Polynomial error approximation of a precision angle measuring system.

Siaudinyte, Lauryna ; Brucas, Domantas ; Rybokas, Mindaugas 等

Introduction

Precision angle measuring systems are widely used for manufacturing various types of machinery embedding them into devices and instruments such as total stations, metal cutting machines, etc. An angle encoder is the main part of an angle measuring system. They may vary in sizes, types and accuracy. Depending on the required accuracy, the whole angle measuring system must be accordingly calibrated. Calibration of such systems is essential to determine errors and increase measurement accuracy.

Calibration process is frequently a very complicated procedure requiring specific equipment, including standards for achieving the best results. Precision angle measurement is compulsory for engineering at various levels of precision, and therefore many artefacts and instruments such as angle gauge blocks, optical polygons and rotary tables are used for angle metrology. To ensure high reliability, they should be calibrated using standard instruments such as an indexing table along with an autocollimator (Just et al. 2003; Kim et al. 2011).

1. Equipment and method used for flat angle calibration

To determine the accuracy of an angle measuring system, a comparative method was chosen. This method is based on a comparison of the measurand and standard angle determining a deviation between those angles. Equipment selected for realisation of this method consists of a precision angle comparator, 36 mirror-sided polygon and an autocollimator. The angle standard as a mirror polygon is an important measurement object used in applied for an angle calibration world wide. When measuring the angle accuracy of a product, many application fields of using the polygon could be used (Watanabe et al. 2003). The angle comparator consists of the basis that includes a precision mechatronicrotary device, a device for detecting circular scale graduation and its relative rotation according to the circular scale measuring system. A section view of the angle comparator is presented in Fig. 1. Another critical element is a calculation system for both the control of the calibration process and data processing. The basic mechatronic system is designed for direct limb and angular encoder calibration. The basis of such system consists of a massive small-grain grey granite brick with aerostatic rotational mechanism mounted on it through a suspension ring. It is rotated using a worm gear. To ensure proper rotation of this gear, combined aerostatic and rolling bearings are installed. The worm gear is mounted on aerostatic bearings and supported radially by rolling bearings. The rotation axis approximately matches the axis of the aerostatic rotational device. For this reason, suspension and radial bearings are mounted on position-correction devices. The whole system is covered with a special cover and a precisely finished top surface protecting the system of damage coming from environmental pollutants. Additional measurement equipment can also be placed on this surface (Kasparaitis, Sukys 2008).

[FIGURE 1 OMITTED]

There are two diverse requirements for the gear of the comparator: it has to ensure constant angular rotation and the possibility of precise positioning while providing a stable angular position within given time.

The gear has not to create any other forces except for torque or tangential rotational force. It is also important that the gear does not generate intensive thermal activity or vibration. To perform accurate measurements, the laboratory should be isolated from any external vibrations under stable temperature.

Important parts of an angle measuring system are shown in the section view in Fig. 1: 1-precision aerostatic shat, 2-reading head, 3-reference circular scale, 4-measured table, 5-rolling bearing, 6-worm gear.

One of the variants of the gear that meets the requirements mentioned before is a worm gear offering autonomous mechanism for the rotation of the gear combined with a precise connection with a rotation device that is rigid along the tangent direction of the spindle and slender in other directions.

The examined comparator has an elastic worm gear connection with the spindle generating pure torque without any radial or axial forces and ensures the stability of the spindle. The worm gear is rotated by an electric motor through a mechanical reducer and can be disconnected to rotate the spindle manually.

The limb of the angle measuring system has a rigid connection avoiding any intermediate sliding mechanical parts. It helps in eliminating negative effects of hysteresis and reverse errors that have a negative effect on precision and are common to mechanical coupling devices (Kasparaitis, Sukys 2008; Sydenham, Thorn 1992).

There are two precise guiding devices made of small-grain granite mounted on the basis of the comparator that have two sliding carriages with stroke-detection microscopes mounted on them. They are placed orthogonally to the axis of the spindle on aerostatic supports.

