A method for analyzing spatial error sensitivity of a five-axis numerical control machine tool

Abstract

The application is suitable for the technical field of machine tool precision design, and provides a five-axis numerical control machine tool whole machine space error sensitivity analysis method, which comprises the following steps: based on the multi-body system theory, the numerical control machine tool is abstracted into a multi-body system, the structure of the machine tool and the correlation between each body are described by a topological structure diagram and a low-order body array table, the geometric error of the numerical control machine tool is analyzed, a generalized coordinate system is established, the position relationship between adjacent bodies is represented by a feature matrix, the mutual relationship between the multi-body systems is represented by a homogeneous transformation matrix, and a space error model of the machine tool is established; combined with the space error model of the machine tool, an improved Sobol global sensitivity analysis method is proposed, the first-order sensitivity coefficient and the global sensitivity coefficient are solved, and the key geometric error of the numerical control machine tool is analyzed. Through the establishment of the space error model of the numerical control machine tool and the improved Sobol global error sensitivity analysis model, a new numerical control machine tool design and improvement concept is proposed, and the machining precision problem of the numerical control machine tool is solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of machine tool precision design, and in particular relates to a method for analyzing the spatial error sensitivity of a five-axis CNC machine tool. Background Art

[0002] CNC machine tools are high-precision, high-efficiency, and high-tech modern electromechanical equipment. As the foundation and core equipment of advanced manufacturing technology, their accuracy directly affects the quality of machined products and constrains the development of manufacturing and various high-tech sectors. The main accuracy indicators of CNC machine tools include machining accuracy, positioning accuracy, and repeatability. Machining accuracy is the ultimate precision pursued by CNC machine tools and reflects the manufacturing capabilities and development level of the machinery manufacturing industry.

[0003] Factors affecting machine tool machining accuracy primarily include geometric errors, thermal errors, servo system errors, and load errors of various machine tool components. Geometric errors have the greatest impact on machining accuracy, accounting for approximately 50%. Geometric errors of machine tools refer to deviations in the actual geometric parameters and positions of various machine tool components from their ideal geometric parameters and positions due to defects in the machine tool's design, manufacturing, and assembly. These errors are generally related to the geometric elements of each component and are inherent to the machine tool itself.

[0004] The geometric error of a machine tool directly affects the position error of the tool processing point. A machine tool has a variety of geometric errors, including positioning error, straightness error, roll error, yaw error, runout error, and perpendicularity and parallelism errors between moving axes. How to effectively identify the key geometric error items that have a greater impact on the processing accuracy of CNC machine tools and effectively control them in the early stages of machine tool design is a key issue in improving machine tool processing accuracy. Summary of the Invention

[0005] The purpose of the embodiments of the present invention is to provide a method for analyzing the spatial error sensitivity of a five-axis CNC machine tool, aiming to solve the problems raised in the above background technology.

[0006] The embodiment of the present invention is implemented as follows: a method for analyzing the spatial error sensitivity of a five-axis CNC machine tool comprises the following steps:

[0007] Based on the multi-body system theory, the CNC machine tool is abstracted as a multi-body system. The structure of the machine tool and the relationship between each body are described using a topological structure diagram and a low-order body array table. The geometric error of the CNC machine tool is analyzed, and a generalized coordinate system is established. The positional relationship is represented by the characteristic matrix between adjacent bodies, and the relationship between the multi-body system is represented by a homogeneous transformation matrix. The spatial error model of the machine tool is established.

[0008] Combining the spatial error model of machine tool, the improved Sobol global sensitivity analysis method is proposed to solve the first-order sensitivity coefficient and global sensitivity coefficient, and to analyze the key geometric error of numerical control machine tool.

[0009] Preferably, the step of describing the structure of the machine tool and the correlation between the individuals by the topological structure diagram and the low-order body array table specifically comprises:

[0010] Taking the machine tool bed as the starting point of the double-branch topological structure chain, the bed B0 is divided into two branches, namely the tool chain and the workpiece chain. Each machine tool component is regarded as a typical body. Starting from the bed, the serial number of each typical body is sequentially calibrated in the natural growth sequence along the direction away from B0, and the calibration is performed according to the tool chain and the workpiece chain respectively.

