A Tolerance Optimization Design Method for Key Components of Multi-Axis CNC Machine Tools

Abstract

A method for optimizing the tolerances of key components of a multi-axis numerical control machine tool belongs to the technical field of optimizing the accuracy design of machine tools. Specifically, it involves a method for modeling the spatial errors of a multi-axis numerical control machine tool based on the theory of multi-body systems and a method for optimizing the design of the tolerance parameters of key components of a multi-axis numerical control machine tool. The present invention establishes a spatial error model of a multi-axis numerical control machine tool and a multi-objective optimization distribution model of the tolerance parameters of key components of a multi-axis numerical control machine tool. At the initial design stage of the machine tool, by optimizing the distribution of the tolerance parameters of the key components of the multi-axis numerical control machine tool, not only can most of the tolerance parameters be relaxed to varying degrees, indirectly reducing the overall manufacturing cost of the numerical control machine tool, but also the machine tool after the optimized distribution of the tolerance parameters can obtain higher machining accuracy.

Description

Technical Field

[0001] The present invention relates to a method for optimizing the tolerances of key components of a multi-axis numerical control machine tool, and belongs to the technical field of optimizing the accuracy design of machine tools. Background Art

[0002] As the mother machine in the industrial field, numerical control machine tools have been widely used in industries such as aerospace, wind power generation, and shipbuilding. Their development plays a very important role in improving the economic level of our country, reflects a country's mechanical manufacturing capacity and development level, and their machining accuracy is an important indicator to measure the performance of machine tools, and also marks a country's scientific and technological level. And reasonable optimization design of the tolerance parameters of key components of numerical control machine tools is an effective method to improve the machining accuracy of machine tools in the initial design stage of machine tools. Therefore, carrying out research on the optimization design of the tolerances of key components of numerical control machine tools has certain theoretical and engineering application values.

[0003] The solution to this key problem is divided into three steps:

[0004] First, based on the theory of multi-body system kinematics, establish the spatial error model of the machine tool;

[0005] First, based on the theory of multi-body system kinematics, the structure of the numerical control machine tool is simplified and analyzed. The machine tool is abstracted into a multi-body system composed of a tool branch and a workpiece branch, the coordinate systems of each key moving component of the machine tool are established, and then based on the principle of homogeneous coordinate transformation, the static and motion transformation matrices between adjacent bodies of the machine tool are established. Finally, the spatial motion error model of the numerical control machine tool is derived.

[0006] Second, establish the tolerance parameter optimization distribution model;

[0007] Taking the tolerance zone width of each tolerance parameter as the design variable; taking the tolerance zone width of each tolerance parameter and the spatial motion error of the machine tool not exceeding the design requirements as the constraint conditions; aiming at the problems of low reliability and applicability of the tolerance-cost model, according to the results of tolerance parameter sensitivity analysis, weights are assigned to each tolerance parameter, and the sum of the products of the tolerance zone width of each tolerance parameter and its corresponding weight factor is maximized and the spatial motion error of the whole machine is minimized as the design goal. Finally, a multi-objective optimization distribution model of the tolerance parameters of key components of the numerical control machine tool is established.

[0008] Third, simulation analysis and experimental verification of the tolerance parameter optimization distribution model;

[0009] Based on the NSGA-II algorithm and using MATLAB R2016b, the tolerance parameters of the key components of the numerically controlled machine tool are optimized and allocated multi-objectively, and finally the Pareto optimal solution set is obtained. At the same time, discussions are carried out with the machine tool design engineers, and the final tolerance optimization and allocation scheme is determined from the Pareto optimal solution set according to the actual production conditions of the enterprise. In order to further verify the accuracy of the tolerance parameter optimization and allocation scheme for the key components of the numerically controlled machine tool, first, the tolerance parameter optimization and allocation scheme considering the actual production conditions of the enterprise is provided to the machine tool factory. Then, the machine tool design engineers replace the relevant components of the numerically controlled machine tool and perform process treatment according to the optimized tolerance parameters. Finally, the improved numerically controlled machine tool is used to process the test specimen and the profile error of the specimen is detected. By comparing the machining errors of the machine tool before and after optimization, the accuracy of the tolerance optimization and allocation results is verified.

[0010] In the invention patent CN108445839A, only the key geometric errors can be identified, and the tolerance parameters of the key components of the machine tool cannot be optimized and allocated. Therefore, it is necessary to propose a method for optimizing the design of the tolerance parameters of the numerically controlled machine tool, so as to realize the precision optimization design of the numerically controlled machine tool, which plays a strong demonstration and leading role for the numerically controlled machine tool research and development units in China to improve the machine tool design with relatively low cost investment, so as to promote the accelerated production and application of more innovative high-end numerically controlled machine tool products of various models in China. Summary of the Invention

[0011] The object of the present invention is to provide a method for optimizing the design of the tolerance of the key components of a multi-axis numerically controlled machine tool. By establishing an optimization allocation model for the tolerance parameters of the key components of the numerically controlled machine tool, the precision optimization design of the numerically controlled machine tool is realized, which has certain practical value and guiding significance for machine tool design engineers.

[0012] In order to achieve the above object, the technical solution adopted by the present invention is a method for predicting the machining error of a multi-axis numerically controlled machine tool. First, the present invention establishes a spatial motion error model of the numerically controlled machine tool, uses the tolerance zone width of each tolerance parameter as the design variable, and uses the spatial motion error not exceeding its design standard and the tolerance zone width of each tolerance parameter not exceeding 2 times its standard value as the constraint conditions. Then, taking the sum of the products of the tolerance zone width of each tolerance parameter and its corresponding weight factor as the maximum and the minimum of the overall spatial motion error as the design goals, finally, a multi-objective optimization allocation model for the tolerance parameters of the key components of the numerically controlled machine tool is established, so as to realize the precision optimization design of the numerically controlled machine tool. Finally, the simulation analysis and experimental verification of the tolerance parameter optimization allocation model of the machine tool are carried out, which proves the correctness of the tolerance parameter optimization allocation model.

