CN110837246A - Method for analyzing geometric error sensitivity of double rotating shafts of five-axis numerical control machine tool - Google Patents

Method for analyzing geometric error sensitivity of double rotating shafts of five-axis numerical control machine tool Download PDF

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CN110837246A
CN110837246A CN201911163064.2A CN201911163064A CN110837246A CN 110837246 A CN110837246 A CN 110837246A CN 201911163064 A CN201911163064 A CN 201911163064A CN 110837246 A CN110837246 A CN 110837246A
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蒋晓耕
崔志威
王量
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Tianjin Polytechnic University
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Abstract

本发明提出了一种五轴数控机床双旋转轴几何误差敏感度分析方法，包括以下步骤：首先基于多体系统理论和齐次变换矩阵构建双旋转轴几何误差的数学模型，然后利用蒙特卡罗方法进行误差参数采样，对数控机床双旋转轴的几何误差进行敏感度分析；最后基于双旋转轴特定转角的位置对误差参数进行敏感度分析，通过设置五组特定转角的位置，计算双旋转轴在此位置的各项几何误差的一阶敏感度和全局敏感度；通过Sobol敏感度分析方法可以识别双旋转轴20项几何误差中的关键几何误差，完成五轴数控机床双旋转轴几何误差的辨识。本发明的检测步骤简洁，测量方便，辨识精度高。

Figure 201911163064

The present invention proposes a method for analyzing the sensitivity of the geometric error of the double rotating axes of a five-axis numerically controlled machine tool. The method conducts error parameter sampling to analyze the sensitivity of the geometric error of the dual rotary axes of the CNC machine tool; finally, based on the position of the specific rotation angle of the dual rotation axis, the sensitivity analysis of the error parameters is carried out. The first-order sensitivity and global sensitivity of the geometric errors at this position; the key geometric errors among the 20 geometric errors of the dual-rotation axis can be identified through the Sobol sensitivity analysis method, and the geometric errors of the dual-rotation axis of the five-axis CNC machine tool can be identified. Identify. The detection steps of the invention are simple, the measurement is convenient, and the identification precision is high.

Figure 201911163064

Description

五轴数控机床双旋转轴几何误差敏感度分析方法Analysis Method of Geometric Error Sensitivity of Double Rotary Axis of Five-axis CNC Machine Tool

技术领域technical field

本发明属于数控机床误差检测技术领域，特别涉及一种基于敏感度分析的五轴数控机床双旋转轴几何误差辨识方法。The invention belongs to the technical field of error detection of numerically controlled machine tools, and particularly relates to a method for identifying geometric errors of five-axis numerically controlled machine tools with double rotation axes based on sensitivity analysis.

技术背景technical background

五轴数控机床相对于三轴数控机床具备加工复杂零件的能力，其原因是五轴数控机床较三轴数控机床增加了两个旋转轴。五轴数控机床进行加工时，生成的刀具路径比传统的三轴机床加工具有更大的灵活性。但是旋转轴在加工时引入了自身误差元素，其中几何误差对总误差的影响极大，在很大程度上影响了机床的加工精度。Compared with three-axis CNC machine tools, five-axis CNC machine tools have the ability to process complex parts. The reason is that five-axis CNC machine tools have two more rotating axes than three-axis CNC machine tools. When machining with a five-axis CNC machine, the resulting toolpaths provide greater flexibility than conventional three-axis machining. However, the rotary axis introduces its own error elements during machining, among which the geometric error has a great influence on the total error, which greatly affects the machining accuracy of the machine tool.

几何误差通常被认为是由于它们的特性而产生的准静态误差，占机床总误差的一半以上，所以研究旋转轴的几何误差对于提高五轴数控机床精度至关重要。旋转轴各项几何误差相互耦合，一般方法对双旋转轴几何误差快速辨识较难，因此提出一种可以快速、简易地辨识五轴数控机床双旋转轴几何误差的方法对提升机床加工精度非常重要。Geometric errors are generally considered to be quasi-static errors due to their characteristics, accounting for more than half of the total error of machine tools, so studying the geometric errors of rotary axes is crucial to improving the accuracy of five-axis CNC machine tools. The geometric errors of the rotary axes are coupled with each other, and it is difficult to quickly identify the geometric errors of the dual rotary axes by the general method. Therefore, it is very important to propose a method that can quickly and easily identify the geometric errors of the dual rotary axes of the five-axis CNC machine tool to improve the machining accuracy of the machine tool. .

发明内容SUMMARY OF THE INVENTION

为解决上述问题，本发明提出一种基于敏感度分析的五轴数控机床双旋转轴几何误差辨识方法，利用多体系统理论和齐次变换矩阵对机床旋转轴A轴、C轴的各项几何误差进行Sobol敏感度分析。该发明可以简便并准确的对几何误差进行辨识，得出各误差元素对总误差的影响程度，进而通过控制关键几何误差提高加工质量。具体步骤如下：In order to solve the above problems, the present invention proposes a method for identifying the geometric errors of the double rotating axes of the five-axis CNC machine tool based on sensitivity analysis. Errors were subjected to Sobol sensitivity analysis. The invention can simply and accurately identify the geometric errors, obtain the influence degree of each error element on the total error, and then improve the processing quality by controlling the key geometric errors. Specific steps are as follows:

步骤1、利用多体系统理论和齐次变换矩阵进行几何误差建模，具体步骤如下：Step 1. Use multi-body system theory and homogeneous transformation matrix for geometric error modeling. The specific steps are as follows:

步骤1.1、基于数控机床部件的拓扑结构关系对数控机床设置参考坐标系和局部坐标系，将Y轴局部坐标系设置为与参考坐标系重合。Step 1.1. Set a reference coordinate system and a local coordinate system for the CNC machine tool based on the topological structure relationship of the CNC machine tool components, and set the Y-axis local coordinate system to coincide with the reference coordinate system.

步骤1.2、对数控机床双旋转轴的几何误差进行辨识，包括12项与位置有关的几何误差(Position-Dependent Geometric Errors，PDGEs)和8项与位置无关的几何误差(Position- Independent Geometric Errors，PIGEs)。Step 1.2. Identify the geometric errors of the dual rotary axes of the CNC machine tool, including 12 Position-Dependent Geometric Errors (PDGEs) and 8 Position-Independent Geometric Errors (PIGEs) ).

