance of the error model output;

f(B) _j represents the tooth profile deviation corresponding to the input error in row j of matrix B;

representation matrix The tooth profile deviation corresponding to the input error in line j;

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

N represents the number of samples for each error.

The beneficial effects of the present invention are:

The invention provides an efficient compensation method for the key errors of gear grinding. First, based on the geometric error distribution of the form grinding machine tool and the actual kinematic chain of the machine tool, the actual forward kinematics model of the gear grinding process is constructed to reflect the position in the tool coordinate system under the influence of the geometric error. The functional relationship between the tool pose and the tool position data in the workpiece coordinate system; then, based on the actual inverse kinematics compensation principle, the analytical expression of the actual motion command of the motion axis after error compensation is derived, revealing the geometric error, ideal tool position data The mapping law between the actual motion command and the actual motion command; finally, according to the principle of conjugate grinding, the geometric error-tooth surface error model is established, the actual tooth profile and tooth direction accuracy are calculated and evaluated, and the key error sources of the tooth profile deviation are identified. The actual inverse kinematics compensation method is simplified to realize efficient error compensation for tooth profile deviation reduction.

Other advantages, objects and features of the present invention will be set forth in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the investigation and research below, or can be obtained from Taught in the practice of the present invention. The objects and other advantages of the invention may be realized and attained by the following specification.

Description of drawings

In order to make the purpose, technical scheme and beneficial effect of the present invention clearer, the present invention provides the following drawings for illustration:

Figure 1 is a schematic diagram of the structure of the profile gear grinding machine.

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

Figure 3 is a flow chart of key error identification.

Fig. 4 is a schematic diagram of error source sensitivity index of tooth profile deviation.

Fig. 5 is a flowchart of an efficient compensation method for key errors of gear grinding.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and implement it, but the examples given are not intended to limit the present invention.

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

Step 1: Modeling the geometric error of the form grinding system;

(1) Analysis of geometric error of form grinding system

The basic structure of the tYAZXRCw five-axis form grinding machine tool shown in Figure 1 includes three linear motion axes of X, Y, and Z and two rotation axes. R represents the base of the machine tool, t represents the tool, that is, the grinding wheel, and w represents the workpiece gear. .

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

Set the X axis as the basic axis without installation error, Z as the auxiliary axis, and the origin of the machine tool coordinate system (reference coordinate system) as the intersection of the centerlines of the A and C rotation axes when each movement axis is at zero. On the basis of configuring the corresponding sub-coordinate systems for the five motion axes, machine base, workpiece, and tool, considering the position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) shown in Table 1, the actual full motion of the machine tool The chain is shown in Figure 2, which shows the full motion chain of the machine tool and 41 geometric errors.

Table 1 Geometric error elements of form gear grinding machine

in,

δ _x (X), δ _y (X), δ _z (X) represent the linear error of the x, y, and z directions of the X-axis movement;

δ _x (Y), δ _y (Y), δ _z (Y) represent the linear error of the Y-axis movement in x, y, and z directions;

δ _x (Z), δ _y (Z), δ _z (Z) represent the linear error of the Z-axis movement in x, y, and z directions;

δ _x (A), δ _y (A), δ _z (A) represent the linear error of A-axis movement in x, y, and z directions;

δ _x (C), δ _y (C), δ _z (C) represent the linear error of C-axis movement in x, y, and z directions;

ε _x (X), ε _y (X), ε _z (X) represent the angular error of the x, y, and z directions of the X-axis movement;

ε _x (Y), ε _y (Y), ε _z (Y) represent the angular error in the x, y, and z directions of the Y-axis movement;

ε _x (Z), ε _y (Z), ε _z (Z) represent the angular error in the x, y, and z directions of the Z-axis movement;

ε _x (A), ε _y (A), ε _z (A) represent the angular error of the A-axis movement in x, y, and z directions;

ε _x (C), ε _y (C), ε _z (C) represent the angular error in the x, y, and z directions of the C-axis movement;

S _YX , S _YZ indicates the verticality error between Y and X axes and the verticality error between Y and Z axes;

S _ZX indicates the verticality error between the Z and X axes;

δ _Ay , δ _Az , β _AZ , γ _AY represent the linear error of A-axis installation in y and z direction and the installation of C-axis in y and z direction angular error;

δ _Cx , δ _Cy , α _CY , β _CX indicate the linear error of C-axis installation in x and y directions and the angular error of C-axis installation in x and y directions;

(2) Build the actual forward kinematics model

If the ideal pose transformation matrix between adjacent component coordinate systems (for example, N to Q coordinate system) can be obtained by multiplying the installation matrix T _pQN and the motion matrix T _{mQN sequentially} .

