直线度误差数字化评定理论与算法研究
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摘要
随着计算机技术、网络技术、信息技术和先进制造技术等的不断发展，制造系统正在由原来的能量驱动型转变为信息驱动型，数字化与精密化已成为先进制造技术的核心与关键；数控机床、三坐标测量机等数字装备已得到广泛应用，加工精度和测量精度都已可达到纳米级。然而，目前对几何误差的评定仍然普遍采用评定精度较低的最小二乘法（LSM）等近似方法。测量精度与评定精度是影响测量结果精度的两个主要因素，迫切需要提高对直线度误差等几何误差的评定精度。同时，直线是构成机械零部件最基本的几何要素，常常作为方向误差、位置误差、跳动误差等的基准要素；空间直线度误差作为最复杂的几何误差之一，其评定理论与算法还很不成熟。
因此，本文针对直线度误差的数字化评定理论与算法进行研究。首先阐述了其研究背景、目的与意义，对平面直线度误差和空间直线度误差评定研究的现状进行了评述，分析了现有研究所存在的问题；说明了对给定平面内直线度误差的评定进行可视化研究的必要性、以及研究空间直线度误差数字化评定理论与算法的重要性。然后，运用几何学原理、最优化理论、凸壳理论、误差理论和可视化技术作了如下研究：
（1）介绍了给定平面内直线度误差评定理论，对其两端点连线算法、LSM算法和最小区域算法进行了可视化设计；并用数字实验验证了三种可视化设计算法的正确性，实现了给定平面内直线度误差评定的可视化。
（2）总结了空间直线度误差评定的基础理论，对评定空间直线度误差的LSM算法进行了深入的理论分析，揭示出其原理缺陷是：不能精确拟合出三维直线，当测点坐标值具有不同的数量级时，其评定结果存在严重失真；并研究了改进的LSM+算法、基于测点集算术平均中心的最小二乘算法（LSABC）、改进的LSABC+算法、三维最小二乘算法（3DLSA）等的数学模型；按这些数学模型进行了数字实验，验证了这些算法是有效的，克服了LSM算法的原理缺陷，实现了精确的三维直线拟合；是具有较高精度的、评定空间直线度误差的好算法。
（3）基于最小区域准则，对空间直线度误差的三点接触情况进行了理论分析，得出了三点接触的最小区域判断准则；对该准则的正确性作了理论分析和验证。在此基础上，研究了评定空间直线度误差的三点接触算法（TPTA）和三点接触的平均中心点求解三维凸壳的新算法(3DCHA),用实例验证表明,3DCHA算法比现有三维凸壳算法的求解效率高2～6倍。利用3DCHA算法能从测点集中提取有效测点、减少计算时间的特点,设计了双基点的空间网格点交叉搜索算法(SCSA),多组测点集的数字实验证明,SCSA算法比现有多种算法具有更高的精度,并能达到纳米级评定精度。
(5)依据新一代GPS标准规定的要素操作技术,在数控三坐标测量机上对两个零件进行了三维数字化检测实验,提取了两个零件圆柱形要素的测点坐标值。研究了求解三维最小二乘拟合圆心、三维最小外接拟合圆心、三维最大内接拟合圆心、三维最小区域拟合圆心的数学模型；按这些数学模型和要素操作技术,编制程序实现了空间直线度误差的数字化精确评定。同时,应用空间直线度误差的数字化精确评定方法,实现了阶梯轴圆柱形零件同轴度误差值的数字化精确评定。
以上研究表明,给定平面内直线度误差评定的可视化研究,以及评定空间直线度误差的LSM+算法、LSABC算法、LSABC+算法和3DLSA算法能满足现代数字制造工程中对直线度误差进行数字化评定的需要。基于最小区域准则的三点接触的最小区域判断准则和TPTA算法、以3DCHA算法为基础的SCSA算法实现了对空间直线度误差的精确评定,并可达到纳米级评定精度；能满足精密/超精密加工和微纳制造等先进制造工程中对空间直线度误差进行数字化精确评定的要求。同时,基于新一代GPS标准规定的要素操作技术,实现了对空间直线度误差和同轴度误差的数字化检测,并用检测数据验证了上述所研究的理论和算法的正确性、有效性和应用效能。因此,本文的研究在数字化制造工程中具有重要的理论意义和应用价值。
With the technology development of the computer, network, information, and advanced manufacturing, the manufacturing systems have been changing from initial energy-driven into current information-driven. Digitization and precision have become the core and key issues of the advanced manufacturing technology. CNC machine tool, coordinate measurement machines and other digital equipments have been got widely used in the manufacturing engineering.The machining accuracy and measuring accuracy have been increased up to nanometer level.However, the least squares method (LSM) and other approximate methods with low evaluation accuracy have been still widely used for the evaluation of geometric error. The measurement accuracy and the evaluation accuracy were two important factors of the measurement result, it is urgent need to improve the evaluation accuracy of geometric error such as straightness error. At the same time, the straight line is the essentialest geometrical factor of mechanical parts, and often used as a datum line of orientation error, or location error, or run-out error. In addition, spatial straightness error is one of the most complex geometric errors, its evaluation theory or algorithm is still immature.
