基于误差分离技术的超精密测量及校正方法研究
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摘要
随着新材料、超精密加工技术的发展，被测对象的尺寸越来越大，超精密加工的加工精度和表面质量越来越高，导致通用的测量仪器所能提供的硬基准的精度不能超过加工精度，测量检测难度比加工难度更大。在此背景下，产生了应用某一特定的测量系统和相应的软件的组合去分离硬基准固有的误差，以此来提供一个比硬基准精度更高的虚拟软基准的测量技术，即误差分离技术。
应用误差分离技术的超精密形状（误差）测量能够使测量精度达到目前所能达到精度的极限。这对提供最高计量标准的国家标准研究所和计量测试技术研究所来说是必不可少的技术。其次，对机床等加工机械进行数值补偿是机械运动的形状测量的另一大目的，可信度高，实施快捷简单的校正方法将是必不可缺的。最后，应用误差分离技术的形状测量能够以更低的硬件投资达到所需要的测量精度，符合建立节约型社会的目标。因此，超精密测量方法中的误差分离技术作为方法论的核心和基础技术，对它的理论研究是现代加工和测量技术的不可或缺的关键技术。而应用此技术的测量方法和校正方法也具有广阔的市场前景。
本文依托国家自然科学基金资助项目“超精密形状误差测量中的误差分离技术研究”（项目编号：50905116）和校企合作项目“电梯轿厢导轨直线度及角度位置关系检测系统”（合同编号：H3007144）开展研究工作，以超精密加工和测量为应用环境，结合项目中电梯轿厢导轨直线度检测的需要，对超精密测量相关的误差分离技术中的若干问题进行系统研究。
首先，论文介绍了本研究的背景、目的和意义。指出了开展以基于误差分离技术的超精密测量方法为研究对象的重要性，分析了包括多步法、自校正法和多探头法的误差分离技术的国内外研究现状及存在的问题，并在此基础上，提出了本论文的主要研究内容。
接着，论文对基于线性方程组的自校正方法进行了系统的研究。从二维工作台定位误差自校正算法的提出，再扩展到三维空间精度自校正法，形成了从二维到三维自校正法的初步体系化。相较原有的基于傅里叶变换的自校正法，基于线性方程组的自校正方法可方便地进行理论误差传播率的推导、具有良好的噪声抑制能力、可对三位姿以上的组合进行自校正计算。
通过本文的自校正研究，取得了以下成果：1）提出了基于线性方程组的二维工作台定位误差自校正算法；推导了测量噪声传递到重构结果的误差传播率，理论表明本算法重构结果的不确定度值小于测量噪声的标准差；2）对五位姿以下的组合测量进行了仿真实验，确认了本算法在不计测量噪声的情况下可实现无误差重构；在考虑测量噪声情况下，重构误差是处于统计学意义上的置信区间内的；3）提出了基于线性方程组的三维空间精度自校正算法，推导了误差传播率，并进行了仿真实验，其理论特性和仿真结果与上述二维的情况一致；4）利用视觉影像仪对镜头误差进行初步的自校正实验表明以上实验结果合理，同时也证明了本文所提出的自校正算法的有效性。
其次，针对超精密长尺度加工件的直线度（轮廓形状）测量问题，论文对多探头法中的变间距2组2点法进行了深入研究。本文在既有的基于傅里叶变换的变间距2组2点法的基础上，重新推导了此方法的计算条件；接着，定量推导了传感器间零点误差对系统造成影响，进而提出了改良的变间距2组2点法，采用蒙特卡罗法计算了该方法的不确定度。与此同时，本文将基于线性方程组的多探头算法应用到变间距2组2点法上，推导了基于线性方程组的变间距2组2点法；并对基于线性方程组的方法与基于傅里叶变换的方法进行了比较。最后，搭建了测量系统进行了实验研究。
通过本文的变间距2组2点法研究，取得了以下成果：1）根据既有的基于傅里叶变换的变间距2组2点法算法，重新推导了参数条件。新的参数条件使传感器间距设置不再直接受限于测量点数，扩大了传感器间距的的实际选择范围。2）定量分析了传感器测头间的零点误差对重构结果造成的影响，提出了改良的傅里叶变换算法；在不计测量噪声的情况下改良算法消除零点误差带来的影响，实现无误差重构；在测量噪声的情况下，推导了一种加权平均最优估计解。3）采用蒙特卡罗法对上述最优估计解的不确定度进行了定量计算。结果表明处于形状中间部分的不确定度与测量噪声是同一数量级的。4）推导了基于线性方程组的变间距2组2点算法，建立了相应的线性方程组模型并计算了该方法的理论不确定度。在不计测量噪声的情况下，本算法可完全消除零点误差带来的影响，实现无误差重构；在考虑测量噪声的情况下，最小二乘解的精度在理论上与零点误差的大小无关，只取决于测量噪声的大小。5）将基于傅里叶变换的方法与基于线性方程组的方法进行了全面比较。6）实验结果表明，对同一测量数据采用2种不同算法得到的重构结果几乎一致，结果表明除计算速度和内存消耗的劣势以外，基于线性方程组的方法在各方面都优于基于傅里叶变换的方法。
再次，针对长导轨直线度测量问题，论文对基于反转和多探头法融合的混合多探头法进行了深入研究。本文在详细说明了GAO的反转六探头法的测量原理的基础上提出的反转四探头法少用两个探头同样实现直线度重构功能。在实验室内搭建了3米长的验证系统，设计了可脱离PC机运行的传感器组数据采集方案，编写了可处理连续扫描数据和异常数据的实用化程序。实验表明，反转六探头法和反转四探头法在不同扫描速度和采样间隔测量条件下的重构结果的重复性在30μm以下。使用法如激光跟踪仪对同一导轨进行了测量，其结果和反转四探头法的重构结果的差在50μm以下，从而有力证明了本论文提出的反转四探头法的有效性。
最后，对整个论文的工作和研究成果进行了总结，并提出了下一步的研究展望。精密测量法中的误差分离技术作为方法论的核心和基础技术，一直以来在多样化的形状测量领域占有一席之地。而随着超精密测量需求越来越多，误差分离技术也将得到越来越多的实际应用。