The actual pose transformation matrix can be obtained by multiplying the installation matrix T _pQN , the installation error matrix T _peQN , the motion matrix T _mQN , and the motion error matrix T _{meQN sequentially} .

Then for the workpiece chain (RCw), the actual transformation matrix of WCS relative to RCS, that is, the actual forward kinematics model of the workpiece chain is:

In the same way, the actual forward kinematics model of the tool chain can be established, that is, the actual transformation matrix of TCS relative to RCS:

in,

By integrating the actual forward kinematics model of the tool chain and the tool chain, the actual forward kinematics model of the whole kinematic chain of the gear grinding machine can be established, that is, the actual transformation matrix of TCS relative to WCS:

From this, it can be seen that the actual transformation matrix of TCS relative to WCS is essentially a matrix function with the ideal motion command of the motion axis and the geometric error as independent variables.

If it is multiplied by the tool nose position P _t and the tool axis vector Q _t in TCS, the tool position data under the influence of geometric error can be obtained:

Among them, P′ _w =[x′,y′,z′] ^T and Q′ _w =[i′,j′,k′] ^T represent the actual tool nose position and the actual tool axis vector, respectively.

Step 2: Geometric error compensation method based on actual inverse kinematics

(1) Actual inverse kinematics compensation principle

Step 1 actually provides an explicit calculation method of actual tool position data considering the influence of geometric error, which can solve the actual tool position data through the known ideal motion command of the motion axis and the measured and calibrated geometric error.

The subsequent error compensation strategy is usually: first calculate the error vector between the actual tool position data and the ideal tool position data, and then add a compensation vector that is equal to the vector and opposite in direction without changing the ideal machining code, and Reversely solve the corresponding compensation processing code, so as to realize error compensation and improve processing accuracy.

In order to simplify the calculation process of error compensation and improve the compensation efficiency, in this embodiment, the goal is to ensure that the ideal tool position data remains unchanged. The motion command of the motion axis will be changed due to the influence of geometric errors. The actual motion command value after error compensation can be calculated according to The above formula is reversely solved, so as to obtain the actual processing code and realize error compensation.

Therefore, the ideal tool position data is given in advance, including tool position data and tool axis vector data, that is, tool nose position P _w =[x,y,z] ^T and tool axis vector Q _w =[i,j,k ] ^T. At the same time, the tool nose position and tool axis vector in TCS are P _t ＝[0,0,0] ^T and Q _t ＝[0,0,1] ^T respectively, and the mapping relationship between them can be expressed as:

That is,

(2) Derivation of the analytical expression of the actual motion command of the rotary axis

Considering that the tool axis vector data is not affected by the motion of the linear axis, first use the mapping relationship between the ideal tool axis vector data and the geometric error of the machine tool to separate and solve the actual motion commands A and C of the rotary axis,

That is,

in,

Indicates the actual transformation matrix from WCS (workpiece coordinate system) to RCS (reference coordinate system);

Indicates the actual transformation matrix from TCS (tool coordinate system) to RCS (reference coordinate system);

If the C-axis motion matrix function is separated and converted to the left side of the equation, it can be simplified to get

in,

T _mCW represents the motion matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

T _meCW represents the motion error matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

T _peCW represents the installation error matrix from WCS (workpiece coordinate system) to C-axis coordinate system;

Therefore, we can get:

(T _mCW T _meCW )[i,j,k,0] ^T ＝[b ₁₃ -b ₃₃ β _CX ,b ₂₃ +b ₃₃ α _CY ,b ₃₃ +b ₁₃ β _CX -b ₂₃ α _CY ,0] ^T

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

Therefore, the calculation formula of the z-direction tool axis vector in the formula can be extracted separately, and the second-order and higher-order items can be omitted to obtain the equation related only to the motion command A,

