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机床误差分量的提取和利用统计分析误差补偿AbstractThe extraction of component errors of a machine tools axes is a critical step for the synthesis of 3D volumetric error mapping,which is a prerequisite to improve the machine tool accuracy by numerical Error compensation.This paper presents a method for the extraction of machine tool component errors from a statistical point of view.First,the B-Spline mathematical model is established to represent the Component error function,and the least-squares tting method to measured data points is presented.Then,statistical analysis is used to select the B-Spline model with proper exibility,so as to separate repeatable errors from random errors in the measured data. Finally, based on the component error extraction method, numerical error compensation experiments were conducted on the XY-plane of a High precision machine tool by using across-grid scale system.According to the statistical analysis of The experimental data,all repeatable errors except the dynamic errors caused by machine tool control system were compensated for.Key words:Machine tool accuracy, Component error ,B-Spline, Statistical analysis, Error mapping翻译：摘要机床的轴组件误差的提取是三维空间误差的映射的合成的关键一步，这是通过数值误差补偿提高机床精度一个先决条件。本文提出了一种从统计的角度来看的机床部件误差提取方法。首先，本文提出了描述元件误差函数的B样条曲线的数学模型和最小二乘拟合的方法来测量数据点。然后，利用统计分析来选择适当的灵活性B样条曲线模型，用来从测量数据的随机误差中分离重复误差。最后，在采用交叉网格尺度系统的高精度机床的XY平面上进行的基于误差分量的提取方法，数值误差补偿实验。根据实验数据的统计分析，所有的重复误差除了由机床控制引起的动态误差都得到了补偿。关键词：机床精度 误差分量 B样条 统计分析 错误映射1. IntroductionThe manufacturing of advanced mechanical parts requires highly accurate machine tools.The development of even more accurate machine tools has been a constant pursuit of the precision engineering community.Due to geometric,kinematic,thermal, cutting-force-induced,and other errors,it is very expensive to further improve the accuracy through design and manufacturing solely,especially at the sub-micron level 1,2.1.介绍先进的机械零件的制造要求精度高的机床。发展更加精密的机床是精密工程社区的不断追求。由于几何，运动学，热，剪切力，和其他的误差，通过进一步改善单独的设计和制造来提高精度是非常昂贵的，特别是在亚微米级 1,2。 Numerical error compensation is a cost-effective way to realize high-precision machining.Since the effort done by Hocken et al. 3, more and more research has been carried out on numerical error compensation for machine tools and coordinate measuring machines. Donmez et al. 4 presented a general methodology for the compensation of quasi-static machine tool errors,including geometric errors and thermally induced errors.In this method, homogeneous transformation matrices were used to describe the relative motion between adjacent machine tool axes,and the volumetric error model was acquired by multiplying a series of homogeneous transformation matrices. Sartori and Zhang 5 reviewed the methods for geometric error measurement and compensation that were developed before 1995.In their paper, they have described the foundations of numerical compensation, including a common language for error measurement and compensation, conditions for carrying out successful error compensation, various error measurement and compensation methods, etc. Recently, Schwenke et al. 6 gave an updated review on the methods for geometric error measurement and compensation from the viewpoint of practical application, including the appearance of new calibration methods, new concepts in international standards, and growing capabilities of machine tool controllers for error compensation.数值误差补偿是一种具有成本效益来实现高精度加工的方式。由于霍肯等人所做的努力3,已经进行了越来越多的基于对机床的数值误差补偿和坐标测量机的研究。Donmez 等人 4 提出了一个通用的方法来进行准静态机床误差补偿，包括几何误差和热引起的误差。在该方法中，用齐次变换矩阵来描述相邻的刀具轴之间的相对运动，并用取得的空间误差模型乘以一系列的齐次变换矩阵。萨托利和张 5 回顾了在1995年之前发现的几何误差测量与补偿方法。在他们的论文中，他们描述了数字补偿的基础，包括误差测量与补偿的共同语言，实施成功误差补偿的条件，各种误差测量和方法等。最近,Schwenke等人6从实际应用的角度给出了一个基于几何误差测量和补偿的更新审查方法,包括新出现的校准方法,国际标准的新概念，和具有增长能力的机床误差补偿控制器。Moreover, most papers focused on the mathematical modeling of the machine tool volumetric error (3D positioning error) and the methods (instruments) to measure the six component errors of the machine tool linear axis, i.e. positioning error, horizontal straightness error, vertical straightness error, roll, pitch, and yaw errors. Realizing the importance of modeling of machine tool component errors to the successful synthesis of 3D volumetric errors, several researchers discussed the mathematical modeling of component errors by processing the raw data measured by instruments. Florussen et al. 7 used ordinary polynomial functions to represent the component errors; the square root of the mean sum of squared errors was analyzed to select the appropriate order of the polynomial function. Kono et al. 8 analyzed machine motion errors in the frequency domain to separate the geometric errors from time-dependent errors. However, due to leakage errors, the truncated Fourier series is not able to describe the component error in both of its ends accurately because the measured error data rarely satisfy the periodic property. In the available methods for the modeling of machine tool component errors 7,8, random errors are coupled with repeatable errors. For high precision applications (sub-micron level), random errors of the machine tool may be comparable to repeatable errors, thus the separation of repeatable errors and random errors becomes desirable. In this paper, a method is proposed to represent machine tool component errors by using a B-Spline model. Statistical analysis is applied to select the B-Spline model with proper exibility, which is used to extract the component error function from the data acquired by repetitive measurements. The extracted error function is treated as repeatable errors only, and the random errors are described by the residual errors, which are subject to a zero mean normal distribution. In order to verify the effectiveness of the component error extraction method, numerical error compensation on the XY-plane of a high precision machine tool is presented, including construction of error mapping, compensation G-Code generation, and analysis of experimental results.此外，大多数论文致力于对机床空间误差的数学建模（三维定位误差）和方法（仪器）测量机床线性轴的六个误差分量，即定位误差，水平直线度误差，垂直的直线度误差，滚动，倾斜，摆动误差。认识到对三维空间误差合成的成功机床部件误差建模的重要性，一些研究人员探讨了通过仪器测量的原始数据的处理对组件误差的数学建模的重要性。Florussen等人【7】使用普通的多项式函数来表示组件误差；对平均值的总和的平方根的平方误差进行了分析，选择合适的多项式函数。Kono等人【8】用频域分析机床运动误差分离几何误差随时间变化的误差。然而，由于泄漏误差，截断的傅里叶级数是无法描述在其两端准确的组件的错误，因为所测量的误差数据很少满足周期性特性。在现有的方法对机床组件误差 7,8 建模，随机误差加上重复错误。对于高精度的应用（亚微米级），机床的随机误差可以媲美重复误差，从而可重复的误差和随机误差的分离变得可取。在本文中，提出了利用B样条模型代表机床部件误差的方法。统计分析方法被用来选择合适的柔性B样条模型，这用来从通过重复测量获得的数据中提取成分的误差函数。提取的误差函数只作为可重复的误差，用残余误差的描述随机误差，这是一个零均值的正态分布。为了验证该误差分量提取方法的有效性，对高精度机床的XY平面数值误差补偿的方法，包括误差映射结构，补偿G代码的生成，并对实验结果的分析。2. Mathematical model of component error and repeatable error extraction method2.1. B-Spline model for the representation of the machine tool component error function The shapes of machine tool component errors vary according to different error types and different machines, thus a versatile mathematical model is needed. In this paper, a B-Spline model is used to represent the machine tool component errors: (1)2 .