Therefore,This research is focused on the digitized evaluation theory and algorithms of the straightness error. Firstly, the research background, the purpose and the meaning of this research were been expounded. The present research situation of the evaluation of the planar straightness error and the spatial straightness error were been reviewed, the problems existing in the current research are analyzed. The research necessity on visualization for evaluation of the planar straightness error and the research importance on digitized and accurate evaluation theory and algorithms of spatial straightness error were been explained. Then, Using geometry theory, optimization theory, convex shell theory, theory of errors and visualization technology, the research is as following:
(1) The evaluation theory of planar straightness error was showed. Visualization design were performed for the two end-points link line algorithm, LSM algorithm and minimum zone algorithm, and the correctness of the three kinds visualization algorithms is verified with digital experiment, the visualization of evaluation for planar straightness error is carried out.
(2) The basic theory for the evaluation of spatial straightness error was summarized. The LSM algorithm for evaluation of the spatial straightness error was analyzed theoretically in depth, its principle defects are revealed as following:with the LSM algorithm,accurate3D fitting straight line could not be obtained, and when the measurement point coordinates with different unit scale, the evaluation results will be in serious distortion. The mathematical models of the improved LSM algorithm, or the Least Square Algorithm Based on the Center of the measured points set (LSABC), or the improved LSABC algorithm, or the three Dimensional Least Square Algorithm (3DLSA) were all deduced. The digital experiment according to these mathematical models shows:these algorithms are effective, the principle defects of the LSM algorithm was overcome, the real3d fitting straight line can be carried out. These algorithms are good algorithms to evaluate the spatial straightness error, and have a higher precision.
(3) Based on the minimum zone criterion, the tangent case with three points of the spatial straightness error was theoretical analyzed, and the minimum zone judgment criterion of the tangent case with three points was obtained, the correctness of the judgment criterion was analyzed and verified theoretically. Based on the analysis, a Three Points Tangent Algorithm (TPTA) and its minimum zone judgment algorithm to evaluate the spatial straightness error were proposed. The TPTA algorithm was verified by some digital experiment, it could guarantee that three measured points are tangent to an inch with the minimum zone circumscribing the measured points set, it is a more effective algorithm than many other algorithms, such as LSM.
(4) According to the tangent characteristics getting touch with four or five points among the measured points set of the spatial straightness error, an new3D Convex Hull Algorithm based on the Center of the measured (3DCHA) was put forward, Some examples show that3DCHA algorithm has higher efficiency to2-6times than other3d convex hull algorithm existing. By the trait of some effective measured points can be extracted from the measured points set and the computational time can be reduced by3DCHA, the Space grid point Cross Search Algorithm based on double basis point (SCSA) was designed. Some digital experiments show that SCSA has higher precision than other algorithms existing, and can obtain the nanometer level evaluation precision in evaluating the spatial straightness error.
(5) According to the geometrical features operating Techniques provided by the new generation Geometrical Product Specification and verification (GPS) standards, the3D digital detect experiments of two parts were performed, the measuring point coordinates of the cylindrical parts were extracted with NC coordinate measuring machine. Some mathematical model for solving the centre of3D least squares fitting circle,3D minimum circumscribed fitting circle,3D maximum inscribed fitting circle,3D minimum zone fitting circle were proposed. By the mathematical model and the element operating Techniques, digital accurate evaluation to the spatial straightness error was come true. At the same time, applying the digital accurate evaluation methods to the spatial straightness error, the digital accurate evaluation the values of the coaxiality error of a ladder cylindrical shaft was come ture.
As above of all research show:the research on the visualization evaluation for planar straightness error, and on the improved LSM algorithm, the LSABC algorithm, and the improved LSABC algorithm, and the3DLSA algorithm which assessing the spatial straightness error could meet the requirement of the digital evaluation to the straightness error in modern digital manufacturing engineering. The minimum zone judgment criterion being tangent with three points and the TPTA algorithm, the SCSA algorithm based on the3DCHA algorithm could realize accurate evaluation to the spatial straightness error, and obtain the nanometer level evaluation precision. It could meet the requirement of the digital accurate evaluation to the spatial straightness error in advanced manufacturing engineering such as precision/ultra-precision processing, micro-nano manufacturing and so on. At the same time, the digital detection of spatial straightness error and the coaxiality error were achieved based on the elements operation Techniques specified by the new generation of GPS standard, and that the correctness, and effectiveness, and the application efficiency of the theory or the algorithm researched above were validated by the digital detected data, Therefore, the research on this topic shows that it has important theoretical significance and application value in the digital manufacturing engineering.