[0011] Each typical body has a neighboring low-order body except for the B0 body. A table is established for the low-order body of each typical body, and L i (j) is represented, which is called the low-order body array table, and j represents the serial number of the typical body, j = 1, 2, 3... i, i represents the number of typical bodies included in the machine tool.

[0012] Preferably, the step of analyzing the geometric error of the numerical control machine tool, establishing a generalized coordinate system, representing the positional relationship between adjacent bodies by a feature matrix, and representing the mutual relationship between the multi-body system by a homogeneous transformation matrix specifically comprises:

[0013] Any object in the spatial coordinate system has 6 degrees of freedom, and 6 errors are generated in the motion process, including 3 linear displacement errors and 3 angular displacement errors. There are 3 non-perpendicularity errors between X, Y and Z guide rails, and 2 perpendicularity errors between C-axis and X, Y axes, so there are a total of 29 errors.

[0014] Determine the reference coordinate system origin position vector of each typical body;

[0015] According to the motion relationship between each component of the numerical control machine tool, the transformation matrix between each adjacent body is established, including the ideal static and actual static homogeneous transformation matrix between adjacent bodies and the ideal motion and actual motion homogeneous transformation matrix between adjacent bodies.

[0016] Preferably, the step of establishing the spatial error model of the machine tool specifically comprises:

[0017] The position vector of the point to be processed in the workpiece coordinate system in the machine tool coordinate system is:

[0018]

[0019] The position vector of the tool center point in the machine tool coordinate system is:

[0020]

[0021] In the actual machining process, precision machining requires that the point to be machined in the workpiece coordinate system coincides with the tool center point, that is:

[0022] p w =p t (3)

[0023] The position vector of the tool center point in the workpiece coordinate system during the actual machining process is:

[0024]

[0025] In formula (4), r wx is the component of the position of the tool center point in the workpiece coordinate system in the X direction; r wy

[0026] is the component of the position of the tool center point in the workpiece coordinate system in the Y direction; r wz is the component of the position of the tool center point in the workpiece coordinate system in the X direction;

[0027] Assuming that all error parameters in equation (1) are 0, the position vector from the tool center point to the workpiece coordinate system under ideal conditions can be obtained:

[0028]

[0029] In formula (5), is the component of the position of the tool center point in the workpiece coordinate system in the X direction under ideal conditions; is the Y-direction component of the position of the tool center point in the workpiece coordinate system under ideal conditions; is the component of the position of the tool center point in the workpiece coordinate system in the Z direction under ideal conditions;

[0030] Subtracting Equation (5) from Equation (4) yields the machine tool spatial motion error model:

[0031] E=(E x E y E z 1) T (6)

[0032] In formula (6), E x is the component of the spatial motion error in the X direction; E y is the component of the spatial motion error in the Y direction; E z is the component of the spatial motion error in the Z direction.

[0033] Preferably, the space error model of the machine tool proposes an improved Sobol global sensitivity analysis method to solve the first-order sensitivity coefficient and the global sensitivity coefficient, and the steps of analyzing the key geometric error of the numerical control machine tool, specifically comprising:

[0034] Error model decomposition:

[0035] According to the decomposition method of the Sobol method, the mathematical model Y=f(h) is decomposed into an incremental order form:

[0036]

[0037] In formula (7), Y0 is the expected value of the overall model obtained by each parameter input; Y i =Y(h i ) is the function value corresponding to the ith input h i ; Y ij =Y(h i ,h j ) is the function value corresponding to the joint action of input h i and h j , and other high-order terms can be obtained in this way;

[0038] The variance calculation of formula (7) is:

[0039]

[0040] Divide both sides of the equation by the total variance of the function and perform an orthogonal transformation to obtain:

[0041]

[0042] First-order sensitivity coefficient solution:

