2011 SLOVAK UNIVERSITY OF TECHNOLOGY18

V. HAŠKOVÁ, M. LIPTÁK

PROCESSING OF CALIBRATION MEASUREMENTS ON EZB-3

KEY WORDS

• Calibration,• correction,• polynomial series,• trigonometric series.

ABSTRACT

The calibration of horizontal circles results in a set of discrete correction values. These corrections are obtained in specific circle positions chosen by the size of the calibration step. For further use it is necessary to know the correction values for any location on a horizontal circle; therefore, it is necessary to know the function of the continuous corrections of a horizontal circle. This could be achieved from the measured values by several methods. In the article two methods are presented for determining this function through the approximation of polynomial and trigonometric series.

Veronika HAŠKOVÁemail:veronika.haskova@stuba.skResearch field: calibration of surveying instruments,calibrationdataprocessingandanalysis,landsurveying.

Address:DepartmentofSurveying,FacultyofCivilEngineering,SlovakUniversityofTechnology,Radlinského13,81368,Bratislava,Slovakia

Miroslav LIPTÁKemail:miroslav.liptak@stuba.skResearch field: land surveying, influence of theenvironmentonterrestrialsurveyingmeasurements.

Address:DepartmentofSurveying,FacultyofCivilEngineering,SlovakUniversityofTechnology,Radlinského13,81368,Bratislava,Slovakia

Vol. XIX, 2011, No. 4, 18 – 23

INTRODUCTION

Toachievetherequiredprecisionofgeodeticworksinallareas,itisnecessary to use such instruments in the measuring process whoseproperties are in compliance with the specified requirements. Oneofthesefeaturesistheconsistency,accuracyandrepeatabilityofthevaluesofthehorizontaldirections.Theprocessfortheverificationofthequalityoftheodolitehorizontalcirclesisbasedonastandard(STNISO 17123-3); some service centres practise the control of dividedcirclesinaccordancewiththemanufacturer’srequirementsundertheDIN and ISO standards (www.geotech.sk). The EZB-3 calibrationequipment of theSlovak Institute ofMetrology (SIM) inBratislavaallows for the calibration of horizontal circle surveying instrumentsusingaprimarystandardoftheplanarangleoftheSlovakRepublic.

EZB-3 STANDARD EQUIPMENT

TheEZB-3standardequipmentisusedforcalibratingapolygonalprismwithacombinedstandarddeviationofcorrectionofuc=0,06''.

The equipment was created using aOKT-315 Zeiss Jena rotationdividing table with aservo-drive, where acalibrated polygonalprismislocated,aswellastwoTA-80Hilger&Wattsphotoelectricautocollimators. Those elements are located on acast iron baseplatewithananti-vibrationdeposition(Mokroš,2001).TheschemeoftheEZB-3standardequipmentisshowninFigure1.Inthebasepositionbothautocollimators(AK)arepointedattheneighbouringfunction surfaces of the traverse; by their stepwise rotation theautocollimatorsmeasure the deviations of the normal positions tothetraversefunctionsurfaceswithregardtotheopticalaxesoftheautocollimators.For calibrating the theodolites a72-edged polygonal prism (PP)from Starett Webber is located on the position of the calibratedtraverse,which is acentric stored calibrated theodolite fittedwithaplane mirror. One of the autocollimators, which also serves asapart of the rotation servo-driving (M) of the dividing table, isfocused on the polygonal prism. The second autocollimator issituatedon thebaseplatewith itsopticalaxisat theheightof thetheodolite’stelescopeaxis.AnexampleoftheodolitecalibrationontheEZB-3standardequipmentisinFigure2.

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19PROCESSING OF CALIBRATION MEASUREMENTS ON EZB-3

Before the measurement it is necessary to fulfill the adjustmentconditions described in (Mokroš, 2001). After adjustment of themeasuring formation, the prism autocollimator is aimed at therequired functional surface of the polygonal prism by turning therotationalpartofthetablewiththeprismandtheodolite.Thenthesecondautocollimatorispointedatthetelescope’ssubsidiarymirrorwiththehelpofthetheodolite’salidadebackrotation.Thevaluesofbothautocollimatorsandthevalueofthetheodolite’sdividingcirclewith agiven number of repetitions are read. The rotation part ofthetablerotatestothenextprism’sfunctionalarea,andtheprocess

is repeated.One calibration step consists of both autocollimatorsscanningdataintothePCofthemeasurementsystemandtheprismandtheodoliteturningabouttherequiredangle(multipleof5°)withadeviationof0.1''.Provided that the dividing circle of the calibrated theodolite andautocollimator scales is oriented so that by pointing at the firsttraversefunctionarea,thereadvalueofthetheodoliteincreasesandthereadvalueoftheautocollimatorsdecreases,thenthecorrectionofthetheodolite’sdividingcircleisdeterminedby:

