CN104950805B - A kind of space error compensation method based on Floyd algorithms - Google Patents

Abstract

A spatial error compensation method based on the Floyd algorithm, based on the error measurement data, using the exponential matrix form of the screw theory, on the basis of the topology of the machine tool, the overall spatial error model of the machine tool is established, and the high-order terms of the error model According to Floyd's shortest distance algorithm, the weight value is continuously adjusted, iterated to the basic range allowed by the accuracy, and the error compensation model is obtained and the compensation effect is achieved with a small amount of calculation. The error compensation can be used in various Real-time compensation of machine tool errors in complex actual machining situations. And compared with the classic error compensation algorithm "ACO-BPN" in the example, it is found through simulation that the Floyd compensation algorithm has the characteristics of better compensation effect, higher execution efficiency and fewer iterations compared with the ACO-BPN compensation algorithm, and it can be used in a variable temperature environment. The robustness of the Floyd compensation algorithm is verified in the paper.

Description

A Spatial Error Compensation Method Based on Floyd Algorithm

technical field

The invention provides a machine tool space error modeling and a space error compensation method based on the model and Floyd algorithm, belonging to the field of CNC machine tool precision compensation model design.

Background technique

High-precision CNC machine tools are often used in modern production, especially for high-efficiency and complex curved parts, which are also an important part of processing and high-performance equipment manufacturing. The spatial error of the machine tool is the most important part affecting the machining accuracy. Thermal error and geometric error account for about 70% of all errors, especially in the case of precision and ultra-precision machining. In mechanical processing, the machining accuracy of the machine tool is ultimately determined by the relative displacement between the tool and the workpiece on the machine tool. The relative displacement between the tool and the workpiece on the machine tool can be calculated using kinematic modeling techniques.

The geometric error of the machine tool mainly comes from the manufacturing accuracy of its guide rail, as well as the installation accuracy and its own straightness and other errors. In order to better improve the accuracy of CNC machine tools, the establishment of error models is also very important. A robust and accurate error model is also the first step in error correction and compensation. Experts and scholars at home and abroad have been unremittingly exploring and researching in the field of establishing the spatial error model of CNC machine tools, and have carried out various work. For example, triangular relationship modeling method, error matrix method, quadratic relationship model method, mechanism modeling method, rigid body kinematics method, etc. However, the actual measured error is the coupling effect of multiple errors such as thermal error and geometric error. In the thermal error compensation method, effective thermal error compensation mainly depends on reliable measuring devices, efficient measuring methods, and a systematic mathematical model that can correctly reflect the internal relationship between the temperature data of key temperature measuring points and the thermal error data of the machine tool. Scholars at home and abroad have done a lot of experiments and research on the error compensation model, revealing the relationship between the temperature of each machine tool component and the thermal error from different angles. Commonly used modeling methods include: least squares fitting modeling, modeling based on time series analysis, artificial neural network modeling, gray theory modeling, least squares support vector machine modeling, etc., among which neural network and gray theory are Two kinds of error compensation models have been widely used in recent years.

At present, both of them have good results in their respective development processes, but there is no systematic solution to the comprehensive compensation method in the case of coupling. Based on this starting point, the patent of the present invention proposes an error compensation model and method in detection, calculation, and prediction errors. The compensation model has the advantages of high compensation accuracy, high calculation efficiency, good time period and closed-loop robustness; based on the error measurement data, using the exponential matrix form of the screw theory, on the basis of the topology of the machine tool, a machine tool is established. The overall spatial error model reduces the high-order terms of the error model to obtain the basic equation of the error model; according to Floyd's shortest distance algorithm, continuously adjusts the weight, iterates to the basic range allowed by the accuracy, and achieves compensation with a small amount of calculation As a result, the error compensation principle technology can be used for real-time compensation of machine tool errors in various complex actual processing situations.

Contents of the invention

The object of the present invention is to provide a compensation method for the spatial comprehensive error of a machine tool, and to establish a spatial comprehensive error compensation model. Propose error compensation models and methods in detection, calculation and prediction errors. The compensation model has the advantages of high compensation accuracy, high calculation efficiency, good time period and closed-loop robustness; based on the error measurement data, using the exponential matrix form of the screw theory, on the basis of the topology of the machine tool, a machine tool is established. For the overall spatial error model, the high-order terms of the error model are reduced to obtain the basic equation of the error model; according to Floyd's shortest distance algorithm, the weight is continuously adjusted, iterated to the basic range allowed by the accuracy, and the error compensation model is obtained. The amount of calculation can achieve the compensation effect. This error compensation principle technology can be used for real-time compensation of machine tool errors in various complex actual processing situations.

In order to achieve the above object, the technical solution adopted in the present invention is a kind of machine tool space error compensation method based on Floyd algorithm, the realization process of this method is as follows, shown in Fig. 1 is the specific implementation steps of this method,

Step 1. Establish the spatial comprehensive error model of the machine tool based on the screw theory

According to the exponential matrix form of the screw theory, each moving part of the machine tool is abstracted into a 6×1 vector form. The motion form and comprehensive error are modularized and expressed in the form of exponential matrix, and the spatial comprehensive error model of the machine tool is established according to the topological structure of the machine tool.

Step 1.1 Exponential matrix form of spinor theory

The motion of any rigid body can be decomposed into two parts: translational motion along the axis and rotational motion around the axis. That is, each component is regarded as a screw. The unit spinor becomes in Plücker coordinates as follows:

To express any form of motion of a rigid body in space, there are:

Among them, υ=[v ₁ , v ₂ , v ₃ ] ^T , Represents an antisymmetric matrix, if ω=[ω ₁ , ω ₂ , ω ₃ ] ^T , can be expressed as:

Rigid body motion generally includes translation and rotation, and the vector q is the same in the rigid body coordinate system and the reference coordinate system. Then the homogeneous transformation matrix of the rigid body is:

The second transformation matrix corresponding to the exponential form of the spinor can be written as: When ω=0, the rigid body only has translational motion, then the homogeneous transformation matrix can be written as:

When ω≠0, there is also rotational motion for rigid bodies, and the exponential matrix at this time is:

in The trigonometric series expansion of is expressed as:

To sum up, the exponential matrix for any form of motion of a rigid body in space can be expressed as:

When $ is the unit screw, when ||ω||≠0, the rotation angle of the mechanical part is expressed as When ||ω||=0, the translation distance is expressed as Points are expressed differently in different coordinate systems, and the difference between them is expressed by a transformation matrix. A screw is also understood as a point in a coordinate system, and its expression in different coordinate systems is also different. Therefore, the form of a transformation matrix is also needed to express the relationship of a screw in different coordinate systems, which is called an adjoint matrix. If the motion screw of a rigid body is The exponential matrix in its transformed form can be expressed as:

Then its adjoint exponential matrix form in this coordinate system:

The adjoint index matrix satisfies the following properties:

For a mechanical structure such as a machine tool, the structure is expressed by an exponential matrix:

T(0) represents its original transformation matrix, which is applied to the error modeling of the machine tool.

