Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 3: variation reduction

In this part of the paper, the uncertainty analysis of coordinate estimation for the case in which the sampled geometric errors are dominated by the random component is investigated. In practice, the uncertainty analysis of coordinate estimation using high-precision datum surfaces often falls into this type. In this paper, a sensitivity matrix, which serves as the theoretical basis of the uncertainty analysis, is presented to establish a linearized relationship between the variations of the coordinate estimation and the geometric errors of the part surface at the measurement points. Based on the sensitivity matrix, quantitative measures are derived for the prediction of coordinate variation. Computer simulation and experiments are conducted to verify the theoretical predictions when the surface geometric errors are small and dominated by the random component.     

结晶器坐标测量机不确定度分析及精度保证

文章对影响结晶器专用坐标测量机测量精度的各项误差进行了详细分析，用齐次坐标理论建立了误差修正模型，对影响测量精度比较大的21项几何误差进行修正，并给出各项几何误差的传递系数。通过对误差修正后的测量机各轴不确定度进行评定，找出影响测量精度的瓶颈因素，分析了存在的问题并给出改进措施，最后通过实验验证了该方法的正确性和实用性，为保证测量机的精度提供了理论指导。     

A unified least-squares approach to the evaluation of geometric errors using discrete measurement data

This paper presents a unified least-squares approach to the best fit of geometric features and evaluation of dimensional errors. The study originated from the need for advanced algorithms for the dimensional measurement of high precision manufactured parts. The proposed algorithm differs from traditional least-squares in that no linearization or approximation is employed and that it is general to all kinds of geometric features. Instead of computing a substitute best fit feature, the algorithm inversely transforms the measurement coordinates to best fit the nominal geometry. The sum of the squared errors in the surface normal direction is thus minimized with respect to the parameters of a rigid body transformation. Form tolerances are then evaluated using the peak-to-valley deviations after the best fit. To examine the uncertainty of the transformation obtained, sensitivity analysis was investigated to relate transformation errors to dimensional errors. A sensitivity measure is used to estimate the joint effect of the measurement locations and the number of measurement data on the accuracy of the coordinate transformation. Computer simulations have been performed on different geometric features to study the robustness and efficiency of the algorithm. Application to the measurement of high precision fuel injectors is also presented.     

Development of a virtual machining system, part 3: cutting process simulation in transient cuts

In Parts 1 and 2 of this three-part paper, a mechanistic cutting force model was developed and machined surface errors for steady cuts under fixed cutting conditions were predicted. The virtual machining system aims to simulate and analyze the machining and the machined states in a general flat end-milling process. This frequently involves transient as well as steady cuts. Therefore, a method for simulating the cutting process of transient cuts needs to be developed to realize the virtual machining system concept. For this purpose, this paper presents a moving edge-node (ME) Z-map model for the cutting configuration calculation. The simulation results of four representative transient cuts in two-dimensional pocket milling and an application of off-line feed-rate scheduling are also given.

In transient cuts, the cutting configurations that are used to predict the cutting force vary during the machining operation. The cutting force model (Part 1) and surface error prediction method (Part 2) were developed for steady cuts; these are extended to transient situations using the ME Z-map model to calculate the varying cutting configurations efficiently. The cutting force and surface errors are then predicted. To validate the feasibility of the proposed scheme, the measured and predicted cutting forces for transient test cuts were compared. The predicted surface error maps for transient cuts were constructed using a computer simulation. Also, off-line feed-rate scheduling is shown to be more accurately performed by applying the instantaneous cutting coefficients that were defined in Part I.     

