Geometric Error Components Research Articles

Micro/nanopositioning coplanar stage with real-time compensation of volumetric error

A micro/nanopositioning coplanar stage is designed with real-time compensation of volumetric error. According to the topological structure of the stage, it has 13 geometric error components. Furthermore, there are positional shifts of a functional point from the displacement feedback point and the measurement point of geometric errors. To improve the positional accuracy, it is crucial to real-time measure, model, and compensate for the volumetric error at the functional point. Therefore, a volumetric error measuring unit (VEMU) is designed to simultaneously real-time measure 13 geometric error components. A volumetric error model is established using the Abbe and Bryan principles. A control unit with two backpropagation neural networks-based PID controllers combines with the VEMU and volumetric error model to realize the real-time compensation of the volumetric error. The positional error of the micro/nanopositioning coplanar stage is reduced approximately 97.6 %, from which the effectiveness and applicability of the proposed methods are verified.

New identification method for computer numerical control geometric errors

To address the problems of the low accuracy of geometric error identification and incomplete identification results of the linear axis detection of computer numerical control (CNC) machine tools, a new 21-item geometric error identification method based on double ball-bar measurement was proposed. The model between the double ball-bar reading and the geometric error term in each plane was obtained according to the three-plane arc trajectory measurement. The mathematical model of geometric error components of CNC machine tools is established, and the error fitting coefficients are solved through the beetle antennae search particle swarm optimization (BAS–PSO) algorithm, in which 21 geometric errors, including roll angle errors, were identified. Experiments were performed to compare the optimization effect of the BAS–PSO and PSO and BAS and genetic particle swarm optimization (GA–PSO) algorithms. Experimental results show that the PSO algorithm is trapped in the local optimum, and the BAS–PSO is superior to the other three algorithms in terms of convergence speed and stability, has higher identification accuracy, has better optimization performance, and is suitable for identifying the geometric error coefficient of CNC machine tools. The accuracy and validity of the identification results are verified by the comparison with the results of the individual geometric errors detected through laser interferometer experiments. The identification accuracy of the double ball-bar is below 2.7 µm. The proposed identification method is inexpensive, has a short processing time, is easy to operate, and possesses a reference value for the identification and compensation of the linear axes of machine tools.

Tolerance analysis of the volumetric error of heavy-duty machine tool based on interval uncertainty

Tolerance design is one of the main stages in the robust design process of machine tool. For heavy-duty machine tools, traditional tolerance analyses based on probabilistic methods may not be suitable because it is generally difficult and costly to obtain the accurate probability distribution of error by the limited geometric error measurements in large dimension. In this paper, a new tolerance analysis method for heavy-duty machine tools is proposed based on the interval uncertainty. Considering independent geometric error components have certain bounds but accurate distributions are unknown, the interval theory is introduced to the kinematic modeling of the static volumetric error of heavy-duty machine tools based on the multi body system (MBS) and homogeneous transformation matrix (HTM). Geometric error components are described in forms of interval numbers to extend the error model to the interval number system. The interval extension of the volumetric error function is further modified so that the variation interval of the volumetric error is thus accurately evaluated by the interval arithmetic. Comparative studies on tolerance analyses of different typical probabilistic/non-probabilistic methods are conducted through numerical simulations. It is found that under the given computational condition, the tolerance analysis based on the interval uncertainty can accurately evaluate the variation range of the volumetric error.

Geometric error sensitivity analysis for a 6-axis welding equipment based on Lie theory

The influence of geometric errors on the accuracy of machine tools tends to attract more attention to the increasing demand for high-precision machining. In this paper, geometric error modeling and sensitivity analysis are employed to quantify the importance of geometric error for a new efficient and automatic 6-axis welding equipment. The geometric error model of the 6-axis welding equipment with 36 geometric error components is established based on Lie theory. Based on the geometric error model, the new sensitivity analysis method, in which the deviation of the welding torch pose is treated as a distance metric in SE(3), is proposed to evaluate the influences of geometric errors on the accuracy of the welding torch. And the sensitivity coefficient of each geometric error is derived by considering the basic value of geometric errors. Numerical simulations of a typical welding trajectory for intersecting pipes are conducted to analyze the sensitivity of geometric errors by the new method. The simulation results verified the validity of the sensitivity analysis method and the dominant geometric errors affecting the accuracy of the welding equipment were identified. Compared with the previous sensitivity analysis method, the proposed sensitivity analysis method considers the orientation error and position error of the welding torch simultaneously, which is more convenient and effective, and can also be applied in precision design and geometric error compensation of machine tools.