[0013] The specific steps of this method are as follows:

[0014] Step 1: Spatial motion error modeling based on multi-body system theory;

[0015] Based on the kinematics theory of multi-body systems, the structure of the machine tool and the correlation between each body are described using a multi-body system schematic diagram and a low-order body array list. Analyze the geometric errors of the CNC machine tool, establish a generalized coordinate system, express the position relationship using the characteristic matrix between adjacent bodies, and represent the mutual relationship between multi-body systems using homogeneous transformation matrices;

[0016] Step 1.1 Establish the topological structure of the CNC machine tool;

[0017] The CNC machine tool is a complex multi-branched system, which divides into two branches at point B d , and except for body B d , each object has an adjacent lower-order body. When deriving kinematics and formulating calculation methods, a table needs to be developed for the lower-order body of each object in the system, denoted by L n (j), called the low-order body array list, as shown in Table 1, where j represents the object number, j = 1, 2, 3... n, and n represents the number of typical bodies included in the machine tool;

[0018] Table 1: Low-order body array of CNC machine tool

[0019] <![CDATA[L 0 (j)]]> 1 2 3 4 5 6 <![CDATA[L 1 (j)]]> 0 1 1 3 4 5 <![CDATA[L 2 (j)]]> 0 0 0 1 3 4 <![CDATA[L 3 (j)]]> 0 0 0 0 1 3 <![CDATA[L 4 (j)]]> 0 0 0 0 0 1 <![CDATA[L 5 (j)]]> 0 0 0 0 0 0

[0020] The numbering rules for typical bodies are as follows:

[0021] First, arbitrarily select a typical body as B d , and then along the direction away from body B d , sequentially label the numbers of each object according to the natural growth sequence, from one branch of the system to the other branch until all objects are labeled;

[0022] Step 1.2 Geometric error analysis of the CNC machine tool

[0023] Any object in the space coordinate system has 6 degrees of freedom, and 6 errors will inevitably occur during the movement, including 3 linear displacement errors and 3 angular displacement errors. These are all errors related to the position points. There are 3 non-perpendicularity errors between the X, Y, and Z guide rails, and a total of 4 perpendicularity errors between the C axis and the X, Y axes, and the A axis and the Y, Z axes. Therefore, a total of 37 errors are shown in Table 2;

[0024] Table 2: Geometric error parameters of CNC machine tool

[0025]

[0026]

[0027] Step 1.3 Establish the characteristic matrix of the CNC machine tool;

[0028] A right-handed Cartesian three-dimensional coordinate system O1-X1Y1Z1 and O j that is fixedly connected to it are established on the bed B1 and all components B j -X j Y j Z j . The set of these coordinate systems is called the generalized coordinate system, and each body coordinate system is called a sub-coordinate system. The three orthogonal bases of each coordinate system are named the X, Y, and Z axes respectively according to the right-hand rule; the corresponding coordinate axes of each sub-coordinate system are respectively parallel; the positive direction of the coordinate axis is the same as the positive direction of its corresponding motion axis;

[0029] According to the motion relationship between the components of the numerically controlled machine tool, the transformation matrix between adjacent bodies is established as shown in Table 3;

[0030] Table 3: Transformation Matrix between Adjacent Bodies

[0031]

[0032]

[0033] where: [Sij] p represents the relative position transformation matrix of body B j relative to body B i ; [Sij] pe represents the relative position error transformation matrix of body B j relative to body B i ; [Sij] s represents the relative motion transformation matrix of body B j relative to body B i ;

[0034] [Sij] se represents the relative motion error transformation matrix of body B j relative to body B i ; x represents the translation distance of the X axis;

[0035] y represents the translation distance of the Y axis;

[0036] z represents the translation distance of the Z axis;

[0037] a represents the rotation angle of the A axis;

[0038] c represents the rotation angle of the C axis;

[0039] Step 1.4 Establish the spatial error model of the machine tool and establish the motion relationship model between adjacent bodies under ideal conditions;

[0040] Let point P be an arbitrary point on body B j and P is on Bi Body coordinate system O i -X i Y i Z i The position matrix expression in it is;

[0041] P ji =[Sij] p [Sij] s r j (1)

[0042] In the formula: P ji is the position matrix expression of point P in the coordinate system O i -X i Y i Z i ;

[0043] r j is the position matrix expression of point P in the coordinate system O j -X j Y j Z j ;

[0044] [Sij] p represents the relative position transformation matrix of body B j relative to body B i ;

[0045] [Sij] s represents the relative motion transformation matrix of body B j relative to body B i ;

[0046] Establishment of the adjacent body motion relationship model under the condition of errors;

[0047] Let point P be any point on body B j The position matrix expression of P in the body coordinate system O i -X i Y i Z i is; i

[0048] P ji =[Sij] p [Sij] pe [Sij] s [Sij] se [Sij]r j (2)

[0049] In the formula: P ji is the position matrix expression of point P in the coordinate system O i -X i Y i Zi The position matrix expression in

[0050] r j is the position of point P in the coordinate system O j -X j Y j Z j The position matrix expression in

[0051] [Sij] p represents the relative position transformation matrix of body B j relative to body B i body;

[0052] [Sij] pe represents the relative position error transformation matrix of body B j relative to body B i body;

[0053] [Sij] s represents the relative motion transformation matrix of body B j relative to body B i body;

[0054] [Sij] se represents the relative motion error transformation matrix of body B j relative to body B i body;

[0055] The coordinates of the tool center point in the tool coordinate system are:

[0056] r t = [0, 0, l, 1] T (3)

[0057] where: l represents the tool length;

[0058] The subscript t represents the tool.

