Traditional approaches about error modeling and analysis of machine tool few consider the probability characteristics of the geometric error and volumetric error systematically. However, the individual geometric error measured at different points is variational and stochastic, and therefore the resultant volumetric error is aslo stochastic and uncertain. In order to address the stochastic characteristic of the volumetric error for multiaxis machine tool, a new probability analysis mathematical model of volumetric error is proposed in this paper. According to multibody system theory, a mean value analysis model for volumetric error is established with consideration of geometric errors. The probability characteristics of geometric errors are obtained by statistical analysis to the measured sample data. Based on probability statistics and stochastic process theory, the variance analysis model of volumetric error is established in matrix, which can avoid the complex mathematics operations during the direct differential. A four-axis horizontal machining center is selected as an illustration example. The analysis results can reveal the stochastic characteristic of volumetric error and are also helpful to make full use of the best workspace to reduce the random uncertainty of the volumetric error and improve the machining accuracy.
Along with rapid progress and development of science, technology and social economy, the machining accuracy of CNC machine tools is increasingly demanding. How to improve the accuracy of CNC machine tools has been gotten great attention [
However, machining accuracy of the multiaxis synchronized machine is mainly affected by the geometric errors of the guide system, structure stiffness, thermal behavior and the dynamic response, and so forth. The geometric errors are those errors that exist in a machine on account of its basic design and those resulting from the inaccuracies built in during assembly and from the components used in the machine [
Geometric error of the machine tool primarily comes from manufacturing or assembly defects misalignment of the machine’s axis and the position and straightness error of each axis. Because the errors of a drive or axis or the outcome of an assembly process are random at some level [
Error modeling technique provides a systematic and suitable way to establish the error model. In recent years, many research work has been done on modeling of multiaxis machine tools to find out the resultant error of individual components in relation to tool and work piece point deviation. And the modeling methods of the geometric errors from different perspectives have experienced several developing phases including geometric modeling methods, error matrix methods (EMMs), quadratic model methods, mechanism modeling methods, rigid body kinematics methods [
Love and Scarr [
In 2003, Lin and Shen [
In recent years, the multibody system (MBS) as for the movement with error of complicated machinery system was presented by Houston in the late 1970s [
The above research work mostly addressed the relationship between the geometric error and volumetric error without considering the stochastic characteristic of geometric error. However, it is not practical to try to manufacture a part to an exact dimension in a production environment because of the inaccuracies associated with machine tools and the apparent randomness of most manufacturing processes, originating from variation in material properties, dimensions, friction, and so forth [
Up to the present, some researchers have become interested in the uncertainty analysis of errors [
Yau [
The rest parts of the paper are organized as follows. In the next section, the modeling of volumetric error for machine tool is given based on MBS theory. The third section presents the stochastic characteristic analysis modeling process of volumetric errors. Section four demonstrates the proposed method with a 4-axis machine tool. The final section contains the conclusions.
It is well known that when an rigid object moves in 3D space, it has six degrees of freedom (DOF); accordingly, its position description has six errors [
Six parametric errors associated with a prismatic joint.
In addition, as a multiaxis machine tool, a rotating shaft also has six geometric errors. As Figure
Six parametric errors associated with a rotary joint.
In this research, a 4-axis precision horizontal machining center, whose 3-dimension digital structure model is shown in Figure
Geometric errors of precision horizontal machining center.
Component | Error term | Symbol | Component | Error term | Symbol |
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Positioning error |
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Axial error |
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Rolling error |
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Around the |
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Britain swing error |
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Positioning error |
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Yaw error |
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Around the |
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Orientation error between joints |
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Positioning error |
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Rolling error |
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Britain swing error |
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Yaw error |
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Rolling error |
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Britain swing error |
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Positioning error |
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Yaw error |
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Structure diagram of precision horizontal machining center. 0 bed; 1
On the basis of the theory of MBS, a multiaxis machine tool can be abstracted into a multibody system. As shown in Figure
Lower body array of precision horizontal machining center.
Topic body |
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1 | 2 | 3 | 4 | 5 | 6 |
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0 | 1 | 2 | 0 | 4 | 5 |
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0 | 0 | 1 | 0 | 0 | 4 |
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0 | 0 | 0 | 0 | 0 | 0 |
Adjacent body freedom of the precision horizontal machining center.
Adjacent body |
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0-1 | 1 | 0 | 0 | 0 | 0 | 0 |
1-2 | 0 | 1 | 0 | 0 | 0 | 0 |
2-3 | 0 | 0 | 0 | 0 | 0 | 1 |
0-4 | 0 | 0 | 1 | 0 | 0 | 0 |
4-5 | 0 | 0 | 0 | 0 | 1 | 0 |
5-6 | 0 | 0 | 0 | 0 | 0 | 0 |
Topological graph of precision horizontal machining center.
In order to facilitate the precision of machine tool modeling, coordinate system is required to be setted specially. Consider the following settings.
The characteristic matrices of precision horizontal machining center.
The adjacent body | Body ideal static, motion characteristic matrix | Body static, kinematic error characteristics matrix |
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2-3 tool |
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0-4 |
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4-5 |
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Supposing that tool forming point in the tool coordinate system coordinate is
Just as aforementioned, geometric errors are caused by the inaccuracies built in during original manufacturing, assembly, and in-service stage of component, and are stochastic variable with uncertainty characteristic [
According to (
With (
When (
Supposing that
Figure
A 4-axis precision horizontal machining center.
Tests of straightness error; (a)
Position dependent on
With the statistical analysis of the gotten sample data, the stochastic characteristic of geometric errors can be obtained. Taking the positioning error
Probabilistic characteristic of
Point1 | Point2 | Point3 | Point4 | Point5 | Point6 | Point7 | Point8 | ||||||||
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1.2 | 0.0120 | 1.3 | 0.3282 | 1.2 | 0.0561 | 1.5 | 0.0930 | 2.1 | 0.0378 | −0.5 | 0.0000 | 1.4 | 0.0112 | 2.5 | 0.0974 |
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Point9 | Point10 | Point11 | Point12 | Point13 | Point14 | Point15 | Point16 | ||||||||
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2.9 | 0.0502 | 3.8 | 0.2430 | 4.0 | 0.1068 | 4.7 | 0.0325 | 3.9 | 0.0707 | 5.9 | 0.0428 | 4.1 | 0.0679 | 5.4 | 0.1735 |
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Point17 | Point18 | Point19 | Point20 | Point21 | Point22 | Point23 | Point24 | ||||||||
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5.5 | 0.0967 | 6.5 | 0.0921 | 6.9 | 0.0165 | 7.3 | 0.0161 | 8.2 | 0.0020 | 8.8 | 0.0000 | 8.2 | 0.1104 | 7.4 | 0.0232 |
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Point25 | Point26 | Point27 | Point28 | Point29 | Point30 | Point31 | Point32 | ||||||||
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6.6 | 0.0342 | 6.1 | 0.0548 | 6.7 | 0.0087 | 7.8 | 0.1096 | 8.2 | 0.1833 | 8.5 | 0.0005 | 7.6 | 0.1540 | 8.7 | 0.0016 |
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Point33 | Point34 | Point35 | Point36 | Point37 | Point38 | Point39 | Point40 | ||||||||
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8.1 | 0.0075 | 8.3 | 0.0237 | 8.8 | 0.0046 | 9.3 | 0.0657 | 9.5 | 0.0000 | 9.8 | 0.0084 | 8.7 | 0.0116 | 8.6 | 0.0016 |
Normal distribution test curve of
With (
Mean value distribution of volumetric error.
As for the 24 position-dependent geometric errors, their relation curves with corresponding
Fitted polynomial curve of
Variance value distribution of volumetric error.
Therefore, with above analysis results of the random uncertainty of volumetric error, on the one hand, we can make full use of the machining performance of machining center by keeping away from the workspace with lower machining accuracy or bigger accuracy fluctuation. For example, advisable adjustment during the clamping process of work piece may be carried out according to the error distribution in Figures
Conventional approaches about error modeling and analysis of machine tool few consider the probability characteristics of the geometric error and volumetric error systematically. However, the individual geometric error is variational and stochastic measured at different points, and therefore the resultant volumetric errors are aslo stochastic and uncertain. In order to investigate the stochastic characteristics of geometric error and volumetric error, further to make rational use of machine tool and improve its performance at lower cost, an analysis method has been developed in this research. Based on MBS theory, an mean value analysis model of the volumetric error of multiaxis machine tool is established. With probability statistics and stochastic process theory, a variance analysis model of volumetric error is established, which can be used to analyze the stochastic characteristic of the volumetric error of the whole 3-dimensional manufacturing workspace. A case study example on four-axis machine tool has been conducted to demonstrate the effectiveness of this method.
Characteristics of this method are summarized as follows. Compared with the traditional analysis methods, the proposed method deems the individual geometric error and volumetric error as uncertainty variables and deals with their random stochastic characteristics with mean and variance. In addition, the proposed probability analysis method is based on MBS theory and in matrix, which can avoid complex mathematics operations caused by direct differential, and is suitable for computer to model and calculation.
Despite the progress, a number of issues need to be further addressed to perfect the currently developed method. Our future research will focus on the following two aspects:
The authors are most grateful to the National Natural Science Foundation of China (no. 51005003), the National Science and Technology Great Special Program (no. 2010ZX04001-041), the Guangdong Provincial Second Batch Leading Figure Program, Rixin Talent Project led by Beijing University of Technology, Beijing Education Committee Scientific Research Project, for supporting the research presented in this paper.