With the development of new materials and ultra-precision machining technology, the size ofthe tested object is turn larger and larger, machining accuracy and surface quality is turn higherand higher. common measuring instruments can not provide hard benchmarks accuracy exceedingthe ultra-precision machining accuracy. Measurement and detect are more difficult than machining.In this background, error separation technology, which is a kind of measurement technique of avirtual soft benchmark accuracy higher than the hard baseline accuracy by application of aparticular measurement system and the appropriate combination of software to separate the errorinherent in the hard benchmark, has produced.
Ultra-precision shape (error) measurements of the application of error separation technologymake the measurement accuracy achieve the current limit of achievable accuracy. To the NationalInstitute of Standards and Measurement and Testing Technology Research Institute, this is theessential technology to provide the highest standards of measurement. Secondly, the numericalcompensation of machine tools and other processing machinery is another major purpose of theshape of the mechanical movement of the measurement. High reliability, quick and simplecalibration method is essential. Finally, the shape of the application of error separation techniquescan to invest lower hardware to achieve the required measurement accuracy, in line with goals toestablish saving society. Therefore, as the core and foundation of the methodology, theoreticalstudy of the error separation techniques in ultra-precision measurement method is the keytechnology of modern processing and measurement techniques indispensable. The methods ofmeasurement and calibration for the application of this technology also have broad marketprospects.
Supported by the Nature Science Fund of China (NSFC)(Project Number:50905116), theCooperation Project of University-enterprise (Contract No. H3007144) under the ultra-precisionmachining and measurement applications environment, combined with the project for the needs ofthe elevator car guide rail detection, ultra-precision measurement error separation techniques arestudied systematically. Theoretical research on the ultra-precise two-dimensional platformpositioning error self-calibration methods are conducted and optimized. The measuring device ofthe elevator car guide rail straightness has been developed.
The main content and contributions of this thesis are summarized as follows:
Firstly, the background, purpose and importance are introduced based on collecting and mastering a large amount of documents and references on error separation technology. Theimportance of research on ultra-precision measurement based on error separation techniques ispointed out. The research status and existing problems of error separation techniques, such asmultistep method, self-calibration method and multiple probe method are analyzed. The mainresearch content is based on this analysis.