Step 1.2 Use the exponential matrix type to model the spatially integrated error of the machine tool

In the case of machine tool operation, the error is mainly the coupling effect of geometric error and thermal error, and the measured error δ includes both thermal error and geometric error. which is:

δ = δ _G + δ _T (14)

Where: δ _G is the geometric error;

_δT is the thermal error;

Generally, each axial movement has 6 directional degrees of freedom, and at the same time, 3 translational errors and 3 rotational errors will be generated. Using the screw theory, Moon et al. defined the error module m _e $ _e .

m _e $ _e = [ε _x ,ε _y ,ε _z ,δ _x ,δ _y ,δ _z ] ^T

Taking the geometric error composition in the X direction as an example, it is mainly divided into three parts. The first part $ _xx contains positioning error and roll error along this direction The second part $ _yx is the linear error and yaw error of the horizontal plane The third part $ _zx is the linear error and yaw error of the vertical plane The thermal deformation of the machine tool is finally reflected on the moving parts of the machine tool. Due to the influence of the thermal deformation of the machine tool, the thermal error caused by the movement track of the machine tool's moving parts deviates from the ideal track. That is to say, when moving in the X direction, similar to the geometric error, there will also be 6 thermal errors. That is, three moving errors: X-direction linear displacement thermal error Y-direction straightness thermal error and Z-direction straightness thermal error Three corner errors: tilt thermal error around the X axis Yaw thermal error around the Z axis and pitch thermal error around the Y axis

The spatial error of the X axis can be expressed as:

The error model of the X-axis is in the form of an exponential matrix, expressed as:

In the same way, the spatial error modules of other axes and the error models of the exponential matrix can be obtained.

Step 1.3 Exponential Matrix Form for Perpendicularity and Parallelism Errors

Due to the difference between the actual axis and the ideal axis, the two adjacent axes are not absolutely 90°; that is to say, there is a verticality error. For the three translational axes, assuming that the Y axis is an ideal axis, there is no verticality error; then the verticality error between the X axis and the Y axis is S _xy , and the verticality error between the Y axis and the Z axis is S _yz , the perpendicularity between X and Z is S _xz . In the plane formed by the Y-axis and the actual installed X-axis, only S _xy exists for the X-axis, and similarly there are two other perpendicularity errors on the actual Z-axis. As shown in Figure 2, since the direction of the actual axis inevitably deviates from the position of the ideal axis, it is considered that a perpendicularity error should be added to the coordinate transformation. For the transformation form of the ideal coordinate axis:

Taking the X direction as an example, the X direction unit screw in the ideal state can be expressed as:

Adding the verticality error in the real state, the actual unit screw in the X direction can be expressed as:

The corresponding exponential matrix can be expressed as:

Another way of writing is to use the form of an adjoint matrix to rotate the ideal X-axis with the Z axis ($ _zr ) at a certain angle to achieve the effect of 90° between the X-axis and the Y-axis, namely:

In the same way, the expression form of the verticality error screw of the Z axis is obtained.

For the rotation of the A axis and the C axis, the deviation in the X and Z directions will occur during installation, that is, the actual parallelism between the A axis and the X axis is the component PY _xA in the Y direction, and the parallelism between the A axis and the X axis The component PZ _xA in the Z direction; similarly, two error terms on the parallelism between the C axis and the Z axis will be obtained. As shown in Figure 3 (the X, Y, and Z axes here are all ideal axes)

Taking the A-axis as an example, the axis unit screw of the A-axis motion in the ideal state is:

Adding the parallelism error components along the Y and Z directions in the real state, the actual unit screw in the A direction is expressed as:

Similar to the matrix transformation method above, the ideal X-axis is rotated by a certain angle along the Y-axis and Z-axis respectively to express the actual position of the A-axis in reality:

In the same way, the expression form of the exponential matrix of the parallelism error of the C axis is obtained.

Step 1.4 Establishment of error model based on topology

The theory of multi-body systems provides a detailed model of the topology of machine tools, which can also be applied in exponential matrices. Ideally, there is no error in the machine tool. The matrix transformation equation in an ideal state can be expressed by T _i :

In practice, due to the error of the machine tool components themselves and position error between parts Added integral part error screw to the screw module. Expressed by T _a :

According to the matrix transformation equation under actual and ideal conditions, the spatial error model of the multi-axis CNC machine tool is obtained:

The components E _x , E _y , and E _z of the corresponding spatial error on the three axes can be expressed as:

[E _x , E _y , E _z ,1] ^T ＝ E·[0,0,0,1] ^T (31)

By omitting the second-order and higher-order terms in the formula, the basic equation of the spatial error can be obtained.

Step 2: Spatial Error Compensation Principle Based on Floyd Algorithm

Generally, there are two error compensation methods for CNC machine tools: ① According to the error data tested in actual processing, manually adjust the NC machining program; The error data is entered in advance to the corresponding error compensation setting items (such as backlash compensation, pitch compensation and tool bar compensation, etc.), and in the actual machining process, the CNC system adds these preset error items to the process calculation for compensation. Reasonable correction of the tool path has also become a way of error compensation in recent years. Seeking the optimal path process in the two points to reduce the deviation will be the main way of error compensation in this paper. The commonly used path planning methods are: parallel shortest path search algorithm, ant colony algorithm, heuristic algorithm based on matrix load balancing, EBSP* algorithm, Dijkstra algorithm, etc. Among them, Dijkstra algorithm is widely used in shortest path planning, but Dijkstra algorithm The implementation form is relatively complicated, and the Floyd algorithm is an algorithm that is easy to understand and easy to design to solve path planning.

