VERTICAL ANGLE CALIBRATION TECHNIQUE
Speaker / Author: F. J. van der Walt
Co-author: O.A. Kruger
National Metrology Institute of South Africa (NMISA)
Private Bag X34, Lynnwood Ridge, Pretoria, 0040, South Africa
e-mail:
Phone: 012841 4799Fax: 012 841 2131
Abstract
Geodetic measuring instruments are widely used in the surveying and civil engineering industry. Theodolites are one of a wide range of instruments used to perform accurate angle measurements in both horizontal and vertical planes. The calibration of the angle measurement in the horizontal plane can be performed relatively easy by using angle reference standards. Although many methods are developed for the calibration of angle measurements performed with a theodolite in the horizontal plane, the most of these methods cannot practically been used for the calibration of angle measurements in the vertical plane. This paper will describe and investigate a practical approach for the calibration of vertical angle measurements.
- Introduction
Theodolites are mainly used in surveying and civil engineering to measure precision vertical and horizontal angular measurements which can be used to determine (calculate) the positions of the measured points in 3D space. Theodolites have been adapted for specialized purposes in fields like metrology and rocket launch technology.Although the precision of the theodolite mainly lies in the direct reading of the micrometer scale in the telescope (sighting scope), it is of utmost importance to have the instrument been calibrated for both the vertical and horizontal angle measurements.
When a target is sight with the telescope, the horizontal angle is read off the circle (scale at the rotating base of the theodolite), and the vertical angle is read directly from the micrometre scale seen in the telescope or from the display of the theodolite. It is therefore necessary to have your theodolite calibrated for both horizontal and vertical angle measurements to ensure traceability to the national standard for angle and to determine a proper measurement uncertainty for such measurements.
Currently a few methods/proposed methods exists for the calibration of the vertical angle measurement component which includes specialized equipment and artifacts (see figure 1 for an example [3]). Independent of which equipment are used, the equipment also needs to be calibrated to ensure traceable measurements.
The aim with this project is to develop (investigate and established) a standard that can be used to calibrate existing calibration equipment (see figure 1) as well as the vertical angle in the context of traceability and calibration as laid down by ISO 17025 [1] and the GUM (Guide to the expression of Uncertainty of Measurement) [2].
Figure 1: Available instrument for vertical angle calibration
- Definition of horizontal and vertical angles
- Horizontal angles are used to determine bearings and directions in control surveys, for locating detail when mapping and for setting out all types of structures.
- Vertical angles are used when determining the heights of points and to calculate slope corrections.
The figures below shows two points A and B and a theodolite T set upon a tripod above a ground point G. Point A is higher than the instrument and isabove the horizontal plane through T, whereas B is lower and below the horizontalplane. At T, the instrument is mounted a vertical distance h above G on its tripod.
Figure 2 shows that the horizontal angle at T between A and B is not the angle in the sloping planecontaining A, T and B, but the angle θ on the horizontal plane through T betweenthe vertical planes containing the lines of sight TA and TB.
Figure 3 shows that the vertical angles to A and B from T are αA (an angle of elevation) and αB (angle ofdepression).Another angle often referred to is the zenith angle. This is defined as the angle inthe vertical plane between the direction vertically above the instrument and the lineof sight for example ZA.
Figure2:Angle in the horizontal plane [4]
Figure3:Angles in the vertical plane [4]
- Theodolite Errors
a.Trunnion axis not perpendicular to the vertical axis.
When the telescope is turning about the trunnion axis (see Figure 4) the line of sight should sweep through a vertical plane. If the trunnion axis is not perpendicular to the vertical axis, then this plane will be deflected. On level ground, the error may be very small, but traversing up or down a steep hill will increase the error.
b.Line of sight is not perpendicular to the trunnion axis.
If the trunnion axis is perpendicular to the vertical axis (see Figure 4), but the line of sight is not perpendicular to the trunnion axis and scope is turning on the trunnion axis, then the line of sight sweeps a cone rather than a plane. This condition is more likely caused by a misalignment of the crosshair reticule inside the scope.
c.The vertical axis is not plumb.
This error is mainly caused by the operator of the theodolite and the easiest to correct. It is a only a matter of proper leveling of the instrument. To reduce the influence of this error, the level vials should be checked frequently to ensure that the vertical axis (see Figure 4) is plumb.
d.The vertical angle collimation is out of adjustment.
Vertical angles are usually measured from the zenith direction. A few seconds, or even minutes of error makes no appreciable difference in horizontal distances, but it can cause great confusion with elevations. Although this error do not changes in the direction of the sight, it is a fairly simple matter to correct the angle without even adjusting the instrument. In fact, electronic instruments typically have an onboard routine that will measure and correct the vertical angle error
Figure 4: Illustration of error components [5]
- Methods investigated
Four(4) different approaches have been investigated and will be discussed in this paper.
4.1Method 1
This method involves the use of a theodolite together with a calibrated line scale with an overall length of 3000 mm.
Due to the difficulty to align the line scale 100% perpendicular to the horizontal measurement plane, the line scale was setup at certain angles from the ideal vertical plane where α= 0,03 degree; 1,85 degree and 4,0 degree (as illustrated in figure 4). The theodolite was setup on the tri-pod at a distance (x = 1630 mm) from the line scale. See figure 5 for an illustration of the setup.
Figure 5: Setup of Line scale and Theodolite
The telescope of the theodolite was adjusted to display a 90 degree reading and a heightreading was recorded from the line scale.
The telescope was then adjusted to an angle of 60 degree (i.e. 30 degree upwards from the horizontal plane) and another height distance was recorded from the line scale.