Secondly, the systematic research on self-calibration method based on linear equations isdone. From the proposing of error self-calibration algorithm for2D workbench location, expandsto3D space precision self-calibration method is deduced, self-calibration method from2D to3D ispreliminarily systematized. Compared with the existing self-calibration method based on Fouriertransform, the self-calibration method based on linear equations has advantages of convenientdeduction of theoretical error propagation rate, good capability of noise elimination and realizationof self-calibration for combination with more than3orientations.
The achievements gained in the research of self-calibration method are summarized as below:1)2D self-calibration algorithm based on linear equations is proposed; error propagation rate thattransmitted to reconfiguration result by measured noise is deduced. Theory indicates that theuncertainty value of reconfiguration result for this algorithm is smaller than the standard deviationof measured noise.2)The realization of error free reconfiguration without considering measurednoise by simulation experiments for combination with less than5orientations; while consideringmeasured noise is confirmed, the reconfiguration error is in the statistic confidence interval.3)3Dspace precision self-calibration algorithm based on linear equations is proposed, the errorpropagation rate is deduced and simulation experiment is proceed; the theoretical character andsimulation result same with2D case stated before is found.4) Preliminary self-calibrationexperiment for shot error with visual image instrument is proved to be rational and also theeffectiveness of the proposed self-calibration algorithm is confirmed.
Thirdly, focused on the measuring problem for ultra-precise long feet products, deeplyresearched the two sets of2-point with changing distance measuring method. From the existingtwo sets of2-point method that based on Fourier transform, the thesis deduces the calculationcondition and expands the parameters choosing area successfully. In this way, application for thismethod became easier. Also, influence to system brought by zero point error is deducedquantificationally. Then optimization of two sets of2-point method is proposed to eliminate theinfluence of zero point error in the situation completely without considering influence ofmeasurement noise. While considering measurement noise, the optimal estimation solution isdeduced to minimize the influence brought by zero point error. And by Monte Carlo Method, theuncertainty of this method is calculated. At the same time, multiple probe method based on linearequations was applied to the two sets of2-point method, and the method based on linear equationswas formed. Two sets of2-point method based on Fourier transform and linear equations arecompared. Finally experiment research is completed.
The achievements gained from research of two sets of2-point method are listed as:1) Fromthe existing two sets of2-point method with changing distance that based on Fourier transform,the thesis deduced the calculation condition. The new condition avoids the situation in which theseparation distance between probes, in order to expand the real choosing interval of probeseparation.2) Influence to system brought by zero point error is deduced quantificationally. Thenoptimization of two sets of2-point method is proposed to completely eliminate the influence ofzero point error in the situation without considering influence of measurement noise. Theoretically,the precision of this solution not only depends on the volume of the measurement noise but alsohas no relation with zero point error.3) By Monte Carlo method, the uncertainty of this method iscalculated. The result, that the uncertainty of mid range on the measurement shape had the sameorder of magnitude with measurement noise, is proved.4) Two sets of2-point method which isbased on linear equations is deduced, linear equation system modeling and calculated theuncertainty of this algorithm has been finished. Without considering influence of measurementnoise, influence brought by zero point error can be eliminated to avoid influence to reconstruction.While considering the influence brought by measurement noise, the theoretical precision of thebest least square solution depends not only on the noise volume but not the zero point error.5) Acomprehensive comparison is made for two sets of2-point algorithm based on Fourier transformand linear equations respectively.6) It’s shown that, the algorithm based on linear equations isbetter than the algorithm based on Fourier transform only except for lower computing speed andlager memory.
Fourthly, aiming at the measuring problems of longer guide rail straightness, further researchon compounded multiple probe method which was based on combination of reversal and multipleprobe method is proposed. In this thesis, based on the measuring principle of reversal6-probemeasuring method proposed by GAO, a reversal4probe measuring method is proposed whichcould also realize straightness reconstruction but using less probes. By modeling a3m longverification system, the sensor data collecting program which could run without PC is designed.Also, the practical program used for dealing with continuous scanning data and unusual data iscompiled. It is proved by experiment that the repeatability of reconstruction result was below30μm when reversal4-probe method and reversal6-probe method are applied under differentsituation of scanning speed and sampling interval. While using laser tracker to measure the sameguide rail, the difference of result was below50μm compared with reversal4-probe measurementmethod, which is also a credible proof of effectiveness of reversal4-probe measurement method.