五轴联动立式磨床几何误差灵敏度研究

以一台五轴联动立式磨床为例,提出了一种分析几何误差灵敏度的方法。基于多体系统理论和齐次坐标变换建立了五轴联动立式磨床的33项几何误差模型,利用求导的方式给出了几何误差灵敏度分析的数学模型。并根据该模型找出了关键几何误差,研究了关键几何误差的灵敏度系数随平移轴运动的变化规律,研究结果为精密机床改进设计和误差的实时补偿提供了理论依据。     

数控车床几何误差建模与应用

以多体系统理论为基础结合齐次坐标变换原理,以i5T5.2型数控车床为例,分析其组成部件及拓扑结构,通过Matlab软件编写程序建立了该型车床的几何误差模型。通过计算机快速准确地对误差模型进行分析,并将分析结果以统计图的形式显示,便于发现误差分布规律及误差作用规律。通过机床实际加工棒料测得加工误差并与机床误差建模分析结果对比,较好地验证了模型的正确性。在后续实际加工中应用误差建模分析数据进行误差补偿,有效地提高了该型机床的加工精度。     

Integrated machining error compensation method using OMM data and modified PNN algorithm

This paper presents an integrated machining error compensation method based on polynomial neural network (PNN) approach and inspection database of on-machine-measurement (OMM) system. To improve the accuracy of the OMM system, geometric errors of the CNC machining center and probing errors are compensated. Machining error distributions of a specimen workpiece are measured to obtain error compensation parameters. To efficiently analyze the machining errors, two machining error parameters, W_err and D_err, are defined. Subsequently, these parameters can be modeled using the PNN approach, which is used to determine machining errors for the considered cutting conditions. Consequently, by using an iterative algorithm, tool path can be corrected to effectively reduce machining errors in the end-milling process. Required programs are developed using Ch language, and modified termination method are applied to reduce computation times. Experiments are carried out to validate the approaches proposed in this paper. The proposed integrated machining error compensation method can be effectively implemented in a real machining situation, producing much fewer errors.     

Evaluation of the measurement uncertainty of a positional error calibrator based on a laser interferometer

A positional error calibrator for evaluating the positioning accuracy of machine tools and coordinate measuring machines (CMM) under dynamic conditions has been developed. It is based on the Hewlett Packard 5529A laser interferometer, which is capable of performing dynamic calibration. This laser interferometer measures the position of the slide as it moves continuously along the machine axis under test. The data are collected on a position basis by triggering the laser interferometer from a position-based reference signal. A software package has been developed for data acquisition and presentation of the positional errors in accordance with ISO 230-2 standard. This paper deals with the evaluation of the measurement uncertainty of this positional error calibrator. In order to assess the total measurement uncertainty of this calibrator, an analysis of the individual errors that make up the accuracy and repeatability error budgets has been carried out. These error budgets consist of the following components: (1) uncertainties intrinsic to the laser system; (2) uncertainties due to environmental effects; (3) measuring uncertainties due to the installation. This uncertainty analysis was carried out when this calibrator was used to assess the positional errors of a moving bridge type CMM at UMIST.     

Validation of volumetric error compensation for a five-axis machine using surface mismatch producing tests and on-machine touch probing

In order to validate volumetric error compensation methods for five-axis machine tools, the machining of test parts has been proposed. For such tests, a coordinate measuring machine (CMM) or other external measurement, outside of the machine tool, is required to measure the accuracy of the machined part. In this paper, a series of machining tests are proposed to validate a compensation strategy and compare the machining accuracy before and after the compensation using only on-machine measurements. The basis of the tests is to machine slots, each completed using two different rotary axes indexations of the CNC machine tool. Using directional derivatives of the volumetric errors, it is possible to verify that a surface mismatch is produced between the two halves of the same slot in the presence of specific machine geometric errors. The mismatch at the both sides of the slot, which materializes the machine volumetric errors is measured using touch probing by the erroneous machine itself and with high accuracy since the measurement of both slot halves can be conducted using a single set of rotary axes indexation and in a volumetric region of a few millimetres. The effect of a compensation strategy is then validated by comparing the surface mismatch value for compensated and uncompensated slots.     

