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.     

Adaptive Monte Carlo applied to uncertainty estimation in five axis machine tool link errors identification with thermal disturbance

Knowledge of a machine tool axis to axis geometric location errors allows compensation and corrective actions to be taken to enhance its volumetric accuracy. Several procedures exist, involving either lengthy individual test for each geometric error or faster single tests to identify all errors at once.This study focuses on the closed kinematic chain method which uses a single setup test to identify the eight link errors of a five axis machine tool. The identification is based on volumetric error measurements for different poses with a non-contact Cartesian measuring instrument called CapBall, developed in house.In order to evaluate the uncertainty on each identified error, a multi-output Monte Carlo approach is implemented. Uncertainty sources in the measurement and identification chain – such as sensors output, machine drift and frame transformation uncertainties – can be included in the model and propagated to the identified errors. The estimated uncertainties are finally compared to experimental results to assess the method. It also reveals that the effect of the drift, a disturbance, must be simulated as a function of time in the Monte Carlo approach.Results shows that the machine drift is an important uncertainty source for the machine tested.     

Identification and compensation of geometric errors of rotary axes on five-axis machine by on-machine measurement

The geometric errors of rotary axes are the fundamental errors of a five-axis machine tool. They directly affect the machining accuracy, and require periodical measurement, identification and compensation. In this paper, a precise calibration and compensation method for the geometric errors of rotary axes on a five-axis machine tool is proposed. The automated measurement is realized by using an on-the-machine touch-trigger technology and an artifact. A calibration algorithm is proposed to calibrate geometric errors of rotary axes based on the relative displacement of the measured reference point. The geometric errors are individually separated and the coupling effect of the geometric errors of two rotary axes can be avoided. The geometry error of the artifact as well as its setup error has little influence on geometric error calibration results. Then a geometric error compensation algorithm is developed by modifying the numeric control (NC) source file. All the geometric errors of the rotary errors are compensated to improve the machining accuracy. The algorithm can be conveniently integrated into the post process. At last, an experiment on a five-axis machine tool with table A-axis and head B-axis structure validates the feasibility of the proposed method.     

Development of a multi-step measuring method for motion accuracy of NC machine tools based on cross grid encoder

Machining accuracy is directly influenced by the quasi-static errors of a machine tool. Since machine errors have a direct effect on both the surface finish and geometric shape of the finished work piece, it is imperative to measure the machine errors and to compensate for them. A revised geometric synthetic error modeling, measurement and identification method of 3-axis machine tool by using a cross grid encoder is proposed in this paper. Firstly a revised synthetic error model of 21 geometric error components of the 3-axis NC machine tools is developed. Also the mapping relationship between the error component and radial motion error of round work piece manufactured on the NC machine tools are deduced. Aiming to overcome the solution singularity shortcoming of traditional error component identification method, a new multi-step identification method of error component by using the cross grid encoder measurement technology is proposed based on the kinematic error model of NC machine tool. Finally the experimental validation of the above modeling and identification method is carried out in the 3-axis CNC vertical machining center Cincinnati 750 Arrow. The entire 21 error components have been successfully measured by the above method. The whole measuring time of 21 error components is cut down to 1–2 h because of easy installation, adjustment, operation and the characteristics of non-contact measurement. It usually takes days of machine down time and needs an experienced operator when using other measuring methods. Result shows that the modeling and the multi-step identification methods are very suitable for ‘on machine’ measurement.     

立式加工中心空间误差验证及补偿

为提高某立式加工中心整机加工精度,借助旋量理论建立完备立式加工中心空间误差模型,在此基础上实现机床空间误差有效补偿.以旋量理论为基础推导并建立机床刀具运动链与工件运动链运动学正解,分析机床21项几何误差原理,在考虑21项几何误差的基础上建立该立式加工中心完备空间误差模型;利用九线法完成各项几何误差辨识;基于旋量运动学正解求解机床运动学逆解后得出运动轴实际运动路径,并通过体对角线实验对比补偿前后的效果.结果表明:所提补偿方法补偿效果显著,验证了机床空间误差模型的准确性,实现了提高机床加工精度的目的.     

