A Novel Error Sensitivity Analysis Method for a Parallel Spindle Head

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.

Global sensitivity analysis has been widely used to detect the relative contributions of input variables to the uncertainty of model output, and then more resources can be assigned to the important input variables to reduce the uncertainty of model output more efficiently. In this paper, a new kind of global sensitivity index based on Gini’s mean difference is proposed. The proposed sensitivity index is more robust than the variance-based first order sensitivity index for the cases with non-normal distributions. Through the decomposition of Gini’s mean difference, it shows that the proposed sensitivity index can be represented by the energy distance, which measures the difference between probability distributions. Therefore, the proposed sensitivity index also takes the probability distribution of model output into consideration. In order to estimate the proposed sensitivity index efficiently, an efficient Monte Carlo simulation method is also proposed, which avoids the nested sampling procedure. The test examples show that the proposed sensitivity index is more robust than the variance-based first order sensitivity index for the cases with non-normal distributions.

Multi-axis machining of a complex surface is not only efficient but also reduces workpiece set-up errors. However, engineers usually rely on experience when determining set-up orientation and often cannot finish complex surfaces in one set-up. In this paper, a rational evaluation method for workpiece orientation and configuration of machine tool before tool path planning is proposed to increase the efficiency and reliability of process planning. By incorporating the visibility cone into workspace analysis, a software module is constructed in this research for optimum workpiece orientation and configuration of machine tool. First, the configurations of three-, four- and five-axis machine tools were synthesised, and the workspace was then analysed using form-shaping function. Both position vector and normal vector of each point along the u and v directions in the machining surface were employed to calculate the visibility range as the point of the visibility cone. In 3-axis machining, all normal vectors of the surface were projected on a unit sphere. All normal vectors were ensured to be within the visibility range of the ball-mill cutter as the workpiece set-up orientation changed. In 4-axis machining, the intersection between the visibility cone of each point and the workspace was obtained by changing the set-up orientation. In 5-axis machining, according to different workspaces of 5-axis machine tools, the visibility range of the machining surface was determined to yield the intersection between the visibility cone and the workspace by changing the workpiece orientation. The proposed method for evaluating manufacturability has been demonstrated to generate optimum workpiece set-up orientation and the suitable configuration of machine tools. Tool interference with workpiece surface can be avoided, which can help improve the planning of tool path.

Computer models are important tools used in scientific investigations of natural phenomena. A computer model may be viewed as an input → output function. In order to assess the influence of each input on the output, formal sensitivity analyses are typically carried out. A specific type of sensitivity analysis addressed in this article is the functional ANOVA, in which an output variance decomposition is used to quantify each input’s importance. Here we propose functional ANOVA for computer models with time series output. The argument given in this article originates in the concept of conditional expectation and variance with respect to time. We also establish a relationship between the proposed sensitivity indices and the global regular sensitivity indices existent in the literature. An application from the automotive industry is presented to illustrate the methods.

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

We present a first-order recursive approach to sensitivity analysis based on the application of the direct differentiation method to the inverse Lagrangian dynamics of rigid multibody systems. Our method is simple and efficient and is characterized by the following features. Firstly, it describes the kinematics of multibody systems using branch connectivity graphs and joint-branch connectivity matrices. For most mechanical systems with an open-tree kinematic structure, this method turns out to be more efficient compared to other kinematic descriptions employing joint or link connectivity graphs. Secondly, a recursive sensitivity analysis is presented for a dynamic system with an open-tree kinematic structure and inverse dynamic equations described in terms of the Lagrangian formalism. Thirdly, known approaches to recursive inverse dynamic and sensitivity analyses are modified to include dynamic systems with external forces and torques acting simultaneously at all joints. Finally, the proposed method for sensitivity analysis is easy to implement and computationally efficient. It can be utilized to evaluate the derivatives of the dynamic equations of multibody systems in gradient-based optimization algorithms. It also allows less experienced users to perform sensitivity analyses using the power of high-level programming languages such as MATLAB. To illustrate the method, simulation results for a human body model are discussed. The shortcomings of the method and possible directions for future work are outlined.

The geometric error of a three-axis vertical machining center is a critical technical indicator for evaluating its machining accuracy, and the precision of error identification plays a decisive role in the engineering effectiveness of compensation strategies. Based on the multibody system topology theory and the Denavit-Hartenberg matrix theory, this study establishes a contribution transfer model to quantify the impact of multi-axis geometric error on the positional errors at the end-effector of the three-axis vertical machining center. Experiments were performed on a VMC650 three-axis vertical machining center platform, where 21 geometric error items were precisely measured using a Renishaw XM-60 laser interferometer. An error spatial distribution function was constructed through polynomial regression analysis, and the optimal fitting coefficients of polynomial basis function combinations were solved to provide an analytical mathematical description for error prediction at arbitrary spatial positions. Based on the geometric error model, this study innovatively combines Monte Carlo sampling with principal component analysis to perform sensitivity analysis on geometric error parameters and analyze the transmission of geometric error under small disturbances, thereby effectively identifying key sensitive geometric error parameters. The research results offer theoretical guidance for structural optimization design and compensation parameter calibration of three-axis vertical machining centers, significantly enhancing their comprehensive machining accuracy.