[0043] The first-order sensitivity coefficient calculation result is obtained:

[0044]

[0045] In formula (10), h ～i is all other input parameter items except h i ; is the expected value calculated for the h ～i input item; S i is the first-order variance ratio corresponding to the input parameter h i ;

[0046] Among them, S i calculated by formula (10) is the first-order sensitivity coefficient of the input error item h i , which reflects the influence degree of the error item on the model output, and S iThe larger the value, the greater the influence of the input parameter on the model;

[0047] Improved Monte Carlo method:

[0048] Generate two independent (K×n) sampling matrices A and B, where K is the number of matrix samples and n is the number of input parameters. Based on the sampling matrices A and B, the matrix and The A matrix is ​​the main body, and the j-th column B in the B matrix is j For the jth column A in matrix A j Perform the overall replacement, and keep the other n-1 columns unchanged. Similarly, we can get the matrix

[0049] Each error term is entered into the calculation;

[0050] According to the matrix A, B, Different combinations between the input variance V i And the total variance V of the model is approximately estimated:

[0051]

[0052] In formulas (11), (12), and (13), m is the mth row corresponding to the sampling matrix;

[0053] Global sensitivity coefficient calculation;

[0054] According to formula (10), the global sensitivity coefficient S can be obtained i The estimation formula is:

[0055]

[0056] An embodiment of the present invention provides a method for analyzing the spatial error sensitivity of a five-axis CNC machine tool. By establishing a spatial error model of the CNC machine tool and an improved Sobol global error sensitivity analysis model, the method analyzes the degree of influence of various geometric errors on the machining accuracy of the machine tool. Finally, the key geometric error items that have a greater impact on the machining accuracy of the CNC machine tool are effectively identified, and a new concept for the design and improvement of CNC machine tools is proposed, fundamentally solving the problem of CNC machine tool machining accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 A flowchart of a method for analyzing spatial error sensitivity of a five-axis CNC machine tool provided by an embodiment of the present invention;

[0058] Figure 2 A schematic diagram of a dual-branch topology chain provided by an embodiment of the present invention;

[0059] Figure 3Improved Monte Carlo method for matrix and schematic diagram is generated. DETAILED DESCRIPTION

[0060] In order to make the purpose, technical scheme and advantages of the present application more clear, the present application is further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and do not limit the present application.

[0061] The specific implementation of the present application is described in detail below in combination with specific examples.

[0062] As Figure 1 shown, a block diagram of a five-axis numerical control machine tool whole machine space error sensitivity analysis method provided by an embodiment of the present application includes the following steps:

[0063] S1, establishing a numerical control machine tool whole machine space error model: based on the multi-body system theory, the structure of the machine tool and the correlation between each adjacent body are expressed through the double-branch topological structure chain and the low-order body array table, the geometric error terms existing between each linear axis and the rotary axis are analyzed, a generalized coordinate system is established, the existing geometric error terms are introduced into the position transformation matrix and the motion transformation matrix of the adjacent body, respectively, to form the position error transformation matrix between adjacent bodies and the motion error transformation matrix between adjacent bodies, to represent the position relationship and the mutual relationship between the multi-body systems, specifically:

[0064] S101, establishing a double-branch topological structure chain of the numerical control machine tool: the numerical control machine tool is a double-branch multi-body system, taking the machine tool bed as the starting point of the double-branch topological structure chain, starting from the bed B0, the bed B0 is divided into two branches, namely the tool chain and the workpiece chain, each machine tool component is regarded as a typical body, the numbering rule of the typical body is that the machine tool bed is taken as the typical body B0, starting from the bed, along the direction away from B0, the serial number of each typical body is calibrated in the natural growth sequence, and the tool chain and the workpiece chain are calibrated respectively, as shown in Figure 2 ; except for the B0 body, each typical body has an adjacent low-order body, a table is prepared for the low-order body of each typical body, and L i (j) is used to represent, which is called a low-order body array table, as shown in Table 1, j represents the serial number of the typical body, j = 1, 2, 3... i, i represents the number of typical bodies included in the machine tool:

[0065] Table 1: Low-order body array table of numerical control machine tool

[0066] Typical body 1 2 3 4 5 6 7 <![CDATA[L 0 (j)]]> 1 2 3 4 5 6 7 L 1 (j)] 0 1 2 3 4 0 6 L 2 (j)] 0 0 1 2 3 0 0 <![CDATA[L 3 (j)]]> 0 0 0 1 2 0 0 L 4 (j)] 0 0 0 0 1 0 0 L 5 (j)] 0 0 0 0 0 0 0

[0067] S102, geometric error analysis of the numerical control machine tool:

[0068] In a spatial coordinate system, any object has six degrees of freedom. Six errors are inevitably generated during motion: three linear displacement errors and three angular displacement errors. These are errors related to the position point. Three non-perpendicularity errors exist between the X, Y, and Z guide rails, and two perpendicularity errors exist between the C-axis and the X and Y axes. Therefore, there are a total of 29 errors, as shown in Table 2:

[0069] Table 2 Geometric error symbols and meanings

[0070]

[0071]

[0072] S103, determine the position vector of the origin of the reference coordinate system of each typical body. The position vector of the origin of the body reference coordinate system of each body is shown in Table 3:

[0073] Table 3

[0074] Origin of the body reference coordinate system of each body Position vector B1 h1 = (h 1x h 1y h 1z 1) T ]]> B2 h2 = (h 2x h 2y h 2z 1) T ]]> B3 h3 = (h 3x h 3y h 3z 1) T ]]> B4 h4 = (h 4x h 4y h 4z 1) T ]]> B5 h5 = (h 5x h 5y h 5z 1) T ]]> B6 h6 = (h 6x h 6y h 6z 1) T ]]> Tool machining point in the tool coordinate system r t = (h tx h ty h tz 1) T ]]> Tool machining point in the workpiece coordinate system r w = (h wx h wy h wz 1) T ]]>

[0075] S104, establishing the adjacent body transformation matrix: Based on the motion relationship between the various components of the CNC machine tool, the transformation matrix between the adjacent bodies is established as shown in Table 4 and Table 5:

[0076] Table 4 Ideal static and actual static homogeneous transformation matrices between adjacent bodies

[0077]

[0078]

[0079] Table 5 Homogeneous transformation matrix of ideal motion and actual motion between adjacent bodies

[0080]

[0081]

[0082] S105. Establishing a spatial error model for the machine tool:

[0083] The position vector of the point to be processed in the workpiece coordinate system in the machine tool coordinate system is:

[0084]

[0085] The position vector of the tool center point in the machine tool coordinate system is:

[0086]

[0087] In the actual machining process, if precision machining is to be achieved, it is necessary to ensure that the machining point in the workpiece coordinate system coincides with the tool center point, that is:

[0088] p w =p t (3)

[0089] The position vector of the tool center point in the workpiece coordinate system in the actual machining process is:

[0090]

[0091] In formula (4), r wx is the component of the position of the tool center point in the workpiece coordinate system in the X direction; r wy

[0092] is the component of the position of the tool center point in the workpiece coordinate system in the Y direction; r wz is the component of the position of the tool center point in the workpiece coordinate system in the X direction;

[0093] Assuming that each error parameter in formula (1) is 0, the position vector of the tool center point in the workpiece coordinate system under ideal conditions can be obtained:

[0094]

[0095] In formula (5), r is the component of the position of the tool center point in the workpiece coordinate system in the X direction under ideal conditions; is the component of the position of the tool center point in the workpiece coordinate system in the Y direction under ideal conditions; is the component of the position of the tool center point in the workpiece coordinate system in the Z direction under ideal conditions;

[0096] Subtracting formula (5) from formula (4), the machine tool space motion error model can be obtained:

[0097] E = (E x E y E z 1) T (6)

[0098] In formula (6), E x is the component of the space motion error in the X direction; E y is the component of the space motion error in the Y direction; and E z is the component of the space motion error in the Z direction.