[0059] In the ideal case, the position matrix expression of the tool center point P branched from the "numerical control machine - workpiece" to the inertial coordinate system:

[0060]

[0061] In the ideal case, the position matrix expression of the tool center point P branched from the "numerical control machine - tool" to the inertial coordinate system:

[0062]

[0063] Precision machining equation of numerical control instructions:

[0064] P w I = P t I(6)

[0065] Ideally, the position matrix expression of the numerical control instruction in the workpiece coordinate system:

[0066]

[0067] In the actual situation, the position matrix expression of the tool center point P from the "machine tool - workpiece" branch to the inertial coordinate system:

[0068]

[0069] In the actual situation, the position matrix expression of the tool center point P from the "machine tool - tool" branch to the inertial coordinate system:

[0070]

[0071] In the actual situation, the position matrix expression of the numerical control instruction in the workpiece coordinate system:

[0072]

[0073] Then the spatial error model of the numerically controlled machine tool is expressed as:

[0074] E=r w -r w I (11)

[0075] Step 2: Construction of the optimization distribution model for the tolerance parameters of the key components of the numerically controlled machine tool;

[0076] Step 2.1 Determination of design variables

[0077] In the present invention, the tolerance zone width of each tolerance parameter of the key components of the numerically controlled machine tool is selected as the design variable, as shown in Equation (12):

[0078] W(T)={W(t1),W(t2),W(t3),...,W(t i )} (12)

[0079] Where W(t i ) represents the tolerance zone width of the i-th tolerance parameter.

[0080] Step 2.2 Determination of constraint conditions

[0081] The constraint condition is a kind of restrictive condition for the value range of the design variable. According to the design requirements of the numerically controlled machine tool, the constraint conditions of the proposed tolerance parameter optimization distribution model are mainly divided into two categories: spatial motion error constraint and tolerance zone width constraint of each tolerance parameter.

[0082] (1) Spatial motion error constraint condition

[0083] According to the design standard of the spatial motion error formulated for this machine tool, the maximum requirements for the spatial motion error components in the X, Y, and Z directions are E bx , E by , E bz . Therefore, in order to make the machine tool meet the design standard requirements, the components of the overall machine spatial motion error in the X, Y, and Z directions must meet the following requirements:

[0084]

[0085] (2) Tolerance zone width constraint condition

[0086] Due to the limitations of actual manufacturing conditions, the tolerance parameter values of the key components of the machine tool cannot reach the ideal state, that is, t i = 0. Therefore, we set the minimum value of the tolerance zone width to W(t i ) > 0. According to the accuracy inspection standard of CNC machine tools, the standard values of each tolerance parameter can be obtained. Since the tolerance parameter optimization distribution method proposed in the present invention needs to maximize the feasible range of each tolerance zone width on the premise of meeting the requirements of the overall machine spatial motion error, if the maximum value of each tolerance zone width is set to infinity, it does not conform to the engineering reality. Therefore, only the maximum value of the tolerance zone width can be appropriately enlarged. In the present invention, twice the standard value of each tolerance parameter is used as its maximum value, and the constraint conditions of each tolerance zone width can be obtained as follows:

[0087] 0 < W(t i ) < 2t ib (14)

[0088] In the formula, t ib represents the i-th tolerance parameter value given as the standard for the gantry five-axis CNC milling machine.

[0089] Step 2.3 Determination of the objective function

[0090] In most engineering design problems, there are multiple objectives that need to be satisfied simultaneously, but they are often conflicting with each other. The optimization problem of multiple objective functions under given constraint conditions is called a multi-objective optimization problem. The multi-objective optimization process considers the coupling relationship between each objective function, can reasonably balance the requirements of each objective function, and obtain a set of optimal solution sets at the same time, providing multiple reasonable design options for design engineers.

[0091] In existing research, although the tolerance-cost model plays an important role in the optimal allocation of tolerance parameters, its applicability and reliability are not strong from the perspective of practical applications. This is because the relationship function between the machining costs of key components of a machine tool and tolerances is obtained by domestic and foreign experts, scholars, and enterprise design engineers through sorting and summarizing a large amount of collected production experience data. Moreover, there are many influencing factors for the machining costs of each component, such as machining processes, the experience of manufacturers, machining procedures, production technologies, and different production enterprises. For the same component, there will be different machining cost experience data. Therefore, different tolerance-cost relationship curves will be generated. By studying various common tolerance-cost models, it can be found that although the tolerance-cost relationship curves are different, there is a certain inverse proportion relationship between them, that is, the larger the tolerance parameter value, the lower the cost.

[0092] To solve the problem of low reliability and applicability of the above tolerance-cost model, the invention first assigns weights to each tolerance parameter, and takes the maximization of the sum of the product of the tolerance zone width of each tolerance parameter and its corresponding weight factor and the minimization of the overall machine spatial motion error as the design goals to establish an optimal allocation model of tolerance parameters.

[0093] The maximization of the sum of the product of the tolerance zone width of each tolerance parameter and its corresponding weight factor is defined as objective function 1, and the specific expression is as follows:

[0094]

[0095] In the formula, SN i represents the weight factor of the i-th tolerance parameter; W(t i ) represents the tolerance zone width of the i-th tolerance parameter.

[0096] Since when actually solving the optimal allocation model of tolerance parameters, it is required to minimize the objective function. Therefore, equation (15) is transformed into the following form:

[0097]

[0098] The minimization of the overall machine spatial motion error is taken as objective function 2, and the specific expression is as follows:

[0099]

[0100] Therefore, by combining equations (16) and (17), the multi-objective optimization design function of the tolerance parameters of the key components of a gantry five-axis CNC milling machine can be obtained as follows:

[0101] min F(W(T))={f1(W(T)),f2(W(T))} (18)

[0102] Combining equations (13), (14), and (18), the standard expression of the tolerance parameter optimization distribution model for the CNC machine tool is as follows:

[0103]

[0104] Thus far, the tolerance parameter optimization distribution model for the CNC machine tool has been established.