步骤1.3、利用齐次变换矩阵构建误差变换矩阵。Step 1.3, using the homogeneous transformation matrix to construct the error transformation matrix.

A轴实际回转中心线在Y轴方向的位置误差EYOA是A轴轴线在Y方向上的偏差量，DYOA是误差变换矩阵：The position error of the actual rotation center line of the A-axis in the Y-axis direction E YOA is the deviation of the A-axis axis in the Y direction, and D YOA is the error transformation matrix:

Figure RE-GSB0000185388090000011
Figure RE-GSB0000185388090000011

A轴实际回转中心线在Z轴方向的位置误差EZOA是A轴轴线在Z方向上的偏差量，DZOA是误差变换矩阵：The position error of the actual rotation center line of the A-axis in the Z-axis direction, E ZOA is the deviation of the A-axis axis in the Z direction, and D ZOA is the error transformation matrix:

Figure RE-GSB0000185388090000021
Figure RE-GSB0000185388090000021

A轴实际回转中心线在XOZ平面的投影与X轴的夹角EBOA是A轴轴线绕Y轴的平行度误差，TB0A是误差变换矩阵：The angle between the projection of the actual rotation center line of the A-axis on the XOZ plane and the X-axis E BOA is the parallelism error of the A-axis axis around the Y-axis, and T B0A is the error transformation matrix:

Figure RE-GSB0000185388090000022
Figure RE-GSB0000185388090000022

A轴实际回转中心线在XOY平面的投影与X轴的夹角ECOA是A轴轴线绕Z轴的平行度误差，TC0A是误差变换矩阵：The angle between the projection of the actual rotation center line of the A-axis on the XOY plane and the X-axis, E COA is the parallelism error of the A-axis axis around the Z-axis, and T C0A is the error transformation matrix:

Figure RE-GSB0000185388090000023
Figure RE-GSB0000185388090000023

其中，DYOA、DZOA、TB0A、TC0A是A轴的4项PIGEs误差变换矩阵。Among them, D YOA , D ZOA , T B0A , and T C0A are the 4-term PIGEs error transformation matrix of the A axis.

C轴实际回转中心线在X轴方向的位置误差EXOC是C轴轴线在X方向上的偏差量，DXOC是误差变换矩阵：The position error of the actual rotation center line of the C-axis in the X-axis direction E XOC is the deviation of the C-axis axis in the X direction, and D XOC is the error transformation matrix:

Figure RE-GSB0000185388090000024
Figure RE-GSB0000185388090000024

C轴实际回转中心线在Y轴方向的位置误差为EYOC是C轴轴线在Y方向上的偏差量，DYOC是误差变换矩阵：The position error of the actual rotation center line of the C-axis in the Y-axis direction is E YOC is the deviation of the C-axis axis in the Y direction, and D YOC is the error transformation matrix:

Figure RE-GSB0000185388090000025
Figure RE-GSB0000185388090000025

C轴实际回转中心线在YOZ平面的投影与Z轴的夹角EAOC是C轴轴线绕X轴的平行度误差，TAOC是误差变换矩阵：The angle between the projection of the actual rotation center line of the C-axis on the YOZ plane and the Z-axis, E AOC is the parallelism error of the C-axis axis around the X-axis, and T AOC is the error transformation matrix:

步骤C轴实际回转中心线在XOZ平面的投影与Z轴的夹角EBOC是C轴轴线绕Y轴的平行度误差，TBOC是误差变换矩阵：Step The angle between the projection of the actual rotation center line of the C-axis on the XOZ plane and the Z-axis E BOC is the parallelism error of the C-axis axis around the Y-axis, and T BOC is the error transformation matrix:

Figure RE-GSB0000185388090000027
Figure RE-GSB0000185388090000027

其中，DXOC、DYOC、TAOC、TBOC是C轴的4项PIGEs误差变换矩阵。Among them, D XOC , D YOC , T AOC , and T BOC are the 4-term PIGEs error transformation matrix of the C-axis.

Figure RE-GSB0000185388090000028
对应的PDGE误差变换矩阵为：
Figure RE-GSB0000185388090000028
The corresponding PDGE error transformation matrix is:

Figure RE-GSB0000185388090000031
Figure RE-GSB0000185388090000031

Figure RE-GSB0000185388090000032
对应的PDGE误差变换矩阵为：
Figure RE-GSB0000185388090000032
The corresponding PDGE error transformation matrix is:

Figure RE-GSB0000185388090000033
Figure RE-GSB0000185388090000033

其中，

Figure RE-GSB0000185388090000034
是基于小角度近似原理的双旋转轴PDGEs误差变换矩阵。in,
Figure RE-GSB0000185388090000034
is the error transformation matrix of dual-rotation axis PDGEs based on the principle of small-angle approximation.

综合上述各误差矩阵，工件相对于刀具的几何误差为：Combining the above error matrices, the geometric error of the workpiece relative to the tool is:

Figure RE-GSB0000185388090000035
Figure RE-GSB0000185388090000035

步骤2、五轴数控机床蒙特卡罗采样的Sobol敏感度分析，具体步骤如下：Step 2. Sobol sensitivity analysis of Monte Carlo sampling of five-axis CNC machine tools. The specific steps are as follows:

步骤2.1、采用蒙特卡罗方法对误差参数进行采样，生成Sobol序列以确定各项误差对机床空间误差的影响。Step 2.1. Use the Monte Carlo method to sample the error parameters, and generate a Sobol sequence to determine the influence of various errors on the machine tool space error.

步骤2.2、基于方差的Sobol敏感度分析需要确定误差数k，两个旋转轴共有20个误差项，所以k设置为20。Step 2.2. The variance-based Sobol sensitivity analysis needs to determine the number of errors k. There are 20 error terms in the two rotation axes, so k is set to 20.