(1)

wherekiisthecorrectionofthedividingcircleofthecalibratedtheodolite at i-point; N is the number of polygonal prismfunction areas;

is the correction of the function area j of

the prism; Bi is the value of the theodolite’sdividing circleat i-point;aj is the reading of the autocollimator scale of thefunctionarea jof thepolygonalprism;bi is thereadingof theautocollimator’sscaleatthei-pointofthedividingcircleofthecalibratedtheodolite.

Corrections determined in this way are important for theodoliteswiththefixedpositionofthedividingcircle.Fortheodolites,whichcanchangethepositionoftheorigin,thesecorrectionsprovideonlyanideaaboutanerrorcourseofahorizontalcircle’sreadingsystemwithoutapossibilityofitscorrection.

CALIBRATION OF A THEODOLITE’S HORIZONTAL CIRCLES ON EZB-3

HorizontalcirclesoftheLeicaTS30andLeicaTCR407electronictotal stations were calibrated on the adapted EZB-3 standardequipment for theodolite calibration. Before the start of thecalibration measurement it is necessary to select the calibrationcycle counts and calibration steps. The calibration step couldbe an integral multiple of the nominal angle value between theneighbouringfunctionareasofthepolygonalprism.Thepolygonalprism, which is apart of the EZB-3 equipment, has 72 functionareas, so the nominal angle value between the neighbouringfunction areas is 5º. Our measurements were realised with a10ºcalibrationstepand4calibrationcyclecounts.Figures3and4showgraphic representations of the horizontal circle corrections fromboth electronic total station calibrations. For the next processingof themeasureddatafileswith10º,20ºand30,ºcalibrationstepsandcyclecountsfrom1to4werecreated.Thecalculationsintheappropriatefilewerethenrealisedfortheaverageofthemeasuredvalues.

Fig. 1 The scheme of the EZB-3 standard equipment.

Fig. 2 Leica TS30 calibration on EZB-3.

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MEASURED DATA PROCESSING

The aim of the measured data processing is to find asufficientfunction type and its parameter values,which indicate the courseof thehorizontalcirclecorrectionvalues.Therearemanyoptionsforsolvingthisapproximation.Inthenextpartwewillfocusonanapproximationbasedonapolynomialfunctionandatrigonometricfunction.

The polynomial function describing the relation between thecorrection values of ahorizontal circle and the changes in thehorizontalcirclereading,couldbewrittenasfollows:

, (2)

where i = 1, 2, ..., m, k is the measurable correction; a is thehorizontalcirclereading;Θjistheunknownpolynomialparameters(j = 0, 1, ..., n). The suitable polynomial degree is determinedpursuant to various characteristics after estimating the unknownparameters.Trigonometric functions are used for approximating periodicalcases,i.e.,caseswitharepetitivecourseatacertainperiod.Thecasecausedbyonesourcecouldbedescribedbythesimplesinusoid:

, (3)

wherei=1,2,...,m,Θ0isasinusoiddriftinayaxisline;P istheperiod;ai is thesinusoidamplitude;ϕ1 is theperiod’soriginshift.Pursuanttotheadditiverelationforthesineoftwoangles,equation(3)couldbymodifiedintheform:

. (4)

Afterimplementingthesubstitution:

, (5)

wecanwritetheequation(4)asfollows:

, (6)

whereΘj are unknown parameters (j = 0, 1, ..., n). The originalparametersaiandϕ1canbecalculatedaccordingtotheformulas:

and . (7)

Each partial periodic error in the divided circle fabrication hasasinusoid course; each trace has adifferent origin, period andamplitude. After their composition in aclosure periodic error,acompositesinusoidarises(Böhm,Radouch,1978). Inhorizontalcirclecorrectionsonlyevenmultiplesoftheargumentareused:

.

(8)

Fig. 3 Graphic representation of the measured horizontal circle correction values of the Leica TS30.

Fig. 4 Graphic representation of the measured horizontal circle correction values of the Leica TCR407.