At the end of the thesis, the research achievements and the work were summarized, and thefollowing research is proposed. In the diversified shape measuring field of precise measuring,error separation technology plays an important role as the core and fundamental technology. Errorseparation technology will have more and more practical usage as the ultra-precision measuringneed grows.
应用误差分离技术的超精密形状（误差）测量能够使测量精度达到目前所能达到精度的极限。这对提供最高计量标准的国家标准研究所和计量测试技术研究所来说是必不可少的技术。其次，对机床等加工机械进行数值补偿是机械运动的形状测量的另一大目的，可信度高，实施快捷简单的校正方法将是必不可缺的。最后，应用误差分离技术的形状测量能够以更低的硬件投资达到所需要的测量精度，符合建立节约型社会的目标。因此，超精密测量方法中的误差分离技术作为方法论的核心和基础技术，对它的理论研究是现代加工和测量技术的不可或缺的关键技术。而应用此技术的测量方法和校正方法也具有广阔的市场前景。
本文依托国家自然科学基金资助项目“超精密形状误差测量中的误差分离技术研究”（项目编号：50905116）和校企合作项目“电梯轿厢导轨直线度及角度位置关系检测系统”（合同编号：H3007144）开展研究工作，以超精密加工和测量为应用环境，结合项目中电梯轿厢导轨直线度检测的需要，对超精密测量相关的误差分离技术中的若干问题进行系统研究。
首先，论文介绍了本研究的背景、目的和意义。指出了开展以基于误差分离技术的超精密测量方法为研究对象的重要性，分析了包括多步法、自校正法和多探头法的误差分离技术的国内外研究现状及存在的问题，并在此基础上，提出了本论文的主要研究内容。
接着，论文对基于线性方程组的自校正方法进行了系统的研究。从二维工作台定位误差自校正算法的提出，再扩展到三维空间精度自校正法，形成了从二维到三维自校正法的初步体系化。相较原有的基于傅里叶变换的自校正法，基于线性方程组的自校正方法可方便地进行理论误差传播率的推导、具有良好的噪声抑制能力、可对三位姿以上的组合进行自校正计算。
通过本文的自校正研究，取得了以下成果：1）提出了基于线性方程组的二维工作台定位误差自校正算法；推导了测量噪声传递到重构结果的误差传播率，理论表明本算法重构结果的不确定度值小于测量噪声的标准差；2）对五位姿以下的组合测量进行了仿真实验，确认了本算法在不计测量噪声的情况下可实现无误差重构；在考虑测量噪声情况下，重构误差是处于统计学意义上的置信区间内的；3）提出了基于线性方程组的三维空间精度自校正算法，推导了误差传播率，并进行了仿真实验，其理论特性和仿真结果与上述二维的情况一致；4）利用视觉影像仪对镜头误差进行初步的自校正实验表明以上实验结果合理，同时也证明了本文所提出的自校正算法的有效性。
其次，针对超精密长尺度加工件的直线度（轮廓形状）测量问题，论文对多探头法中的变间距2组2点法进行了深入研究。本文在既有的基于傅里叶变换的变间距2组2点法的基础上，重新推导了此方法的计算条件；接着，定量推导了传感器间零点误差对系统造成影响，进而提出了改良的变间距2组2点法，采用蒙特卡罗法计算了该方法的不确定度。与此同时，本文将基于线性方程组的多探头算法应用到变间距2组2点法上，推导了基于线性方程组的变间距2组2点法；并对基于线性方程组的方法与基于傅里叶变换的方法进行了比较。最后，搭建了测量系统进行了实验研究。
通过本文的变间距2组2点法研究，取得了以下成果：1）根据既有的基于傅里叶变换的变间距2组2点法算法，重新推导了参数条件。新的参数条件使传感器间距设置不再直接受限于测量点数，扩大了传感器间距的的实际选择范围。2）定量分析了传感器测头间的零点误差对重构结果造成的影响，提出了改良的傅里叶变换算法；在不计测量噪声的情况下改良算法消除零点误差带来的影响，实现无误差重构；在测量噪声的情况下，推导了一种加权平均最优估计解。3）采用蒙特卡罗法对上述最优估计解的不确定度进行了定量计算。结果表明处于形状中间部分的不确定度与测量噪声是同一数量级的。4）推导了基于线性方程组的变间距2组2点算法，建立了相应的线性方程组模型并计算了该方法的理论不确定度。在不计测量噪声的情况下，本算法可完全消除零点误差带来的影响，实现无误差重构；在考虑测量噪声的情况下，最小二乘解的精度在理论上与零点误差的大小无关，只取决于测量噪声的大小。5）将基于傅里叶变换的方法与基于线性方程组的方法进行了全面比较。6）实验结果表明，对同一测量数据采用2种不同算法得到的重构结果几乎一致，结果表明除计算速度和内存消耗的劣势以外，基于线性方程组的方法在各方面都优于基于傅里叶变换的方法。
再次，针对长导轨直线度测量问题，论文对基于反转和多探头法融合的混合多探头法进行了深入研究。本文在详细说明了GAO的反转六探头法的测量原理的基础上提出的反转四探头法少用两个探头同样实现直线度重构功能。在实验室内搭建了3米长的验证系统，设计了可脱离PC机运行的传感器组数据采集方案，编写了可处理连续扫描数据和异常数据的实用化程序。实验表明，反转六探头法和反转四探头法在不同扫描速度和采样间隔测量条件下的重构结果的重复性在30μm以下。使用法如激光跟踪仪对同一导轨进行了测量，其结果和反转四探头法的重构结果的差在50μm以下，从而有力证明了本论文提出的反转四探头法的有效性。
最后，对整个论文的工作和研究成果进行了总结，并提出了下一步的研究展望。精密测量法中的误差分离技术作为方法论的核心和基础技术，一直以来在多样化的形状测量领域占有一席之地。而随着超精密测量需求越来越多，误差分离技术也将得到越来越多的实际应用。
With the development of new materials and ultra-precision machining technology, the size ofthe tested object is turn larger and larger, machining accuracy and surface quality is turn higherand higher. common measuring instruments can not provide hard benchmarks accuracy exceedingthe ultra-precision machining accuracy. Measurement and detect are more difficult than machining.In this background, error separation technology, which is a kind of measurement technique of avirtual soft benchmark accuracy higher than the hard baseline accuracy by application of aparticular measurement system and the appropriate combination of software to separate the errorinherent in the hard benchmark, has produced.
