Error Analysis and Correction of a High Precision Co-ordinate Table

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1 Error Analysis an Correction of a High Precision Co-orinate Table Gyula Hermann BrainWare Lt. Völgy utca 13/A, H-1021 Buapest Hungary hermgy@iif.hu Abstract: A small scale, high precision co-orinate table was evelope as part of a project to solve measurement problems originating from microtechnic. Construction with a single reference surface was selecte an vacuum preloae air bearing applie in orer to enable smooth motion an eliminate stick slip. Laser interferometers were applie to measure both the isplacement an the angular error. The paper escribes the mathematical evaluation of the measuring system s error an how that is taken into account in the evaluation software. Keywors: Co-orinate table, measuring system, evaluation software, error computation 1 Introuction Co-orinate metrology is toay a firmly establishe technique in inustry. The universal applicability an high egree of automation accounts for it s succes. Two other factors are equally important: tolerances are becomming closer as a result of interchangeable manufacturing an the increase of subcontracting is obliging manufacturers to assume responsibility for the imensional integrity of their parts. With the appearance of micromachine mechanical elements increase eman can be observe for higher resolution an more accurate inspection of these parts. 2 The Table Construction For many reasons ultra precise X-Y tables are guie by a flat reference surface. These tables usually employ vacuum or magnetically preloae air bearings. Their avantage is that only one flat guie surface is necessary, whereas the oppose

2 bearing preloaing requires two flat guie surfaces that are parallel. Other avantages of the air bearings are the zero static friction, which makes infinite resolution an the very high repeatability possible. Moreover porous air bearings average the errors of the guie surface finish an irregularities. In our case, the table has four legs an rests on four vacuum preloae flat porous air bearings. The porous carbon in the bearing prouces a uniform air pressure across the face of the bearing while automatically restricting an amping the air flow at the same time. As a consequence the possibility of collapse is reuce an a higher pitch moment stiffness is reache. As a reference surface a 000 class granite[1] control surface plate, having an overall flatness of less then 2 µm, is use an the axes are riven by linear motors. The winow in the moving part of the table enables the installation of a back lighting LED panel. Guieway Linear motors Winow Reference surface Preloae airbearings Carriage Figure 1 The mechanical construction

3 3 The Measuring System The table position is etermine by two plane mirror laser interferometers, a single- an a ualbeam, relative to the measuring frame fixe to the granite base. The laser sources are couple to the sensor heas via fiberoptic cables. The single beam interferometer enables a isplacement measurement along one axis with a resolution of 1 nm (uner ieal environmental conitions) while the ualbeam interferometer allows the simultaneous isplacement an angular measurement. The angular resolution is approximately 0.05 arcsec. The measuring range is 100 mm an ±2 arcmin respectively. singlebeam laser interferometer ualbeam laser interferometer electronic unit X off Y off fiberoptic link the moving part of the table Figure 2 The outline of the measuring system

4 4 The Geometric Moel of the Measuring System Y ref Y table T size T size L x1 Y off L x2 X c,y c + (p x,p y ).θ X table L y 0,0 X off X ref Figure 3 The geometric moel Where: X ref an Y ref are X the Y axis of the measuring frame co-orinate system X table an Y table are X the Y axis of the table co-orinate system X c an Y c are the co-orinates of the table coorinate system s origo X off an Y off are the positions of the interferometers along the measuring frame axis is the separation between the two beams in the ualbeam interferometer L x1, L x2 an L y are the isplacements measure by the interferometer T size is the table size Let us in the first instance assume that the roll an pitch error of the table is negligable. From Fig. 3 the following two equation for the origo of the table coorinate system axis can be seen: X c -L x2 Y off -Y c = tgθ (1)

5 L y - Y c X off - X c = tgθ (2) Where tgθ = L x1-l x2 (3) Hereout follows for the co-orinates of the origo: X c = ( X off - L x2 )tg 2 θ 1+tg 2 θ Y c = Y off.tg 2 θ 1+tg 2 θ + tgθ(l x2 - X off ) Ly 1+tg 2 θ + 1+tg 2 θ + (Y off -L y )tgθ 1+tg 2 θ + L x2 (4) The co-orinates of an arbitrary point on the table can be expresse by using the following equation: P x P y = cos -sin sin cos p x p + y X c Y ===> c cos sin -sin cos P x -X c P y -Y = c p x p (6) y (5) 5 The Error Moel In the above outline measuring system we have to take into account the following error sources: isplacement measurement error as a result of wavelength variation angle measurement error perpenicularity error of the laserbeams an the reflecting mirrors flatness error of the mirrors reference surface flatness error resulting in role an pitch error on the table Let us consier these error components one by one: The isplacement of the reflecting mirror is given by the following equation: D x = Where: c 2f (N + φ 2π ) (7) c is the spee of light f is the laser frequency

6 N is the number 2πs an Φ is the phase angle The laser frequency stability, after warmup is better than 3 x The spee of light is influence by the temperature, the humiity, the turbulence an the contamination of the air. The frequency of the laser can be stabilize an the effect of temperature an humiity can be compensate igitaly by measuring both air an object temperature an the humiity. The error equation for the angle is: Δtgθ Δ(L x2-l x1 ) + Δ tgθ (8) Hereout follows that the angular error is etermine by the relative error of the beamistance of the ualbeam interferometer. The eviation from the prepenicularity of the reflecting mirrors an that of the irections of the interferometers contribute to the overall error of the system. Also the perpenicularity error of the reference system (X ref,y ref ) has a similar effect. The angle between the mirrors is φ, whereas the angles between the reference axis an the axis of the interferometers are given by α an β respectively. From Fig. 4 it can be immeiatly seen that: Δ = 90 - φ = θ - ψ (9) Where Δ = 90º- φ is the perpenicularity error of the table (the reflecting mirrors)