步骤2.3、生成Sobol序列A、B两个参数样本矩阵，记为：Step 2.3. Generate two parameter sample matrices of Sobol sequence A and B, denoted as:

Figure RE-GSB0000185388090000036
Figure RE-GSB0000185388090000036

其中xij表示第j个(i＝1，2，3...k)误差元素的第i个(j＝1，2，3...n)样本。where x ij represents the ith (j=1, 2, 3...n) sample of the jth (i=1, 2, 3...k) error element.

步骤2.4、将矩阵B的第i列换到矩阵A的第i列，矩阵A的其余列不变，得到矩阵记为ABi，如下所示：Step 2.4. Change the i-th column of matrix B to the i-th column of matrix A, and the remaining columns of matrix A remain unchanged, and the obtained matrix is denoted as AB i , as shown below:

Figure RE-GSB0000185388090000037
Figure RE-GSB0000185388090000037

通过上述方法构造出矩阵A、B、ABi，共有(k+2)*N组旋转轴误差参数，因此得到(k+2)*N组

Figure RE-GSB0000185388090000038
值。对于每组
Figure RE-GSB0000185388090000039
存在唯一的矩阵A、B、ABi相对应，记作f(A)， f(B)，f(ABi)。The matrices A, B, AB i are constructed by the above method, and there are (k+2)*N groups of rotation axis error parameters, so the (k+2)*N groups are obtained
Figure RE-GSB0000185388090000038
value. for each group
Figure RE-GSB0000185388090000039
There are unique matrices A, B, AB i corresponding to each other, denoted as f(A), f(B), f(AB i ).

步骤2.5、通过系统响应的方差计算公式可以计算各项误差元素的一阶敏感度和全局敏感度，计算公式如下：Step 2.5. The first-order sensitivity and global sensitivity of each error element can be calculated by the variance calculation formula of the system response. The calculation formula is as follows:

Var(Y)＝Var(YA+YB) (3)Var(Y)=Var(Y A +Y B ) (3)

Figure RE-GSB0000185388090000046
Figure RE-GSB0000185388090000046

Figure RE-GSB0000185388090000047
Figure RE-GSB0000185388090000047

Y＝(ya1 ya2 ... yan yb1 yb2 ... ybn)T (6)Y=(y a1 y a2 ... y an y b1 y b2 ... y bn ) T ( 6 )

Figure RE-GSB0000185388090000041
Figure RE-GSB0000185388090000041

其中yj1为输入矩阵对应的输出值，Var(Y)为Y的标准差。where y j1 is the output value corresponding to the input matrix, and Var(Y) is the standard deviation of Y.

步骤2.6、误差元素的一阶敏感度和全局敏感度计算公式：Step 2.6. The first-order sensitivity and global sensitivity calculation formula of the error element:

Figure RE-GSB0000185388090000042
Figure RE-GSB0000185388090000042

Figure RE-GSB0000185388090000043
Figure RE-GSB0000185388090000043

Figure RE-GSB0000185388090000044
Figure RE-GSB0000185388090000044

Si表示误差元素xi的一阶敏感度，STi表示误差元素xi的全局敏感度。误差元素xi的一阶敏感度表示该误差对机床空间误差的直接影响，全局敏感度表示该误差对机床空间误差的耦合影响。S i represents the first-order sensitivity of the error element xi , and S Ti represents the global sensitivity of the error element xi . The first-order sensitivity of the error element xi represents the direct influence of the error on the spatial error of the machine tool, and the global sensitivity represents the coupled effect of the error on the spatial error of the machine tool.

目标五轴数控机床的PDGEs和PIGEs的采样比是3∶7，其中位置误差和角度误差采样范围分别是(0，1)μm和(0，1)″。The sampling ratio of PDGEs and PIGEs of the target five-axis CNC machine tool is 3:7, and the sampling ranges of position error and angle error are (0, 1) μm and (0, 1)”, respectively.

步骤3、基于旋转轴特定转角位置对误差参数进行敏感度分析。由于旋转轴所转到的角度不同，旋转轴各项误差对机床精度的影响也随之改变。针对五轴数控机床旋转轴在不同转角位置的敏感度问题，设置五组特定转角位置和一组在其加工范围内转角位置随机变化对五轴数控机床双旋转轴进行敏感度分析。Step 3. Perform a sensitivity analysis on the error parameters based on the specific rotational angle position of the rotating shaft. Due to the different rotation angles of the rotary axis, the influence of various errors of the rotary axis on the accuracy of the machine tool also changes. Aiming at the sensitivity of the rotary axis of the five-axis CNC machine tool at different corner positions, five groups of specific corner positions and a group of random changes in the position of the rotation angle within the processing range are set to analyze the sensitivity of the dual rotation axis of the five-axis CNC machine tool.

步骤3.1、设置A轴转角为0°、C轴转角为0°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.1. Set the rotation angle of the A-axis to 0° and the rotation angle of the C-axis to 0°. Perform a sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.2、设置A轴转角为45°、C轴转角为45°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.2. Set the rotation angle of the A axis to 45° and the rotation angle of the C axis to 45°. Perform a sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.3、设置A轴转角为90°、C轴转角为90°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.3. Set the rotation angle of A-axis to 90° and the rotation angle of C-axis to 90°. Perform sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.4、设置A轴转角为0°、C轴转角为90°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.4. Set the rotation angle of the A axis to 0° and the rotation angle of the C axis to 90°. Perform a sensitivity analysis on the five-axis CNC machine tool at this position, and calculate its first-order sensitivity and global sensitivity.

步骤3.5、设置A轴转角为90°、C轴转角为0°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.5. Set the rotation angle of the A axis to 90° and the rotation angle of the C axis to 0°. Perform a sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.6、设置A轴转角在0°-90°内随机变化，C轴转角在0°-360°内随机变化，对五轴数控机床进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.6, set the A-axis rotation angle to vary randomly within 0°-90°, and the C-axis rotation angle to vary randomly within 0°-360°, perform sensitivity analysis on the five-axis CNC machine tool, and calculate its first-order sensitivity and global sensitivity .