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21PROCESSING OF CALIBRATION MEASUREMENTS ON EZB-3

The formulation of equation (8) as afunction of parameters Θj(j=0,1, ...,n) is ananalog to the formulationof theelementarysinusoid.Thedifferenceisonlymoreunknownparameters.The estimation of the unknown parameters of polynomial andtrigonometric series is solved by the least squares method. Thetheoretical model described by equations (2) and (8) can beexpressedinamatrixform:

, (9)

andafterthelinearizationasfollows:

, (10)

where is an-dimensional vector of unknown parameters;is an-dimensional vector with the approximate values of ;

isanm ×n-dimensionalmatrixof thefirstderivates of thevector function in accordancewith , andcalculatedfortheapproximatevaluesoftheunknownparameters;and is an- dimensional vector of the increments of theunknownparameter .Thestochasticmodelrelativetotheoncerealizedmeasurementofeachdirectlymeasurableparametercouldbewrittenasfollows:

, (11)

where is am-dimensional observation vector, whose ithcomponentmodels themeasurement of the ith component of thevectork; isam-dimensionalvectoroftherandomerrors; is

anm ×m-dimensionalcovariancematrixofvector .Because isestimatedinaform ,thesubjectofthecalculationis the vector of the increments in pursuance of realizationx of arandom vector , . Using theminimizingofaquadraticformforestimating ofthevectorofincrements ,weget(Kubáčková,1990):

(12)

The solution is conditional on the existence of all the necessarymatrixinversionsthatoccurredinequation(12).

EVALUATION OF RESULTS

The calculations necessary for the approximation based on thepolynomial function and trigonometric function were realized byMicrosoftExcel,PTCMathCADandOriginLabOriginsoftware.Bothfunctionsweredeterminedforeachofthecorrectionvaluefiles,whichwereachievedbythecalibrationofthehorizontalcirclesofbothelectronictotalstations.ThesuitabledegreeofthepolynomialfunctionandthetypeoftrigonometricserieswasrealizedaccordingtothevalueofthecoefficientofadeterminationR2andtheadjustedcoefficientofdetermination .Bothcoefficientsindicatetherateofcomplianceoftheempiricallymeasureddatawiththeproposedmodels. The coefficient of adetermination R2 value closer to 1means that the proposed model better represents the empiricallymeasured data. The number of estimated parameters comes onlyinto the estimation of the adjusted coefficient of determination

Tab. 1 Coefficients of determination for Leica TS30.

Step/CyclePolynomial series Trigonometric series

Degree R2 R2adj Multiple R2 R2

adj

10º/1 6(0) 0.770 0.733 2 0.549 0.52310º/2 6(0) 0.778 0.743 2 0.576 0.55110º/3 6(0) 0.762 0.723 2 0.563 0.53710º/4 6(0) 0.731 0.688 2 0.572 0.54720º/1 6(0) 0.872 0.823 2 0.583 0.53120º/2 8(0,2) 0.914 0.870 2 0.626 0.58020º/3 8(0,1) 0.904 0.856 2 0.601 0.55120º/4 8(0,1,2) 0.860 0.806 2 0.614 0.56630º/1 10(0,1,2,5,6) 0.858 0.756 2 0.661 0.59330º/2 10(0,1,2,5,6) 0.839 0.724 2 0.684 0.62030º/3 10(0,1,2,8,9) 0.788 0.363 2 0.675 0.61030º/4 10(0,1,2,8,9) 0.725 0.529 2 0.675 0.611

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. It follows that by the addition of aparameter, which doesnot increase themodel’squality, thevalueof thecoefficientR2 isnot decreasing; the coefficient could decrease and achieveanegativevaluetoo.In Table1 the values of coefficients R2 and , which wereestimatedasmaximalforthenumberofcalibrationstepsandcyclecounts from the calibration results of the LeicaTS30 in amodelbased on polynomial and trigonometric series, are summarized.Also,thedegreesofthepolynomialseriesandthemultiplesoftheargumentofthetrigonometricseriesassociatedwiththecoefficientR2and aregiven.Inthebracketstheinsignificantparametersarewritten.Theseparameterswereeliminatedfromtheadjustmentresultsandtheremainingparameterswereestimatedagain.Foreachvalueofthecalibrationstepthenumberofcyclecountswithahighvalueof ishighlighted.Ascanbeseenfromthetable,thebestresultsachievedfortheLeicaTS30inthecaseofapolynomialseriesisbya20ºcalibrationstepwith2cyclecounts,andtheequation,whichdefinesthecourseofthehorizontalcircle’scorrectionvalues,isintheformofapolynomialfunctionofthe8thdegreewithouttheparameters and :

(13)

inthecaseofthetrigonometricseries,thebestresultsachievedbythe30ºcalibrationstepwith2cyclecountsandtheequationishereintheformofatrigonometricfunctionwithmultiplesofargument2:

(14)

Agraphic representation of the curves determined by equations(13)and(14)definethecourseofthehorizontalcircle’scorrectionvaluesfortheLeicaTS30electronictotalstationisinFigure5.ThevaluesofcoefficientsR2and ,whichwereestimatedfromthecalibrationresultsoftheLeicaTCR407electronictotalstation

Tab. 2 Coefficients of determination for Leica TCR407.

