- Research
- Open Access
Three-axes error modeling based on second order dual numbers
- Michal Holub^{1},
- Jaroslav Hrdina^{2},
- Petr Vašík^{2}Email author and
- Jan Vetiška^{1}
https://doi.org/10.1186/s13362-015-0016-y
© Holub et al. 2015
- Received: 11 May 2015
- Accepted: 3 November 2015
- Published: 17 November 2015
Abstract
The aim of the paper is to employ the dual numbers in the multi axes machine error modelling in order to apply the algebraic methods in computations. The calculus of higher order dual numbers allows us to calculate with the appropriate geometric parametrization effectively. We test the model on the phantom data based on the real machine tool. The results of our analysis are used for the geometric manufacturing accuracy description of the work space, together with the reduction of the measuring time.
Keywords
- matrices
- dual numbers
- kinematics
- error modeling
MSC
- 20H25
- 15B10
- 70B10
1 Introduction
Machining centres (MCs) belong to the group of production machines, which require high performance, manufacturing accuracy, reliability, safety, etc. With growing demands on manufacturing quality of components for the aerospace, power and pharmaceutical (medical) industry, an increasingly greater emphasis is placed on control and growth of manufacturing eligibility. This is influenced by manufacturing accuracy of the machine tool, i.e. mainly by its geometric accuracy. Geometric accuracy belongs to the group of quasi-static errors that constitute 60-70% of the total error of the machine tool [1]. Apart from geometric errors, these quasi-static errors also include kinematic and thermal errors, [2].
For long-term sustainability of manufacturing accuracy (manufacturing eligibility), it is necessary to pay attention to the mount of MT on a suitably rigid base, to set up the machine to the required geometric accuracy and next, considering the needs and requirements, to apply the appropriate software compensations. In the phase of machine tool use, various technologies for verification of machine tool eligibility are then deployed. For these measurements, the emphasis laid is on adequate measurement accuracy with the highest possible interpretation of the results and especially on short measurement times associated with the necessary MT shutdowns.
Different approaches to measurement, data processing and creation of models for generating compensation tables are described in a number of scientific papers. For verification of the proposed models the most often used measuring instruments are the Ballbar-type apparatus [3], the laser interferometer [4–6], the laser tracker [7–9], and the laser tracer [10–14]. Modelling of errors of machine tools is closely associated with the used instrumentation. This implies a further procedure for the development of individual mathematical models. A currently researched topic is modelling of volumetric accuracy of machine tools with various kinematic structures. The aim of this research is to use mathematical models to create the so-called map of machine tool accuracy and to utilize the acquired deviations suitable for the relevant software compensation of machine tools. Error modelling of machine tools has been for long time subjected to intensive research. The most common approach to the model creation, based on rigid body kinematics and transformations of coordinates between the individual rigid bodies, is a homogeneous transformation matrix (HTM). HTM modelling is described for various kinematic chains of machining centres in publications [5, 15–20].
To achieve relevant results, it is necessary to obtain measurement data free of thermal influence because this influence caused by the change in internal and external ambient conditions may invalidate the measurement by thermal deformation of the machine, but also by the very process of measurement. For this reason, it is necessary to complete the measurements in the shortest possible time interval under stable conditions. In manufacturing facilities, especially in large machining centres, it is almost unrealistic to maintain constant ambient conditions. Therefore the obtained results on the state of the machine may be distorted both due to thermal deformations of the machine and also due to temperature change of ambient air [21].
Information on “geometric behaviour” of small and large CNC MTs can be obtained by various approaches. One of the variants is to machine a test component; this will enable an assessment of machine tool eligibility. The second variant is based on direct measurement of geometrical deviations. This variant is less desirable from the perspective of users; the reason is a necessary machine tool shutdown.
If we intend to analyse the entire working space of MT and obtain an error map, the measurements will be very time-consuming. To meet the requirement on minimum time demands with the aim to create a map of machine tool errors, it is recommended to propose a suitable methodology of measuring and processing of the obtained data. This methodology is not only dependent on the size of MT working space, but also on the kinematic chain between the workpiece and the tool and also on the measuring device used.
2 Algebraic approach
Theorem 2.1
The theory of matrices over the algebra \(\mathbb{D}_{1}^{2}\) is much more sophisticated, some results can be found in [26], e.g. the description of the relevant \(2 \times2\) matrix class equivalent to the one mentioned in Theorem 2.1 is the following:
Theorem 2.2
3 Virtual three-axis machine
Parametric errors for three-axis MT
Axis | Axis error (ISO/paper symbol) | |||||
---|---|---|---|---|---|---|
X-axis | EXX/\(\delta_{xx}\) | EYX/\(\delta_{yx}\) | EZX/\(\delta _{zx}\) | EAX/\({\varepsilon }_{xx}\) | EBX/\({\varepsilon }_{yx}\) | ECX/\({\varepsilon }_{zx}\) |
Y-axis | EXY/\(\delta_{xy}\) | EYY/\(\delta_{yy}\) | EZY/\(\delta _{zy}\) | EAY/\({\varepsilon }_{xy}\) | EBY/\({\varepsilon }_{yy}\) | ECY/\({\varepsilon }_{zy}\) |
Z-axis | EXZ/\(\delta_{xz}\) | EYZ/\(\delta_{yz}\) | EZZ/\(\delta _{zz}\) | EAZ/\({\varepsilon }_{xz}\) | EBZ/\({\varepsilon }_{yz}\) | ECZ/\({\varepsilon }_{zz}\) |
Axis | Squareness error (ISO/paper symbol) | ||
---|---|---|---|
X-axis | \(\mathrm{B0Z}\)/\(S_{xy}\) | \(\mathrm{C0Y}\)/\(S_{xz}\) | |
Y-axis | \(\mathrm{A0Z}\)/\(S_{yz}\) | ||
Z-axis | \(\mathrm{C0Y}\)/\(S_{xz}\) |
Adjusted space in the machine coordinate system with the start-CP
Axis | Start-CP [mm] | End [mm] | Length [mm] |
---|---|---|---|
X-axis | 300 | 450 | 150 |
Y-axis | 200 | 300 | 100 |
Z-axis | −200 | −50 | 150 |
4 Error modelling
5 Conclusion
We used the phantom data based on the technical parameters and measured errors of the demonstrator to check the proposed method of the calculation of the geometric errors linear parametrization. To avoid the discontinuity in the error evolution, we focused on such part of the MS space where the errors are approximately linear. Consequently, we calculate the linear parameters in question classically by means of the Moore-Penrose inverse matrix with the modification based on the calculus of the dual numbers formulated in [22]. Our aim is to both predict the machine tool behaviour and to use the measured data for the correct description of its possible states. The main contribution of our calculation proposal is the reduction if the number of inputs that are necessary to measure within the machine working space. This leads to significant reduction of the machines shutdown time needed for the data collection. Furthermore, it is possible to use more elementary measurement tools like laser interferometer, the price of which is remarkably lower than that of the Laser TRACER, which was used within this publication. The disadvantage of the proposed algorithm is the assumption on the errors linearity behaviour. This questions the suitability for small and middle-sized machines. On the other hand, the error linearity of particular geometric errors can be expected for large machines. We conclude that, based on the provided graphs, for such machines our results are highly applicable.
Declarations
Acknowledgements
The authors were supported by the project NETME CENTRE PLUS (LO1202). The results of the project NETME CENTRE PLUS (LO1202) were co-funded by the Ministry of Education, Youth and Sports within the support programme “National Sustainability Programme I”.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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