[0105] Step 3: Simulation analysis and experimental verification of the tolerance parameter optimization distribution model for the CNC machine tool;

[0106] Step 3.1 Simulation analysis

[0107] Based on the NSGA-II algorithm and using MATLAB R2016b, multi-objective optimization distribution of the tolerance parameters of the key components of the CNC machine tool was carried out, and finally a Pareto optimal solution set was obtained. After discussing with the machine tool design engineers, the final tolerance optimization distribution plan was determined from the Pareto optimal solution set according to the actual production conditions of the enterprise. To more intuitively reflect the advantages of the finally determined tolerance optimization distribution plan, the tolerance parameter values before and after optimization were compared in the form of a histogram. It can be intuitively seen from the histogram that most of the tolerance parameter items have been relaxed to varying degrees. Since there is an inverse relationship between tolerance and cost, it indirectly shows that the manufacturing cost of the whole machine has been reduced. Therefore, the comparison results initially verify the feasibility and effectiveness of the tolerance parameter optimization distribution method for the key components of the CNC machine tool proposed in the present invention.

[0108] Step 3.2 Experimental verification

[0109] To further verify the accuracy of the tolerance parameter optimization distribution plan for the key components of the CNC machine tool, first, the tolerance parameter optimization distribution plan considering the actual production conditions of the enterprise was provided to the machine tool factory. Then, the machine tool design engineers replaced and processed the relevant components of the CNC machine tool according to the optimized tolerance parameters. Finally, the improved CNC machine tool was used to mill the test specimen and the contour error of the specimen was detected. By comparing the machining errors of the machine tool before and after optimization, the accuracy of the tolerance optimization distribution results was verified.

[0110] Compared with the prior art, the present invention has the following beneficial effects.

[0111] In existing research methods, most researchers use precision optimization allocation methods based on design experience and tolerance-cost models to allocate tolerances for the key components of machine tools. The implementation of these methods highly depends on the design experience and level of machine tool design engineers. Moreover, most tolerance-cost models are obtained based on design experience or through a large number of experiments under specified conditions, with low reliability and generality. Therefore, precision optimization design is easily affected by subjective factors, resulting in unreasonable tolerance optimization allocation results and thus being unable to correctly guide the innovative design of machine tools. In the present invention, the tolerance zone widths of various tolerance parameters are used as design variables, and the constraints are that the spatial motion error does not exceed its design standard and the tolerance zone widths of various tolerance parameters do not exceed twice their standard values. Then, with the maximization of the sum of the products of the tolerance zone widths of various tolerance parameters and their corresponding weight factors and the minimization of the overall machine spatial motion error as the design objectives, a multi-objective optimization allocation model for the tolerance parameters of the key components of CNC machine tools is established, thereby achieving precision optimization design for CNC machine tools. Description of the Drawings

[0112] Figure 1 is the implementation flowchart of the method of the present invention;

[0113] Figure 2 is the structural schematic diagram of a five-axis machine tool;

[0114] Figure 3 is the topological structure diagram of a five-axis machine tool;

[0115] Figure 4a ) is the comparison of tolerance parameter values before and after optimization - the comparison of tolerance parameter values corresponding to linear deviation;

[0116] Figure 4b ) is the comparison of tolerance parameter values before and after optimization - the comparison of tolerance parameter values corresponding to angular deviation

[0117] Figure 5 is the three-dimensional model diagram of an "S"-shaped test specimen;

[0118] Figure 6a ) Profile error of the "S"-shaped test specimen before and after tolerance optimization - at the detection line L1;

[0119] Figure 6b ) Profile error of the "S"-shaped test specimen before and after tolerance optimization - at the detection line L2;

[0120] Figure 6c ) Profile error of the "S"-shaped test specimen before and after tolerance optimization - at the detection line L3. Detailed Implementation Manner

[0121] Taking the five-axis overhead beam moving gantry CNC milling machine as an example, the above-mentioned method for optimizing the tolerance parameters of the key components of the five-axis CNC milling machine is verified.

[0122] Specifically, it includes the following steps:

[0123] Step 1: Taking the five-axis CNC machine tool as an example, establish the spatial error model of the machine tool;

[0124] Based on the theory of multi-body system kinematics, use the topological structure diagram and the low-order body array table to describe the structure of the machine tool and the connection relationship between each body, analyze the geometric error of the CNC machine tool, establish a generalized coordinate system, express the position relationship with the characteristic matrix between adjacent bodies, and represent the mutual relationship between multi-body systems with homogeneous transformation matrices;

[0125] Step 1.1 Establish the topological structure of the five-axis CNC machine tool;

[0126] The structure of this machine tool is as Figure 2 shown. It includes the bed, workbench, tool, workpiece, X-axis, Y-axis, Z-axis, A-axis, C-axis, and spindle;

[0127] The five-axis CNC machine tool is a multi-branch complex system. The topological structure of this machine tool is as Figure 3 shown. It branches into two branches from point B d . Except for body B d , each object has a neighboring lower-order body. When deriving kinematics and formulating calculation methods, a table needs to be formulated for the lower-order body of each object in the system, denoted by L n (j), called the low-order body array table, as shown in Table 1. Here, j represents the serial number of the object (j = 1, 2, 3... n), and n represents the number of typical bodies included in the machine tool;

[0128] Table 1: Low-order body array of CNC machine tool

[0129] <![CDATA[L 0 (j)]]> 1 2 3 4 5 6 <![CDATA[L 1 (j)]]> 0 1 1 3 4 5 <![CDATA[L 2 (j)]]> 0 0 0 1 3 4 <![CDATA[L 3 (j)]]> 0 0 0 0 1 3 <![CDATA[L 4 (j)]]> 0 0 0 0 0 1 <![CDATA[L 5 (j)]]> 0 0 0 0 0 0

[0130] The numbering rule of typical bodies is as follows:

[0131] First, arbitrarily select a typical body as B d , and then along the direction away from body B d , sequentially label the serial numbers of each object according to the natural growth sequence, from one branch of the system to the other branch until all objects are labeled;

[0132] Step 1.2 Analyze the geometric error of the five-axis CNC machine tool;

[0133] In a spatial coordinate system, any object has six degrees of freedom. During the movement process, six errors will inevitably occur, including three linear displacement errors and three angular displacement errors. These are all errors related to the position points. There are three non-perpendicularity errors between the X, Y, and Z guide rails, and a total of four perpendicularity errors between the C axis and the X and Y axes, and between the A axis and the Y and Z axes. Therefore, a total of 37 errors are shown in Table 2;

[0134] Table 2: Geometric error parameters of a five-axis CNC machine tool

[0135]

[0136] Step 1.3 Establish the characteristic matrix of the five-axis CNC machine tool;

[0137] A right-handed rectangular Cartesian three-dimensional coordinate system O1-X1Y1Z1 and O j fixedly connected to it are established on the bed B1 and all components B j -X j Y j Z j . The set of these coordinate systems is called the generalized coordinate system, and each body coordinate system is called a sub-coordinate system. The three orthogonal bases of each coordinate system are named the X, Y, and Z axes according to the right-hand rule; the corresponding coordinate axes of each sub-coordinate system are respectively parallel; the positive direction of the coordinate axis is the same as the positive direction of the corresponding motion axis;

[0138] According to the motion relationship between the components of the CNC machine tool, the transformation matrix between adjacent bodies can be established as shown in Table 3;

[0139] Table 3: Transformation matrix between adjacent bodies

[0140]

[0141]

[0142] Step 1.4 Establish the spatial error model of the machine tool;

[0143] The coordinates of the tool center point in the tool coordinate system are:

[0144] r t =[0,0,l,1] T (1)

[0145] l represents the tool length;

[0146] The subscript t represents the tool

[0147] In the ideal case, the position matrix expression of the tool center point P branched from the "machine tool - workpiece" to the inertial coordinate system:

[0148] P wI = [S12] p [S12] s r w (2)

[0149] Ideally, the position matrix expression of the tool center point P along the "machine - tool" branch in the inertial coordinate system is:

[0150] P t I = [S13] p [S13] s [S34] p [S34] s [S45] p [S45] s [S56] p [S56] s r t (3)

[0151] CNC instruction precision machining equation:

[0152] P w I = P t I (4)

[0153] Ideally, the position matrix expression of the CNC instruction in the workpiece coordinate system is:

[0154] r w I = ([S12] p [S12] s ) -1 [S13] p [S13] s [S34] p [S34] s [S45] p [S45] s [S56] p [S56] s r t (5)

[0155] In the actual situation, the position matrix expression of the tool center point P along the "machine - workpiece" branch in the inertial coordinate system is:

[0156] P w = [S12] p [S12] pe [S12] s [S12] se r w (6)

[0157] In the actual situation, the position matrix expression of the tool center point P in the inertial coordinate system according to the "machine tool - tool" branch is as follows:

[0158]

[0159] In the actual situation, the position matrix expression of the numerical control instruction in the workpiece coordinate system is as follows:

[0160]

[0161] Then the spatial error model of the machine tool is expressed as:

[0162] E = r w - r w I (9)

[0163] Step 2: Prediction modeling of machining accuracy based on tolerance;

[0164] Step 2.1 Determination of design variables

[0165] The gantry five-axis CNC milling machine studied in the present invention includes a total of 25 tolerance parameters, as shown in Table 4. Therefore, the present invention selects the tolerance zone width of the 25 tolerance parameters as the design variables, as shown in Equation (1):

[0166] W(T) = {W(t1), W(t2), W(t3),..., W(t 25 )} (10)

[0167] In the formula, W(t i ) represents the tolerance zone width of the i-th tolerance parameter.

[0168] Table 4 Definitions of various tolerance parameters

[0169]

[0170] Step 2.1 Determination of constraint conditions

[0171] The constraint condition is a kind of restrictive condition for the value range of the design variable. According to the design requirements of the gantry five-axis CNC milling machine, the constraint conditions of the proposed tolerance parameter optimization distribution model are mainly divided into two categories: spatial motion error constraint and tolerance zone width constraint of each tolerance parameter.

[0172] (1) Spatial motion error constraint condition

[0173] According to the design standard of the spatial motion error formulated for this machine tool, it is known that the maximum requirements for the spatial motion error components in the X, Y, and Z directions are all 0.02 mm. Therefore, in order to make the machine tool meet the design standard requirements, the components of the overall machine spatial motion error in the X, Y, and Z directions must meet the following requirements:

[0174]

[0175] (2) Tolerance zone width constraint condition

[0176] Due to the limitations of actual manufacturing conditions, the tolerance parameter values of the key components of the machine tool cannot reach the ideal state, that is, t i = 0. Therefore, we set the minimum value of the tolerance zone width as W(t i ) > 0. According to the precision inspection standard BS ISO 8636-2:2007 of the gantry five-axis CNC milling machine, the standard values of each tolerance parameter can be obtained. Since the optimization distribution method of the tolerance parameters proposed in the present invention needs to maximize the feasible range of each tolerance zone width on the premise of meeting the requirements of the overall machine spatial motion error, if the maximum value of each tolerance zone width is set to infinity, it does not conform to the engineering reality. Therefore, only the maximum value of the tolerance zone width can be appropriately enlarged. In the present invention, twice the standard value of each tolerance parameter is used as its maximum value, and the constraint conditions for each tolerance zone width can be obtained as follows:

[0177] 0 < W(t i ) < 2t ib (12)

[0178] In the formula, t ib represents the value of the i-th tolerance parameter given in the standard of the gantry five-axis CNC milling machine.

[0179] Finally, the constraint conditions for each tolerance zone width are determined as shown in Table 5.