The Floyd algorithm is a method realized by calculating the weight matrix. Its main idea is to start from the weighted adjacency matrix D ⁽⁰⁾ representing the distance between any two nodes w _i and w _j , and first calculate D ⁽¹⁾ , that is, calculate After comparing all the possible paths from w _i to w _j , find the optimal path to replace the corresponding path in D ⁽⁰⁾ , iteratively list the distance matrix D ⁽¹⁾ , each of the paths in D ⁽¹⁾ The element represents the optimal path between any two points in the network after one iteration, that is, the optimal path between any two points in the network directly or only passing through an intermediate point, which is the shortest. Secondly, in order to improve the reliability of optimization, the iteration matrix is constructed Insert node w _r into the two nodes for path length comparison, if there is or It means that after inserting node w _r , from w _i to w _j will not be shorter than the original one.

Generally, in machine tool compensation, one of the results of machine tool errors is that the travel point is offset and an invalid distance is generated. Seeking the shortest path is a compensation work that needs to be completed as a travel out-of-tolerance. In the above cases, the target point w _t is known, and the actual arrival point w _j has Next, another way of error generation is that due to manufacturing defects, the stroke does not reach the predetermined position point and the forming point needs to be extended. That is, the target point w _t is known, the actual arrival point w _j is known, and all possible paths from w _i to w _t are known to be calculated once. After comparison, the iterative matrix is constructed again Insert node w _r into the two nodes for path length comparison, if there is or Then it means that after inserting node w _r , the length from w _i to w _j will not be longer than the original one. This method is the core method of the compensation principle of Floyd algorithm. The specific implementation process is shown in Figure 4.

If the machine tool operates in the XY plane, as shown in Figure 5. Its coordinate point is ( _xi , y _i ), and there are 8 forms of movement. One unit Δ of movement becomes ( _xi + Δ, y _i + Δ), ( _xi , y _i + Δ), ( _xi -Δ,y _i +Δ), ( _xi -Δ,y _i ), ( _xi -Δ,y _i -Δ), (x _i ,y _i -Δ), ( _xi +Δ, y _i -Δ), the distance to be moved may be This is the standard for rasterizing the grid.

The implementation steps of the Floyd error compensation algorithm are as follows:

Step 1: Rasterize the path as n×n. From the perspective of the weight matrix, use the vertical line method to select nodes w ₀ , w ₁ , w ₂ , w ₃ , w ₄ , and w _j for the path, and get the mutual The weight relationship and direction relationship between them.

Step 2: Calculate the shortest weight matrix when there is one intermediate node between w _i and w _j . Suppose w _i passes through an intermediate point w _r to w _j , then the shortest distance from w _i to w _j is: The shortest weight matrix is

Step 3: Calculate the shortest weight matrix when there are k intermediate nodes between w _i and w _j . Let w _i pass through the intermediate point w _r to w _j , the shortest distance from w _r to point w _j through kr intermediate points Then the shortest distance for w _i to reach point w _j through k intermediate points is The shortest weight matrix is:

Step 4: compare |w _r | _y ≤ L _limit , and output the compensation result if it is established. If not, n=n×n, and return to the first step to continue running until the range meets the interval condition.

Description of drawings

Fig. 1. This method is based on the flow chart of the implementation of the spatial error compensation principle of the Floyd algorithm.

Fig. 2. Schematic diagram of the verticality error decomposition of the machine tool.

Fig. 3. Schematic diagram of the parallelism error decomposition of the machine tool.

Figure 4. Flowchart of the implementation of the Floyd compensation algorithm.

Figure 5. Diagram of the standard grid rasterization method.

Figure 6. Schematic diagram of a five-axis CNC machining machine.

Figure 7. Schematic diagram of the topology of a five-axis CNC machine tool.

Figure 8. X-direction error distribution path rasterization diagram.

Figure 9. X-direction error distribution path node setting diagram.

Figure 10. Directed weighted graph of error distribution paths.

Figure 11. Compensation effect diagram of Floyd compensation algorithm.

Figure 12. Comparison of compensation effects between the Floyd compensation algorithm and the ACO-BPN compensation algorithm.

Figure 13. Path diagram of error distribution under variable temperature.

Figure 14. Effect diagram of error compensation under variable temperature conditions based on Floyd compensation algorithm.

detailed description

Calculation example: Take the five-axis linkage CNC machining machine tool as an example (Figure 6)

Step 1. Establish the spatial comprehensive error model of the machine tool based on the screw theory

According to the exponential matrix form of the screw theory, each moving part of the machine tool is abstracted into a 6×1 vector form. The motion form and error are modularized and expressed in the form of exponential matrix, and the spatial error model of the machine tool is established according to the topological structure of the machine tool (Fig. 7).

Step 1.1 Exponential matrix form of spinor theory

The motion of any rigid body can be decomposed into two parts: translational motion along the axis and rotational motion around the axis. Each component can be regarded as a screw. The unit spinor in Plücker coordinates is as follows:

To express any form of motion of a rigid body in space, there are:

Among them, υ=[v ₁ , v ₂ , v ₃ ] ^T , Represents an antisymmetric matrix, if ω=[ω ₁ , ω ₂ , ω ₃ ] ^T , can be expressed as:

Rigid body motion generally includes translation and rotation, and the vector q is the same in the rigid body coordinate system and the reference coordinate system. Then the homogeneous transformation matrix of the rigid body is:

The second transformation matrix corresponding to the exponential form of the spinor can be written as: When ω=0, the rigid body only has translational motion, then the homogeneous transformation matrix is:

When ω≠0, there is also rotational motion for rigid bodies, and the exponential matrix is:

in The trigonometric series expansion of can be expressed as:

To sum up, the exponential matrix for any form of motion of a rigid body in space can be expressed as:

When $ is the unit screw, when ||ω||≠0, the rotation angle of the mechanical part is expressed as When ||ω||=0, the translation distance is expressed as Points are expressed in different ways in different coordinate systems, and the difference between them can be expressed by a transformation matrix. A spinor can also be understood as a point in a coordinate system, and its expression in different coordinate systems is also different. Therefore, the form of a transformation matrix is also needed to express the relationship of the spinor in different coordinate systems, which can be called an adjoint matrix. If the motion screw of a rigid body is The index matrix of its transformed form is expressed as:

Then its adjoint exponential matrix form in this coordinate system:

The adjoint index matrix satisfies the following properties:

For a mechanical structure such as a machine tool, the structure is expressed by an exponential matrix:

T(0) represents its original transformation matrix, which can be applied to error modeling of machine tools.