组件的数学模型误差和重复误差提取方法2.1b样条模型表示机床组件的误差函数机床组件的形状误差根据不同的错误类型和不同的机器,因此需要一个通用的数学模型。在本文中用b样条模型用于表示机床组件错误: (1)where xsxe is the nominal movement range of the machine axis, Vi are ordinates, Ni,p(u) are basis functions dened on a knot vector U=u0,u1,.,un + p + 1, and p is the degree of basis functions 9. Compared with the nominal movement range, the magnitude of ordinates Vi is very small, thus uniform B-Spline basis functions are adequate, and the knot vector is chosen as (2)The basis functions are defined recursively as (3)其中xsxe是机床轴的名义移动范围，Vi是纵坐标，Ni，P（U）的基础功能去连接定义在节点矢量U =U0，U1，.，UN+ P+1，p是基函数9的程度。与额定运动范围相比，纵坐标Vi的幅度非常小，从而均匀B样条基函数是充分的，并且所述节点矢量被选择为：基函数被递归的定义为：2.2. Algorithms for component error extraction Since random errors exist in the motion of the machine tool and measurement process of the measuring device, the measured machine tool error data are not directly applicable for numerical error compensation. However, random errors usually are subject to a zero mean normal distribution 10, and thus can be separated from the repeatable errors by performing repetitive measurements and applying statistical data analysis. In order to extract the repeatable error from the measured data, the B-Spline mathematical model will be fitted to the measured data 11. 11. Once the order p is selected(the common choice is3,which provides a C2 continuity and good local support property),the number of ordinates n+1 determines the flexibility of the B-Spline mathematical model. In order to carry out statistical analysis, the B-Spline model fitting algorithm produces not only the values of ordinates but also the statistical information of the residual errors, i.e. the mean and standard deviation. In this section, the B-Spline function fitting algorithm is presented, followed by the method for component error extraction with statistical analysis2.2。算法提取组件误差由于随机误差存在于机床的运动和测量过程的测量装置,测量机床误差数据并不直接适用于数值误差补偿。然而,随机误差通常受到零意味着正态分布10,可以分开可重复的错误,从而执行重复的测量和统计数据分析的应用。为了从测量数据中提取的可重复的错误,将b样条数学模型拟合测量数据11,11。一旦订单选择p(is3共同的选择,它提供了一种C2连续性和良好的本地支持属性),纵坐标n + 1的数量决定了b样条数学模型的灵活性。为了进行统计分析,b样条模型拟合的算法产生不仅纵坐标的值,而且剩余的统计信息错误,即平均值和标准偏差。在本节中,给出了b样条函数拟合的算法,其次是组件的方法误差提取和统计分析(1) B-Spline model tting algorithm: BSpline_Fitting (n, p, S, V, m, s).Input: degree of B-Spline basis functions p, number of control ordinates n + 1, and measured data set S=(xi,yi), i=0,1,.,m. Output: ordinates V =V0 ,V1 ,. . .,Vn T , mean of residual errors m, and standard deviation of residual errors .The detailed steps are as follows:Step 1. Calculate the parameter values of data points by ui=(xi-xs)/(xe-xs)Step 2. Calculate the observation matrix of the data points byStep 3.Calculate the ordinates by solving NTNV=NTY, where V =V0,V1,. . .,VnT , Y =y0,y1,. . .,ymT . Step 4. Calculate the residual errors of each data point byStep 5. Calculate the mean and the standard deviation of the residual errors of the data set by(2) Method for component error extraction with statistical Analysis.Step 1. Perform a series of BSpline_Fitting (i, p, S, V, , ), i=p + 1, p+ 2,.q (q is a reasonably large number for the application under discussion). Step 2. Check the variation of mean m and standard deviation s with respect to the number of ordinates i and determine the proper number of ordinates iproper beyond which no further signicant decreases on m and s can be achieved by increasing the number of ordinates i in the tting.Step 3. Select the B-Spline function tted by B-Spline_Fitting (iproper, p, S, V, , ) as the extracted component error function.