步骤3的Sobol敏感度分析结果显示，影响加工精度的误差主要分布在PIGEs部分。PIGEs对五轴数控机床空间误差的影响程度较大，而PDGEs对五轴数控机床空间误差影响较小，通过Sobol敏感度分析方法可以识别20项几何误差中的关键几何误差。几何误差中影响机床精度的关键几何误差来自PIGEs中的EY0A、EY0C、EA0C，影响加工精度的一阶敏感度占 20项几何误差的67.42％，考虑相互作用情况下，影响加工精度的全局敏感度占20项几何误差的26.97％。The Sobol sensitivity analysis results in step 3 show that the errors affecting the machining accuracy are mainly distributed in the PIGEs part. PIGEs have a greater influence on the spatial error of five-axis CNC machine tools, while PDGEs have less impact on the spatial errors of five-axis CNC machine tools. The key geometric errors among the 20 geometric errors can be identified by the Sobol sensitivity analysis method. Among the geometric errors, the key geometric errors affecting machine tool accuracy come from E Y0A , E Y0C , and E A0C in PIGEs, and the first-order sensitivity affecting machining accuracy accounts for 67.42% of the 20 geometric errors. The global sensitivity accounts for 26.97% of the 20-item geometric errors.

通过上述Sobol敏感度分析结果，对五轴数控机床的几何误差进行辨识，控制关键误差可以提高数控机床的精度。Through the above Sobol sensitivity analysis results, the geometric errors of five-axis CNC machine tools are identified, and the control of key errors can improve the accuracy of CNC machine tools.

以上完成了五轴数控机床双旋转轴几何误差敏感度分析，包括8项与位置无关的几何误差和12项与位置有关的几何误差的敏感度分析。The above completes the sensitivity analysis of the geometric errors of the five-axis CNC machine tool and the dual rotating axes, including the sensitivity analysis of 8 geometric errors independent of position and 12 geometric errors related to position.

本发明有效的解决了五轴数控机床中双旋转轴几何误差的辨识与检测，提出了有效的五轴数控机床双旋转轴的几何误差敏感度分析。The invention effectively solves the identification and detection of the geometric error of the double rotating shafts in the five-axis numerical control machine tool, and proposes an effective sensitivity analysis of the geometric error of the double rotating shafts of the five-axis numerical control machine tool.

附图说明Description of drawings

图1为本发明方法实施例中五轴数控机床的参考坐标系图。FIG. 1 is a reference coordinate system diagram of a five-axis CNC machine tool in a method embodiment of the present invention.

图2为本发明方法实施例中A轴的PIGEs示意图。Figure 2 is a schematic diagram of the PIGEs of the A-axis in the method embodiment of the present invention.

图3为本发明方法实施例中C轴的PIGEs示意图。FIG. 3 is a schematic diagram of the PIGEs of the C-axis in the method embodiment of the present invention.

图4是本发明方法实施例中一阶敏感度结果示意图。其中，序号1表示A轴0°时， C轴0°时的一阶敏感度。2表示A轴45°时，C轴45°时的一阶敏感度。3表示A轴0°时，C轴90°时的一阶敏感度。4表示A轴90°时，C轴0°时的一阶敏感度。5表示A轴 90°时，C轴90°时的一阶敏感度。6表示A轴0°-90°时，C轴0°-360°时的一阶敏感度。误差项的表示：1-EYOA、2-EZOA、3-EXOC、4-EYOC、5-EBOA、6-ECOA、7-EAOC、8-EBOC、9- δx(a)、10-δy(a)、11-δz(a)、12-εx(a)、13-εy(a)、14-εz(a)、15-δx(c)、16-δy(c)、17-δz(c)、 18-δx(c)、19-εy(c)、20-εz(c)。FIG. 4 is a schematic diagram of a first-order sensitivity result in an embodiment of the method of the present invention. Among them, No. 1 indicates the first-order sensitivity when the A-axis is 0° and the C-axis is 0°. 2 represents the first-order sensitivity when the A-axis is 45° and the C-axis is 45°. 3 represents the first-order sensitivity when the A-axis is 0° and the C-axis is 90°. 4 represents the first-order sensitivity when the A-axis is 90° and the C-axis is 0°. 5 represents the first-order sensitivity when the A-axis is 90° and the C-axis is 90°. 6 represents the first-order sensitivity of the A-axis at 0°-90° and the C-axis at 0°-360°. Representation of error terms: 1-E YOA , 2-E ZOA , 3-E XOC , 4-E YOC , 5-E BOA , 6-E COA , 7-E AOC , 8-E BOC , 9- δ x ( a), 10-δ y (a), 11-δ z (a), 12-ε x (a), 13-ε y (a), 14-ε z (a), 15-δ x (c) , 16-δ y (c), 17-δ z (c), 18-δ x (c), 19-ε y (c), 20-ε z (c).

图5是本发明方法实施例中全局敏感度结果示意图。其中，序号1表示A轴0°时，C轴0°时的一阶敏感度。2表示A轴45°时，C轴45°时的一阶敏感度。3表示A轴0°时，C轴90°时的一阶敏感度。4表示A轴90°时，C轴0°时的一阶敏感度。5表示A轴 90°时，C轴90°时的一阶敏感度。6表示A轴0°-90°时，C轴0°-360°时的一阶敏感度。误差项的表示：1-EYOA、2-EZOA、3-EXOC、4-EYOC、5-EBOA、6-ECOA、7-EAOC、8-EBOC、9- δx(a)、10-δy(a)、11-δz(a)、12-εx(a)、13-εy(a)、14-εz(a)、15-δx(c)、16-δy(c)、17-δz(c)、 18-εx(c)、19-εy(c)、20-εz(c)。FIG. 5 is a schematic diagram of a global sensitivity result in an embodiment of the method of the present invention. Among them, No. 1 represents the first-order sensitivity when the A-axis is 0° and the C-axis is 0°. 2 represents the first-order sensitivity when the A-axis is 45° and the C-axis is 45°. 3 represents the first-order sensitivity when the A-axis is 0° and the C-axis is 90°. 4 represents the first-order sensitivity when the A-axis is 90° and the C-axis is 0°. 5 represents the first-order sensitivity when the A-axis is 90° and the C-axis is 90°. 6 represents the first-order sensitivity of the A-axis at 0°-90° and the C-axis at 0°-360°. Representation of error terms: 1-E YOA , 2-E ZOA , 3-E XOC , 4-E YOC , 5-E BOA , 6-E COA , 7-E AOC , 8-E BOC , 9- δ x ( a), 10-δ y (a), 11-δ z (a), 12-ε x (a), 13-ε y (a), 14-ε z (a), 15-δ x (c) , 16-δ y (c), 17-δ z (c), 18-ε x (c), 19-ε y (c), 20-ε z (c).