Machining accuracy improvement for a dual-spindle ultra-precision drum roll lathe based on geometric error analysis and calibration

This paper performs a comprehensive analysis and calibration on the geometric error of the ultra-precision drum roll lathe with dual-spindle symmetrical structure and cross slider layout. Firstly, the volumetric error model which contains all geometric errors of the dual-spindle ultra-precision drum roll lathe (DSUPDRL) is developed based on the combination of the homogenous transfer matrix (HTM) and multi-body system (MBS) theory. Secondly, sensitivity analysis for the volumetric error model is conducted to identify the sensitive geometric error components of the DSUPDRL using an improved Sobol method based on the quasi-Monte Carlo algorithm. The result of sensitivity analysis laid the foundation for the subsequent geometric error calibration. Then, some sensitive error components along the X and Z directions are calibrated using a laser interferometer and a pair of inductance displacement probes. Besides the volumetric error model, the concentricity error caused by dual-spindle symmetrical structure is proposed and calibrated by the on-machine measurement using a classic reversal method. Finally, a large-scale roller mold with a diameter of 250 mm and a length of 600 mm is machined using the DSUPDRL after calibration. The experimental result shows that 1.4 μm/600 mm generatrix accuracy is obtained, which validate the effectiveness of the geometric error analysis and calibration.

Dynamic and static error identification and separation method for three-axis CNC machine tools based on feature workpiece cutting

CNC machine tools play an extremely important role in the equipment manufacturing industry. Because a machine tool has multiple geometric errors which are highly coupled, the error identification process is difficult. Most direct and indirect measurement methods do not consider the dynamic errors in the actual cutting process. This study aims to optimize the error identification process of three-axis CNC machine tools, and proposes an efficient identification and separation method for the dynamic error and the quasi-static error based on feature workpiece cutting. In this study, a stepped feature workpiece that can reflect geometric errors is designed, and the mapping relationship between the geometric error components and the features is established. After the feature workpiece is cut, it performs two tests of on-machine measurement and CMM calibration sequentially. It can realize the identification and separation of 7 geometric errors and corresponding dynamic errors based on the two sets of test data. This method takes the actual cutting workpiece as the error source and can identify and separate the dynamic error and the quasi-static error simultaneously. Thus, the measurement efficiency of this method is higher than direct measurements. The geometric error identification result is highly consistent with the laser interferometer measurement, which proves the method in this study is feasible, efficient, and accurate.

Identification and compensation of geometric errors of rotary axes in five-axis machine tools through constructing equivalent rotary axis (ERA)

This paper proposes a volumetric accuracy enhancement method of rotary axes in five-axis machine tools through constructing equivalent rotary axis (ERA). The influences of translational axes, setup errors of table ball and tool ball are simultaneously considered. A new concept named ERA, originated from the definition of axis of rotation, is proposed to globally describe the instantaneous position and orientation of rotary axes. Based on this concept, only six parameters are needed to describe the influences of geometric errors rather than ten parameters required by the existing concept of PDGEs and PIGEs. At the same time, the coupled relationship between the compensated motion commands of rotary axes and geometric error components is eliminated, and the geometric error compensation is realized by formulating analytical expressions. The typical characteristics of the proposed method lie in the following three aspects. First, the axis twists in the identification method naturally follow the constraint of rotary axis, and thus the normalization and iteration procedure required by existing literatures are avoided. Second, compact form of geometric error compensation is established and explicit equations are given to generate the compensated motion commands. Specially, the inverse kinematic model of five-axis machine tools can be well integrated into the proposed method. Third, besides machine tools with non-orthogonal rotary axes, the proposed method can also be easily applied to five-axis machine tools with non-orthogonal rotary axes just by modifying the ERA parameters. Simulation results and blade machining tests verify the effectiveness of the proposed method, and more than 70% machining precision improvement is observed.