（1）B样条模型拟合的算法：B样条拟合(n, p, S, V, m, s）。输入：B样条程度的基函数p，控制坐标n+1和测量数据集S=(xi,yi), i=0,1,.,m。输出：坐标V =V0，V1，。 。 。，VN T，平均误差，剩余误差的标准偏差。详细步骤如下：步骤1： 计算数据点的参数值ui=(xi-xs)/(xe-xs)步骤2：通过计算数据点的观测矩阵步骤3：通过求解NTNV=NTY得到大量计算的坐标，其中V =V0,V1,. . .,VnT , Y =y0,y1,. . .,ymT . 步骤4：计算每个数据点的残余误差通过步骤5：计算平均值和设定的数据误差的标准偏差通过（2）用统计分析的误差分量的提取方法。步骤1。进行一系列的B样条拟合(i, p, S, V, , )，i=p + 1, p+ 2,.q （q在应用讨论中是一个相当大的量）。步骤2.检查的平均值和标准差 的变化，以纵坐标的数量i和确定坐标的适当数目iproper观察其有没有随着减小而减小，并且可以通过增加在纵坐标的数量来实现该拟合。步骤3。选择B样条函数通过B样条拟合（iproper，p, S, V, , )作为提取的分量误差函数2.3. An example of machine tool component error extractionThe approach is explained by using an example of extracting straightness error function. In this example, straightness error data from the X-axis of a high precision machine tool was measured by using the KGM cross-grid scale system 12, as shown in Fig. 1. In order to separate repetitive machine tool errors from random noise by using a statistical method, ve repetitive measurements were carried out to measure the horizontal straightness error of the X-axis. This number of repetitions is also suggested by ISO 230-2 standard for the measurement of positioning errors of machine axis 13. The measured data of the ve measurements and their combination (10,000 data points) are shown in Fig. 2.2.3。机床误差分量例提取该方法是利用提取的直线度误差函数的一个例子说明。在这个例子中，利用KGM交叉网格尺度系统12来测量高精度机床X轴直线度误差的数据，如图1所示。为了分离机床重复误差的随机噪声，利用统计方法，通过五次重复测量来测量X轴的水平直线度误差。这种重复次数也由ISO230-2对机床轴定位误差的测量采用 13 。五次的测量以及它们的组合的测量数据（10000个数据点）如图2所示。In the least-squares tting of the B-Spline model to the measured data, the knot vector, the degree of basis functions, and the number of ordinates need to be determined beforehand. As mentioned in Section 2, a uniform knot vector and third-degree basis functions are appropriate for the representation of machine tool component errors. The only factor that controls the exibility of the B-Spline model is the number of ordinates. As the number of ordinates increases, the exibility of the B-Spline model (the ability to represent a more complex shape) increases.在B样条模型实测数据最小二乘拟合，节点矢量，基函数的程度，和坐标数需要事先确定。在2节中提到的，一个统一的节点矢量和三度的基函数是适当的机床部件误差的表示。控制的B样条模型灵活性的唯一因素是坐标的数目。随着坐标的数目增加，B样条模型灵活性（代表一个更复杂的形状的能力）增加。In order to select the B-Spline model with proper exibility, a series of B-Spline models with increase in number of ordinates was tted to the combined data of ve repetitive measurements. The statistical information, i.e. the means and standard deviations of residual errors from the ttings with B-Spline function are shown in Figs. 3 and 4, respectively. The selection of model exibility (number of ordinates) is based on the observation. A model with insufcient exibility cannot fully represent the repeatable errors; on the other hand, a model with too much exibility will introduce random errors into the extracted error function. Therefore, the model with proper exibility is important for the full representation of repeatable errors and ltering out random errors in the extracted component error function.为了选择合适的灵活性B样条模型，一系列的坐标的数目增加使B样条模型是符合五次结合的重复测量数据的。这一统计信息，从装置与B样条函数方法得到的残差标准差如图3和4分别所示。灵活性模型的选择（坐标的数目）是基于这样的观察。一个有效的灵活性不足的模型不能完全代表重复性误差；另一方面，有太多的灵活性的模型会引入随机误差到将提取的误差函数中。因此，适当的灵活性的模型在重复错误和筛选出随机误差在提取元件误差函数随机误差的完整的表示是很重要的。