具体实施方式Detailed ways

为解决上述问题，本发明提出一种基于敏感度分析的五轴数控机床双旋转轴几何误差辨识方法，利用多体系统理论和齐次变换矩阵对机床旋转轴A轴、C轴的各项几何误差进行Sobol敏感度分析。该发明可以简便并准确的对几何误差进行辨识，得出各误差元素对总误差的影响程度，进而通过控制关键几何误差提高加工质量。具体步骤如下：In order to solve the above problems, the present invention proposes a method for identifying the geometric errors of the double rotating axes of the five-axis CNC machine tool based on sensitivity analysis. Errors were subjected to Sobol sensitivity analysis. The invention can simply and accurately identify the geometric errors, obtain the influence degree of each error element on the total error, and then improve the processing quality by controlling the key geometric errors. Specific steps are as follows:

步骤1、利用多体系统理论和齐次变换矩阵进行几何误差建模，具体步骤如下：Step 1. Use multi-body system theory and homogeneous transformation matrix for geometric error modeling. The specific steps are as follows:

步骤1.1、基于数控机床部件的拓扑结构关系对数控机床设置参考坐标系和局部坐标系，将Y轴局部坐标系设置为与参考坐标系重合，数控机床的参考坐标系如图1所示。Step 1.1. Set the reference coordinate system and local coordinate system for the CNC machine tool based on the topological structure relationship of the CNC machine tool components, and set the Y-axis local coordinate system to coincide with the reference coordinate system. The reference coordinate system of the CNC machine tool is shown in Figure 1.

步骤1.2、对数控机床双旋转轴的几何误差进行辨识，包括12项与位置有关的几何误差(Position-Dependent Geometric Errors，PDGEs)和8项与位置无关的几何误差(Position- Independent Geometric Errors，PIGEs)，A轴的PIGEs如图2所示，C轴的PIGEs如图3所示。Step 1.2. Identify the geometric errors of the dual rotary axes of the CNC machine tool, including 12 Position-Dependent Geometric Errors (PDGEs) and 8 Position-Independent Geometric Errors (PIGEs) ), the PIGEs of the A-axis are shown in Fig. 2, and the PIGEs of the C-axis are shown in Fig. 3.

步骤1.3、利用齐次变换矩阵构建误差变换矩阵。Step 1.3, using the homogeneous transformation matrix to construct the error transformation matrix.

A轴实际回转中心线在Y轴方向的位置误差EYOA是A轴轴线在Y方向上的偏差量，DYOA是误差变换矩阵：The position error of the actual rotation center line of the A-axis in the Y-axis direction E YOA is the deviation of the A-axis axis in the Y direction, and D YOA is the error transformation matrix:

Figure RE-GSB0000185388090000061
Figure RE-GSB0000185388090000061

A轴实际回转中心线在Z轴方向的位置误差EZOA是A轴轴线在Z方向上的偏差量，DZOA是误差变换矩阵：The position error of the actual rotation center line of the A-axis in the Z-axis direction, E ZOA is the deviation of the A-axis axis in the Z direction, and D ZOA is the error transformation matrix:

Figure RE-GSB0000185388090000062
Figure RE-GSB0000185388090000062

A轴实际回转中心线在XOZ平面的投影与X轴的夹角EBOA是A轴轴线绕Y轴的平行度误差，TB0A是误差变换矩阵：The angle between the projection of the actual rotation center line of the A-axis on the XOZ plane and the X-axis E BOA is the parallelism error of the A-axis axis around the Y-axis, and T B0A is the error transformation matrix:

A轴实际回转中心线在XOY平面的投影与X轴的夹角ECOA是A轴轴线绕Z轴的平行度误差，TC0A是误差变换矩阵：The angle between the projection of the actual rotation center line of the A-axis on the XOY plane and the X-axis, E COA is the parallelism error of the A-axis axis around the Z-axis, and T C0A is the error transformation matrix:

Figure RE-GSB0000185388090000064
Figure RE-GSB0000185388090000064

其中，DYOA、DZOA、TB0A、TC0A是A轴的4项PIGEs误差变换矩阵。Among them, D YOA , D ZOA , T B0A , and T C0A are the 4-term PIGEs error transformation matrix of the A axis.

C轴实际回转中心线在X轴方向的位置误差EXOC是C轴轴线在X方向上的偏差量，DXOC是误差变换矩阵：The position error of the actual rotation center line of the C-axis in the X-axis direction E XOC is the deviation of the C-axis axis in the X direction, and D XOC is the error transformation matrix:

Figure RE-GSB0000185388090000065
Figure RE-GSB0000185388090000065

C轴实际回转中心线在Y轴方向的位置误差为EYOC是C轴轴线在Y方向上的偏差量，DYOC是误差变换矩阵：The position error of the actual rotation center line of the C-axis in the Y-axis direction is E YOC is the deviation of the C-axis axis in the Y direction, and D YOC is the error transformation matrix:

C轴实际回转中心线在YOZ平面的投影与Z轴的夹角EAOC是C轴轴线绕X轴的平行度误差，TAOC是误差变换矩阵：The angle between the projection of the actual rotation center line of the C-axis on the YOZ plane and the Z-axis, E AOC is the parallelism error of the C-axis axis around the X-axis, and T AOC is the error transformation matrix:

步骤C轴实际回转中心线在XOZ平面的投影与Z轴的夹角EBOC是C轴轴线绕Y轴的平行度误差，TBOC是误差变换矩阵：Step The angle between the projection of the actual rotation center line of the C-axis on the XOZ plane and the Z-axis E BOC is the parallelism error of the C-axis axis around the Y-axis, and T BOC is the error transformation matrix:

Figure RE-GSB0000185388090000073
Figure RE-GSB0000185388090000073

其中，DXOC、DYOC、TAOC、TBOC是C轴的4项PIGEs误差变换矩阵。Among them, D XOC , D YOC , T AOC , and T BOC are the 4-term PIGEs error transformation matrix of the C-axis.