非球面超精密磨削误差建模与补偿研究

为提高课题组自研的超精密磨床加工精度,基于多体系统理论,运用齐次坐标变换原理,分析该超精密磨床37项几何误差来源,对非球面超精密磨削的综合误差建模。超精密磨床的多项几何误差元素已在制造阶段标定、补偿,取砂轮对刀误差和砂轮轮廓半径磨损误差作为主要面形误差来源,分别推导其对综合误差的传递函数,分析误差辨识方法,建立误差修正补偿模型,提出基于直接补偿的点补修正法。试验结果表明：建立的综合误差模型正确,根据误差辨识方法和修正补偿模型,修正误差后面形误差显著降低,有效提高面形精度。     

Derivation of machine tool error models and error compensation procedure for three axes vertical machining center using rigid body kinematics

Volumetric positional accuracy constitutes a large portion of the total machine tool error during machining. In order to improve machine tool accuracy cost-effectively, machine tool geometric errors as well as thermally induced errors have to be characterized and predicted for error compensation. This paper presents the development of kinematic error models accounting for geometric and thermal errors in the Vertical Machining Center (VMC). The machine tool investigated is a Cincinnati Milacron Sabre 750 3 axes CNC Vertical Machining Center with open architecture controller. Using Rigid Body Kinematics and small angle approximation of the errors, each slide of the three axes vertical machining center is modeled using homogeneous coordinate transformation. By synthesizing the machine's parametric errors such as linear positioning errors, roll, pitch and yaw etc., an expression for the volumetric errors in the multi-axis machine tool is developed. The developed mathematical model is used to calculate and predict the resultant error vector at the tool–workpiece interface for error compensation.     

Accuracy improvement of miniaturized machine tool: Geometric error modeling and compensation

A novel capacitance–sensor based multi-degree-of-freedom (DOF) measurement system has been developed for measuring geometric errors of a miniaturized machine tool (mMT) overcoming the size limitations. In the present work five geometric error components of a three-axis mMT are measured simultaneously along each axis and the squareness errors are determined by the slopes of straightness error profiles. Least-squares fitting method is used to represent the analytical models of geometric errors. A kinematic chain consisting of various structural members of mMT is introduced to establish the positional relationships among its coordinate frames. Based on this kinematic chain a general volumetric error model has been developed to synthesize all geometric error components of a miniaturized machine tool. Then, a recursive compensation method is proposed to achieve error compensation efficiently. Test results show that the positioning accuracy of miniaturized machine tool has been improved with compensation.     

Analysis of geometric errors associated with five-axis machining centre in improving the quality of cam profile

A five-axis machine is presently one of the most versatile machine tools available and they are becoming increasingly common. To increase the accuracy capabilities of such machines, it is crucial to be able to study the geometric errors of the components and its effect on the quality of machined products. In five-axis machine tools, all linear axes are theoretically perpendicular (dot product, cos 90°=0) to each other and directed along or around the X, Y and Z of the cartesian coordinate system; but in working machines, the axes are nearly perpendicular (cos89.90°≠0) because of manufacturing error and assembly error or quasi-static error. The present paper discusses the development of a generalised error model for the effects of geometric errors of the components of the kinematic chain of a machine in the workspace and the results obtained by this model have been verified experimentally. The effect of geometric error has been studied further for cam profile generation using a five-axis machining centre and an improvement in the profile has been obtained.     

A new geometric error modeling approach for multi-axis system based on stream of variation theory

This paper introduces a new geometric error modeling approach for multi axes system (MAS) based on stream of variation (SOV) theory, especially for multi-axis precision stage. SOV is used for measuring product quality for some complicated multi operations system, which is widely used in error propagation in engineering field. This paper introduces SOV concept into geometric error modeling for MAS. Instead of different process in manufacture, the new error modeling approach regards each axis as a station in MAS, and calculates the deviations after each station which is considered as upstream factor to next station. It is clear to observe how geometric errors give influence and how deviations accumulate. Different with conventional methods which are only used for error compensation in machine tools, the new error model is beneficial for sensitive error control and optimal configuration selection in design part. In addition, the new error modeling has some merits such as debugging easily due to observe the deviations after every station. A case study of new error modeling procedure for six-axis stage (SAS) in optoelectronic packaging system (OPS) is developed, and applications related to error reduction order and optimal configuration selection are processed based on the new error model.     