The application of a regularization method to the estimation of geometric errors of a three-axis machine tool using a double ball bar

Geometric accuracy is crucially important for machine tools. Identification of geometric errors, especially position-dependent geometric errors, is still a challenging issue. This paper presents a systematic and fast approach to identify the geometric error components of a precision machine tool using double ball bar (DBB). The approach can be implemented in three steps: (1) polynomial based error modeling that relates the DBB radius error directly to the geometric error parameters of machine tool; (2) spatial measurement trajectory planning with a single installation of DBB in order to avoid producing extra setup errors; (3) error identification with regularization method that can solve the ill-posed identification problem effectively. Simulations and experiments show the accuracy and effectiveness of the proposed identification approach. The results of the DBB test show that, utilizing the proposed identification method, the roundness errors of the three circular paths in xy-, yz- and xz-plane are reduced from 27.3 μm, 20.7 μm and 24.1 μm to 9.2 μm, 12.3 μm and 7.8 μm, respectively, with error compensation.

F test-based automatic modeling of single geometric error component for error compensation of five-axis machine tools

This paper presents one automatic modeling method of position-dependent geometric error components based on F test in statistics and one error compensation method of five-axis machine tools to ensure textures of the workpiece. At first, according to the definition of error components, the polynomials with zero constant term are chosen as the expressions. The calculation of corresponding coefficients is presented in details. Then, F test is introduced to evaluate the overall significance of polynomials. The automatic modeling of position-dependent geometric error components is expressed as seeking the polynomial with the best overall significance among a series of the polynomials with different orders. It is automatic and programmatic to improve the efficiency and precision of modeling. It can overcome the difficulty in determining the order of the polynomials. Next, geometric error compensation by limiting ideal tool positions of tool poses is proposed to ensure machining requirements of textures of the workpiece. The rotation angles of two rotary axes are optimized by particle swarm optimization (PSO) with the mathematical expressions of integrated geometric errors of five-axis machine tools. The movements of linear axes are calculated with inverse kinematics of the machine tool by inputting ideal tool positions and optimized rotation angles. Finally, the experiments are carried out on one SmartCNC500_DRTD five-axis machine center to testify the precision of the automatic modeling and the effectiveness of the error compensation.

Investigation of sensitivity analysis and compensation parameter optimization of geometric error for five-axis machine tool

To improve the accuracy of five-axis machine tool with a swiveling head, an approach for optimizing compensation values taking into account the sensitivity of position-independent geometric error is proposed in this paper. At first, the synthetic volumetric error model of five-axis machine tool has been established with homogeneous transformation method and multibody system theory, which describes the effect of geometric error components on position and orientation error vector intuitively. Second, the probability distributions of geometric errors in workspace of machine tool are sufficiently taken into account in order to overcome the defects of the analysis at certain locations, and then, global quantitative sensitivity analysis is introduced for determining the effect of each geometric error on precision of machine tool. Next, the optimum values are obtained by multiobjective quality loss and precision robustness trade-offs based on archive-based microgenetic algorithm (AMGA) for geometric error compensation. Finally, the geometric error compensation experiments were carried out, and the results show that the accuracy of measuring trajectories are improved significantly, which reaches 73.7% after error compensation with optimum values based on sensitivity analysis. Hence, the proposed methodology of analysis and compensation are effective for analyzing the effect of geometric errors and improving the precision of machine tool.

Geometric error modeling and compensation for five-axis CNC gear profile grinding machine tools

The relative position and orientation deviation between grinding wheel and workpiece caused by geometric errors affect the machining accuracy of five-axis CNC gear profile grinding machine tools directly. Therefore, geometric error modeling and compensation are presented according to homogeneous transformation and differential motion matrix based on the multi-body system theory for the accuracy enhancement of the machine tools. Firstly, the open kinematic chain and the ideal homogeneous transformation matrix from workpiece to grinding wheel are established according to the topological structure of the CFXZAY-type, five-axis gear profile grinding machine tools and the basic homogeneous transformation matrix between coordinate frames. Secondly, the homogenous transformation matrices of linear pairs and rotary pairs with geometric errors are calculated, and the relationships of the error propagation from workpiece to grinding wheel are acquired as well on the basis of 37 geometric error components analysis. The geometric error models including position and orientation errors of grinding wheel in workpiece coordinate system are obtained with matrix multiplication by using small-angle approximation and ignoring the second-order and high-order error terms. Then, the Jacobian is obtained by using transforming differential motion matrix of each motion axis to compensate the integrated error components of the grinding wheel, which can make the compensation effective and convenient with using the corresponding homogeneous transformation matrix. Finally, error measurement, error compensation, and machining experiments are carried out on a five-axis CNC gear profile grinding machine tool SKMC-1200W/10 to verify the applicability and effectiveness of the proposed error modeling, error compensation, and research approach.

A geometric error identification method for the swiveling axes of five-axis machine tools by static R-test

This paper proposes a new method to identify the 12 geometric error components of the swiveling axes of five-axis machine tools by static R-test. First, the geometric error modeling of a five-axis machine tool including the error sources of the C-A swiveling head is briefly introduced based on the MBS method. With the assistance of an auxiliary fixture, the measuring strategy of the swiveling axes that consists of six inspection modes by static R-test is proposed. Then, a geometric error identification algorithm for the C- and A-axis is respectively proposed based on the least square method. Numerical simulation is conducted and the results show that the accuracy of the identification algorithm can reach 89 %. An optimization method for the auxiliary fixture is proposed by minimizing the condition number of the identification matrixes to further improve the identification accuracy. Finally, numerical simulation shows that the identification accuracy can be increased to 97 % after optimization. The theoretical analysis and numerical simulation results indicate that the proposed swiveling axes geometric error identification method is both efficient and effective.

An RBF neural network approach to geometric error compensation with displacement measurements only

A novel radial basis function (RBF) neural network-based geometric error compensation method with displacement measurements only is proposed in this paper. The individual geometric error components are formulated mathematically based on laser interferometer calibration with displacement measurements only and modeled using RBF neural network for error compensation in motion controller. Only 4 and 15 displacement measurements are required to identify the error components for XY and XYZ table, respectively. The experiment results on two XY tables illustrate the effectiveness of the proposed method. The overall errors can be reduced significantly after compensation, and different data intervals can be selected to reduce calibration time but maintain a high level of accuracy. The proposed methodology can be extended to other types of precision machine and is more suitable for precision machines requiring a relative low level of accuracy, but fast calibration like those used for acceptance testing and periodic checking.

A New Error Compensation Method for a Four-Axis Horizontal Machine Tool

A new method compensating geometric error components of a four-axis horizontal machine tool is presented in this paper. Homogeneous transformation matrices (HTMS) and error conversion are integrated in the compensation strategy. A mathematical model which contains 29 geometric error components is established based on HTMS and the errors in X, Y and Z directions are obtained through calculation. The errors in three directions are compensated by shifting the corresponding axis. But the configuration of this machine tool is X-axis, Z-axis, B-axis, C-axis without Y-axis, so the errors in X, Y and Z directions need to be converted into X, Z and C directions which is different from the traditional machine tools. The errors after conversion are used to compensation directly. This approach is significant for the error modeling and compensation which is an easy and efficient way to improve the precision of the four-axis machine tools.

A New Algorithm for the Identification of CNC Geometric Errors

This paper proposes a geometric error model for the three-axis machine tool using the homogeneous transformation matrix. An error-prediction model for the three-axis CNC machine tools is presented. This model takes the geometric errors into consideration, using the homogeneous coordinate transformation matrix(HTM) to analyze the geometric error model, which can change the tool's complex position and orientation changes in space into matrix calculations to make the model simpler and more intuitive. A simple method using the cubic polynomial functions to fit the geometric error components is proposed. Using this method, the high-order, non-linear spatial geometric problem can be transformed into algebraic equation. To large extent, the difficulty of calculation is reduced. The QC20 ball-bar is used to measure, analyze and separate all the geometric error components. A new kind of particle swarm algorithm based on the genetic algorithm is developed to solve the algebra equation.

A Modified Sequential Multilateration Scheme and its Application in Geometric Error Measurement of Rotary Axis

Geometric error compensation becomes an essential technology for improving the accuracy of multi-axes machine tools. Laser tracker has been has been considered as an enabling instrument which can be used to measure the geometric error of machine tools due to its high accurate and feasible measurement capability. This paper presents a modified sequential multilateration method for measuring the geometric errors of rotary axes of the machine tools. The method is developed based on a single laser tracker and three targets fixing on the rotary stage. The coordinates of the three targets are measured based on the modified sequential multilateration method using a single laser tracker when the rotary stage is rotated. Then geometric error components are identified based on established geometric error model which takes the positioning error of the targets as input. The validity of the proposed method is verified through both computer simulation and experiments conducted on a multi-axis machine tool.

Integrated geometric and thermal error modeling and compensation for vertical machining centers

The accuracy of machine tools is significantly affected by the geometric defect and thermal deformation of the mechanical components. This paper intends to provide a comprehensive compensation method for the integrated geometric and thermal errors of machine tools. Firstly, a synthesized volumetric model is established with homogeneous transformation matrix method, considering both the geometric and thermal effects. Then, in order to improve the modeling accuracy and efficiency of the geometric error components, an automatic modeling algorithm is proposed with the Chebyshev polynomial-based orthogonal least squares regression. Also, to improve the robustness of the thermal error models for the feed axes and the spindle system, the thermal effects caused by external ambient variation and internal heat sources are identified and modeled separately. Finally, an intelligent virtual compensation system is developed for machine tools based on the function of external machine original coordinate shift and fast Ethernet data interaction technique, and compensation tests on a vertical machining center showed that the position accuracy of the machine tool could be significantly improved after compensation.

Time-varying positioning error modeling and compensation for ball screw systems based on simulation and experimental analysis

Thermal expansion of ball screw systems affects the machining accuracy of machine tools significantly. This paper intends to provide a comprehensive error compensation method for the time-varying positioning error of machine tools. To confirm the thermal deformation mechanism of ball screw systems, experiments have been designed to study the thermal behaviors of a ball screw system under varying temperature conditions. An exponential algorithm is proposed to predict the temperature variation pattern of the ball screw based on finite element analysis, and the actual thermal boundary conditions of the ball screw system are exactly defined according to the proposed algorithm and the experimental results. Then, a comprehensive compensation model is established based on the decomposition of the initial geometric error and thermal error components. Finally, a real-time error compensation system is developed for machine tools based on the function of external machine original coordinate shift and fast Ethernet data interaction, and satisfactory results have been achieved for the compensation experiments on a machining center.

Measurement and compensation of geometric errors of three-axis machine tool by using laser tracker based on a sequential multilateration scheme

A three-point method based on sequential multilateration is applied to measuring the geometric error components (GECs) of three-axis machine tools (MTs). To meet the accuracy requirement of geometric error mapping, a sequential multilateration scheme is developed for high-accuracy point measurement by introducing four additional targets into the measuring system, and the uncertainty of point measurement is verified by simulation. Three independent targets fixed on the MT’s spindle assembly move along with each axis step by step and their coordinates at each step can be determined by the distance data acquired by laser tracker at all steps based on sequential multilateration. Then the volumetric errors of the three target points can be obtained by comparing the actual coordinates and the corresponding desired coordinates, and nine equations can be established by substituting volumetric errors into the error model of linear axis, so that the six GECs of each axis can be obtained by solving these equations. The three squareness errors can be determined by computing the angles between the average lines of the three axes which are achieved by linear curve fitting. Experiments are conducted to measure these 21 GECs, and the volumetric errors in the three-axis MT’s workspace, which are determined by these measured GECs based on the error model of three-axis MT, are compensated. Finally, the positioning errors of the MT with compensation and without compensation are evaluated by laser interferometer, respectively, the experimental results of which demonstrate that the positioning errors are significantly reduced by the error compensation.

Three-point method for measuring the geometric error components of linear and rotary axes based on sequential multilateration

The linear and rotary axes are fundamental parts of multi-axis machine tools. The geometric error components of the axes must be measured for motion error compensation to improve the accuracy of the machine tools. In this paper, a simple method named the three-point method is proposed to measure the geometric error of the linear and rotary axes of the machine tools using a laser tracker. A sequential multilateration method, where uncertainty is verified through simulation, is applied to measure the 3D coordinates. Three non-collinear points fixed on the stage of each axis are selected. The coordinates of these points are simultaneously measured using a laser tracker to obtain their volumetric errors by comparing these coordinates with ideal values. Numerous equations can be established using the geometric error models of each axis. The geometric error components can be obtained by solving these equations. The validity of the proposed method is verified through a series of experiments. The results indicate that the proposed method can measure the geometric error of the axes to compensate for the errors in multi-axis machine tools.