Complete machine space error sensitivity analysis method for five-axis numerical control machine tool

Abstract

The invention is suitable for the technical field of machine tool precision design, and provides a five-axis numerical control machine tool complete machine space error sensitivity analysis method, which comprises the following steps: abstracting a numerical control machine tool into a multi-body system based on a multi-body system theory, describing the incidence relation between the structure of the machine tool and each individual by using a topological structure chart and a low-sequence body array list, and calculating the overall space error sensitivity of the five-axis numerical control machine tool. The geometric error of the numerical control machine tool is analyzed, a generalized coordinate system is established, the position relation is expressed by using the characteristic matrix between adjacent bodies, the mutual relation between multiple body systems is expressed by using a homogeneous transformation matrix, and a space error model of the machine tool is established; and in combination with a space error model of the machine tool, an improved Sobol global sensitivity analysis method is provided, a first-order sensitivity coefficient and a global sensitivity coefficient are solved, and key geometric errors of the numerical control machine tool are analyzed. By establishing the space error model of the numerical control machine tool and the improved Sobol global error sensitivity analysis model, a new design and improvement concept of the numerical control machine tool is provided, and the problem of machining precision of the numerical control machine tool is solved.

Description

Technical Field

[0001] The 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 a kind of modern electromechanical equipment with high precision, high efficiency and high technology. As the foundation and core equipment of advanced manufacturing technology, their precision index directly affects the quality of mechanically processed products and restricts the development of the manufacturing field and various high-tech technologies. The precision indexes of CNC machine tools mainly include processing accuracy, positioning accuracy and repeated positioning accuracy. Among them, processing accuracy is the ultimate accuracy pursued by CNC machine tools, reflecting the manufacturing capacity and development level of the machinery manufacturing industry.

[0003] The factors that affect the machining accuracy of machine tools mainly include the geometric errors, thermal errors, servo system errors and load errors of various machine tool components. Among them, geometric errors have the greatest impact on machining accuracy, accounting for about 50%. The geometric error of machine tools refers to the deviation of the actual geometric parameters and positions of various components or parts in the machine tool from the ideal geometric parameters and positions due to defects in the design, manufacturing, assembly, etc. of the machine tool. This error is generally related to the geometric elements of various components or parts of the machine tool and is an inherent error of 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 verticality and parallelism errors between moving axes. How to effectively identify the key geometric error items that have a greater impact on the machining 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 machining 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-mentioned 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 by a topological structure diagram and a low-order body array table. The geometric error of the CNC machine tool is analyzed, a generalized coordinate system is established, the position relationship is represented by the characteristic matrix between adjacent bodies, the relationship between multi-body systems is represented by a homogeneous transformation matrix, and the spatial error model of the machine tool is established.

[0008] Combined with the spatial error model of machine tools, an improved Sobol global sensitivity analysis method is proposed to solve the first-order sensitivity coefficient and the global sensitivity coefficient and analyze the critical geometric errors of CNC machine tools.

[0009] Preferably, the step of describing the structure of the machine tool and the association relationship between the various entities using the topological structure diagram and the low-order entity array table specifically includes:

[0010] The machine bed is taken as the starting point of the double-branch topological structure chain. It is divided into two branches starting from the bed B0, 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 marked in turn according to the natural growth sequence, and the marking is carried out according to the tool chain and the workpiece chain respectively.

[0011] 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, which is called the low-order body array table, j represents the serial number of the typical body, j=1, 2, 3...i, i represents the number of typical bodies contained in the machine tool.

[0012] Preferably, the steps of analyzing the geometric errors of the CNC machine tool, establishing a generalized coordinate system, using a characteristic matrix between adjacent bodies to represent the positional relationship, and using a homogeneous transformation matrix to represent the mutual relationship between the multi-body system specifically include:

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

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

[0015] 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.

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

[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 center point of the tool, 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 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;

[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 steps of combining the spatial error model of the machine tool, proposing an improved Sobol global sensitivity analysis method, solving the first-order sensitivity coefficient and the global sensitivity coefficient, and analyzing the critical geometric errors of the CNC machine tool specifically include:

[0034] Error model decomposition:

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

[0036]

[0037] 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 i =Y(h i ,h j ) is the input item h i and h j The corresponding function value under the joint action, and other high-order terms can be obtained by analogy;

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

[0039]

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

[0041]

[0042] First-order sensitivity coefficient solution:

[0043] Solve and obtain the calculation result of the first-order sensitivity coefficient:

[0044]

[0045] In formula (10), h ～i To divide h i All other input parameters except For h ～i The expected value is calculated from the input items; S i The input parameter h i The corresponding first-order variance ratio;