Step 1.2 Use the exponential matrix type to model the spatially integrated error of the machine tool

In the case of machine tool operation, the error is the coupling effect of geometric error and thermal error, and the measured error δ includes both thermal error and geometric error. which is:

δ = δ _G + δ _T (45)

Where: δ _G is the geometric error;

_δT is the thermal error;

Generally, each axial movement has 6 directional degrees of freedom, and at the same time, 3 translational errors and 3 rotational errors will be generated. Using the screw theory, define the error module m _e $ _e .

m _e $ _e = [ε _x ,ε _y ,ε _z ,δ _x ,δ _y ,δ _z ] ^T

Taking the geometric error composition in the X direction as an example, it is mainly divided into three parts. The first part $ _xx contains positioning error and roll error along this direction The second part $ _yx is the linear error and yaw error of the horizontal plane The third part $ _zx is the linear error and yaw error of the vertical plane The thermal deformation of the machine tool is finally reflected on the moving parts of the machine tool. Due to the influence of the thermal deformation of the machine tool, the thermal error caused by the movement track of the machine tool's moving parts deviates from the ideal track. That is to say, when moving in the X direction, similar to the geometric error, there will also be 6 thermal errors. That is, three moving errors: X-direction linear displacement thermal error Y-direction straightness thermal error and Z-direction straightness thermal error Three corner errors: tilt thermal error around the X axis Yaw thermal error around the Z axis and pitch thermal error around the Y axis

The spatial error of the X axis is expressed as:

The error model of the X-axis is in the form of an exponential matrix, expressed as:

Similarly, the exponential matrix form of the error model of the Y axis and the Z axis can be expressed as follows:

The error model index matrix form of A axis and C axis:

Step 1.3 Exponential Matrix Form for Perpendicularity and Parallelism Errors

Due to the difference between the actual axis and the ideal axis, the two adjacent axes are not absolutely 90°; that is to say, there is a verticality error. For the three translational axes, the Y axis is an ideal axis, and there is no verticality error; then the verticality error between the X axis and the Y axis is S _xy , and the verticality error between the Y axis and the Z axis is S _yz , the perpendicularity between X and Z is S _xz . In the plane formed by the Y-axis and the actual installed X-axis, there is only S _xy for the X-axis, and there are other two perpendicularity errors on the actual Z-axis similarly.

Taking the X direction as an example, the X direction unit screw in the ideal state can be expressed as:

Adding the verticality error in the real state, the actual unit screw in the X direction can be expressed as:

The corresponding exponential matrix can be expressed as:

Another way of writing is to use the form of an adjoint matrix to rotate the ideal X-axis with the Z axis ($ _zr ) at a certain angle to achieve the effect of 90° between the X-axis and the Y-axis, namely:

The second item "r" in the subscript means to rotate with the first item as the axis;

Similarly, the Z-direction unit screw can be expressed as:

Adding the verticality error in the real state, the actual unit screw in the Z direction can be expressed as:

Another writing method is similar to the above-mentioned matrix transformation method. The ideal Z-axis is rotated by a certain angle along the X-axis and Y-axis respectively to express the actual position of the Z-axis in the real state:

For the rotation of the A axis and the C axis, the deviation in the X and Z directions will occur during installation, that is, the actual parallelism between the A axis and the X axis is the component PY _xA in the Y direction, and the parallelism between the A axis and the X axis The component PZ _xA in the Z direction; similarly, two error terms on the parallelism between the C axis and the Z axis will be obtained.

Taking the A-axis as an example, the axis unit screw of the A-axis motion in an ideal state can be written as:

Adding the parallelism error components along the Y and Z directions in the real state, the actual unit screw in the A direction can be expressed as:

Similar to the matrix transformation method above, the ideal X-axis is rotated by a certain angle along the Y-axis and Z-axis respectively to express the actual position of the A-axis in reality:

In an ideal state, the axis unit screw of C-axis motion can be written as:

Adding the parallelism error components along the X and Y directions in the real state, the actual unit screw in the A direction can be expressed as:

Similar to the matrix transformation method above, the ideal Z-axis is rotated by a certain angle along the X-axis and Y-axis respectively to express the actual position of the A-axis in the real state:

Step 1.4 Establishment of error model based on topology

The theory of multi-body systems provides a detailed model of the topology of machine tools, which can also be applied in exponential matrices. Ideally, there is no error in the machine tool; therefore, the order of the model can be expressed as: C _i →X _i →Y _i →Z _i →A _i . The matrix transformation equation in an ideal state can be expressed by T _i :

The error index matrix model of the overall machine tool can be written as:

in, represents the screw of the foundation;

The spatial error model of the five-axis machine tool represented by the exponential matrix based on the screw can be expressed as:

The components E _x , E _y , and E _z of the corresponding spatial error on the three axes can be expressed as:

[E _x , E _y , E _z ,1] ^T ＝ E·[0,0,0,1] ^T (78)

For large five-axis CNC machine tools, the installation error of the tool and the workpiece is ignored here because it is very small.

Therefore, the components of the spatial error of the machine tool in the three axes can be written as:

E _x ＝γPY _zC +δ _xA -δ _xx -δ _xy +δ _zx ε _yx ε _xy -δ _yy ε _yx ε _xy -γPX _zC +xS _xy ε _yy

-δ _yy ε _xA ε _zx +αPY _xA ε _xA +(z+δ _zz +δ _zA +δ _yz ε _xC )(ε _yx +ε _yy -ε _xy ε _zx )

(γ+δ _zC +δ _xC ε _yA )(ε _zC +ε _zz -ε _xy ε _zx )+ε _zx (ε _xy ε _yy -ε _zy ε _zA )-ε _yx (ε _yy +ε _xy ε _zy )

-y(ε _zx +ε _zC +ε _yz (1-ε _yx ε _yy )+ε _xy (ε _yx ε _yC +ε _yy ε _zx ε _zy ))

+(-zS _yz +δ _yz -δ _zz ε _xz )(ε _zx +ε _zy (ε _zA -ε _yx ε _yy )+ε _xy (ε _yx +ε _yy ε _zx ε _zy ))

E _y ＝-δ _yx -δ _zx ε _xx +α(ε _yA +PY _xA )-γ(ε _xA +ε _xC ε _yC ε _zC )+ε _xx (-δ _zy +δ _yy ε _xy )+