[0099] S2, in combination with the space error model of the machine tool, an improved Sobol global sensitivity analysis method is proposed to solve the first-order sensitivity coefficient and the global sensitivity coefficient, and to analyze the key geometric error of the numerical control machine tool:

[0100] Sensitivity analysis indicates that each attribute varies in the possible value range, studies and predicts the influence degree of the variation of these attributes on the model output value, and the size of the influence degree is called the sensitivity coefficient of the attribute, the greater the sensitivity coefficient, the greater the influence of the attribute on the model output, the core purpose of sensitivity analysis is to obtain the size of the sensitivity coefficient of each attribute through the analysis of the attributes of the model, and in practical application, according to experience, the attributes with small sensitivity coefficients are removed, and the attributes with large sensitivity coefficients are mainly considered, therefore, the embodiment of the present application adopts the improved Sobol global sensitivity analysis method to analyze the key geometric errors of the numerical control machine tool;

[0101] The Sobol global sensitivity analysis method is a Monte Carlo method based on variance, the uncertainty of the input parameters is transmitted through the model to cause the uncertainty of the whole system response, the core idea of the method is to decompose the system function into the sum of 2n incremental terms, the total variance and the partial variance of each term are calculated through sampling to obtain the sensitivity, since the geometric error is not a fixed value, but follows a certain probability characteristic distribution and fluctuates randomly in a certain interval, therefore, the 29 geometric errors are defined as a 29-dimensional unit body as the spatial domain of the input factors, ΔS xy 、ΔS xz 、ΔS yz 、Δα xc 、Δβ yc are geometric errors irrelevant to the motion amount, which are taken as fixed values, only the sensitivity of the remaining 24 geometric errors to the machining precision needs to be studied, and the step specifically includes the following processes:

[0102] S201, error model decomposition:

[0103] According to the decomposition method of the Sobol method, the mathematical model Y=f(h) is decomposed into the form of incremental order:

[0104]

[0105] In formula (7), Y0 is the expected value of the overall model obtained by inputting each parameter; Y i =Y(h i ) is the function value corresponding to the ith input term h i ; Y ij =Y(h i ,h j ) is the function value corresponding to the joint action of input terms h i and h j , and other high-order terms can be obtained in this way;

[0106] The variance of formula (7) is calculated to obtain:

[0107]

[0108] Divide the total variance of the function by both sides of the equation and make an orthogonal transformation to get:

[0109]

[0110] S202, first-order sensitivity coefficient solving:

[0111] Further solving obtains the first-order sensitivity coefficient calculation result:

[0112]

[0113] In formula (10), h ～i is all input parameter items except h i ; is the expected value calculated for h ～i input item; S i is the first-order variance ratio corresponding to the input parameter h i ;

[0114] Where S i calculated by formula (10) is the first-order sensitivity coefficient of the input error item h i , which reflects the influence degree of the error item on the model output, and the larger the S i value, the greater the influence degree of the input parameter on the model;

[0115] S203, improved Monte Carlo method:

[0116] The improved Monte Carlo method has better calculation convergence than the conventional Monte Carlo method, generates two independent (Kx n) sampling matrices A and B, where K is the number of matrix sampling, and n is the number of input parameters. On the basis of the sampling matrices A and B, the matrix and is based on the A matrix, the jth column B j in the B matrix is replaced in the whole jth column A j in the A matrix, and the other n-1 columns are unchanged, and the matrix is obtained in the same way. Figure 3

[0117] S204, each error item input calculation:

[0118] According to different combinations between the matrices A, B, , the variance V i of each input item and the total variance V of the model are approximately estimated:

[0119] ​

[0120] In formula (11), (12), (13), m is the mth row corresponding to the sampling matrix;

[0121] Global sensitivity coefficient calculation;

[0122] According to formula (10), the global sensitivity coefficient S i The estimation formula is:

[0123]