[0046] 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. 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 matrices and The A matrix is ​​taken as the main body, and the j-th column B in the B matrix is j For the jth column A in the matrix A j Perform overall replacement, and keep the other n-1 columns unchanged. Similarly, we 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 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 influence of various geometric errors on the machining accuracy of the machine tool is analyzed. Finally, the key geometric error terms 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 the CNC machine tool is proposed, thereby fundamentally solving the problem of the machining accuracy of the CNC machine tool. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 A flowchart of a method for analyzing the 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 double-branch topology structure chain provided by an embodiment of the present invention;

[0059] Figure 3The matrix in the improved Monte Carlo method provided by the embodiment of the present invention is and Generate a schematic diagram. DETAILED DESCRIPTION

[0060] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0061] The specific implementation of the present invention is described in detail below in conjunction with specific embodiments.

[0062] like Figure 1 As shown, a block diagram of a method for analyzing spatial error sensitivity of a five-axis CNC machine tool provided by an embodiment of the present invention includes the following steps:

[0063] S1. Establish the spatial error model of the whole CNC machine tool: Based on the multi-body system theory, the structure of the machine tool and the relationship between each adjacent body are described through the double-branch topological structure chain and the low-order body array table. The geometric error terms of each linear axis and rotary axis are analyzed, and a generalized coordinate system is established. The existing geometric error terms are introduced into the position transformation matrix and motion transformation matrix of the adjacent bodies, respectively, to form the position error transformation matrix between adjacent bodies and the motion error transformation matrix between adjacent bodies, which represent the position relationship and mutual relationship between the multi-body system, specifically:

[0064] S101. Establish a double-branch topological structure chain of CNC machine tools: CNC machine tools are double-branch multi-body systems. The machine bed is taken 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. The numbering rule of the typical body is as follows: take the machine bed as the typical body B0, start from the bed, and in the direction away from B0, calibrate the serial number of each typical body in turn according to the naturally increasing sequence, and calibrate according to the tool chain and the workpiece chain respectively, such as Figure 2 As shown; except for B0, each typical body has an adjacent low-order body. A table is prepared for the low-order body of each typical body, using L i (j) indicates, which is called the low-order body array table, as shown in Table 1, j indicates the serial number of the typical body, j = 1, 2, 3...i, i indicates the number of typical bodies contained in the machine tool:

[0065] Table 1 Low-order array table of CNC machine tools

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

[0067] S102, Geometric error analysis of CNC machine tools:

[0068] In the spatial coordinate system, any object has 6 degrees of freedom. During the movement, 6 errors will inevitably be generated, including 3 linear displacement errors and 3 angular displacement errors. These are errors related to the position point. There are 3 non-perpendicularity errors between the X, Y, and Z guide rails, and 2 perpendicularity errors 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, determining 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] The origin of the body reference coordinate system of each body Position vector B1 <![CDATA[h1=(h 1x h 1y h 1z 1) T ]]> B2 <![CDATA[h2=(h 2x h 2y h 2z 1) T ]]> B3 <![CDATA[h3=(h 3x h 3y h 3z 1) T ]]> B4 <![CDATA[h4=(h 4x h 4y h 4z 1) T ]]> B5 <![CDATA[h5=(h 5x h 5y h 5z 1) T ]]> B6 <![CDATA[h6=(h 6x h 6y h 6z 1) T ]]> Tool processing point is in the tool coordinate system <![CDATA[r t =(h tx h ty h tz 1) T ]]> Tool processing point is in the workpiece coordinate system <![CDATA[r w =(h wx h wy h wz 1) T ]]>

[0075] S104, establishing adjacent body transformation matrix: according to 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 matrix 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 the spatial error model of 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 satisfy the requirement that the point to be machined in the workpiece coordinate system coincides with the center point of the tool, that is:

[0088] p w =p t (3)

[0089] The position vector of the tool center point in the workpiece coordinate system during 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 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:

[0094]

[0095] 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 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 equation (5) from equation (4) yields the machine tool spatial motion error model:

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

[0098] 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.

[0099] S2. 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, and analyze the critical geometric errors of the CNC machine tool:

[0100] Sensitivity analysis indicates that each attribute changes within a possible range of values, and studies and predicts the degree of influence of the changes in these attributes on the model output value. The magnitude of the influence is called the sensitivity coefficient of the attribute. The larger the sensitivity coefficient, the greater the influence of the attribute on the model output. The core purpose of sensitivity analysis is to obtain the magnitude of the sensitivity coefficient of each attribute by analyzing the attributes of the model. In practical applications, attributes with very small sensitivity coefficients are removed based on experience, and attributes with larger sensitivity coefficients are focused on. Therefore, the embodiment of the present invention adopts an improved Sobol global sensitivity analysis method to analyze the critical geometric errors of CNC machine tools.