(δ _xy -δ _zy ε _yA )(ε _xx ε _xC -ε _yA )+x(-ε _xx ε _yA +ε _zx )+xS _xy (1-ε _xx ε _yy ε _zx )

-(δ _yy +δ _zy ε _xy )(1+ε _xx ε _yy ε _zx )+(z+δ _zz +δ _yz ε _xz )(ε _xy +ε _yy ε _xA +ε _xx (1+ε _yx (- ε _yy +ε _xy ε _zx )))

+(-zS _xz +δ _xz +δ _zz ε _zy +δ _zA ε _yy )(ε _xx ε _yx -ε _zx +(1+ε _xx ε _yx ε _zx )(ε _xy ε _yy -ε _zy )

+ε _xx (ε _yy +ε _xy ε _zC ))-y(ε _zy (ε _xx ε _yx -ε _zx )+ε _xx (-ε _xy +ε _yy ε _zA )+(1+ε _xx ε _yC ε _zy )(1+ε _xy ε _xC ε _zx ))

+(-zS _yz +δ _yz -δ _zz ε _xz )((ε _xA ε _yA -ε _zA )ε _zy +ε _xx (-ε _xy +ε _yy ε _zy )+(1+ε _xx ε _yx ε _zx ) (1+ε _xy ε _yy ε _zy ))

E _z ＝δ _zz -δ _zx -δ _zy +δ _yx ε _xx +α(ε _zC +ε _xC ε _yC ε _xz )-x(ε _yx +ε _xx ε _zx )+(-δ _xy +δ _zy ε _xA )(ε _yx +ε _xx ε _zA )

+xS _xy (-ε _xx +ε _yx ε _zA )+(z+δ _zz +δ _yz ε _xz )(1+ε _yC (ε _yy +ε _xy ε _zx )-ε _xx (ε _xy +ε _xA ε _yy ))

+(-zS _xz +γPZ _xA (δ _xz +δ _zC ε _yz ))(ε _yx +ε _yy +ε _xx ε _zx +(-ε _xx +ε _xy ε _zx )(ε _xy ε _yy -ε _zy )+ ε _xy ε _zC )

-y(-ε _xy +ε _yy ε _zy +ε _xC (ε _yA +ε _xx ε _zx )+(-ε _xx +ε _yx ε _zx )(1+ε _xy ε _yy ε _zy ))

+(-zS _yz +δ _yz -δ _zC ε _xz )(-ε _xy +ε _yy ε _zy +ε _zC (ε _yA +ε _xx ε _zx )+(-ε _xx +ε _yx ε _zx )(1+ε _xy ε _yy ε _zy ))

After omitting the higher-order terms, the spatial error equation is obtained:

E _x ＝δ _xA -δ _xx -δ _xy +γPY _zC +γε _zC +γε _zz -zS _xz -zε _yx -zε _yy -yε _zC -yε _zx

E _y ＝δ _yz -δ _yx -δ _yy +α(ε _yA +PY _xA )-γε _xA +xS _xy -zS _yz +zε _xx +zε _xy +xε _zx (79)

E _z ＝δ _zz -δ _zx -δ _zy +αε _zC -xε _yx +yε _xx +yε _xy

Step 2: Spatial Error Compensation Principle Based on Floyd Algorithm

Generally, there are two error compensation methods for CNC machine tools: ① According to the error data tested in actual processing, manually adjust the NC machining program; The error data is entered in advance to the corresponding error compensation setting items (such as backlash compensation, pitch compensation and tool bar compensation, etc.), and in the actual machining process, the CNC system adds these preset error items to the process calculation for compensation. Reasonable correction of the tool path has also become a way of error compensation in recent years. Seeking the optimal path process in the two points to reduce the deviation will be the main way of error compensation in this paper. The commonly used path planning methods are: parallel shortest path search algorithm, ant colony algorithm, heuristic algorithm based on matrix load balancing, EBSP* algorithm, Dijkstra algorithm, etc. Among them, Dijkstra algorithm is widely used in shortest path planning, but Dijkstra algorithm The implementation form is relatively complicated, and the Floyd algorithm is an algorithm that is easy to understand and easy to design to solve path planning.

The Floyd algorithm is a method realized by calculating the weight matrix. Its main idea is to start from the weighted adjacency matrix D ⁽⁰⁾ representing the distance between any two nodes w _i and w _j , and first calculate D ⁽¹⁾ , that is, calculate After comparing all the possible paths from w _i to w _j , find the optimal path to replace the corresponding path in D ⁽⁰⁾ , iteratively list the distance matrix D ⁽¹⁾ , each of the paths in D ⁽¹⁾ The element represents the optimal path between any two points in the network after one iteration, that is, the optimal path between any two points in the network directly or only passing through an intermediate point, which is the shortest. Secondly, in order to improve the reliability of optimization, the iteration matrix is constructed Insert node w _r into the two nodes for path length comparison, if there is or It means that after inserting node w _r , from w _i to w _j will not be shorter than the original one.

Generally, in machine tool compensation, one of the results of machine tool errors is that the travel point is offset and an invalid distance is generated. Seeking the shortest path is a compensation work that needs to be completed as a travel out-of-tolerance. In the above cases, the target point w _t is known, and the actual arrival point w _j has Next, another way of error generation is that due to manufacturing defects, the stroke does not reach the predetermined position point and the forming point needs to be extended. That is, the target point w _t is known, the actual arrival point w _j is known, and all possible paths from w _i to w _t are known to be calculated once. After comparison, the iterative matrix is constructed again Insert node w _r into the two nodes for path length comparison, if there is or Then it means that after inserting node w _r , the length from w _i to w _j will not be longer than the original one. This method is the core method of the compensation principle of Floyd algorithm.

Taking the X-axis movement as an example, rasterize the X-Ex image into a 100x100 work area. As shown in Figure 8.

From the starting point w ₀ to the target point w _j , there is an offset due to machine error. In order to obtain higher precision of the machine tool, according to the principle of Floyd compensation, firstly, node selection is carried out on the error path, and the connection line or normal tangent line is used as the vertical line by using the vertical line method, and the intersection point of the vertical line and the error block is used as the node, such as As shown in FIG. 9 , the nodes S={w ₀ , w ₁ , w ₂ , w ₃ , w ₄ , w _j } are obtained.