To estimate influence of velocity on kinematic accuracy for a cross-linked Stewart type Parallel Machine Tool, position and orientation errors of the moving tool platform are researched. Based on rigid body dynamics, inertial and friction forces and moments are considered. Firstly, analytical expression of driving force is derived for each link. Secondly, change of length is derived using Hooke’s Law for each link. Then mapping matrix between change of link length and change of platform position and orientation is derived based on both Euler angle and revolving angle around a spatial axis. Finally, analytical expression of position and orientation errors of the moving platform is derived. Figures of distribution of position and orientation errors in workspace under two velocities are obtained respectively. The results show that, in frequently used workspace, all of the three components of position error are less than 3.5μm. All of them increase with z coordinate of platform center. Position error is influenced slightly by velocity. The difference of position error between two velocities is less than 2%.

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.

This paper proposes a recognition methodology for key geometric errors using the feature extraction method and accuracy retentivity analysis and presents the approach of optimization compensation of the geometric error of a multiaxis machine tool. The universal kinematics relations of the multiaxis machine tool are first modelled mathematically based on screw theory. Then, the retentivity of geometric accuracy with respect to the geometric error is defined based on the mapping between the constitutive geometric errors and the time domain. The results show that the variation in the spatial error vector is nonlinear while considering the operation time of the machine tool and the position of the motion axes. Based on this aspect, key factors are extracted that simultaneously consider the correlation, similarity, and sensitivity of the geometric error terms, and the results reveal that the effect of the position‐independent geometric errors (PIGEs) on the error vectors of the position and orientation is greater than that of the position‐dependent geometric errors (PDGEs) of the linear and rotary axes. Then, the fruit fly optimization algorithm (FOA) is adopted to determine the compensation values through multiobjective tradeoffs between accuracy retentivity and fluctuation in the geometric errors. Finally, an experiment on a four‐axis horizontal boring machine tool is used to validate the effectiveness of the proposed approach. The experimental results show that the variations in the precision of each test piece are lower than 25.0%, and the maximum variance in the detection indexes between the finished test pieces is 0.002 mm when the optimized parameters are used for error compensation. This method not only recognizes the key geometric errors but also compensates for the geometric error of the machine tool based on the accuracy retentivity analysis results. The results show that the proposed methodology can effectively enhance the machining accuracy.

Kinematic accuracy is a key reason which influence workpiece's geometric error precision on traditional working process of precisely CNC（Computerized Numerical Control）P3G（polygon profile with 3 lobes） grinding machine. A systematic geometric error model has been presented for CNC P3G grinding machine， proposed multi-body system theory integrate with the structure of CNC P3G grinding machine tools, researched on the machine's space geometric errors. By means of separate geometric errors from the machine tools, build geometric mathematical error model. Then, identify 21 error parameters through method of 9 lines, analysis and calculate the total space geometric errors of the workpiece and wheel. Finally, formed a parameter-list and applied software error compensational technique ， achieved real-time control to the motions of workpiece and wheel. Experimental results shown that the geometrical error modeling technique is accurate and efficient, and the precision of CNC P3G grinding machine is highly raised 70%.

The objective of this research is to develop a geometric error model for multi-axis machine tools based on a closed-loop configuration. In this study, the geometric error model of each axis is derived by using 4×4 homogeneous transformation matrix. The ideal cutter location of the sculptured workpiece surface is calculated using the Bezier bicubic parametric surface representation method and machine geometric errors. The actual cutter locations is calculated by considering runout error of the cutting tool and machine geometric errors. Then, the step-by-step volumetric error analysis method is suggested on the basis of the closed-loop configuration of the multi-axis machine tools. The simulation study shows the simplicity and effectiveness of the proposed strategy.

PurposeThe objectives of this paper are the application of sensitivity analysis (SA) methods in atmospheric dispersion modeling to the emission dispersion model (EDM) to study the prediction of atmospheric dispersion of NO2 generated by an industrial fire, whose results are useful for fire safety applications. The EDM is used to predict the level concentration of nitrogen dioxide (NO2) emitted by an industrial fire in a plant located in an industrial region site in Algeria.Design/methodology/approachThe SA was defined for the following input parameters: wind speed, NO2 emission rate and viscosity and diffusivity coefficients by simulating the air quality impacts of fire on an industrial area. Two SA methods are used: a local SA by using a one at a time technique and a global SA, for which correlation analysis was conducted on the EDM using the standardized regression coefficient.FindingsThe study demonstrates that, under ordinary weather conditions and for the fields near to the fire, the NO2 initial concentration has the most influence on the predicted NO2 levels than any other model input. Whereas, for the far field, the initial concentration and the wind speed have the most impact on the NO2 concentration estimation.Originality/valueThe study shows that an effective decision-making process should not be only based on the mean values, but it should, in particular, consider the upper bound plume concentration.

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

The article is devoted to increasing the volumetric accuracy of multi-axis machine tools. Based on the model of measurement and calculation of the volumetric geometric error of machine tools presented by the authors, a method has been developed which allows to signifi cantly improve the accuracy of the movement of the operating unit of a three-axis metal-cutting machine tool to a given point. The astatic law and the calculation of the coordinate correction according to the differential geometric model of the volumetric error are used to control the movement of the operating unit. The developed method is implemented programmatically and allows to perform the correction in the control program represented in G-code. As a result of experiments on software correction of volumetric errors according to the developed method, conducted at the State Engineering Centre of the Moscow State University of Technology “STANKIN” on a threeaxis CNC machine tool, a signifi cant reduction (up to 90 %) of geometric volumetric errors was shown. The developed method can be used to improve the accuracy of metal-cutting machine tools by creating postprocessors that calculate corrections to the coordinates set in the control system based on the results of interferometric measurements.Keywords: volumetric accuracy, interference measurements, three-axis machine tools, machine tool accuracy, geometric errors, astatic control law, mathematical modelling, differential geometry