[0180] Table 5 Constraint conditions for each tolerance zone width

[0181]

[0182]

[0183] Step 2.3 Determination of the objective function

[0184] In most engineering design problems, there are multiple objectives that need to be satisfied simultaneously, but they often conflict with each other. The optimization problem of multiple objective functions under given constraints is called a multi-objective optimization problem. The multi-objective optimization process takes into account the coupling relationship between each objective function, can reasonably balance the requirements of each objective function, and obtain a set of optimal solution sets at the same time, providing multiple reasonable design options for design engineers.

[0185] In existing research, although the tolerance-cost model plays an important role in the optimal allocation of tolerance parameters, considering its applicability and reliability from the perspective of practical application is not strong. Because the relationship function between the machining cost of each key component of the machine tool and the tolerance is obtained by domestic and foreign experts, scholars and enterprise design engineers sorting out and summarizing a large amount of collected production experience data, and there are many influencing factors for the machining cost of each component, such as machining process, the experience of the manufacturer, machining process, production technology and different production enterprises, etc. For the same component, there will be different machining cost experience data. Therefore, different tolerance-cost relationship curves will be generated. By studying various common tolerance-cost models, it can be found that although the tolerance-cost relationship curves are different, there is a certain inverse proportional relationship between them, that is, the larger the tolerance parameter value, the lower the cost.

[0186] To solve the problem of low reliability and applicability of the above tolerance-cost model, the present invention takes the maximization of the sum of the product of the tolerance zone width of each tolerance parameter and its corresponding weight factor and the minimization of the overall machine space motion error as the design objectives, and establishes an optimal allocation model of tolerance parameters.

[0187] The maximization of the sum of the product of the tolerance zone width of each tolerance parameter and its corresponding weight factor is defined as objective function 1, and the specific expression is as follows:

[0188]

[0189] In the formula, SN i represents the sensitivity coefficient of the i-th tolerance parameter; W(t i ) represents the tolerance zone width of the i-th tolerance parameter.

[0190] Since when actually solving the optimal allocation model of tolerance parameters, it is required to minimize the objective function. Therefore, formula (13) is transformed into the following form:

[0191]

[0192] The minimization of the overall machine space motion error is taken as objective function 2, and the specific expression is as follows:

[0193]

[0194] Therefore, by combining equations (14) and (15), the multi-objective optimization design function of the tolerance parameters of the key components of the gantry five-axis CNC milling machine can be obtained as follows:

[0195] min F(W(T))={f1(W(T)),f2(W(T))} (16)

[0196] By combining equations (11), (12) and (16), the standard expression form of the tolerance parameter optimization distribution model can be obtained as:

[0197]

[0198] So far, the tolerance parameter optimization distribution model of the gantry five-axis CNC milling machine has been established.

[0199] Step 3: Simulation analysis and experimental verification of the tolerance parameter optimization distribution model of the CNC machine tool;

[0200] Step 3.1 Simulation analysis

[0201] Based on the NSGA-II algorithm and using MATLAB R2016b, the multi-objective optimization distribution of the tolerance parameters of the key components of the gantry five-axis CNC milling machine is carried out, and finally the Pareto optimal solution set is obtained. For the specific example of the tolerance parameter optimization design of the key components of the gantry five-axis CNC milling machine, discussions are carried out with the machine tool design engineers, and the final tolerance optimization distribution plan is determined from the Pareto optimal solution set according to the actual production conditions of the enterprise. Table 6 shows the comparison between the tolerance parameter values before and after optimization. A positive difference between the two indicates that the tolerance parameter item has been relaxed, and on the contrary, a negative difference between the two indicates that the tolerance parameter item has been tightened.

[0202] To more intuitively reflect the advantages of the finally determined tolerance optimization distribution plan, the tolerance parameter values before and after optimization are compared in the form of a histogram. Since the tolerance parameters corresponding to the linear deviation are generally in millimeters, and the tolerance parameters corresponding to the angular deviation are generally expressed in the form of a ratio, and the representation methods of the two tolerance parameters are different, therefore, the two tolerance parameters are compared separately, as shown in Figure 4a ) and Figure 4b ) respectively. According to Figure 4a ) and Figure 4b ), it can be intuitively seen that most of the tolerance parameter items have been relaxed to varying degrees. Since there is an inverse relationship between tolerance and cost, it indirectly shows that the overall manufacturing cost has been reduced. Therefore, the comparison results initially verify the feasibility and effectiveness of the tolerance parameter optimization distribution method for the key components of the gantry five-axis CNC milling machine proposed in the present invention.

[0203] Comparison of Tolerance Parameter Values of Key Components of Gantry Five-Axis CNC Milling Machine before and after Optimization

[0204]

[0205] 3.2 Experimental Verification

[0206] To further verify the accuracy of the optimized distribution scheme of tolerance parameters for key components of the gantry five-axis CNC milling machine, first, the optimized distribution scheme of tolerance parameters considering the actual production conditions of the enterprise is provided to Beijing First Machine Tool Works. Then, the machine tool design engineers replace and process the relevant components of the gantry five-axis CNC milling machine according to the optimized tolerance parameters. Finally, the improved gantry five-axis CNC milling machine is used to mill the test specimen and detect the contour error of the specimen. By comparing the machining errors of the machine tool before and after optimization, the accuracy of the optimized tolerance distribution result is verified. The present invention selects the "S"-shaped test specimen as the research object. As Figure 5 shown, it is machined using the gantry five-axis CNC milling machine before and after the optimization of tolerance parameters, and then the contour error of the "S"-shaped test specimen is measured using a coordinate measuring machine. The above two "S"-shaped test specimens are the same in all other conditions except for the machining machine tool.

[0207] The test results are compared and analyzed. As Figure 6a)-6c) shown, the contour errors of the "S"-shaped test specimens on L1, L2, and L3 before and after the optimization of tolerance parameters are respectively shown. To more intuitively verify the advantages of the finally determined optimized tolerance distribution scheme, the average contour error values of the "S"-shaped test specimens obtained before and after the optimization of tolerance parameters are compared, as shown in Table 7.