Ultra-precision shape (error) measurements of the application of error separation technologymake the measurement accuracy achieve the current limit of achievable accuracy. To the NationalInstitute of Standards and Measurement and Testing Technology Research Institute, this is theessential technology to provide the highest standards of measurement. Secondly, the numericalcompensation of machine tools and other processing machinery is another major purpose of theshape of the mechanical movement of the measurement. High reliability, quick and simplecalibration method is essential. Finally, the shape of the application of error separation techniquescan to invest lower hardware to achieve the required measurement accuracy, in line with goals toestablish saving society. Therefore, as the core and foundation of the methodology, theoreticalstudy of the error separation techniques in ultra-precision measurement method is the keytechnology of modern processing and measurement techniques indispensable. The methods ofmeasurement and calibration for the application of this technology also have broad marketprospects.
Supported by the Nature Science Fund of China (NSFC)(Project Number:50905116), theCooperation Project of University-enterprise (Contract No. H3007144) under the ultra-precisionmachining and measurement applications environment, combined with the project for the needs ofthe elevator car guide rail detection, ultra-precision measurement error separation techniques arestudied systematically. Theoretical research on the ultra-precise two-dimensional platformpositioning error self-calibration methods are conducted and optimized. The measuring device ofthe elevator car guide rail straightness has been developed.
The main content and contributions of this thesis are summarized as follows:
Firstly, the background, purpose and importance are introduced based on collecting and mastering a large amount of documents and references on error separation technology. Theimportance of research on ultra-precision measurement based on error separation techniques ispointed out. The research status and existing problems of error separation techniques, such asmultistep method, self-calibration method and multiple probe method are analyzed. The mainresearch content is based on this analysis.
Secondly, the systematic research on self-calibration method based on linear equations isdone. From the proposing of error self-calibration algorithm for2D workbench location, expandsto3D space precision self-calibration method is deduced, self-calibration method from2D to3D ispreliminarily systematized. Compared with the existing self-calibration method based on Fouriertransform, the self-calibration method based on linear equations has advantages of convenientdeduction of theoretical error propagation rate, good capability of noise elimination and realizationof self-calibration for combination with more than3orientations.