Figure RE-GSB0000185388090000074
对应的PDGE误差变换矩阵为：
Figure RE-GSB0000185388090000074
The corresponding PDGE error transformation matrix is:

Figure RE-GSB0000185388090000076
对应的PDGE误差变换矩阵为：
Figure RE-GSB0000185388090000076
The corresponding PDGE error transformation matrix is:

Figure RE-GSB0000185388090000077
Figure RE-GSB0000185388090000077

其中，

Figure RE-GSB0000185388090000078
是基于小角度近似原理的双旋转轴PDGEs误差变换矩阵。in,
Figure RE-GSB0000185388090000078
is the error transformation matrix of dual-rotation axis PDGEs based on the principle of small-angle approximation.

综合上述各误差矩阵，工件相对于刀具的几何误差为：Combining the above error matrices, the geometric error of the workpiece relative to the tool is:

Figure RE-GSB0000185388090000079
Figure RE-GSB0000185388090000079

步骤2、五轴数控机床蒙特卡罗采样的Sobol敏感度分析，具体步骤如下：Step 2. Sobol sensitivity analysis of Monte Carlo sampling of five-axis CNC machine tools. The specific steps are as follows:

步骤2.1、采用蒙特卡罗方法对误差参数进行采样，生成Sobol序列以确定各项误差对机床空间误差的影响。Step 2.1. Use the Monte Carlo method to sample the error parameters, and generate a Sobol sequence to determine the influence of various errors on the machine tool space error.

步骤2.2、基于方差的Sobol敏感度分析需要确定误差数k，两个旋转轴共有20个误差项，所以k设置为20。Step 2.2. The variance-based Sobol sensitivity analysis needs to determine the number of errors k. There are 20 error terms in the two rotation axes, so k is set to 20.

步骤2.3、生成Sobol序列A、B两个参数样本矩阵，记为：Step 2.3. Generate two parameter sample matrices of Sobol sequence A and B, denoted as:

Figure RE-GSB0000185388090000081
Figure RE-GSB0000185388090000081

其中xij表示第j个(i＝1，2，3...k)误差元素的第i个(j＝1，2，3...n)样本。where x ij represents the ith (j=1, 2, 3...n) sample of the jth (i=1, 2, 3...k) error element.

步骤2.4、将矩阵B的第i列换到矩阵A的第i列，矩阵A的其余列不变，得到矩阵记为ABi，如下所示：Step 2.4. Change the i-th column of matrix B to the i-th column of matrix A, and the remaining columns of matrix A remain unchanged, and the obtained matrix is denoted as AB i , as shown below:

Figure RE-GSB0000185388090000082
Figure RE-GSB0000185388090000082

通过上述方法构造出矩阵A、B、ABi，共有(k+2)*N组旋转轴误差参数，因此得到 (k+2)*N组

Figure RE-GSB0000185388090000083
值。对于每组
Figure RE-GSB0000185388090000084
存在唯一的矩阵A、B、ABi相对应，记作f(A)， f(B)，f(ABi)。The matrices A, B, AB i are constructed by the above method, and there are (k+2)*N groups of rotation axis error parameters, so the (k+2)*N groups are obtained
Figure RE-GSB0000185388090000083
value. for each group
Figure RE-GSB0000185388090000084
There are unique matrices A, B, AB i corresponding to each other, denoted as f(A), f(B), f(AB i ).

步骤2.5、通过系统响应的方差计算公式可以计算各项误差元素的一阶敏感度和全局敏感度，计算公式如下：Step 2.5. The first-order sensitivity and global sensitivity of each error element can be calculated by the variance calculation formula of the system response. The calculation formula is as follows:

Var(Y)＝Var(YA+YB) (3)Var(Y)=Var(Y A +Y B ) (3)

Figure RE-GSB00001853880900000810
Figure RE-GSB00001853880900000810

Figure RE-GSB00001853880900000811
Figure RE-GSB00001853880900000811

Y＝(ya1 ya2 ... yan yb1 yb2 ... ybn)T (6)Y=(y a1 y a2 ... y an y b1 y b2 ... y bn ) T (6)

Figure RE-GSB0000185388090000085
Figure RE-GSB0000185388090000085

其中yj1为输入矩阵对应的输出值，Var(Y)为Y的标准差。where y j1 is the output value corresponding to the input matrix, and Var(Y) is the standard deviation of Y.

步骤2.6、误差元素的一阶敏感度和全局敏感度计算公式：Step 2.6. The first-order sensitivity and global sensitivity calculation formula of the error element:

Figure RE-GSB0000185388090000086
Figure RE-GSB0000185388090000086

Figure RE-GSB0000185388090000087
Figure RE-GSB0000185388090000087

Figure RE-GSB0000185388090000089
Figure RE-GSB0000185388090000089

Si表示误差元素xi的一阶敏感度，STi表示误差元素xi的全局敏感度。误差元素xi的一阶敏感度表示该误差对机床空间误差的直接影响，全局敏感度表示该误差对机床空间误差的耦合影响。S i represents the first-order sensitivity of the error element xi , and S Ti represents the global sensitivity of the error element xi . The first-order sensitivity of the error element xi represents the direct influence of the error on the spatial error of the machine tool, and the global sensitivity represents the coupled effect of the error on the spatial error of the machine tool.

目标五轴数控机床的PDGEs和PIGEs的采样比是3∶7，其中位置误差和角度误差采样范围分别是(0，1)μm和(0，1)″。The sampling ratio of PDGEs and PIGEs of the target five-axis CNC machine tool is 3:7, and the sampling ranges of position error and angle error are (0, 1) μm and (0, 1)”, respectively.