A 36 sided polygon is placed on the top of the precision angle measuring system with the embedded Renishaw angle encoder. The system is precisely centred and levelled. The autocollimator Hilger & Watts is pointed directly to the mirror face of the polygon and the system is set to the position in which the readout of the autocollimator is 0.00 arc seconds as shown in Fig. 2.

[FIGURE 2 OMITTED]

2. Data processing

A systematic error along with the evaluation of its possible cause is one of the key procedures in data processing (Rabinovich 2010). One of the main features influencing the precision of any rotary device is the eccentricity of mounting a rotating or measuring element (eccentricity of bearing or the measuring scale etc.). Having obtained scale calibration results, the eccentricity of the scale (or a disc mounting) can be calculated. Data on encoder systematic errors allow determining the mechanical systematic errors of the elements of the test rig such as the eccentricity of bearings or the scale itself (Kim et al. 2011, Stone et al. 2004). The encoder was calibrated using one reading head and the first harmonic was noticed in the chart. The results of calibrating the encoder are displayed in Fig. 3.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The measurement results are displayed in Fig. 4. The measurement errors were calculated following the evaluation of the results of polygon calibration and the accuracy of the autocollimator. Two reading sensors were embedded in the measuring system to eliminate eccentricity errors. Fig. 4 displays the results of twelve circles rotating the angle measuring system forwards and backwards.

Having the smoothened data and using Fourier analysis, the harmonics of measurements could be calculated (Brucas, Giniotis 2008; Brucas et al. 2006). Finite Fourier series can be expressed applying the following formula:

[delta][phi]([phi]) = [A.sub.0] + 2 [n-1.summation over (m=1)] ([A.sub.m] cos 2[pi] m [f.sub.1][phi] + [B.sub.m] sin 2[phi] m [f.sub.1][phi]) + [A.sub.n] cos 2[phi] m [f.sub.1][phi]. (1)

The coefficients of the equation for each type of harmonics can be calculated as

[A.sub.m] = 1 / N [n-1.summation over (r=-n)] [delta][[phi].sub.r] cos 2[pi]mr / N; (2)

[B.sub.m] = 1 / N [n-1.summation over (r=-n)] [delta][[phi].sub.r] cos 2[pi]mr / N; (3)

where: m--the number of harmonics, N--the total number of measurements, r--the measurement number.

The amplitude of each type of harmonics is

[R.sub.m] = [square root of ([A.sup.2.sub.m] + [A.sup.2.sub.m])]. (4)

The phase shift of each type of harmonics (regarding the zero point) is

[[phi].sub.m] = arctg (-[B.sub.m] / [A.sub.m]) (5)

The received results indicate that the polynomial curve of the fourth order reflects the view of the present measurement data. The equation for the fourth order polynomial curve is

y = (7 x [10.sup.-4])[x.sup.4]--(4 x [10.sup.-7])[x.sup.3] + (6 x [10.sup.-5])[x.sup.2] + 0,0022x--0,1824. (6)

The calculation of a standard deviation is shown in Fig. 5.

[FIGURE 5 OMITTED]

The equation for the polynomial curve of the standard deviation can be expressed as

y = (6 x [10.sup.-11])[x.sup.4]--(2 x [10.sup.-8])[x.sup.3]--[10.sup.-6][x.sup.2] + 0,0011x + 0,1003. (7)

Conclusions

Systematic errors of the angle measuring system have been determined. Calculating the errors of the polygon and the accuracy of the evaluated autocollimator and encoder has displayed the results of systematic errors that tend to have the highest values in the range of the 36 sided polygon of 100[degrees]-170[degrees]. This leads to the assumption that obvious systematic errors may be caused by other components of the angle measuring system (such as bearings) or the accuracy of installing the measurement system.

doi:10.3846/20296991.2013.786879

Acknowledgement

This research was funded by the European Social Fund under the Global Grant measure.

References

Brucas, D.; Giniotis, V. 2008. Circular scale eccentricity analysis, Mechanika 2(57): 48-53. ISSN 1392-1207.

Brucas, D.; Giniotis, V.; Petroskevicius, P. 2006. The construction of the test bench for calibration of geodetic instruments. Geodesy and Cartography 32(3): 66-70. ISSN 13921541.

Just, A.; Krause, M.; Probst, R.; Wittekopf, R. 2003. Calibration of high-resolution electronic autocollimators against an angle comparator, Metrologia 40: 288-294. http://dx.doi.org/10.1088/0026-1394/40/5/011

Kasparaitis, A.; Sukys, A. 2008. Metrologinio ukio ir matavimu organizavimas. Vilnius: Technika. 192 p. ISBN 978-9-95528306-5. http://dx.doi.org/10.3846/991-S

Kim, J. A., et al. 2011. Precision angle comparator using self-calibration of scale errors based on the equal-division-averaged method, in Proc. of MacroScale 2011. Wabern, 1-4.

Rabinovich, S. G. 2010. Evaluating measurement accuracy. New York: Springer. 278 p. ISBN 978-1-4419-1455-2. http://dx.doi.org/10.1007/978-1-4419-1456-9

Stone, J. A.; Amer, M.; Faust, B.; Zimmerman, J. 2004. Uncertainties in small-angle measurement systems used to calibrate angle artifacts, Journal of Research of the National Institute of Standards and Technology 109(3): 319-333. http://dx.doi.org/10.6028/jres.109.024

Sydenham, P. H.; Thorn, R. 1992. Handbook of Measurement science: Elements of Change, vol. 3. Chichester. New York. Brisbane. Toronto. Singapore: John Wiley & Sons. 600 p.

Watanabe, T.; Fujimoto, H.; Nakayama, K.; Kajitani, M.; Masuda, T. 2003. Calibration of polygon mirror by the rotary encoder calibration system, in Proc. of XVIIIMEKO World Congress, June 22-27, 2003, Dubrovnik, Croatia, 18901893.

Lauryna Siaudinyte (1), Domantas Brucas (2), Mindaugas Rybokas (3), Genadijus Kulvietis (4)

(1,2) Institute of Geodesy, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(3,4) Department of Information Technologies, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

E-mails: (1) lauryna.siaudinyte@vgtu.lt (corresponding author); (2) domka@ktv.lt; (3) mindaugas.rybokas@vgtu.lt; 4gk@vgtu.lt

Received 05 February 2013; accepted 26 February 2013

Lauryna SIAUDINYTE. Junior research fellow at the Institute of Geodesy. Sauletekio al. 11, LT-10223 Vilnius, Lithuania, Ph. +370 5 274 4705, e-mail: lauryna.siaudinyte@vgtu.lt

MSc from VGTU in 2010. Currently, PhD student at the Department of Geodesy and Cadastre, Vilnius Gediminas Technical University.

Research interests: calibration of geodetic instruments, angle measurements, measurement accuracy and precision.

Domantas BRUCAS. Head of Space Science and Technology Institute, Sauletekio al. 15, LT-10223 Vilnius, Lithuania. Doctor (2008). The author of two educational books and more than 30 scientific papers. Participated in a number of international conferences, owns three registered patents.

Research interests: comparator development for angular measurements, automation of processing measurement results.

Mindaugas RYBOKAS. Assoc. Prof., Dr at the Department of Information Technologies, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania (Ph. +370 5 274 4832), e-mail: mindaugas.rybokas@vgtu.lt.

The author of more than 35 scientific papers; participated in a number of International conferences.

Research interests: analysis of information measuring systems.

Genadijus KULVIETIS. Prof., Dr Habil at the Department of Information Technologies, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania. E-mail: gk@vgtu.lt.

The author of two educational books and more than 50 scientific papers. Participated in a number of international conferences.

Research interests: analysis of information measuring systems.