The achievements gained in the research of self-calibration method are summarized as below:1)2D self-calibration algorithm based on linear equations is proposed; error propagation rate thattransmitted to reconfiguration result by measured noise is deduced. Theory indicates that theuncertainty value of reconfiguration result for this algorithm is smaller than the standard deviationof measured noise.2)The realization of error free reconfiguration without considering measurednoise by simulation experiments for combination with less than5orientations; while consideringmeasured noise is confirmed, the reconfiguration error is in the statistic confidence interval.3)3Dspace precision self-calibration algorithm based on linear equations is proposed, the errorpropagation rate is deduced and simulation experiment is proceed; the theoretical character andsimulation result same with2D case stated before is found.4) Preliminary self-calibrationexperiment for shot error with visual image instrument is proved to be rational and also theeffectiveness of the proposed self-calibration algorithm is confirmed.
Thirdly, focused on the measuring problem for ultra-precise long feet products, deeplyresearched the two sets of2-point with changing distance measuring method. From the existingtwo sets of2-point method that based on Fourier transform, the thesis deduces the calculationcondition and expands the parameters choosing area successfully. In this way, application for thismethod became easier. Also, influence to system brought by zero point error is deducedquantificationally. Then optimization of two sets of2-point method is proposed to eliminate theinfluence of zero point error in the situation completely without considering influence ofmeasurement noise. While considering measurement noise, the optimal estimation solution isdeduced to minimize the influence brought by zero point error. And by Monte Carlo Method, theuncertainty of this method is calculated. At the same time, multiple probe method based on linearequations was applied to the two sets of2-point method, and the method based on linear equationswas formed. Two sets of2-point method based on Fourier transform and linear equations arecompared. Finally experiment research is completed.
The achievements gained from research of two sets of2-point method are listed as:1) Fromthe existing two sets of2-point method with changing distance that based on Fourier transform,the thesis deduced the calculation condition. The new condition avoids the situation in which theseparation distance between probes, in order to expand the real choosing interval of probeseparation.2) Influence to system brought by zero point error is deduced quantificationally. Thenoptimization of two sets of2-point method is proposed to completely eliminate the influence ofzero point error in the situation without considering influence of measurement noise. Theoretically,the precision of this solution not only depends on the volume of the measurement noise but alsohas no relation with zero point error.3) By Monte Carlo method, the uncertainty of this method iscalculated. The result, that the uncertainty of mid range on the measurement shape had the sameorder of magnitude with measurement noise, is proved.4) Two sets of2-point method which isbased on linear equations is deduced, linear equation system modeling and calculated theuncertainty of this algorithm has been finished. Without considering influence of measurementnoise, influence brought by zero point error can be eliminated to avoid influence to reconstruction.While considering the influence brought by measurement noise, the theoretical precision of thebest least square solution depends not only on the noise volume but not the zero point error.5) Acomprehensive comparison is made for two sets of2-point algorithm based on Fourier transformand linear equations respectively.6) It’s shown that, the algorithm based on linear equations isbetter than the algorithm based on Fourier transform only except for lower computing speed andlager memory.
Fourthly, aiming at the measuring problems of longer guide rail straightness, further researchon compounded multiple probe method which was based on combination of reversal and multipleprobe method is proposed. In this thesis, based on the measuring principle of reversal6-probemeasuring method proposed by GAO, a reversal4probe measuring method is proposed whichcould also realize straightness reconstruction but using less probes. By modeling a3m longverification system, the sensor data collecting program which could run without PC is designed.Also, the practical program used for dealing with continuous scanning data and unusual data iscompiled. It is proved by experiment that the repeatability of reconstruction result was below30μm when reversal4-probe method and reversal6-probe method are applied under differentsituation of scanning speed and sampling interval. While using laser tracker to measure the sameguide rail, the difference of result was below50μm compared with reversal4-probe measurementmethod, which is also a credible proof of effectiveness of reversal4-probe measurement method.
At the end of the thesis, the research achievements and the work were summarized, and thefollowing research is proposed. In the diversified shape measuring field of precise measuring,error separation technology plays an important role as the core and fundamental technology. Errorseparation technology will have more and more practical usage as the ultra-precision measuringneed grows.
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