-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

According to the auxiliary angle formula and trigonometric inverse function, the analytical expression of the actual A-axis motion command is:

in,

Similarly, if the A-axis motion matrix function is separated and converted to the right side of the equation, it can be simplified to get:

in,

T _peZA represents the installation error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _mZA represents the motion matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _meZA represents the motion error matrix from the A-axis coordinate system to the Z-axis coordinate system;

Indicates the actual transformation matrix from TCS (tool coordinate system) to A-axis coordinate system;

Indicates the actual transformation matrix from WCS (workpiece coordinate system) to RCS (reference coordinate system);

Since the right side of the equation is a matrix function related only to the rotation of the A axis, according to the theory of rotation invariance, it can be known that the x-direction data of the tool axis vector is not affected by the rotation of the A axis. Therefore, the calculation formula of the x-direction tool axis vector in the formula can be extracted separately, and the second-order and higher-order terms can be omitted to obtain the equation related only to the motion command C,

[(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 trigonometric inverse function,

sin(C+φ)＝ε _y (Y)+ε _y (A)-k(β _CX -ε _y (X)-ε _y (Z)-S _ZX -β _AZ )

in,

Thus, the analytic expression of the actual C-axis motion command can be obtained as:

(3) Derivation of the analytical expression of the actual motion command of the linear axis

Based on the actual motion command of the rotary axis, the actual motion command of the linear axis is solved by using the mapping relationship between the ideal tool position data and the geometric error of the machine tool:

That is,

in,

T _pRX represents the installation matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _peRX represents the installation error matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _mRX represents the motion matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _meRX represents the motion error matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _pXZ represents the installation matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _peXZ represents the installation error matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _mXZ represents the motion matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _meXZ represents the motion error matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _pZA represents the installation matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _peZA represents the installation error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _mZA represents the motion matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _meZA represents the motion error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _pAY represents the installation matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _peAY represents the installation error matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _mAY represents the motion matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _meAY represents the motion error matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _pYT represents the installation matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _peYT represents the installation error matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _mYT represents the motion matrix from TCS (tool coordinate system) to Y-axis coordinate system;

T _meYT represents the motion error matrix from TCS (tool coordinate system) to Y-axis coordinate system.

Based on the translational feature separation theory of block calculation, it can be transformed into:

T _meRX represents the motion error matrix from the X-axis coordinate system to the RCS (reference coordinate system);

T _peXZ represents the installation error matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _meXZ represents the motion error matrix from the Z-axis coordinate system to the X-axis coordinate system;

T _peZA represents the installation error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _mZA represents the motion matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _meZA represents the motion error matrix from the A-axis coordinate system to the Z-axis coordinate system;

T _peAY represents the installation error matrix from the Y-axis coordinate system to the A-axis coordinate system;

T _meAY represents the motion error matrix from the Y-axis coordinate system to the A-axis coordinate system;

If set

in,

f ₁₁ ~ f ₃₄ represent the value of each matrix element after T _meRX T _peXZ matrix operation;

g ₁₁ ~ g ₃₄ represent the value of each matrix element after T _meRX T _peXZ T _meXZ T _peZA T _mZA T _meZA T _peAY matrix operation;

h ₁₁ ~ h ₃₄ represent the value of each matrix element obtained after multiplying the above matrices;

Available,

Expanding it into a system of polynomial equations, we get:

According to the above formula, the analytical expression of the error-containing linear axis motion command (X, Y, Z) can be obtained.

Since f ₃₃ ＝1, f ₂₃ ＝-ε _x (X), then ③×f ₂₃ -②, and omitting higher-order items above the second order, we can get:

x(f ₂₃ a ₃₁ -a ₂₁ )+y(f ₂₃ a ₃₂ -a ₂₂ )+z(f ₂₃ a ₃₃ -a ₂₃ )+(f ₂₃ a ₃₄ -a ₂₄ )-f ₂₃ h ₃₄ +h ₂₄ ＝Y(f ₂₃ g ₃₂ -g ₂₂ )

Therefore, the analytical expression of the actual motion command of the Y-axis 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)