步骤3、基于旋转轴特定转角位置对误差参数进行敏感度分析。由于旋转轴所转到的角度不同，旋转轴各项误差对机床精度的影响也随之改变。针对五轴数控机床旋转轴在不同转角位置的敏感度问题，设置五组特定转角位置和一组在其加工范围内转角位置随机变化对五轴数控机床双旋转轴进行敏感度分析。Step 3. Perform a sensitivity analysis on the error parameters based on the specific rotational angle position of the rotating shaft. Due to the different rotation angles of the rotary axis, the influence of various errors of the rotary axis on the accuracy of the machine tool also changes. Aiming at the sensitivity of the rotary axis of the five-axis CNC machine tool at different corner positions, five groups of specific corner positions and a group of random changes in the position of the rotation angle within the processing range are set to analyze the sensitivity of the dual rotation axis of the five-axis CNC machine tool.

步骤3.1、设置A轴转角为0°、C轴转角为0°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.1. Set the rotation angle of the A-axis to 0° and the rotation angle of the C-axis to 0°. Perform a sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.2、设置A轴转角为45°、C轴转角为45°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.2. Set the rotation angle of the A axis to 45° and the rotation angle of the C axis to 45°. Perform a sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.3、设置A轴转角为90°、C轴转角为90°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.3. Set the rotation angle of A-axis to 90° and the rotation angle of C-axis to 90°. Perform sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.4、设置A轴转角为0°、C轴转角为90°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.4. Set the rotation angle of the A axis to 0° and the rotation angle of the C axis to 90°. Perform a sensitivity analysis on the five-axis CNC machine tool at this position, and calculate its first-order sensitivity and global sensitivity.

步骤3.5、设置A轴转角为90°、C轴转角为0°，对五轴数控机床在此位置进行敏感度分析，计算其一阶敏感度和全局敏感度。Step 3.5. Set the rotation angle of the A axis to 90° and the rotation angle of the C axis to 0°. Perform a sensitivity analysis on the position of the five-axis CNC machine tool to calculate its first-order sensitivity and global sensitivity.

步骤3.6、设置A轴转角在0°-90°内随机变化，C轴转角在0°-360°内随机变化，计算五轴数控机床的一阶敏感度和全局敏感度。Step 3.6. Set the A-axis rotation angle to vary randomly within 0°-90°, and the C-axis rotation angle to vary randomly within 0°-360°, and calculate the first-order sensitivity and global sensitivity of the five-axis CNC machine tool.

步骤3的Sobol敏感度分析结果(如图4、图5)显示，影响加工精度的误差主要分布在PIGEs部分。PIGEs对五轴数控机床空间误差的影响程度较大，而PDGEs对五轴数控机床空间误差影响较小，通过Sobol敏感度分析方法可以识别20项几何误差中的关键几何误差。几何误差中影响机床精度的关键误差来自PIGEs中的EY0A、EY0C、EA0C，影响加工精度的一阶敏感度占20项几何误差的67.42％，考虑相互作用情况下，影响加工精度的全局敏感度占20项几何误差的26.97％。The Sobol sensitivity analysis results in step 3 (Fig. 4 and Fig. 5) show that the errors affecting the machining accuracy are mainly distributed in the PIGEs part. PIGEs have a greater influence on the spatial error of five-axis CNC machine tools, while PDGEs have less impact on the spatial errors of five-axis CNC machine tools. The key geometric errors among the 20 geometric errors can be identified by the Sobol sensitivity analysis method. The key errors affecting machine tool accuracy in geometric errors come from E Y0A , E Y0C , and E A0C in PIGEs, and the first-order sensitivity affecting machining accuracy accounts for 67.42% of the 20 geometric errors. Considering the interaction, the global impact of machining accuracy The sensitivity accounts for 26.97% of the 20-item geometric errors.

通过上述Sobol敏感度分析结果，对五轴数控机床的几何误差进行辨识，控制关键几何误差可以提高数控机床的精度。Through the above Sobol sensitivity analysis results, the geometric errors of five-axis CNC machine tools are identified, and the control of key geometric errors can improve the accuracy of CNC machine tools.

以上完成了五轴数控机床双旋转轴几何误差敏感度分析，包括8项与位置无关的几何误差和12项与位置有关的几何误差的敏感度分析。The above completes the sensitivity analysis of the geometric errors of the five-axis CNC machine tool and the dual rotating axes, including the sensitivity analysis of 8 geometric errors independent of position and 12 geometric errors related to position.

本发明有效的解决了五轴数控机床中双旋转轴几何误差的辨识与检测，提出了有效的五轴数控机床双旋转轴的几何误差敏感度分析。The invention effectively solves the identification and detection of the geometric error of the double rotating shafts in the five-axis numerical control machine tool, and proposes an effective sensitivity analysis of the geometric error of the double rotating shafts of the five-axis numerical control machine tool.

本发明对五轴数控机床双旋转轴几何误差进行敏感度分析，最后得到五轴数控机床双旋转轴的关键几何误差。附图只是一个优选实例，上述的实施例只是为了描述本发明，并不用以限制本发明，凡在本发明的精神和原则之内，所做的任何修改、等同替换、改进等，均应涵盖在本发明的保护范围之内。The invention conducts sensitivity analysis on the geometric errors of the double rotating axes of the five-axis numerical control machine tool, and finally obtains the key geometric errors of the double rotating axes of the five-axis numerical control machine tool. The accompanying drawing is only a preferred example, the above-mentioned embodiment is only for describing the present invention, not for limiting the present invention, any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall cover within the protection scope of the present invention.