Step/CyclePolynomial series Trigonometric series

Degree R2 R2adj Multiple R2 R2

adj

10º/1 6(0) 0.464 0.377 2,4 0.555 0.49910º/2 6(0) 0.490 0.408 2,4 0.629 0.59510º/3 6(0) 0.499 0.418 2,4,6 0.642 0.58410º/4 6(0) 0.493 0.412 2,4 0.631 0.58520º/1 10(0) 0.777 0.554 2,4 0.687 0.62420º/2 10(0) 0.811 0.622 2,4,6 0.827 0.76120º/3 10(0) 0.815 0.630 2,4,6 0.828 0.74220º/4 10(0) 0.803 0.606 2,4,6 0.817 0.72530º/1 10(0) 0.956 0.824 2,4 0.771 0.65630º/2 10(0) 0.950 0.800 2,4 0.849 0.77330º/3 10(0) 0.935 0.741 2,4 0.812 0.74930º/4 10(0) 0.945 0.780 2,4 0.829 0.771

Fig. 5 Curves determining the corrections course of the horizontal circle for the Leica TS30.

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23PROCESSING OF CALIBRATION MEASUREMENTS ON EZB-3

inthemodelbasedonthepolynomialandtrigonometricseries,arepresentedinTable2.Thenumberofcyclecountswithahighvalueof foreachvalueofthecalibrationstepishighlighted.FromtheresultswecanseethatforLeicaTCR407thebestresultsachievedinthecaseofapolynomialseriesisbythe30ºcalibrationstep with a1 cycle count, and the equation, which defines thecourse of the horizontal circle’scorrection values, is in the formofapolynomialfunctionofthe10thdegreewithoutparameter :

(15)inthecaseof trigonometricseries thebestresultsachievedbythe30ºcalibrationstepwith2cyclecountsand theequation is in theformofatrigonometricfunctionwithmultiplesofargument2and4:

(16)

Agraphic representation of the curves determined by equations(15) and (16), which define the course of the horizontalcircle’scorrection values for the Leica TCR407 electronic totalstation,isinFigure6.

CONCLUSION

In the paper the procedure for calibrating the horizontal circle ofthe LeicaTS30 and LeicaTCR407 electronic total stations on theEZB-3 standard equipment of the Slovak Institute ofMetrology inBratislavawasdescribed.Forbothinstrumentsthesuitablefunction,

which determines the course of the horizontal circle’scorrectionvaluesdependingon thevalueof thehorizontalcircle reading,wasdetermined.Thefunctionapproximationwasbasedonthepolynomialfunction and trigonometric function.According to the value of thecoefficientsofthedetermination,thesuitabledegreeofthepolynomialfunctionandthe typeof the trigonometricseriesforbothelectronictotalstationswasselected.Fromtheestimatedresultsitcanbesaidthat for determining the course of the horizontal circle’scorrectionvalues,thecalibrationofbothinstrumentswitha30ºor20ºcalibrationstepupto2calibrationcyclecountsissufficient.

ACKNOWLEDGEMENT

Theauthorsaregrateful to theSlovakGrantAgencyVEGAfor itsfinancial support of this project.The project registration number is1/0142/10.

REFERENCES

[1] Böhm, J., Radouch, V.: Vyrovnávací počet. Adjustmentcomputation.Prague:Kartografie,1978.510pp.(inCzech).

[2] <http://www.geotech.sk/Sluzby/Kalibracie.html>.[3] Kubáčková,L.:Metódyspracovaniaexperimentálnychúdajov.

MethodsofExperimentalDataProcessing.Bratislava:VEDA,1990.324pp.ISBN80-224-0104-8(inSlovak).

[4]Mokroš, J. Kalibrácia horizontálneho kruhu teodolitov.

Calibration of theodolite horizontal circle. In: Metrológiavgeodézii. Bratislava: Katedra geodetických základov SvF,2001.ISBN80-227-1617-0,pp.123-130(inSlovak).

[5] STNISO17123-3. Optics and optical Instruments-Fieldproceduresfortestinggeodeticandsurveyinginstruments-Part3:Theodolites.2001.

Fig. 6 Curves determining the corrections course of the horizontal circle for the Leica TCR407.