From ③×g ₂₂ -②×g ₃₂ , and omitting higher-order terms above the second order, we can get:

x(-cosA(ε _y (C)+β _CX cosC+(ε _x (X)+ε _x (Z)+ε _x (A)+S _YZ -α _CY )sinC)-sinA(sinC+ε _z (C ) cos C))

+y(cosA(ε _x (C)+β _CX sinC-(ε _x (X)+ε _x (Z)+ε _x (A)+S _YZ -α _CY )cosC)-sinA(cosC-ε _z ( C) sin C))

+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)

＝Z(cosA-(ε _x (X)+ε _x (Z)+ε _x (A)+S _YZ -ε _x (X))sinA)

Therefore, the analytical expression of the Z-axis actual motion command is:

Z＝{x(-cosA(ε _y (C)+β _CX cosC+(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 _CX sinC-(ε _x (X)+ε _x (Z)+ε _x (A)+S _YZ -α _CY )cosC)-sinA(cosC-ε _z ( C) sin C))

+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)

Based on the obtained actual motion commands of the Y and Z axes, according to formula ①, it can be obtained:

X＝xa ₁₁ +ya ₁₂ +za ₁₃ +a ₁₄ -h ₁₄ -Zf ₁₃ -Yg ₁₂

Therefore, the analytical expression of the actual motion command of the X-axis 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)

-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

So far, the mapping rules of the actual motion command, geometric error and ideal tool position data of the five motion axes of the NC form gear grinding machine have been clearly expressed. Under the premise that the ideal tool position data is given and the geometric error of the machine tool is calibrated effectively, the actual NC machining instruction with error compensation can be directly obtained according to the above analytical expression, so as to realize error compensation.

Step 3: Key error identification and compensation model simplification

(1) Geometric error-tooth surface error model

The forward kinematics model of CNC form grinding gear describes the transformation relationship between the position and attitude of the gear and the grinding wheel coordinate system. If the actual situation considering the geometric error is compared with the ideal state ignoring the influence of the error, and then multiplied by the homogeneous coordinates of the center position and attitude of the grinding wheel in TCS, the geometric error-tool space position of the gear grinding machine can be established The pose error model, that is, the relative pose error between the grinding wheel and the gear blank in WCS:

Considering that the geometric error-tool space error model ignores the influence of the cutting motion interference between the tool and the workpiece surface on the machining effect, it is necessary to further construct the model from the tool space error to the tooth surface pose error based on the material removal mechanism in the actual grinding process. In order to establish a geometric error-tooth surface pose error model that can more directly reflect the influence of geometric errors on the tooth surface pose error.

According to the principle of conjugate grinding, the axial profile parameter of the grinding wheel is represented by η, and the axial profile parameter equation of the grinding wheel is established:

Let φ represent the rotation angle of the axial profile of the grinding wheel, and r ^wt represent the parametric equation of the sand profile, where the first superscript refers to the grinding wheel, and the second superscript represents TCS, then the two-parameter equation of the grinding wheel surface in TCS can be established :

in,

From this, it can be obtained that the unit normal vector of the grinding wheel surface in TCS is:

Based on the transformation relationship between the gear and the grinding wheel coordinate system established by the forward kinematics model, the parameter equation of the grinding wheel surface in WCS under ideal conditions and actual conditions can be obtained as follows:

Among them, the first superscript of r and n refers to the grinding wheel, the second superscript indicates WCS, and the third superscript indicates ideal state i or actual state a.

When the grinding wheel is known, the conjugate grinding contact condition between the grinding wheel and the gear can be expressed as the radial vector r ^ww from the origin of WCS to a point on the surface of rotation of the grinding wheel, if this point makes a spiral motion around the gear axis k ^gw The velocity vector is perpendicular to its normal line n ^ww on the surface of the grinding wheel, then this point is the grinding contact point.

f＝(k ^gw ×r ^ww +pk ^gw )·n ^ww ＝0

Bringing r ^ww and n ^ww under ideal and actual conditions into the above formula respectively, the numerical solution of the grinding contact line can be obtained by discretizing the grinding trajectory and using the dichotomy method. If the tooth surface formed by grinding is composed of λ contact lines, and each contact line is composed of n discrete contact points, the numerical model of the tooth surface can be expressed as:

The difference between the actual state and the ideal state can be compared, and the tooth surface pose error model can be established.