[0124] The above merely describes preferred embodiments of the present application and is not used to limit the present application, and any modification, equivalent replacement, and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for analyzing the spatial error sensitivity of a five-axis CNC machine tool, characterized in that: The following steps are involved: Based on the multi-body system theory, the CNC machine tool is abstracted as a multi-body system. The structure of the machine tool and the relationship between each body are described using a topological structure diagram and a low-order body array table. The geometric error of the CNC machine tool is analyzed, and a generalized coordinate system is established. The positional relationship is represented by the characteristic matrix between adjacent bodies, and the relationship between the multi-body system is represented by a homogeneous transformation matrix. The spatial error model of the machine tool is established. Combined with the spatial error model of the machine tool, an improved Sobol global sensitivity analysis method is proposed to solve the first-order sensitivity coefficient and the global sensitivity coefficient, including: Error model decomposition: According to the decomposition method of the Sobol method, the mathematical model Y = f(h) is decomposed into the form of increasing order: In formula (7), Y0 is the expected value of the overall model obtained by inputting various parameters; Y i =Y(h i ) is the i-th input item h i The corresponding function value; Y ij =Y(h i ,h j ) is the input item h i and h j The corresponding function values ​​under the joint action, and other high-order terms can be obtained by analogy; Calculating the variance of formula (7) yields: Divide both sides of the equal sign by the total variance of the function and perform an orthogonal transformation to obtain: First-order sensitivity coefficient solution: Solve and obtain the calculation results of the first-order sensitivity coefficient: In formula (10), h ～i To divide h i All other input parameters except For h ～i The expected value is calculated by input; S i The input parameter h i The corresponding first-order variance ratio; The S calculated by formula (10) i That is the input error term h i The first-order sensitivity coefficient reflects the influence of the error term on the model output, S i The larger the value, the greater the influence of the input parameter on the model; Improved Monte Carlo method: Generate two independent (K×n) sampling matrices A and B, where K is the number of matrix samples and n is the number of input parameters. Based on the sampling matrices A and B, the matrix and The A matrix is ​​the main body, and the j-th column B in the B matrix is j For the jth column A in matrix A j Perform the overall replacement, and keep the other n-1 columns unchanged. Similarly, we can get the matrix Each error term is entered into the calculation; According to the matrix A, B, Different combinations between the input variance V i And the total variance V of the model is approximately estimated: In formulas (11), (12), and (13), m is the mth row corresponding to the sampling matrix; Global sensitivity coefficient calculation; According to formula (10), the global sensitivity coefficient S can be obtained i The estimation formula is:

2. The method for analyzing spatial error sensitivity of a five-axis CNC machine tool according to claim 1, wherein: The step of using a topological structure diagram and a low-order body array table to describe the structure of the machine tool and the relationship between the various bodies specifically includes: The machine bed is used as the starting point of the double-branch topological structure chain. Starting from the bed B0, it is divided into two branches, namely the tool chain and the workpiece chain. Each machine tool component is regarded as a typical body. Starting from the bed, along the direction away from B0, the serial number of each typical body is calibrated in sequence according to the naturally increasing sequence. The calibration is carried out separately for the tool chain and the workpiece chain. Except for B0, each typical body has an adjacent low-order body. A table is prepared for the low-order bodies of each typical body, using L i (j) represents a low-order body array table, where j represents the serial number of the typical body, j=1, 2, 3...i, and i represents the number of typical bodies contained in the machine tool.

3. The method for analyzing spatial error sensitivity of a five-axis CNC machine tool according to claim 2, wherein: The steps of analyzing the geometric errors of the CNC machine tool, establishing a generalized coordinate system, expressing the positional relationship using a characteristic matrix between adjacent bodies, and expressing the mutual relationship between the multi-body system using a homogeneous transformation matrix specifically include: In a spatial coordinate system, any object has six degrees of freedom, generating six errors during motion: three linear displacement errors and three angular displacement errors. There are three non-perpendicularity errors between the X, Y, and Z guide rails, and two perpendicularity errors between the C axis and the X and Y axes, resulting in a total of 29 errors. Determine the origin position vector of the reference coordinate system of each typical body; According to the motion relationship between the various components of the CNC machine tool, the transformation matrix between adjacent bodies is established, including the ideal static and actual static homogeneous transformation matrix between adjacent bodies and the ideal motion and actual motion homogeneous transformation matrix between adjacent bodies.