[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, resulting in the uncertainty of the entire system response. The core idea of ​​this method is to decompose the system function into the sum of 2n increasing terms, and calculate the total variance of the model response and the partial variance of each sub-term by sampling, so as to obtain the sensitivity. Since the geometric error is not a fixed value, but obeys a certain probability characteristic distribution and fluctuates randomly within a specific interval, it is necessary to define the 29 geometric errors as a 29-dimensional unit body as the spatial domain of the input factors, ΔS xy , ΔS xz , ΔS yz , Δα xc , Δβ yc All of them are geometric errors that are independent of the amount of motion. They are taken as constants. It is only necessary to study the sensitivity of the remaining 24 geometric errors to the machining accuracy. This 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 increasing order:

[0104]

[0105] 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 i =Y(h i ,h j ) is the input item h i and h j The corresponding function value under the joint action, and other high-order terms can be obtained by analogy;

[0106] The variance calculation of formula (7) is as follows:

[0107]

[0108] Divide both sides of the equal sign by the total variance of the function and perform an orthogonal transformation to obtain:

[0109]

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

[0111] Further solving the problem, we can get the calculation result of the first-order sensitivity coefficient:

[0112]

[0113] In formula (10), h ～i To divide h i All other input parameters except For h ～i The expected value is calculated from the input items; S i The input parameter h i The corresponding first-order variance ratio;

[0114] 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. i The larger the value, the greater the influence of the input parameter on the model;

[0115] S203, Improved Monte Carlo method:

[0116] The improved Monte Carlo method has better computational convergence than the conventional Monte Carlo method. It generates 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. The matrix is ​​obtained based on the sampling matrices A and B. and The A matrix is ​​taken as the main body, and the j-th column B in the B matrix is j For the jth column A in the matrix A j Perform overall replacement, and keep the other n-1 columns unchanged. Similarly, we get the matrix like Figure 3 As shown;

[0117] S204, inputting each error term into calculation;

[0118] 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:

[0119]

[0120] In formulas (11), (12), and (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 description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

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 by a topological structure diagram and a low-order body array table. The geometric error of the CNC machine tool is analyzed, a generalized coordinate system is established, the position relationship is represented by the characteristic matrix between adjacent bodies, the relationship between multi-body systems is represented by a homogeneous transformation matrix, and the spatial error model of the machine tool is established. Combined with the spatial error model of machine tools, an improved Sobol global sensitivity analysis method is proposed to solve the first-order sensitivity coefficient and the global sensitivity coefficient and analyze the critical geometric errors of CNC machine tools.

2. The method for analyzing the spatial error sensitivity of a five-axis CNC machine tool according to claim 1 is characterized in that: 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 association relationship between the various bodies specifically includes: The machine bed is taken as the starting point of the double-branch topological structure chain. It is divided into two branches starting from the bed B0, 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 marked in turn according to the natural growth sequence, and the marking is carried out according to the tool chain and the workpiece chain respectively. 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, which is called the low-order body array table, j represents the serial number of the typical body, j=1, 2, 3...i, 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, characterized in that: The steps of analyzing the geometric errors of the CNC machine tool, establishing a generalized coordinate system, using a characteristic matrix between adjacent bodies to represent the positional relationship, and using a homogeneous transformation matrix to represent the mutual relationship between the multi-body system specifically include: In the spatial coordinate system, any object has 6 degrees of freedom, and 6 errors are generated during the movement, including 3 linear displacement errors and 3 angular displacement errors. There are 3 non-perpendicularity errors between the X, Y, and Z guide rails, and 2 perpendicularity errors between the C axis and the X and Y axes, so there are 29 errors in total. 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 the spatial error sensitivity of a five-axis CNC machine tool according to claim 3 is characterized in that: 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: The position vector of the tool center point in the machine tool coordinate system is: In the actual machining process, precision machining requires that the point to be machined in the workpiece coordinate system coincides with the center point of the tool, 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 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; 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.

5. The method for analyzing the spatial error sensitivity of a five-axis CNC machine tool according to claim 1, characterized in that: The improved Sobol global sensitivity analysis method is proposed in combination with the spatial error model of the machine tool, and the steps of solving the first-order sensitivity coefficient and the global sensitivity coefficient and analyzing the critical geometric errors of the CNC machine tool are specifically as follows: 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 j =Y(h i ,h j ) is the input item h i and h j The corresponding function value under the joint action, and other high-order terms can be obtained by analogy; The variance calculation of formula (7) is as follows: 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 result 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 from the input items; 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. 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 matrices and The A matrix is ​​taken as the main body, and the j-th column B in the B matrix is j For the jth column A in the matrix A j Perform overall replacement, and keep the other n-1 columns unchanged. Similarly, we 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 i The estimation formula is:

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