Use the directed graph to calculate the adjacency matrix, and calculate the weights between nodes according to the positions of w ₀ , w ₁ , w ₂ , w ₃ , w ₄ , and w _j . S ₀₁ refers to the weight between w ₀ and w ₁ , The size of the weight represents the moving distance between two points. If there is an error module between two nodes, the weight between the two points is ∞. The form of error compensation is to point to the direction of j along the connection line, that is, the direction between the nodes on the same side of the connection line j, and the points on different sides of j do not directly limit the mutual direction. Therefore, there are only sequential directions. If there is an obstacle between two points, the weight is still ∞. According to the node position, the error module area to be avoided, and the direction of the target point _wj , the directed weighted graph 10 of the error distribution path is obtained. Then the following formula shows the adjacency matrix of a directed graph.

Compensation Algorithm Implementation

Step 1: Rasterize the path as n×n. From the perspective of the weight matrix, use the vertical line method to select nodes w ₀ , w ₁ , w ₂ , w ₃ , w ₄ , and w _j for the path, and get the mutual The weight relationship and direction relationship between them. Weight matrix A ₀ :

Step 2: Calculate the shortest weight matrix when there is one intermediate node between w _i and w _j . Suppose w _i passes through an intermediate point w _r to w _j , then the shortest distance from w _i to w _j is: The shortest weight matrix is

Step 3: Calculate the shortest weight matrix when there are k intermediate nodes between w _i and w _j . Let w _i pass through the intermediate point w _r to w _j , the shortest distance from w _r to point w _j through kr intermediate points Then the shortest distance for w _i to reach point w _j through k intermediate points is The shortest weight matrix is: The weight matrix passing through 2 nodes in the axial direction is:

Step 4: compare |w _r | _y ≤ L _limit , and output the compensation result if it is established. If not, n=n×n, and return to the first step to continue running until the range meets the interval condition.

Due to the motion error of the machine tool, it is caused by the geometric accuracy of the part and the ambient temperature at the same time. The error can be reduced to zero only under ideal conditions. The axis moves through 3 nodes, and the weight matrix of 4 nodes is the same as the weight matrix of 2 nodes. The axial movement is from w ₀ to w _j , and the distance through w ₂ and w ₄ is shorter; repeat this process, the purpose of compensation can be achieved. In the case of simulation iterations of the algorithm, the termination condition is |E _x |≤0.01mm; the error change distribution diagram after compensation of the X-Ex diagram is obtained, as shown in Figure 11. Even if the temperature changes, the Floyd algorithm is used to compensate the error, and the error can still be adjusted to control the machining accuracy.

There are many kinds of compensation methods, such as the commonly used algorithms such as B-P neural network algorithm, ACO ant colony algorithm, etc. Since the error is composed of multiple errors such as geometric error and thermal error, the influence caused by geometric error and thermal error accounts for about 70% of the result. In order to prove the superiority of the Floyd compensation algorithm, the present invention will compare the principles of the ACO-BPN compensation algorithm. And get the corresponding contrast effect.

When the laser interferometer is used for measurement, the measured data is the result of multiple errors at the same time. The ACO-BPN algorithm and the Floyd algorithm compensation method are simulated and compared. From the data, we can clearly see that the Floyd algorithm is better than ACO-BPN algorithm. From the perspective of compensation effect, it is shown in Figure 12. The Floyd compensation algorithm can compensate the original average error of 68 μm to only 4 μm, which is reduced by 94.2%, while the ACO-BPN algorithm can be compensated to the range of 6 μm, which is reduced by 91.2%. From the perspective of the compensation effect, the Floyd compensation algorithm is better than the ACO-BPN algorithm.

Secondly, in terms of algorithm, the Floyd algorithm is also obviously better than the ACO-BPN algorithm. Table 1 shows the number of iterations and the execution frequency of statements.

Table 1. Floyd algorithm and ACO-BPN algorithm

Compared with the ACO-BPN algorithm, the frequency of the Floyd algorithm changes with the number of iterations n. With the increase of the n value, the execution frequency of the statement increases. The Floyd algorithm is obviously lower than the ACO-BPN algorithm statement execution frequency, that is, the Floyd algorithm. It takes less time, which fully proves the superiority and rationality of the Floyd algorithm in error compensation.

Compensation Effect of Floyd Compensation Algorithm under Variable Temperature

In order to further prove the superiority of the Floyd algorithm, the compensation effect of the Floyd compensation algorithm is introduced below under variable temperature conditions. When the ambient temperature remains unchanged, the continuous operation of the machine will result in an increase in the temperature of the body. When measuring on the machine tool, the errors measured by the instruments used are the coupling values of thermal errors and geometric errors. In the above, the spatial error model of the machine tool has been expressed. This part will focus on how to separate and establish the relationship between the thermal error and the temperature source of the machine tool. Separate out the thermal error term, i.e. dwell the tool point to a preset position. When measuring each axis, three sensors are used in the machine tool, which are respectively installed in the key parts of the active part, driven part and environmental parts. Taking the X-axis as an example, sensors are respectively arranged on the X-axis screw nut, the X-axis screw seat and the machine body. The temperature of the parts was measured within 100 minutes as shown in Table 2. The measured results are shown in Figure 13. According to the compensation principle of the Floyd algorithm in the above part, the compensation effect in terms of temperature can also be predicted and simulated. As shown in Figure 14.

Table 2. Temperature Statistics

Through the compensation of Floyd algorithm, the error range is controlled in [-6μm, 7μm]. The Floyd algorithm still has an obvious effect on error compensation in a variable temperature environment.

This method provides a machine tool space error modeling and error compensation technology method based on the model and Floyd algorithm, and establishes a space comprehensive error compensation model. Propose error compensation models and methods in detection, calculation and prediction errors. The compensation model has the advantages of high compensation accuracy, high calculation efficiency, good time period and closed-loop robustness; based on the error measurement data, using the exponential matrix form of the screw theory, on the basis of the topology of the machine tool, a machine tool is established. For the overall spatial error model, the high-order terms of the error model are reduced to obtain the basic equation of the error model; according to Floyd's shortest distance algorithm, the weight is continuously adjusted, iterated to the basic range allowed by the accuracy, and the error compensation model is obtained and reduced The amount of calculation can achieve the compensation effect. This error compensation principle technology can be used for real-time compensation of machine tool errors in various complex actual processing situations. And compared with the classic error compensation algorithm "ACO-BPN" in the example, it is found through simulation that the Floyd compensation algorithm has the characteristics of better compensation effect, higher execution efficiency and fewer iterations compared with the ACO-BPN compensation algorithm, and it can be used in a variable temperature environment. The robustness of the Floyd compensation algorithm is verified in the paper.