Based on the geometric error-tooth surface pose error model, the tooth surface position error and the ideal tooth surface normal vector can be calculated by dot product, so as to establish the tooth surface normal error model, that is,

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

Since the ground tooth surface is composed of λ contact lines containing n discrete points, if the normal error rotation of the kth contact line is transformed to the gear end section, the corresponding tooth profile deviation curve can be obtained, and further analysis Evaluation indicators such as the total tooth profile deviation (F _α ), tooth profile shape deviation (f _fα ) and tooth profile slope deviation (f _Hα ) can be obtained; if the normal errors of all the jth points on the λ contact line are extracted separately Then the corresponding helix deviation curve can be obtained, and further analysis can obtain the evaluation indexes such as the total deviation of the helix (F _β ), the deviation of the helix shape (f _fβ ) and the deviation of the slope of the helix (f _Hβ ).

(2) Key error source identification process

As shown in Figure 3, Figure 3 is the key error identification process. According to the Sobol method, the identification and analysis of key error sources can be carried out for tooth profile deviation and helix deviation respectively. Taking the analysis of key error sources of tooth profile deviation as an example.

1) A random sampling matrix H _N×2m is generated based on the sampling sequence, which is constructed considering the probability distribution of the input error geometric items. Among them, N represents the sampling number of each error, and m represents the number of errors.

2) Construct the geometric error input matrices A _N×m , B _N×m and A _B ⁱ _N×m according to the random sampling matrix H _N×2m . Take the first m columns of H _N×2m as matrix A, and the last m columns as matrix B, and construct m derived matrices A _B ⁱ , except that the i-th column is equal to the i-th column of B, and the rest of the columns come from A.

3) Each row of the input matrix is used as a set of geometric error parameters, and there are N (m+2) sets in total. And bring each group of errors dismantled from A, B and A _B ⁱ into the tooth profile deviation model F _α =f(G), G represents the geometric error set, calculate N(m+2) times, and get the corresponding The tooth profile deviation f(A), f(B), f(A _B ⁱ ). Finally, the sensitivity indices of geometric error elements are calculated by Monte Carlo estimation formula, including the first-order sensitivity index S _i and the global sensitivity index S _Ti .

The key error source of the total tooth profile deviation can be judged by comparing the global sensitivity index of the geometric error elements of the tooth profile deviation.

(3) Efficient compensation method for key errors

As shown in Figure 4, the base figure 4 is the error source sensitivity index of the tooth profile deviation, which is based on the identified key error source, ignoring the influence of other geometric errors, and simplifying the error compensation model. For example, the geometric error source sensitivity index for tooth profile deviation. The abscissa of the figure represents the error serial number, but the Y-axis geometric error is not counted; the ordinate represents the error sensitivity index.

If 0.05 is used as the sensitive threshold, the key error numbers for preliminary judgment of tooth profile deviation are: 4, 10, 17, 27, 28, 29, 32, 33, that is, the geometric error items ε _x (X), ε _x (Z ), ε _x (A), ε _x (C), ε _y (C), ε _z (C), α _CY , β _CX .

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

in,

in,

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 _CX cosC+(ε _x (X)+ε _x (Z)+ε _x (A)-α _CY )sinC)-sinA(sinC+ε _z (C ) cos C))

+y(cosA(ε _x (C)+β _CX sinC-(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 above-mentioned simplified analytical expression of the actual motion command of the motion axis, the NC code after the corresponding key error compensation can be obtained, and the original code can be replaced to run, and the efficient error compensation for tooth profile deviation reduction can be realized.

The above-mentioned embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or transformations made by those skilled in the art on the basis of the present invention are all within the protection scope of the present invention. The protection scope of the present invention shall be determined by the claims.

Claims (9)

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

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:

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,

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;

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

wherein,

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

T_ZAto show the axis AA practical transformation matrix from a standard 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_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_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_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,

b₁₁～b₃₄after representing matrix operationsThe 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,

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

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

wherein,

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;

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.

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

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.

5. The method of claim 1, wherein: 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,

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

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:

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,

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,

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;

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,

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_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,

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;

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,

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;

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,

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

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,

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,

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,

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 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,

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,

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,

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;

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,

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；

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

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, F_αRepresenting tooth profile deviation;

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,

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.