Claims (4)

1. The method for analyzing the geometric error sensitivity of the double rotating shafts of the five-axis numerical control machine tool comprises the following steps of:
step 1, constructing a geometric error mathematical model of a double-rotation shaft by utilizing a multi-body system theory and a homogeneous transformation matrix, and identifying each geometric error;
2, performing Sobol sensitivity analysis based on Monte Carlo sampling on the double rotating shafts of the five-axis numerical control machine tool;
and 3, carrying out sensitivity analysis on the error parameters based on the positions of the specific corners of the double rotating shafts, and carrying out sensitivity analysis on the double rotating shafts by setting five groups of positions of the specific corners.
2. The method for analyzing the geometric error sensitivity of the double rotating shafts of the five-axis numerical control machine tool according to claim 1, wherein in the step 1, a mathematical model of the geometric errors of the double rotating shafts is constructed by using a multi-body system theory and a homogeneous transformation matrix so as to identify the geometric errors of the double rotating shafts, and the method comprises the following steps:
step 1.1, setting a reference coordinate system and a local coordinate system for a numerical control machine tool, and setting a Y-axis local coordinate system to be coincident with the reference coordinate system;
step 1.2, identifying geometric errors of double rotating shafts of the numerical control machine tool;
step 1.3, constructing an error transformation matrix by utilizing the homogeneous transformation matrix;
taking the axis A as an example, obtaining a geometric error transformation matrix of the double rotating shafts through a homogeneous transformation matrix;
position-independent error transformation matrix for axis a:
Figure FSA0000195650160000011
Figure FSA0000195650160000012
Figure FSA0000195650160000014
position-dependent error transformation matrix for the a-axis:
Figure FSA0000195650160000015
likewise, the position-independent error transformation matrix for the C-axis:
Figure FSA0000195650160000016
Figure FSA0000195650160000021
Figure FSA0000195650160000022
the C-axis position-dependent error transformation matrix:
Figure FSA0000195650160000024
and integrating the error matrixes, wherein the geometric error of the workpiece relative to the cutter is as follows:
Figure FSA0000195650160000025
the establishment of the double-rotating-axis mathematical model of the five-axis numerical control machine tool is completed.
3. The method for analyzing geometrical error sensitivity of double rotation axes of a five-axis numerical control machine tool according to claim 1, wherein in the step 2, the Sobol sensitivity analysis based on Monte Carlo sampling comprises the steps of:
step 2.1, sampling the error parameters based on a Monte Carlo method, and generating a Sobol sequence to determine the influence of each error on the machine tool space error;
step 2.2, determining the error number k based on the variance Sobol sensitivity analysis, wherein the dual rotation axes have 20 error items in total, so that k is set to be 20;
step 2.3, generating two parameter sample matrixes of the Sobol sequence A, B, and recording as follows:
Figure FSA0000195650160000026
wherein xijAn ith (j ═ 1, 2, 3.. n) sample representing a jth (i ═ 1, 2, 3.. k) error element;
step 2.4, the ith column of the matrix B is changed to the ith column of the matrix A, and the rest columns of the matrix A are unchanged, so that the matrix AB is obtainediAs follows:
Figure FSA0000195650160000027
the matrix A, B, AB is constructed by the method described aboveiThe total (k +2) N sets of rotation axis error parameters are obtained, thus obtaining (k +2) N sets
Figure FSA0000195650160000031
The value is obtained. For each group
Figure FSA0000195650160000032
There is a unique matrix A, B, ABiCorresponding to f (A), f (B), f (AB)i)；
Step 2.5, calculating the first-order sensitivity and the global sensitivity of each error element by a variance calculation formula of system response, wherein the calculation formula is as follows:
Var(Y)＝Var(YA+YB) (3)
Figure FSA0000195650160000038
Figure FSA0000195650160000039
Y＝(ya1ya2... yanyb1yb2... ybn)T(6)
Figure FSA0000195650160000033
wherein y isj1Var (Y) is the standard deviation of Y for the output values corresponding to the input matrix;
step 2.6, calculating a formula of the first-order sensitivity and the global sensitivity of the error element:
Figure FSA0000195650160000034
Figure FSA0000195650160000035
Figure FSA0000195650160000037
wherein SiIs the first order sensitivity of the i term error, STiIs the global sensitivity of the i-th error, each error element xiThe first order sensitivity of (a) represents the direct effect of the error on the machine space error, and the global sensitivity represents the coupled effect of the error on the machine space error.
4. The method for analyzing geometric error sensitivity of dual rotational axes of a five-axis numerical control machine tool according to claim 1, wherein in the step 3, the sensitivity analysis of the error parameters is performed based on the positions of the specific rotation angles of the dual rotational axes, and the sensitivity analysis of the dual rotational axes is performed by setting five groups of the positions of the specific rotation angles, including the steps of:
step 3.1, setting the rotation angle of the shaft A to be 0 degrees and the rotation angle of the shaft C to be 0 degrees, carrying out sensitivity analysis on the position of the five-axis numerical control machine tool, and calculating the first-order sensitivity and the global sensitivity of the five-axis numerical control machine tool;
step 3.2, setting the rotation angle of the shaft A to be 45 degrees and the rotation angle of the shaft C to be 45 degrees, carrying out sensitivity analysis on the position of the five-axis numerical control machine tool, and calculating the first-order sensitivity and the global sensitivity of the five-axis numerical control machine tool;
3.3, setting the rotation angle of the shaft A to be 90 degrees and the rotation angle of the shaft C to be 90 degrees, carrying out sensitivity analysis on the position of the five-axis numerical control machine tool, and calculating the first-order sensitivity and the global sensitivity of the five-axis numerical control machine tool;
step 3.4, setting the rotation angle of the shaft A to be 0 degrees and the rotation angle of the shaft C to be 90 degrees, carrying out sensitivity analysis on the position of the five-axis numerical control machine tool, and calculating the first-order sensitivity and the global sensitivity of the five-axis numerical control machine tool;
step 3.5, setting the rotation angle of the shaft A to be 90 degrees and the rotation angle of the shaft C to be 0 degree, carrying out sensitivity analysis on the position of the five-axis numerical control machine tool, and calculating the first-order sensitivity and the global sensitivity of the five-axis numerical control machine tool;
step 3.6, setting the shaft A rotation angle to randomly change within 0-90 degrees and the shaft C rotation angle to randomly change within 0-360 degrees, carrying out sensitivity analysis on the five-axis numerical control machine tool, and calculating the first order sensitivity and the global sensitivity of the five-axis numerical control machine tool;
through the Sobol sensitivity analysis result, the geometric errors of the five-axis numerical control machine tool are identified, and the accuracy of the numerical control machine tool can be improved by controlling the key geometric errors.
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