To nd the proper number of ordinates of the B-Spline model for the representation of machine tool component error, statistical information can be utilized. From Fig. 3, it can be seen that the mean of residual errors is extremely small (10 17 mm) regardless of the number of ordinates. Theoretically, the mean of the residual errors is zero, which can be explained as follows:为了找到适当数量的B样条模型的纵坐标，以得到机床误差分量的表示，可以利用统计信息。从图3可以看出，残余的均是非常小的（1017毫米）忽略坐标数。从理论上讲，残余误差的均值为零，这可以解释如下：Let R denote the vector of residual errors, and it can be represented as （7）Multiply both sides of Eq.(7) with NT, then we have （8）Add the individual equations in Eq.(8), we have （9）R表示残差向量，可表示为： （7）两边同乘以 NT, 我们可以得到 （8）在方程8两边加上单独的方程，我们得到 （9）Due to the property of partition of unity of B-Spline basis functions i Ni,p（uj）=1 9, Eq(9) can be simplied to rj=0. Thus the mean of the residual errors is also zero.由于对B样条基函数统一分区的属性Ni,p（uj）=1 9 ，公式（9）可以简化为rj = 0。这样的残余误差的均值也为零From Fig. 4, it can be seen that the standard deviation is strongly related to the number of ordinates. The standard deviation almost reaches the minimum with 20 ordinates, and there is no obvious improvement with the further increase of ordinates. Thus, the straightness error function is acquired by tting the B-Spline model with 20 ordinates. To show the importance of tting with the proper number of ordinates, three cases tting with insufcient ordinates, proper number of ordinates, and excessive ordinates are given below:从图4可以看出，标准偏差和纵坐标的数量密切相关的。标准差几乎达到最小20个坐标，随着坐标的进一步增加并没有明显好转。因此，直线度误差的函数的拟合可以通过B样条模型的20个坐标获取。为了显示设置与适当坐标数量的重要性，三个例子的纵坐标不足的拟合，适当数量的坐标，和过度的坐标如下：(1) Fitting with insufcient ordinates:when a model with insufcient exibility is tted to the data, it cannot represent the inherent complexity of repeatable errors. Some of the repeatable errors are recognized as random errors, and thus the standard deviation is big and the distribution of residual errors is far from a zero mean normal distribution. Refer to Figs. 5 and 6 for B-Spline tting with 4 ordinates.（1） 有效坐标的拟合：当一个柔性不足的模型做数据拟合，它不能代表重复错误的固有的复杂性。一些重复误差是随机误差，因此标准偏差大，残余误差的分布是一个远离零均值的正态分布。参考图5和6的B样条拟合的四个坐标。 (2) Fitting with proper number of ordinates:in order to distinguish random error from repeatable error, it is critical to select a model with proper exibility to t the measured data. The straightness error function extracted with a model of proper number of ordinates (20 ordinates) is shown in Fig. 7. The histogram of the residual errors of the extracted error function is shown in Fig. 8, which is very close to a zero mean normal distribution（2）与适当数量坐标的拟合：为了区分随机误差的重复性误差，选择了一个适当的灵活性的模型来适应测量数据。有适当数量的坐标模型中提取出的直线度误差函数（20个坐标）如图7所示。对所提取的误差函数残差的直方图如图8所示，这是非常接近一零均值的正态分布。(3) Fitting with excessive ordinates:when a model with too much exibility (excessive ordinates) is used in the tting, although repeatable errors can be represented, random errors will also be tted by the model and the extracted component error function will be contaminated by random errors. For example, when tting with 50 ordinates, no further improvement in the magnitude and distribution of the residual errors can be seen, as shown in Figs. 9 and 10.（3）多坐标拟合：当一个用于拟合的模型过于灵活（过度坐标），虽然重复性误差可以表示，随机误差也可以用于模型拟合和通过随机误差提取的分量误差函数。例如，当安装在50个坐标，残余误差的大小和分布没有进一步的改进。从图9和10可以看出。3. Numerical error compensation on a high precision machine ToolBased on the proposed machine tool component error extraction method, numerical error compensation on the XY-plane of the linear-driven high precision machine tool was studied. First, the 2D error mapping between the