4. The method for analyzing spatial error sensitivity of a five-axis CNC machine tool according to claim 3, wherein: The step of establishing the spatial error model of the machine tool specifically includes: The position vector of the point to be processed in the workpiece coordinate system in the machine tool coordinate system is: Where, is the ideal stationary homogeneous transformation matrix between adjacent bodies along the Z axis 0-4, is the actual static homogeneous transformation matrix between adjacent bodies on the Z axis 0-4, is the ideal motion homogeneous transformation matrix between adjacent bodies along the Z axis 0-4, is the actual motion homogeneous transformation matrix between adjacent bodies on the Z axis 0-4, is the ideal stationary homogeneous transformation matrix between adjacent bodies along the C axis 4-5, is the actual static homogeneous transformation matrix between adjacent bodies on the C axis 4-5, is the ideal motion homogeneous transformation matrix between adjacent bodies on the C axis 4-5, is the ideal stationary homogeneous transformation matrix between adjacent bodies of workpiece 5-6, is the actual static homogeneous transformation matrix between adjacent bodies of workpiece 5-6, is the ideal motion homogeneous transformation matrix between adjacent bodies of workpiece 5-6, is the actual motion homogeneous transformation matrix between adjacent bodies of workpiece 5-6, r w is the position vector of the tool processing point in the workpiece coordinate system; The position vector of the tool center point in the machine tool coordinate system is: Where, is the ideal stationary homogeneous transformation matrix between adjacent bodies on the X axis 0-1, is the actual static homogeneous transformation matrix between adjacent bodies on the X axis 0-1, is the ideal motion homogeneous transformation matrix between adjacent bodies on the X axis 0-1, is the actual motion homogeneous transformation matrix between adjacent bodies on the X axis 0-1, is the ideal stationary homogeneous transformation matrix between adjacent bodies on the Y axis 1-2, is the actual static homogeneous transformation matrix between adjacent bodies on the Y axis 1-2, is the ideal motion homogeneous transformation matrix between adjacent bodies on the Y axis 1-2, is the actual motion homogeneous transformation matrix between adjacent bodies on the Y axis 1-2, is the ideal stationary homogeneous transformation matrix between adjacent bodies of tool 2-3, is the actual static homogeneous transformation matrix between adjacent bodies of tool 2-3, is the ideal motion homogeneous transformation matrix between adjacent bodies of tool 2-3, is the actual motion homogeneous transformation matrix between adjacent bodies of tool 2-3, r t is the position vector of the tool processing point in the tool coordinate system; In the actual machining process, precision machining requires that the point to be machined in the workpiece coordinate system coincide with the tool center point, that is: p w =p t (3) The position vector of the tool center point in the workpiece coordinate system during the actual machining process is: In formula (4), r wx is the component of the position of the tool center point in the workpiece coordinate system in the X direction; r wy is the component of the position of the tool center point in the workpiece coordinate system in the Y direction; r wz is the component of the position of the tool center point in the workpiece coordinate system in the X direction; Assuming that all error parameters in equation (1) are 0, the position vector from the tool center point to the workpiece coordinate system under ideal conditions can be obtained: In formula (5), is the component of the position of the tool center point in the workpiece coordinate system in the X direction under ideal conditions; is the Y-direction component of the position of the tool center point in the workpiece coordinate system under ideal conditions; is the component of the position of the tool center point in the workpiece coordinate system in the Z direction under ideal conditions; Subtracting Equation (5) from Equation (4) yields the machine tool spatial motion error model: And=(And x AND y AND z 1) T (6) In formula (6), E x is the component of the spatial motion error in the X direction; E y is the component of the spatial motion error in the Y direction; E z is the component of the spatial motion error in the Z direction.

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