[0208] Table 7 Comparison of Average Contour Errors of "S"-Shaped Test Specimens before and after Optimization of Tolerance Parameters

[0209]

[0210] As can be seen from Table 7, compared with before the optimization of tolerance parameters, the average contour errors of the optimized "S"-shaped test specimens are reduced by 0.018 mm, 0.013 mm, and 0.019 mm at L1, L2, and L3 respectively. That is, the machining accuracies of the gantry five-axis CNC milling machine after the optimization of tolerance parameters are improved by 31.6%, 27.1%, and 28.8% respectively. Therefore, the multi-objective optimized distribution model of tolerance parameters for key components of the gantry five-axis CNC milling machine proposed in the present invention can not only relax most tolerance parameters to varying degrees, but also enable the machine tool to obtain higher machining accuracy after the optimization of tolerance parameters, further confirming that the actual application effect of this model is significantly effective and solving the problem that the existing accuracy optimization design method cannot correctly guide the innovative design of machine tools.

Claims

1. A tolerance optimization design method for key components of a multi-axis CNC machine tool, characterized in that, The method specifically includes the following steps: Step 1. Based on the kinematics theory of multi-body systems, establish the spatial error model of the machine tool; First, based on the kinematics theory of multi-body systems, a simplified analysis of the structure of the CNC machine tool is carried out. The machine tool is abstracted into a multi-body system composed of a tool branch and a workpiece branch. The coordinate systems of key moving components of the machine tool are established. Then, based on the principle of homogeneous coordinate transformation, the static and motion transformation matrices between adjacent bodies of the machine tool are established. Finally, the spatial motion error model of the CNC machine tool is deduced; Step 2. Establish the optimization distribution model of tolerance parameters; The tolerance zone widths of various tolerance parameters are used as design variables; the tolerance zone widths of various tolerance parameters and the spatial motion error of the machine tool not exceeding the design requirements are used as constraint conditions; aiming at the problems of low reliability and applicability of the tolerance-cost model, according to the sensitivity analysis results of tolerance parameters, weights are assigned to various tolerance parameters. The sum of the products of the tolerance zone widths of various tolerance parameters and their corresponding weight factors is maximized and the spatial motion error of the whole machine is minimized as the design objectives. Finally, a multi-objective optimization distribution model of tolerance parameters for key components of the CNC machine tool is established; Step 3. Simulation analysis and experimental verification of the optimization distribution model of tolerance parameters.

2. The tolerance optimization design method for key components of a multi-axis CNC machine tool according to claim 1, characterized in that, The specific content of Step 1 is as follows: Step 1.1 Establish the topological structure of the CNC machine tool; The numerically controlled machine tool is a multi-branched complex system, which divides into two branches at B d Except for B d Each object outside has an adjacent lower-order object. A table is formulated for the lower-order objects of each object in the system, denoted by L n (j), called the lower-order object array table, as shown in Table 1. Here, j represents the serial number of the object, j = 1, 2, 3... n, and n represents the number of typical objects included in the machine tool; Table 1: Low-order body array of the CNC machine tool The numbering rules of typical bodies are as follows: First, arbitrarily select a typical object as B d , and then, along the direction away from B d object, sequentially label the numbers of each object according to a naturally increasing sequence, from one branch of the system to another, until all objects are labeled; Step 1.2 Geometric error analysis of the CNC machine tool Any object in the space coordinate system has 6 degrees of freedom, and 6 errors will inevitably occur during the movement process, including 3 linear displacement errors and 3 angular displacement errors. These are all errors related to the position points. There are 3 non-perpendicularity errors between the X, Y, and Z guide rails, and there are 4 perpendicularity errors between the C axis and the X and Y axes, and between the A axis and the Y and Z axes. Therefore, a total of 37 errors are shown in Table 2; Table 2: Geometric error parameters of the CNC machine tool Step 1.3 Establish the characteristic matrix of the CNC machine tool; On the bed body B1 and all components B j a right-handed Cartesian three-dimensional coordinate system O1-X1Y1Z1 and O j -X j Y j Z j are established and fixedly connected thereto. The set of these coordinate systems is called the generalized coordinate system, and each body coordinate system is called a sub-coordinate system. The three orthogonal bases of each coordinate system are respectively named the X, Y, and Z axes according to the right-hand rule; the corresponding coordinate axes of each sub-coordinate system are respectively parallel; the positive direction of the coordinate axis is the same as the positive direction of the corresponding motion axis. According to the motion relationship between components of the CNC machine tool, the transformation matrices between adjacent bodies are established as shown in Table 3; Table 3: Transformation matrix between adjacent bodies Where: [Sij] p represents the relative position transformation matrix of body B j relative to body B i ; [Sij] pe represents the relative position error transformation matrix of body B j relative to body B i ; [Sij] s Denote B j The relative motion transformation matrix of the body with respect to B i for the body; [Sij] se Denote B j The relative motion error transformation matrix of the body with respect to B i of the body; x represents the translation distance of the X axis; y represents the translation distance of the Y axis; z represents the translation distance of the Z axis; a represents the rotation angle of the A axis; c represents the rotation angle of the C axis; Step 1.4 Establish the spatial error model of the machine tool Establishment of the motion relationship model between adjacent bodies under ideal conditions; Let point P be any point on body B j and the position matrix expression of P in the body coordinate system O i -X i Y i Z i is as follows; i ​ P ji = [Sij] p [Sij] s r j (1) Where: P ji is the position matrix expression of point P in the coordinate system O i -X i Y i Z i ; r j is the position matrix expression of point P in the coordinate system O j -X j Y j Z j ; [Sij] p Indicates B j The body with respect to B i The relative position transformation matrix of the body; [Sij] s Denote B j The body relative to B i The relative motion transformation matrix of the body; Establishment of the motion relationship model between adjacent bodies under error conditions; Let point P be any point on body B j In the body coordinate system O i -X i Y i Z i The position matrix expression of P in it is; i ​ P ji = [Sij] p [Sij] pe [Sij] s [Sij] se r j (2) Where: P ji is the position matrix expression of point P in the coordinate system O i -X i Y i Z i ; r j is the position matrix expression of point P in the coordinate system O j -X j Y j Z j ; [Sij] p Indicates B j The body relative to B i Relative position transformation matrix of the body; [Sij] pe Denote B j The body with respect to B i The relative position error transformation matrix of the body; [Sij] s Denote B j The body relative to B i The relative motion transformation matrix of the body; [Sij] se Indicates B j The relative motion error transformation matrix of the body with respect to B i of the body; The coordinates of the tool center point in the tool coordinate system are: r t ＝[0,0,l,1] T (3) In the formula: l represents the tool length; The subscript t represents the tool; The position matrix expression of the tool center point P in the inertial coordinate system according to the "CNC machine tool - workpiece" branch under ideal conditions: The position matrix expression of the tool center point P in the inertial coordinate system according to the "CNC machine tool - tool" branch under ideal conditions: Precision machining equation of CNC instructions: The position matrix expression of CNC instructions in the workpiece coordinate system under ideal conditions: The position matrix expression of the tool center point P in the inertial coordinate system according to the "machine tool - workpiece" branch under actual conditions: The position matrix expression of the tool center point P in the inertial coordinate system according to the "machine tool - tool" branch under actual conditions: In the actual situation, the position matrix expression of the numerical control instruction in the workpiece coordinate system: Then the spatial error model of the numerically controlled machine tool is expressed as:

3. A method for optimizing the tolerances of key components of a multi-axis CNC machine tool according to claim 1, characterized in that, The specific content of the second step is as follows: Step 2.1 Determination of design variables Select the tolerance zone width of each tolerance parameter of the key components of the numerically controlled machine tool as the design variable, as shown in Equation (12): W(T) = {W(t1), W(t2), W(t3),..., W(t i )} (12) where W(t i ) represents the tolerance zone width of the i-th tolerance parameter; Step 2.2 Determination of constraint conditions According to the design requirements of the numerically controlled machine tool, the constraint conditions of the proposed tolerance parameter optimization distribution model are divided into two categories: spatial motion error constraints and tolerance zone width constraints of each tolerance parameter; (1) Spatial motion error constraint conditions According to the design standard of the spatial motion error formulated for this machine tool, the maximum requirements for the spatial motion error components in the X, Y, and Z directions are E bx , E by , E bz respectively; therefore, in order to make the machine tool meet the design standard requirements, the components of the overall machine spatial motion error in the X, Y, and Z directions must meet the following requirements: (2) Tolerance zone width constraint conditions Set the minimum value of the tolerance zone width to W(t i ) > 0; According to the accuracy inspection standard of CNC machine tools, the standard values given for each tolerance parameter can be obtained. Appropriately expand the maximum value of the tolerance zone width, and take twice the standard value of each tolerance parameter as its maximum value. The constraint conditions for each tolerance zone width can be obtained as follows: 0 < W(t i ) < 2t ib In equation (14), t ib represents the value of the i-th tolerance parameter given as standard for the gantry-type five-axis CNC milling machine; Step 2.3 Determination of the objective function First, assign weights to each tolerance parameter. Taking the sum of the products of the tolerance zone width of each tolerance parameter and its corresponding weight factor to be maximized and the minimum of the overall spatial motion error as the design objectives, establish a tolerance parameter optimization distribution model; Define the sum of the products of the tolerance zone width of each tolerance parameter and its corresponding weight factor to be maximized as objective function 1, and the specific expression is as follows: where $S_N$ i represents the weighting factor of the $i$-th tolerance parameter; $W(t$ i ) represents the tolerance zone width of the $i$-th tolerance parameter; Transform Equation (15) into the following form: Take the minimum of the overall spatial motion error as objective function 2, and the specific expression is as follows: Therefore, combining Equations (16) and (17), the multi-objective optimization design function of the tolerance parameters of the key components of the gantry five-axis numerically controlled milling machine can be obtained as follows: min F(W(T))={f1(W(T)),f2(W(T))} (18) Combining Equations (13), (14), and (18), the standard expression form of the tolerance parameter optimization distribution model can be obtained as: So far, the tolerance parameter optimization distribution model of the numerically controlled machine tool has been established.

4. A method for optimizing the tolerances of key components of a multi-axis CNC machine tool as described in claim 1, characterized in that, The specific content of the third step is as follows: Step 3.1 Simulation analysis Based on the NSGA-II algorithm and using MATLAB R2016b, perform multi-objective optimization distribution on the tolerance parameters of the key components of the numerically controlled machine tool, and finally obtain the Pareto optimal solution set. Determine the final tolerance optimization distribution plan from the Pareto optimal solution set according to the actual production conditions of the enterprise; Use the form of a histogram to compare the tolerance parameter values before and after optimization; it can be intuitively seen from the histogram that most tolerance parameter items are relaxed to varying degrees. Since there is a certain inverse relationship between tolerance and cost, it indirectly shows that the manufacturing cost of the whole machine is reduced; Step 3.2 Experimental verification In order to further verify the accuracy of the tolerance parameter optimization distribution plan of the key components of the numerically controlled machine tool, first provide the tolerance parameter optimization distribution plan considering the actual production conditions of the enterprise to the machine tool factory. Then, the machine tool design engineer replaces and processes the relevant components of the numerically controlled machine tool according to the optimized tolerance parameters. Finally, use the improved numerically controlled machine tool to mill the test specimen and detect the profile error of the specimen. Verify the accuracy of the tolerance optimization distribution result by comparing the machining errors of the machine tool before and after optimization.

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