Claims (1)

1. A spatial error compensation method based on Floyd algorithm, is characterized in that: the realization process of this method is as follows, Step 1: Establish the spatial comprehensive error model of the machine tool based on the screw theory According to the exponential matrix form of the screw theory, each moving part of the machine tool is abstracted into a 6×1 vector form; the movement form and comprehensive error are modularized and expressed in the form of an exponential matrix, and the machine tool is established according to the topology of the machine tool The spatial comprehensive error model of ; Step 1.1 Exponential matrix form of screw theory The motion of any rigid body is decomposed into two parts: translational motion along the axis and rotational motion around the axis; that is, each component is regarded as a screw; the unit screw becomes as follows in Plücker coordinates: $＝[ω ^T v ^T ] ^T ＝[ω ₁ ,ω ₂ ,ω ₃ ,v ₁ ,v ₂ ,v ₃ ] ^T (1) To express any form of motion of a rigid body in space, there are: $\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}& \mathrm{<span\; class="notranslate">\; v</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ Among them, υ=[v ₁ ,v ₂ ,v ₃ ] ^T , Represents an antisymmetric matrix, if ω＝[ω ₁ ,ω ₂ ,ω ₃ ] ^T , Then it is expressed as: $\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ {\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ Rigid body motion includes translation and rotation, and the vector q is the same in the rigid body coordinate system and the reference coordinate system; then the homogeneous transformation matrix of the rigid body is: $\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; R</span>}& \mathrm{<span\; class="notranslate">\; q</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ The second transformation matrix corresponding to the exponential form of the spinor is written as: When ω=0, the rigid body only has translational motion, then the homogeneous transformation matrix is: $\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 3</span>}}& \mathrm{<span\; class="notranslate">\; v</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ When ω≠0, there is also rotational motion for rigid bodies, and the exponential matrix is: in The trigonometric series expansion of is expressed as: ${\mathrm{<span\; class="notranslate">\; e</span>}}^{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\frac{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; +</span>\frac{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ In summary, the exponential matrix for any form of motion of a rigid body in space is expressed as: When $ is the unit screw, when ‖ω‖≠0, the rotation angle of the mechanical part is expressed as When ‖ω‖=0, the translation distance is expressed as Points are expressed in different ways in different coordinate systems, and the difference between them is expressed by a transformation matrix; a screw is also understood as a point in a coordinate system, and the way it is expressed in different coordinate systems is also different, so it is also required The form of the transformation matrix to express the relationship of the screw in different coordinate systems is called the adjoint matrix; if the motion screw of the rigid body is θ$, the exponential matrix of the transformation form is expressed as: ${\mathrm{<span\; class="notranslate">\; e</span>}}^{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; R</span>}& \mathrm{<span\; class="notranslate">\; q</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$ Then its adjoint exponential matrix form in this coordinate system: $\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; R</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}& \mathrm{<span\; class="notranslate">\; R</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$ The adjoint exponential matrix satisfies the following properties: ${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$ For a mechanical structure such as a machine tool, an exponential matrix is used to express its structure as follows: $\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; ...</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$ T(0) represents its original transformation matrix, which is applied to the error modeling of the machine tool; Step 1.2 Use the exponential matrix type to model the spatially integrated error of the machine tool In the case of machine tool operation, the error is mainly the coupling effect of geometric error and thermal error, and the measured error δ includes both thermal error and geometric error; namely: δ = δ _G + δ _T (14) Where: δ _G is the geometric error; _δT is the thermal error; Each axial movement will have 6 directional degrees of freedom, and will produce 3 translational errors and 3 rotational errors at the same time; use the screw theory to define the error module m _e $ _e ; m _e $ _e = [ε _x ,ε _y ,ε _z ,δ _x ,δ _y ,δ _z ] ^T The geometric error in the X direction is mainly divided into three parts; the first part $ _xx includes the positioning error and the roll error along this direction The second part $ _yx is the linear error and yaw error of the horizontal plane The third part $ _zx is the linear error and yaw error of the vertical plane The thermal deformation of the machine tool is finally reflected on the moving parts of the machine tool. Due to the influence of the thermal deformation of the machine tool, the thermal error caused by the motion trajectory of the moving parts of the machine tool deviates from the ideal trajectory; that is, the movement in the X direction is similar to the geometric error. There are 6 thermal errors; that is, 3 movement errors: X-direction linear displacement thermal error Y-direction straightness thermal error and Z-direction straightness thermal error Three corner errors: tilt thermal error around the X axis Yaw thermal error around the Z axis and pitch thermal error about the Y axis ${\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; G</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$ The spatial error of the X axis is expressed as: ${\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$ The error model of the X-axis is in the form of an exponential matrix, expressed as: ${\mathrm{<span\; class="notranslate">\; T</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; x</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$ In the same way, the spatial error modules of other axes and the error model of the exponential matrix are obtained; Step 1.3 Exponential matrix form for squareness and parallelism errors Due to the difference between the actual axis and the ideal axis, the two adjacent axes are not absolutely 90°; that is to say, there is a verticality error; for the three translational axes, it is assumed that the Y axis is For an ideal axis, there is no perpendicularity error; then the perpendicularity error between the X axis and the Y axis is S _xy , the perpendicularity error between the Y axis and the Z axis is S _yz , and the perpendicularity error between X and Z is S _xz ; In the plane formed by the Y-axis and the actual installed X-axis, there is only S _xy for the X-axis. Similarly, there are other two verticality errors in the actual Z-axis; because the actual axis direction inevitably deviates from the ideal The position of the axis, so it is considered that the verticality error should be added to the coordinate transformation. For the transformation form of the ideal coordinate axis: The ideal X-direction unit screw is expressed as: $ _xi = [0,0,0,1,0,0] ^T (20) Adding the verticality error in the real state, the actual unit screw in the X direction is expressed as: $ _xs ＝[0,0,0,cos(S _xy ),-sin(S _xy ),0] ^T (21) The corresponding exponential matrix is expressed as: ${\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; x</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; two</span>}<span\; class="notranslate">\; )</span>$ Another way of writing is to use the form of an adjoint matrix to