7 YBrefB TBsizeB Y table T size L x1 L x2.φ X table.θ Y off.α X c,y c.ψ ß 0,0 X off L y X ref Figure 4 Perpenicularity error moel X c - L x2 cosα Y off - L x2 sinα - Y c = tg ψ (10) L y cosβ - Y c X off - L y sinβ - X c = tg θ (11) (L x2 - L x1 ) cosα + (L x2 - L x1 ) sinα = tg ψ (12) From hereout follows by substituting sinα = α, cosα = 1, sin β = β, cos β = 1, an neglecting the higher orer small terms we get: tg θ meas = L x2-l x1 an tg θ = tg(90º-φ) + tgψ 1 -tg(90º-φ)tgψ (L x2 -L x1 )cosα θ = 90 - ( φ + ψ) = Δ - arctg + (L x2 -L x1 )sinα = Δ - arctg tgθ meascosα 1+tgθ meas sinα (13) L x2 X c = 1+αtgθ + tgθ meas (Δ+tgθ meas )(X off -L x2 ) meas 1+(3α+Δ)tgθ meas +tg 2 θ meas + - (1+2αtgθ meas)tgθ meas L y 1+(3α+Δ)tgθ meas +tg 2 θ meas + αtg 3 θ meas X off 1+(3α+Δ)tgθ meas +tg 2 θ meas - (1+(2α+Δ)tgθ meas-δtgθ meas )tgθ meas Y off 1+(3α+Δ)tgθ meas +tg 2 θ meas (14)

8 (1+2αtgθ meas ) L y Y c = ( 1+(2α+Δ)tgθ meas +tg 2 θ meas ) + (Δ+tgθ meas )(L x2 - X off ) (1+(2α+Δ)tgθ meas +tg 2 θ meas ) + αtg 2 θ meas X off + ( 1+(2α+Δ)tgθ meas +tg 2 θ meas ) + tgθ meas (Δ+tgθ meas ) Y off (1+(2α+Δ)tgθ meas +tg 2 θ meas ) Using these results the co-orinates of a point in the table co-orinate system can be exprese using equation [6]. The geometric constant values Δ, α an β use in the computation can be measure by conventional optical techniques. If we consier the laserbeam as ray with nearly zero iameter then the flatness error of the mirrors results in an error of the isplacement measurement. However the finite iameter of the beam has an integration property, so the local flatness error can be compensate to a certain exten. Waveness errors larger then the beam iameter have to be taken into account. A map of the mirror surface is the necessary an sufficient information for the compensation of this type of errors. The map can be capture by interferometric techniques. The applie technologies in manufacturing optical surfaces ensure that the functions escribing the surface is continoues into the secon erivatives. Therefore a spline moel of the surface is space an time saving. (15) Y ref Y table Best fit straightline L x1 Angle between the best fit straightlines L x2 X table Y off X c,y c Best fit straightline L y 0,0 X off X ref Figure 5 Moel of the straightness (flatness) error

9 6 The Software Implementation The interferometer elivers through USB ports two current co-orinate an a current angle value. These signals are connecte to a personal computer responsible for the calculations. As high presion measurements are connecte with relavtively slow isplacements the computational time is not a critical factor an as we have seen the amount of ata involve is also small. The compensation for the spee of light eviation is performe by the electronic unit of the interferometer. There is no nee for any aitional harware. The program performing the above mentione calculations is written in C++ an runs on the Winow XP operating system. Base on an analysis of the formulas at some point multilenght arithmetic is use in orer to ensure the require accuracy. The measurement result wil be publishe in separate paper. Conclusions The research a part of which presente here prove the usefulness of software error compensation in co-orinate measurement. Accoring to our expectation combine with new construction the accuracy can be improve with an orer of magnitue. The initial results are promising an the technique can extene into the thir imension. Acknowlegment This research was supporte by the Hungarian Scientific Research Foun through the contract T References [1] G. Belforte, B. Bona, E. Canuto, F. Donati, F. Ferraris, I. Gorini, S. Morei, M. Peisino, S. Sartori Coorinate Measuring Machines an Machine Tools Selfcalibration an Error Correction Annals of the CIRP Vol. 36/1/1987, pp [2] B. P. Demiovich, I. A. Maron Computational Mathematics MIR Publishers Moscow 1973 [3] K. F. Eman, B. T. Wu, A Generalize Geometric Error Moel for Multi- Axis Machines Annals of the CIRP Vol. 36/1/1987, pp [4] P. M. Ferreire, C. R. Liu An Analitical Quaratic Moel for the Geometric Error of a Machine Tool Journal of Manufacturing Systems Vol. 5, No. 1, pp [5] B. Meulenbel, Analyse Uitgeverij Stam N. V. Haarlem 1962 [6] F. C. e Nijs Moelling of a Coorinate Measuring Machine for Analysis of Its Dynamic Behaviour Annals of the CIRP Vol. 37/2/1988, pp

10 [7] J. W. M. C. Teeuiwsen, J. A. Soons, P. H. J Schellekens A General Methoe for Error Description of CMMs Using Polinomial Fitting Proceures Annals of the CIRP Vol. 38/1/1989, pp [8] M. M. Weck, Geometric an kinematic errors. Technology of Machine Tools, 1980, 5, 9-12 [9] M. M. Weck, Werkzeugmaschinen Ban 2 VDI-Verlag Düsselorf 1981 [10] G. Zhang, R. Veale, B. Borchart an R. Hocken Error Compensation of Coorinate Measuring Machines Annals of the CIRP Vol. 34/1/1985, pp

6. Friction and viscosity in gasses

6. Friction and viscosity in gasses IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner