At the moment in Lithuanian there is no existing reglamentation dealing with calibration or testing of geodetic angle measuring instrumentation (as theodolites, tacheometers, laser scanners, laser trackers etc.) in full range of its angular measurements. Though calibration and testing of horizontal angle measurements can be performed by multiple methods, calibration of vertical angle measurements still is a serious problem. In the paper a review on possible methodology of calibration and testing of vertical angle measuring instrumentation (like tacheometers, laser scanners etc.) is given together with proposed principles and technical solutions for the calibration

Domantas Bručas

Genadijus Kulvietis

Gintaras Dmitrijev

Keneth Grattan

Lauryna Šiaudinytė

Mindaugas Rybokas

English

Journal of Measurements in Engineering

 8 JOURNAL OF MEASUREMENTS IN ENGINEERING. MARCH 2013, VOLUME 1, ISSUE 1  
2. Methodology of calibration of vertical angle 
measurements 
Domantas Bručas1, Lauryna Šiaudinytė2, Mindaugas Rybokas3, Keneth Grattan4, 
Genadijus Kulvietis5, Gintaras Dmitrijev6 
Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius-40, Lithuania 
1Corresponding author 
E-mail: 1domka@ktv.lt, 2lauryna@inbox.lt, 3mindaugas.rybokas@vgtu.lt, 4k.t.v.grattan@city.ac.uk, 
5gk@vgtu.lt, 6gintaras.dmitrijev@dok.vgtu.lt 
(Received 22 October 2012; accepted 4 March 2013) 
Abstract. At the moment in Lithuanian there is no existing reglamentation dealing with 
calibration or testing of geodetic angle measuring instrumentation (as theodolites, tacheometers, 
laser scanners, laser trackers etc.) in full range of its angular measurements. Though calibration 
and testing of horizontal angle measurements can be performed by multiple methods, calibration 
of vertical angle measurements still is a serious problem. In the paper a review on possible 
methodology of calibration and testing of vertical angle measuring instrumentation (like 
tacheometers, laser scanners etc.) is given together with proposed principles and technical 
solutions for the calibration. 
Keywords: vertical angle, calibration, systematic errors, random error, calibration method. 
1. Introduction 
Testing and calibration of angle measurement instrumentation can provide the knowledge of 
both random and systematic (biases) errors of the measures. If determination of random errors of 
measurements defines the uncertainty of measurements, the determination of systematic errors (or 
biases) can provide valuable information for increasing the accuracy of measurements by 
providing (physically or numerically) the corrections of results. Therefore precise determination 
of systematic errors of measurements is crucial and capable of increasing the accuracy of 
measurements considerably. 
Generally there are several groups of plane angle measurement principles (methods) which 
could be applied for testing and (especially) calibration of angle measurement 
instrumentation [1, 2]: 
1. Solid angular gauge method: 
– polygons (multiangular prisms); 
– angular prisms; 
– angle gauges, etc.; 
2. Trigonometric method (angle determination by means of linear measurements); 
3. Goniometric method (plane angle determination by means of a circular scale): 
– full circle (limb, circular code scales etc.); 
– non-full circle (sector scales). 
Nowadays in geodesy, surveying and machine engineering lots of instrumentation relying on 
precise angle measurements is implemented. These instruments are: theodolites, tacheometers, 
total stations, laser scanners, laser trackers etc. The accuracy of measurements of these instruments 
relies on precise determination of both horizontal and vertical angle measures. Since angle 
encoders implemented in mentioned instruments (same as the entire system) can produce 
significant errors in measurement, these angle measuring instruments should be thoroughly tested 
and calibrated. Calibration and testing of angle measuring instruments has always been a serious 
problem and if calibration of the horizontal angle measurements could be quite efficiently 
accomplished using standard precise turn tables (quite widely implemented in metrology and 
industry), calibration of vertical angle measures required some special instrumentation [3-5]. 
Additionally since modern angle encoders implemented in the measuring instrumentation produce 
2. METHODOLOGY OF CALIBRATION OF VERTICAL ANGLE MEASUREMENTS.  
DOMANTAS BRUČAS, LAURYNA ŠIAUDINYTĖ, MINDAUGAS RYBOKAS, KENETH GRATTAN, GENADIJUS KULVIETIS, GINTARAS DMITRIJEV 
 © VIRBROENGINEERING. ISSN PRINT 2335-2124, ISSN ONLINE 2424-4635, KAUNAS, LITHUANIA 9 
a huge number of angle measures a calibration in the entire range of angle measures is also 
required. 
Presently a standard outlining the testing of angle measuring instrumentation is available. Such 
testing determines only the uncertainty of angle measures in certain very limited angular positions, 
thus practically determining only the random error of measures by testing the repeatability. Such 
testing cannot ensure the reliability of measures. The equipment capable of determining the 
precise systematic errors of the measures and therefore determining the true angle values is usually 
operated by large instrumentation producing or implementing companies and is not available for 
wide public. 
Further in this paper we present several cost-effective methods of possible vertical angle 
calibration, with some of the results of the real measurements.  
2. Principle of calibration of vertical angle measures 
Calibration of vertical angle measures of the geodetic instruments was usually performed by a 
special bench composed of autocollimators attached at the different vertical angles to the 
calibrated instrument and mirror (which autocollimators pointed to) attached to the telescope of 
the instrument, in this case the entire test bench used to be extremely bulky and able to measure 
only very limited number of vertical angles [4]. A new approach to the problem was 
implementation of the precise angle encoder for the creation of vertical angle reference, in this 
case it was possible to create unlimited number of reference angle values, but the equipment was 
extremely expensive [6-8]. One of the projects of similar equipment allowing high accuracy 
calibration of vertical angle measures is shown in Fig. 1. 
Bench control 
unit 
Control PC 
Calibrated 
instrument 
Autocollimator 
control unit 
Pointing 
autocollimator Precise angle encoder  
Fig. 1. Arrangement of instrumentation for calibration of vertical angle measures of geodetic instruments  
Test bench shown in Fig. 1 can be implemented for testing and calibration of vertical angle 
measures with high precision (depending on the mechanical structure, encoder and positioning 
autocollimator used), nonetheless such arrangement is extremely expensive due to high cost of 
instrumentation and manufacturing of high accuracy mechanical equipment [9-10]. 
To decrease the complexity of equipment required (thus increasing the needed input from the 
operator) a method of calibration of vertical angles of geodetic equipment implementing 
trigonometric methods was proposed (Fig. 2). 
In case shown in Fig. 2 the vertical angle measure is derived from the precise linear bar, with 
telescope being collimated manually by operator. That way the precise calibration of vertical angle 
measurements can be done implementing relatively inexpensive and widely available equipment. 
Nonetheless the greatest difficulty in case of such calibration is precise determination of the 
distance from rotation angle of the telescope to linear bar. This distance cannot be measured by 
simple means due to practically unknown nature of the mechanical attachment of telescope [11]. 
To solve the problem a method of determination of distance by means of precise movement of 
instrument under testing to determined position with repeated scale measurements was developed. 
2. METHODOLOGY OF CALIBRATION OF VERTICAL ANGLE MEASUREMENTS.  
DOMANTAS BRUČAS, LAURYNA ŠIAUDINYTĖ, MINDAUGAS RYBOKAS, KENETH GRATTAN, GENADIJUS KULVIETIS, GINTARAS DMITRIJEV 
10 JOURNAL OF MEASUREMENTS IN ENGINEERING. MARCH 2013, VOLUME 1, ISSUE 1  
The principle of proposed vertical angle calibration is based on the trigonometric angle 
determination. The arrangement for calibration is shown in Fig. 3.  
Control PC 
Calibrated 
instrument 
Precise linear 
bar 
 
Fig. 2. Arrangement for calibration of vertical angle measures implementing trigonometric methods 
lm 
h 1
 
h 3
 
h 2
 φ  2
 
φ  3
 
φ 1 
φ  4
 
scale 
collimation lines 
calibrated 
instrument 
h 4
 
 
Fig. 3. Arrangement for vertical angle calibration for geodetic instruments 
As can be seen from the picture an instrument to be calibrated is placed at a certain known 
distance (݈௠, Fig. 3) from the precise linear scale. The spyglass of the instrument is declined at 
the angle ߮ (߮ଵ, ߮ଶ, ߮ଷ or ߮ସ) accuracy of which must be calibrated. The reading ℎ (ℎଵ, ℎଶ, ℎଷ 
or ℎସ) from the scale is taken. The angle of interest is expressed: 
߮ = arctan ℎ݈௠, (1) 
where ℎ – the reading from the scale, ݈௠ – distance from the instrument’s center to the reading 
surface of the scale.  
It is difficult to measure with high accuracy, so additional measures must be taken for the 
calculations. It is accomplished by adding the known length measure ݈௘ of high accuracy. It can 
be the end gauge, linear incremental transducer, etc. On the second step the instrument is replaced 
on the distance (݈ + ݈௘) from the scale preserving the same value ߱ of instrument’s view angle. 
The position of horizontal axis is replaced from the point ଵܱ to the position ܱଶ. 
For the precise vertical angle measurements the distance from calibrated instrument 
(tacheometer) and the reference measure (linear scale) has to be determined quite precisely (down 
to 0.01 mm). 
The determination of the distance from calibrated instrument to the reference measure can be 
determined using the layout shown in Fig. 4. The distance ݈௠ from instrument calibration position 
(0) to the reference measure (scale) can be calculated by moving the instrument linearly to the 
2. METHODOLOGY OF CALIBRATION OF VERTICAL ANGLE MEASUREMENTS.  
DOMANTAS BRUČAS, LAURYNA ŠIAUDINYTĖ, MINDAUGAS RYBOKAS, KENETH GRATTAN, GENADIJUS KULVIETIS, GINTARAS DMITRIJEV 
 © VIRBROENGINEERING. ISSN PRINT 2335-2124, ISSN ONLINE 2424-4635, KAUNAS, LITHUANIA 11 
subsidiary positions (1, 2) while the same vertical angle (ߙଵ or ߙଶ) of the calibrated instrument 
spyglass is being kept. Therefore the readouts of the vertical scale at the same vertical angle and 
different length to the scale can be made and distances ݈ଵ, ݈ଶ and thus ݈ଷ can be determined. The 
calibrated instrument should move from position 0 to positions 1 and 2 by means of some kind of 
guide-skid, and the linear movement (݈௣ଵ and ݈௣ଶ) determined using the linear photoelectrical scale, 
laser interferometer or even precise linear optical scale (with microscope). 
lp1 
lm 
l 1 
l 3 
l 2 
α 2 α 2 
α 1 α 1
 
scale 
collimation lines 
2 
0 
lp2 1 
calibrated 
instrument 
 
Fig. 4. Determination of the distance from calibrated instrument to the reference measure; 0 – instrument 
calibration position, 1, 2 – subsidiary calibrated instrument position used for distance determination 
To determine the distance ݈௠  calibrated instrument at the calibration position 0 should be 
pointed at the certain scale stroke with the vertical angle ߙଵ and scale readings taken. After that 
instrument is moved to the position 1 keeping the same vertical angle ߙଵ  till the spyglass 
collimated to different scale stroke, thus scale distance ݈ଵ determined. After that instrument moved 
back to the position 0 and collimated to the different scale stroke with vertical angle ߙଶ, the 
measurement procedure is repeated thus the distance ݈ଶ determined. Distances ݈௣ଵ and ݈௣ଶ will 
usually be different since it is almost impossible (in case of visual scale) to collimate exactly to 
the scale stroke (having different vertical angles ߙଵ and ߙଶ) having the same horizontal movement. 
The procedure of two measurements (݈௣ଵ and ݈௣ଶ) is essential since it is also almost impossible 
(again in case of simple visual scale) to collimate the tacheometer at the horizontal collimation 
line to a certain scale stroke. 
Having the mentioned measurements performed the vertical angles can be determined (using 
linear measures, Fig. 3): 
tanߙଵ =
݈ଵ
݈௣ଵ, (2) 
and: 
tanߙଶ =
݈ଵ
݈௣ଶ, (3) 
where ݈ଵ , ݈ଶ  – scale measures at the same vertical angle but different linear distance from 
calibrated instrument to the reference scale; ݈௣ଵ , ݈௣ଶ  – linear movement of instrument during 
measurements. 
Also the total measured length of the scale could be expressed (Fig. 1): 
݈ଵ + ݈ଶ + ݈ଷ = ݈௠ ⋅ tanߙଵ + ݈௠ ⋅ tanߙଶ, (4) 
where ݈௠ – distance from the calibrated instrument to the reference scale. 
2. METHODOLOGY OF CALIBRATION OF VERTICAL ANGLE MEASUREMENTS.  
DOMANTAS BRUČAS, LAURYNA ŠIAUDINYTĖ, MINDAUGAS RYBOKAS, KENETH GRATTAN, GENADIJUS KULVIETIS, GINTARAS DMITRIJEV 
12 JOURNAL OF MEASUREMENTS IN ENGINEERING. MARCH 2013, VOLUME 1, ISSUE 1  
Therefore the real distance form calibrated instrument to the scale: 
݈௠ =
݈ଵ + ݈ଶ + ݈ଷ
tanߙଵ + tanߙଶ. (5) 
Using equations (2) and (3) it could be written: 
݈௠ =
݈ଵ + ݈ଶ + ݈ଷ
݈ଵ݈௣ଵ +
݈ଶ݈௣ଶ
. (6) 
Therefore the real distance from calibrated instrument to the reference scale can be determined 
using only linear scale. 
After the precise distance to the linear reference (linear bar) was determined a simple 
trigonometric method of angle measurements can be performed. This way practically unlimited 
number of vertical angle measures can be tested (calibrated) overviewing entire range of vertical 
angle encoder values of the instrument. The accuracy of calibration therefore depends on the 
accuracy of reference bar and accuracy of pointing of telescope to the specific dash of the bar 
(therefore the qualification of human operator). 
Such arrangement allows simple and cost-effective calibration of vertical angle measures with 
accuracy high enough. 
3. Laboratory arrangement 
The described principle of vertical angle measurements was tested at Institute of Geodesy, 
VIilnius Gediminas Technical University. The arrangement of instrumentation for measurements 
is shown in Fig. 5. 
For precise measurements of horizontal displacements of the instrument high accuracy two 
axis horizontal table (Fig. 5) was used (ensuring accuracy of movement of up to 1 µm). The 
industrial laboratory linear scale of high accuracy of 1 m in length with the scale strokes at every 
1 mm was used. After the measurement and calculation of linear distance from tacheometer to the 
scale performed according to the previous chapter, it was determined that the distance ݈ equals to 
2.4215 m. 
The main objective of experiment was to test the calibration method and obtain preliminary 
results of the systematic errors (biases) of the vertical angle measurements using Trimble 5503 
tacheometer [12]. 
The calibration was performed using 1 m precise linear scale collimating the spyglass to the 
scale strokes at a pitch of 10 mm and 12 mm. The resulting calculated deviations of tacheometer 
readings are shown if Fig. 6 (10 mm pitch). 
As can be seen from Fig. 6 some of the systematic errors of vertical angle measures can be 
noted, though more serious and thorough analysis of results is needed. By determination of 
accurate systematic errors across the range of vertical angle measurements and later numerical 
removal of determined errors (biases) the accuracy of instrument can be substantially increased. 
4. Possibilities for further development of the instrumentation 
The method of vertical angle calibration described above due to purely manual nature of 
measurements is quite time consuming and additionally imposing human (operator) errors, which 
are often quite hard to overcome. For further implementation of described method automation 
should be developed. Several types of possible automation of the equipment are given in Fig. 7. 
2. METHODOLOGY OF CALIBRATION OF VERTICAL ANGLE MEASUREMENTS.  
DOMANTAS BRUČAS, LAURYNA ŠIAUDINYTĖ, MINDAUGAS RYBOKAS, KENETH GRATTAN, GENADIJUS KULVIETIS, GINTARAS DMITRIJEV 
 © VIRBROENGINEERING. ISSN PRINT 2335-2124, ISSN ONLINE 2424-4635, KAUNAS, LITHUANIA 13 
 
Fig. 5. Arrangement for automated vertical angle measurements calibration 
 
Fig. 6. Deviations of vertical angle readings taken at the linear pitch of 10 mm 
 
a) 
 
b) 
Fig. 7. Possible arrangement for automated/semi-automated vertical angle measurements calibration:  
a) arrangement with automated pointing mark positioning,  
b) arrangement with automated telescope movement 
In Fig. 7a, possible arrangement with minimal level of automation is given, in this case the 
linear (vertical) reference is realized by means of mechanically movable dash linear movement of 
which is determined by linear encoder. This way the errors due to pointing accuracy to multiple 
dashes can be eliminated. 
Further, in Fig. 7b, an arrangement with automated movement of the telescope is given. In this 
case a special scanning CCD camera (or CCD line) tracking the movement of the dash is attached 
to the telescope, with telescope movement being realized by special mechanical arm. This way a 
full automation of the calibration process with elimination of operator errors could be guaranteed. 
2. METHODOLOGY OF CALIBRATION OF VERTICAL ANGLE MEASUREMENTS.  
DOMANTAS BRUČAS, LAURYNA ŠIAUDINYTĖ, MINDAUGAS RYBOKAS, KENETH GRATTAN, GENADIJUS KULVIETIS, GINTARAS DMITRIJEV 
14 JOURNAL OF MEASUREMENTS IN ENGINEERING. MARCH 2013, VOLUME 1, ISSUE 1  
5. Conclusions 
1. A simple method of calibration of vertical angle measurements accuracy of geodetic 
instruments was proposed, similar method of angle measurement calibration could be 
implemented for all angle measurements, both vertical and horizontal of collimated instruments 
such as theodolites, tacheometers, etc.; 
2. Possibilities of calibration of vertical angle measures were demonstrated implementing 
available cost-efective instrumentation; 
3. Initial results of calibration of vertical angle measurements are given; 
4. Due to manual nature of present measurements some operator caused errors can be expected. 
Therefore a further development of automation of process is essential. 
Acknowledgements 
This research is funded by the European Social Fund under the Project No.  
VP1-3.1-ŠMM-07-K-01-102. 
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Methodology of calibration of vertical angle measurements

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Methodology of calibration of vertical angle measurements

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