基于SAPSO-GA算法的高速曲轴随动数控磨床几何误差辨识方法

以某型数控曲轴磨床作为研究对象,对其结构和运动进行分析,推导出曲轴磨削时理想的砂轮轨迹方程。根据多体系统理论建立含有误差参数的模型,并推导出机床-工件和机床-刀具的运动链位置矩阵,得出机床精密加工的约束方程。对磨床的几何误差进行研究,建立几何误差模型。为快速、准确辨识出各项几何误差,提出一种混合SAPSO-GA算法。通过对比球杆仪测量补偿前后的运动轨迹,分析补偿效果。结果表明:所提方法提高了辨识准确性,通过补偿大大提高了曲轴随动磨床的加工精度。     

A general approach for error modeling of machine tools

This paper presents a general and systematic approach for geometric error modeling of machine tools due to the geometric errors arising from manufacturing and assembly. The approach can be implemented in three steps: (1) development of a linear map between the pose error twist and source errors within machine tool kinematic chains using homogeneous transformation matrix method; (2) formulation of a linear map between the pose error twist and the error intensities of a machine tool; (3) combination of these two models for error separation. The merit of this approach lies in that it enables the source errors affecting the compensatable and uncompensatable pose accuracy of the machine tool to be explicitly separated, thereby providing designers and/or field engineers with an informative guideline for the accuracy improvement by suitable measures, i.e. component tolerancing in design, manufacturing and assembly processes, and error compensation. Two typical multi-axis machine tools are taken as examples to illustrate the generality and effectiveness of this approach.     

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.     

Parameter estimation and model verification of first order quasistatic error model for three-axis machining centers

Quasistatic error sources, which include thermal, mechanical loading and geometric error sources, are responsible for a very large proportion (typically, 70%) of the volumetric errors of a numerically controlled machine-tool. This paper, the first in a set of three, discusses the development of a very general model for the effects of geometric errors of the components of the kinematic chain (structural members and axes) of a machine on the volumetric errors in the work space. The effects of the other two sources are modeled as changes in the geometric error. The generality of the model arises from the fact that the errors along an axis of the machine can be characterized by any polynomial functions and the model is recursive in the order of these polynomials. This model can be used as the basis of a compensation scheme as well as in budgeting of errors on a machine-tool.     

Computer-aided accuracy enhancement for multi-axis CNC machine tool

A computer-aided error compensation scheme has been developed to enhance the accuracy of multi-axis CNC machine tools by compensating for machine geometric and thermal errors in software way. Stationary geometric errors including the coupling effect of linkage errors between machine slides are calibrated off line. Dynamic thermal errors are predicted on line by an artificial neural network model. Because machine errors are variant with the cutting time and slide positions, a PC based compensation controller has been developed to upgrade commercial CNC controllers for real-time error compensation. The real-time compensation capability is achieved by digital I/0 communication between the compensation controller and CNC controller without the need of any hardware modification to the machine servo-drive loops. The compensation scheme implemented on a horizontal machining center has been proven to improve the machine accuracy by one order of magnitude using a laser interferometer and cutting test.     

High-precision machining by measurement and compensation of motion error

This paper describes a systematic method to model and compensate geometric errors of machine tools. In order to separate geometric errors from other errors, measured errors are analyzed in the frequency domain by using the Fourier series. Then, the frequency components corresponding to geometric errors are selected based on the repeatability of their wavelength. Finally, the components are reconstructed and forwarded for the compensation by a fine motion drive. A CNC machine tool with a fine motion mechanism on the Z-axis was developed to compensate the error components in the Z direction on the XY plane. A flat surface machining with non-rotational cutting tools was tested to validate our approach. On the plane of 45 mm×70 mm, the fluctuation of the relative displacement was reduced from 1.3 to 0.5 μm P-V. Machining experiments with a single-crystal diamond tool were also carried out and the straightness of the profile curve was reduced from 1.0 to 0.4 μm. The result of the experiments showed that the geometric errors were compensated separately from the vibration due to the bending mode of the machine column.     

An application of real-time error compensation on a turning center

A real-time error compensation (RTEC) system is developed to correct thermally-induced and geometric errors on a four-axis dual-spindle turning center. These errors vary with different cutting tool positions as well as different thermal conditions, thus real-time correction is required. Two problems in the current RTEC approach are addressed: (1) the difficulty in the actual measurement of error components according to the defined coordinates and (2) the selection of a small set of appropriate temperature variables from numerous candidate thermal sensors on a machine structure. A flexible error measurement method and an optimal temperature variable selection process are proposed to overcome these difficulties. After machine errors were characterized and modeled, the effectiveness of the developed RTEC system was evaluated by using laser inspection and an actual cutting test. The maximal diagonal displacement error is corrected from 75.0 to 7.5 μm. In the cutting test, the part diameter error of a car steering joint is reduced from 60 to 10 μm.     

Tool path accuracy enhancement through geometrical error compensation

Kinematic and geometric errors of CNC machine tools, introduce large deviations in the real path traveled by the cutting tool. Tool path deviation reduces geometrical and dimensional accuracy of the machined features of the component. Tool path modification is an effective strategy to increase accuracy of the machined features. An improved error estimation model based on kinematic transformation concepts has been developed and used to calculate the volumetric overall error. These calculations are applicable for each arbitrary target positions of the machine's work space. Also a NC Program editor software has been developed in order to manage the calculations, modifications and to generate the new compensated NC program. The compensation procedure includes: fragmentation of nominal tool path to small linear elements, translating nominal position of elements to real positions using the Kinematics error model, finding compensated positions using the error compensation algorithm, converting newly generated elements to new tool paths using the packing algorithms and finally editing old NC program using NC code generator algorithm. Experimental tests showed 4-8 times accuracy improvement for linear, and S-pline tool paths deviations.     

基于多体系统理论的车铣中心空间误差模型分析

数控机床的误差建模是进行机床运动设计、精度分析和误差补偿的关键技术,也是保证机床加工精度的重要环节.本文利用多体系统理论来构建超精密数控机床的几何误差模型,该模型简便、明确,不受机床结构和运动复杂程度的限制,为计算机床误差、实现误差补偿和修正控制指令提供了理论依据.在机床实际应用中,可以利用由精密机床误差建模所推导出的几何位置误差来修正理想加工指令,控制机床的实际运动,从而实现几何误差补偿,提高机床加工精度.     

Accuracy test of five-axis CNC machine tool with 3D probe-ball. Part II: errors estimation

To improve the accuracy of CNC machine tools, error sources and its effects on the overall position and orientation errors must be known. Most motional errors in the error model of five-axis machine tool can be measured with modern laser interferometer devices, but there are still some not measurable geometric errors. These not measurable errors include constant, inaccurate link errors of components such as rotary axes block, main spindle block and tool holder. After setting all measured errors in the error model, a reduced error model is defined, which describes the influence of each unknown and not measurable link error on the overall position errors of the five-axis machine tool. On the other hand, the newly developed probe-ball device can measure the overall position errors of five-axis machine tools directly. Based on the reduced model and the overall position errors, the link errors can be estimated very accurately with the least square estimation method. The error model is then fully known and can be used for advanced purposes such as error prediction and compensation.     

Accuracy enhancement of five-axis machine tool based on differential motion matrix: Geometric error modeling,identification and compensation

This paper presents the precision enhancement of five-axis machine tools according to differential motion matrix, including geometric error modeling, identification and compensation. Differential motion matrix describes the relationship between transforming differential changes of coordinate frames. Firstly, differential motion matrix of each axis relative to tool is established based on homogenous transformation matrix of tool relative to each axis. Secondly, the influences of errors of each axis on accuracy of tool are calculated with error vector of each axis. The sum of these influences is integration of error components of machine tool in coordinate system of tool. It endows the error modeling clear physical meaning. Moreover, integrated error components are transformed to coordinate frame of working table for integrated error transformation matrix of machine tools. Thirdly, constructed Jacobian is established using differential motion matrix of each axis without extra calculation to compensate the integrated error components of tool. It makes compensation easy and convenient with reuse of intermediate. Fourthly, six-circle method of ballbar is developed based on differential motion matrix to identify all ten error components of each rotary axis. Finally, the experiments are carried out on SmartCNC500 five-axis machine tool to testify the effectiveness of proposed accuracy enhancement with differential motion matrix.     

Mapping the effects of positioning errors on the volumetric accuracy of five-axis CNC machine tools

Fixe-axis capabilities on machine tools are becoming increasingly common. However, the problem of modeling the propagation of errors of individual axes through the kinematic chain, and their effect on the position and orientation errors of the cutting tool in the machine's work space has not been addressed. To increase the accuracy capabilities of such machines it is crucial to be able to study this propagation and develop approaches to minimize their effects on the errors at the tool tip. This paper discusses an approach to model the effects of the positioning errors of a machine's axes on the accuracy (positioning and orientation) of the cutting tool in its work space. Computer programs are developed for implementing the models and generating error contours or maps showing the variation of the different components of a machine's volumetric errors in its work space. This is a useful tool that can be used in machine tool design for the budgeting of errors and for optimization of a machine's accuracy.     

Real-time cutting force induced error compensation on a turning center

A real-time error compensation system has been developed to reduce the cutting force induced planar error of a two-axis turning center by using sensing, metrology, modeling, and computer control techniques. Ten error components are formulated as a two-dimensional error field. A piezoelectric force sensor mounted in the pocket under the tool turret is used to characterize the cutting forces. A compensation controller based on an IBM/PC has been linked with the existing computer numerical controller (CNC) to correct machine errors in real time. Three different types of cutting tests were performed and the results showed that the maximum diameter error in the workpiece was reduced by 67–85% using this compensation system.     

A new compensation method for geometry errors of five-axis machine tools

The present study aims to establish a new compensation method for geometry errors of five-axis machine tools. In the kinematic coordinate translation of five-axis machine tools, the tool orientation is determined by the motion position of machine rotation axes, whereas the tool tip position is determined by both machine linear axes and rotation axes together. Furthermore, as a nonlinear relationship exists between the workpiece coordinates and the machine axes coordinates, errors in the workpiece coordinate system are not directly related to those of the machine axes coordinate system. Consequently, the present study develops a new compensation method, the decouple method, for geometry errors of five-axis machine tools. The method proposed is based on a model that considers the tool orientation error only related to motion of machine rotation axes, and it further calculates the error compensations for rotation axes and linear axes separately, in contrast to the conventional method of calculating them simultaneously, i.e. determines the compensation of machine rotation axes first, and then calculates the compensation associated with the machine linear axes. Finally, the compensation mechanism is applied in the postprocessor of a CAM system and the effectiveness of error compensation is evaluated in real machine cutting using compensated NC code. In comparison with previous methods, the present compensation method has attributes of being simple, straightforward and without any singularity point in the model. The results indicate that the accuracy of positioning was improved by a factor of 8–10. Hence, the new compensation mechanism proposed in this study can effectively compensate geometry errors of five-axis machine tools.     

Total differential methods based universal post processing algorithm considering geometric error for multi-axis NC machine tool

Multi-axis numerical control machining for free-form surfaces needs CAD/CAM system for the cutter location and orientation data. Since these data are defined with respect to the coordinate of workpiece, they need converting for machine control commands in machine coordinate system, through a processing procedure called post processing. In this work, a new universal post processing algorithm considering geometric error for multi-axis machine tool with arbitrary configuration. Firstly, ideal kinematic model and real kinematic model of the multi-axis NC machine tool are built respectively. Difference between the two kinematic models is only whether to consider the machine tool's geometric error or not. Secondly, a universal generalized post processing algorithm containing forward and inverse kinematics solution is designed to solve kinematic models of multi-axis machine tool. Specially, the inverse kinematics solution is used for the ideal kinematic model, while the forward kinematics solution is used for the real kinematic model. Then, a total differential algorithm is applied to improve the calculation speed and reduce the difficulty of inverse kinematics solution. Realization principle of the total differential algorithm is to transform the inverse kinematics solution problem into that one of solving linear equations based on spatial relationship of adjacent cutter locations. Thirdly, to reduce the complexity of geometric error calibration experiment, effect weight of geometric error components is determined by the sensitivity analysis based on orthogonal method, and then the real kinematic model considering geometric error is established. Finally, the universal post processing algorithm based on total differential methods is implemented and demonstrated experimentally in a five-axis machine tool. The results show that the maximum error value can be decreased to one-fifth using the proposed method in this paper.