Development of a virtual machining system, part 2: prediction and analysis of a machined surface error

In part 2 of this three-part paper, a newly developed method that predicts the three-dimensional machined surface errors generated during the peripheral end milling process is presented. From the cutting force prediction system of Part 1, since the uncut chip thickness is calculated by tracing the movement of the cutter, the positions at which the cutting edges pass over the workpiece surface can readily be obtained. In this part of the paper these positions are used to construct surface error maps. In addition, by using the estimated locations of the peak and valley values of the cutting force component normal to the machined surface, a quantitative analysis of the machined surface error is given and followed by theoretical explanations. A series of machining tests on aluminum workpieces were conducted to validate the effectiveness of the model. The predicted cutting forces and surface errors were found in good agreement with their measured counterparts.     

Compensation for form error of end-milled sculptured surfaces using discrete measurement data

In this research, improvement of the form accuracy of parts having end-milled sculptured surfaces is studied. The proposed scheme uses the discrete measurement data from coordinate measuring machines and consists of three steps: optimal match, decomposition, and compensation. The optimal macth is used to best fit the measurement data with the design form so as to determine sampled form errors. The decomposition is developed to characterize the sampled form errors into the deterministic and the random components, which are associated with the waviness and the roughness of the form errors, respectively. Then, in the compensation step, compensation for the waviness is applied to improve the form accuracy of an end-milled sculptured surface. An experiment is conducted to demonstrate the effectiveness of the proposed scheme.     

Parametric process optimization to improve the accuracy of rapid prototyped stereolithography parts

The functional requirements of a rapid prototyping system are speed and accuracy, and they are both functions of vendor defaulted and user selected manufacturing parameters. Accuracy is evaluated by dimensional errors, form errors and surface roughness of manufactured parts. A specially designed specimen with 20 dimensional, geometrical, and surface roughness features has been used in the inspection of RP manufacturing processes. In terms of Taguchi experimental design techniques, an orthogonal array of experiments has been developed which has the least number of experimental runs and desired process parameter settings. Using a 3-D coordinate measuring machine and surface profilometer, a series of measurements in evaluating the SLA parts quality has been conducted to find the functional relationships between the output part quality and input manufacturing process parameters. Two analysis tools, response surface methodology and Analysis of Variance (ANOVA), have been used to evaluate the SLA RP process and to perform the product optimization. The optimal setups of SLA manufacturing parameters for both individual features and a general part with various features have been concluded from this study.     

基于多体系统理论的矫直机几何误差的建模与补偿

针对矫直机的几何误差建模和补偿技术进行研究。将矫直机床视为一个多体系统结构,完成对矫直机的拓扑结构描述,并进行压头、夹头和跨距调整等关键运动部件的误差分析。通过其次变换矩阵体系统推导机床几何误差模型,基于建立的误差模型,采用激光干涉仪识别机床运动轴的几何位置误差。提出矫直机的几何误差补偿方法并设计与开发补偿软件,对矫直机定位精度进行补偿。通过实验进行了误差补偿效果评价,实验结果证明误差模型的正确性以及软件补偿的可行性。     

Investigation of accuracy of aerostatic guideways

Aerostatic guideways are often used in machines requiring very high motion accuracy such as coordinate measuring machines. Currently, positioning error analysis for such machines focuses on the relationship between volumetric errors on one hand and axes’ motion errors and axes’ relative location errors on the other. The internal mechanisms causing motion errors are rarely considered. In order to gain a deeper understanding of aerostatic guideways, this paper investigates the relationship between the motion errors of the axis’ carriage and the guideways’ geometric errors both mathematically and experimentally. The analytical model uses bearings location and stiffness, guideway geometry and static equilibrium to produce a model in matrix form. Validation experiments are conducted on a machine axis moving on aerostatic guideways with and without preload.     

加工中心几何误差建模分析与误差补偿策略

运用齐次坐标变换原理和刚体假设，建立了加工中心从刀点到工作台的总体误差传递矩阵，推导出了通用的加工中心几何误差计算模型；用此模型对TH6350卧式加工中心进行了几何误差计算，并对误差模型进行了校验和分析，提出了基于PC机的加工中心误差补偿策略。