因此，本文针对直线度误差的数字化评定理论与算法进行研究。首先阐述了其研究背景、目的与意义，对平面直线度误差和空间直线度误差评定研究的现状进行了评述，分析了现有研究所存在的问题；说明了对给定平面内直线度误差的评定进行可视化研究的必要性、以及研究空间直线度误差数字化评定理论与算法的重要性。然后，运用几何学原理、最优化理论、凸壳理论、误差理论和可视化技术作了如下研究：
（1）介绍了给定平面内直线度误差评定理论，对其两端点连线算法、LSM算法和最小区域算法进行了可视化设计；并用数字实验验证了三种可视化设计算法的正确性，实现了给定平面内直线度误差评定的可视化。
（2）总结了空间直线度误差评定的基础理论，对评定空间直线度误差的LSM算法进行了深入的理论分析，揭示出其原理缺陷是：不能精确拟合出三维直线，当测点坐标值具有不同的数量级时，其评定结果存在严重失真；并研究了改进的LSM+算法、基于测点集算术平均中心的最小二乘算法（LSABC）、改进的LSABC+算法、三维最小二乘算法（3DLSA）等的数学模型；按这些数学模型进行了数字实验，验证了这些算法是有效的，克服了LSM算法的原理缺陷，实现了精确的三维直线拟合；是具有较高精度的、评定空间直线度误差的好算法。
（3）基于最小区域准则，对空间直线度误差的三点接触情况进行了理论分析，得出了三点接触的最小区域判断准则；对该准则的正确性作了理论分析和验证。在此基础上，研究了评定空间直线度误差的三点接触算法（TPTA）和三点接触的平均中心点求解三维凸壳的新算法(3DCHA),用实例验证表明,3DCHA算法比现有三维凸壳算法的求解效率高2～6倍。利用3DCHA算法能从测点集中提取有效测点、减少计算时间的特点,设计了双基点的空间网格点交叉搜索算法(SCSA),多组测点集的数字实验证明,SCSA算法比现有多种算法具有更高的精度,并能达到纳米级评定精度。
(5)依据新一代GPS标准规定的要素操作技术,在数控三坐标测量机上对两个零件进行了三维数字化检测实验,提取了两个零件圆柱形要素的测点坐标值。研究了求解三维最小二乘拟合圆心、三维最小外接拟合圆心、三维最大内接拟合圆心、三维最小区域拟合圆心的数学模型；按这些数学模型和要素操作技术,编制程序实现了空间直线度误差的数字化精确评定。同时,应用空间直线度误差的数字化精确评定方法,实现了阶梯轴圆柱形零件同轴度误差值的数字化精确评定。
以上研究表明,给定平面内直线度误差评定的可视化研究,以及评定空间直线度误差的LSM+算法、LSABC算法、LSABC+算法和3DLSA算法能满足现代数字制造工程中对直线度误差进行数字化评定的需要。基于最小区域准则的三点接触的最小区域判断准则和TPTA算法、以3DCHA算法为基础的SCSA算法实现了对空间直线度误差的精确评定,并可达到纳米级评定精度；能满足精密/超精密加工和微纳制造等先进制造工程中对空间直线度误差进行数字化精确评定的要求。同时,基于新一代GPS标准规定的要素操作技术,实现了对空间直线度误差和同轴度误差的数字化检测,并用检测数据验证了上述所研究的理论和算法的正确性、有效性和应用效能。因此,本文的研究在数字化制造工程中具有重要的理论意义和应用价值。
With the technology development of the computer, network, information, and advanced manufacturing, the manufacturing systems have been changing from initial energy-driven into current information-driven. Digitization and precision have become the core and key issues of the advanced manufacturing technology. CNC machine tool, coordinate measurement machines and other digital equipments have been got widely used in the manufacturing engineering.The machining accuracy and measuring accuracy have been increased up to nanometer level.However, the least squares method (LSM) and other approximate methods with low evaluation accuracy have been still widely used for the evaluation of geometric error. The measurement accuracy and the evaluation accuracy were two important factors of the measurement result, it is urgent need to improve the evaluation accuracy of geometric error such as straightness error. At the same time, the straight line is the essentialest geometrical factor of mechanical parts, and often used as a datum line of orientation error, or location error, or run-out error. In addition, spatial straightness error is one of the most complex geometric errors, its evaluation theory or algorithm is still immature.