rotate the ideal X-axis by a certain angle with Z as the axis $ _zr to achieve the effect of 90° between the X-axis and the Y-axis, namely: ${\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$ $ _zr = [0,0,1,0,0,0] ^T In the same way, the expression form of the verticality error screw of the Z axis is obtained; For the rotation of the A axis and the C axis, the deviation in the X and Z directions will occur during installation, that is, the actual parallelism between the A axis and the X axis is the component PY _xA in the Y direction, and the parallelism between the A axis and the X axis The component PZ _xA in the Z direction; similarly, two error terms on the parallelism between the C axis and the Z axis will be obtained; In an ideal state, the axis unit screw of A-axis motion is: $ _Ai = [1,0,0,0,0,0] ^T (24) Adding the parallelism error components along the Y and Z directions in the real state, the actual unit screw in the A direction is expressed as: $ _As ＝[cos(PY _xA )cos(PZ _xA ),-sin(PY _xA )cos(PZ _xA ),-sin(PZ _xA ),0,0,0] ^T (25) ${\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; A</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; py</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; py</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; py</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; py</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 26</span>}<span\; class="notranslate">\; )</span>$ Similar to the above-mentioned matrix transformation method, the ideal X-axis is rotated by a certain angle along the Y-axis and Z-axis respectively to express the actual position of the A-axis in the real state: ${\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; py</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; PZ</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}\mathrm{<span\; class="notranslate">\; A</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \$</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 27</span>}<span\; class="notranslate">\; )</span>$ In the same way, the expression form of the exponential matrix of the parallelism error of the C axis is obtained; Step 1.4 Establishment of error model based on topology The multi-body system theory provides a very detailed topological model of the machine tool, which is also applied in the exponential matrix; under ideal conditions, there is no error in the machine tool; under ideal conditions, the matrix transformation equation is represented by T _i : ${\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; c</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; e</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{{\mathrm{<span\; class="notranslate">\; e</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 28</span>}<span\; class="notranslate">\; )</span>$ In practice, due to the error of the machine tool components themselves and position error between parts Add the overall component error screw to the screw module; denoted by T _a : ${\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; b</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; c</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; e</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; e</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \$</span>}}}_{\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 29</span>}<span\; class="notranslate">\; )</span>$ According to the matrix transformation equation under actual and ideal conditions, the spatial error model of the multi-axis CNC machine tool is obtained: E＝T _i ^-1 T _a (30) The components E _x , E _y , and E _z of the corresponding spatial error on the three axes are expressed as: [E _x , E _y , E _z ,1] ^T ＝ E·[0,0,0,1] ^T (31) By omitting the second-order and higher-order terms in the formula, the basic equation of the spatial error can be obtained; Step 2: Spatial Error Compensation Principle Based on Floyd Algorithm Generally, there are two error compensation methods for CNC machine tools: ① According to the error data tested in actual processing, manually adjust the NC machining program; The data is entered in advance to the corresponding error compensation setting items. In the actual processing process, the CNC system adds these preset error items into the process calculation for compensation; reasonable correction of the tool path has also become a way of error compensation in recent years; Seeking the optimal path process in the two points to reduce the deviation will be the main method of error compensation in this paper; The Floyd algorithm is a method realized by calculating the weight matrix. Its main idea is to start from the weighted adjacency matrix D ⁽⁰⁾ representing the distance between any two nodes w _i and w _j , and first calculate D ⁽¹⁾ , that is, calculate After comparing all the possible paths from w _i to w _j , find the optimal path to replace the corresponding path in D ⁽⁰⁾ , iteratively list the distance matrix D ⁽¹⁾ , each of the paths in D ⁽¹⁾ The element represents the optimal path between any two points in the network after one iteration, that is, the optimal path between any two points in the network directly or only passing through an intermediate point, which is the shortest; secondly, in order to improve the reliability of optimization , to construct an iterative matrix Insert node w _r into the two nodes for path length comparison, if there is or It means that after inserting node w _r , from w _i to w _j will not be shorter than the original one; In machine tool compensation, one of the results of the error of the machine tool is to make the travel point offset and produce an invalid distance; seeking the shortest path is the compensation work that needs to be completed as a travel over-tolerance. The above situations are all at the known target point w _t , actually reach the point w _j and have Next, another way of error generation is that due to manufacturing defects, the travel does not reach the predetermined position point and the formation point needs to be extended; that is, the target point w _t is known, the actual arrival point w _j is known, and the calculation is known from w _i to w _t go through all possible paths of one revolution, after comparison, build the iteration matrix again Insert node w _r into the two nodes for path length comparison, if there is or It means that after inserting node w _r , from w _i to w _j will not be longer than the original one, this method is the core method of the compensation principle of Floyd algorithm; If the machine tool tool operates in the XY plane; its coordinate point is ( _xi , y _i ), there are 8 forms of movement, and one unit of movement Δ becomes ( _xi + Δ, y _i + Δ), ( x _i ,y _i +Δ), ( _xi -Δ,y _i +Δ), ( _xi -Δ,y _i ), ( _xi -Δ,y _i -Δ), ( _xi ,y _i - Δ), ( _xi +Δ, y _i -Δ), the distance to be moved may be In this way as a standard for rasterizing the grid; The implementation steps of the Floyd error compensation algorithm are as follows: Step 1: Rasterize the path as n×n. From the perspective of the weight matrix, use the vertical line method to select nodes w ₀ , w ₁ , w ₂ , w ₃ , w ₄ , and w _j for the path, and get the mutual The weight relationship and direction relationship among them; Step 2: Calculate the shortest weight matrix when there is one intermediate node between w _i and w _j ; suppose w _i passes through an intermediate point w _r to w _j , then the shortest distance from w _i to w _j is: The shortest weight matrix is Step 3: Calculate the shortest weight matrix when there are k intermediate nodes between w _i and w _j ; suppose w _i passes through the intermediate point w _r to w _j , and w _r passes through kr intermediate points to reach point w _j . distance Then the shortest distance for w _i to reach point w _j through k intermediate points is The shortest weight matrix is: Step 4: Compare |w _r | _y ≤ L _limit , if true, output the compensation result; if not, n=n×n, and return to the first step to continue running until the range meets the interval condition.