Therefore,This research is focused on the digitized evaluation theory and algorithms of the straightness error. Firstly, the research background, the purpose and the meaning of this research were been expounded. The present research situation of the evaluation of the planar straightness error and the spatial straightness error were been reviewed, the problems existing in the current research are analyzed. The research necessity on visualization for evaluation of the planar straightness error and the research importance on digitized and accurate evaluation theory and algorithms of spatial straightness error were been explained. Then, Using geometry theory, optimization theory, convex shell theory, theory of errors and visualization technology, the research is as following:
(1) The evaluation theory of planar straightness error was showed. Visualization design were performed for the two end-points link line algorithm, LSM algorithm and minimum zone algorithm, and the correctness of the three kinds visualization algorithms is verified with digital experiment, the visualization of evaluation for planar straightness error is carried out.
(2) The basic theory for the evaluation of spatial straightness error was summarized. The LSM algorithm for evaluation of the spatial straightness error was analyzed theoretically in depth, its principle defects are revealed as following:with the LSM algorithm,accurate3D fitting straight line could not be obtained, and when the measurement point coordinates with different unit scale, the evaluation results will be in serious distortion. The mathematical models of the improved LSM algorithm, or the Least Square Algorithm Based on the Center of the measured points set (LSABC), or the improved LSABC algorithm, or the three Dimensional Least Square Algorithm (3DLSA) were all deduced. The digital experiment according to these mathematical models shows:these algorithms are effective, the principle defects of the LSM algorithm was overcome, the real3d fitting straight line can be carried out. These algorithms are good algorithms to evaluate the spatial straightness error, and have a higher precision.
(3) Based on the minimum zone criterion, the tangent case with three points of the spatial straightness error was theoretical analyzed, and the minimum zone judgment criterion of the tangent case with three points was obtained, the correctness of the judgment criterion was analyzed and verified theoretically. Based on the analysis, a Three Points Tangent Algorithm (TPTA) and its minimum zone judgment algorithm to evaluate the spatial straightness error were proposed. The TPTA algorithm was verified by some digital experiment, it could guarantee that three measured points are tangent to an inch with the minimum zone circumscribing the measured points set, it is a more effective algorithm than many other algorithms, such as LSM.
(4) According to the tangent characteristics getting touch with four or five points among the measured points set of the spatial straightness error, an new3D Convex Hull Algorithm based on the Center of the measured (3DCHA) was put forward, Some examples show that3DCHA algorithm has higher efficiency to2-6times than other3d convex hull algorithm existing. By the trait of some effective measured points can be extracted from the measured points set and the computational time can be reduced by3DCHA, the Space grid point Cross Search Algorithm based on double basis point (SCSA) was designed. Some digital experiments show that SCSA has higher precision than other algorithms existing, and can obtain the nanometer level evaluation precision in evaluating the spatial straightness error.
(5) According to the geometrical features operating Techniques provided by the new generation Geometrical Product Specification and verification (GPS) standards, the3D digital detect experiments of two parts were performed, the measuring point coordinates of the cylindrical parts were extracted with NC coordinate measuring machine. Some mathematical model for solving the centre of3D least squares fitting circle,3D minimum circumscribed fitting circle,3D maximum inscribed fitting circle,3D minimum zone fitting circle were proposed. By the mathematical model and the element operating Techniques, digital accurate evaluation to the spatial straightness error was come true. At the same time, applying the digital accurate evaluation methods to the spatial straightness error, the digital accurate evaluation the values of the coaxiality error of a ladder cylindrical shaft was come ture.
As above of all research show:the research on the visualization evaluation for planar straightness error, and on the improved LSM algorithm, the LSABC algorithm, and the improved LSABC algorithm, and the3DLSA algorithm which assessing the spatial straightness error could meet the requirement of the digital evaluation to the straightness error in modern digital manufacturing engineering. The minimum zone judgment criterion being tangent with three points and the TPTA algorithm, the SCSA algorithm based on the3DCHA algorithm could realize accurate evaluation to the spatial straightness error, and obtain the nanometer level evaluation precision. It could meet the requirement of the digital accurate evaluation to the spatial straightness error in advanced manufacturing engineering such as precision/ultra-precision processing, micro-nano manufacturing and so on. At the same time, the digital detection of spatial straightness error and the coaxiality error were achieved based on the elements operation Techniques specified by the new generation of GPS standard, and that the correctness, and effectiveness, and the application efficiency of the theory or the algorithm researched above were validated by the digital detected data, Therefore, the research on this topic shows that it has important theoretical significance and application value in the digital manufacturing engineering.
引文
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