The telescope was then adjusted to an angle of 120 degree (i.e. 30 degree downwardsfrom the horizontal plane) and another height distance was recorded from the line scale.
All measurements were repeated ten times (10x) and the averages are reported in table 1.
The angular deviations from the measured angles to the calculated theoretical anglesare listed in table 2.
Nominal Angle(deg) / Measured Angle @ Position 1
(deg) / Measured Angle @ Position 2
(deg) / Measured Angle @ Position 3
(deg)
60 / 60,0073 / 60,0006 / 60,0011
90 / 90,0008 / 90,0003 / 90,0004
120 / 120,0007 / 120,0001 / 119,9976
Table 1: Averages from ten measurements
Table 2: Deviation from theoretical angle
Angle from Vertical(α) / Theodolite Angle / Theoretical Angle / Deviation(deg) / (deg) / (deg) / (arc sec)
Position 1 / 0,03 / 60,00725 / 60,03284 / -92,1
120,00072 / 120,02547 / -89,1
Position 2 / 1,850 / 59,992 / 60,090 / -352,8
120,001 / 120,240 / -860,4
Position 3 / 4,010 / 60,018 / 60,030 / -43,2
119,998 / 120,160 / -583,2
*Position 1 where angle of line scale from vertical (α) = 0,03 degree
*Position 2 where angle of line scale from vertical (α) = 1,85 degree
*Position 3 where angle of line scale from vertical (α) = 4,00 degree
4.2Method 2
The next approach was based on a publication [6] where the arrangement is based on the trigonometric angle determination using the reference scale of the length for vertical readings and another reference measure of length. Due to the complexity of this method, the calibration of only one angle will be discussed.
With this procedure a line scale is placed vertically at a distance from the theodolite. With the theodolite at the initial position (P1), the telescope was first adjusted to 90 degree and a height (H0) was measured and recorded from the line scale. A calibrated laser distance meter was used to measure the distance (L1) between the rotation axis of the telescope and the line scale. The telescope was then set to the angle (α1) needed to be calibrated (eg. 60 degree from the vertical axis of the theodolite) and another height reading (H1) was measured and recorded from the line scale.
The telescope was adjusted back to 90 degree and the theodolite was moved to position 2 (P2) at a known distance (L2) from the initial position. The telescope was set to the angle in question (α1) and another height reading from the line scale was recorded (H2).
By using normal trigonometric equations and substitution functions, the actual angle of interest can be calculated.The setup is illustrated in figure6.
Figure 6: Setup for vertical angle calibration
4.3Method 3
A holder was manufactured and mounted onto the telescope of the theodolite. A traceable electronic clinometer with a resolution of 1 arc second was then placed in position on the holder.
The telescope was tilted to the various angles in question and the actual tilting angles were directly recorded from the clinometer. Figure 7 illustrates the setup.
Figure 7: Theodolite with clinometer attached.
4.4Method 4
The last method which was investigated involves the use of a traceable angle standard, an electronic clinometer and a collimator with the ability to focus the theodolite telescope to infinity. The angle standard used is a high precision Moore index table with an accuracy of less than 0,1 arc sec.The Moore index table have 1440 indexing positions with an indexing increment of 0,25 degree = 15 minutes. The standard is mounted in a vertical position onto a secondary rotary table. A beam is mounted horizontally to the secondary rotary table and is used to rigidly mount the collimator. The theodolite is placed next to the secondary rotary table with the telescope facing the collimator. The clinometer is mounted horizontally onto the angle standard. See figure 8.
Figure 8: Setup with Angle standard and collimator
The angle standard was set to zero degree (horizontal) and the clinometer was set to zero. The theodolite was set to 90 degree and focused to infinity through the collimator. The angle reading from the theodolite was recorded. The scope was moved vertically to an angle bigger and smaller than 90 degree and back again to 90 degree to determine the repeatability.
The angle standard was indexed to +30 degree while the secondary index table was adjusted to position the collimator to -30 degree from the horizontal plane (120 degree). The angleangular position of the collimator was measured and recorded with the theodolite. This procedure was repeated ten times to determine the repeatability of measurement.
The angle standard was indexed to -30 degree from the horizontal plane while the secondary index table was adjusted to position the collimator to +30 degree from the horizontal plane (60 degree). The angle angular position of the collimator was measured and recorded with the theodolite. This procedure was repeated ten times to determine the repeatability of measurement. Table 3 lists the measured averages and the deviation from the standard angles and table 4 lists the standard deviations calculated from the groups of ten measurements.
Table 3: Deviations from standard
Standard Angle(deg) (min) (sec) / Average angle
(deg) (min) (sec) / Deviation
(sec)
90 / 0 / 0 / 90 / 0 / 6,1 / 6,1
60 / 0 / 0 / 59 / 59 / 57,2 / -2,8
120 / 0 / 0 / 120 / 0 / 7,6 / 7,6
Table 4: Standard deviations
Average angle(deg) (min) (sec) / Standard Deviation
(sec)
90 / 0 / 6,1 / 0,9831
59 / 59 / 57,2 / 0,7888
120 / 0 / 7,6 / 1,2649
4.4.1 Uncertainty of measurement calculation
- Conclusion
The above methods were investigated in a laboratory environment and the following were found:
- While using method 1 it is clear that the out of vertical error of the line scalehave a magnificent influence on the measured angles. When the measurements were performed in position 1 with the line scale only 0,03 degree from vertical, huge errors are still experienced. The reason for this is the difficulty to measure (determine) the exact distance of the trunnion axis to the line scale. It was found that 1,0 mm deviation in distance can result in an error >1 minute.
- Although the perpendicularity error of the line scale, as in method 2, has less influence on the accuracy of the angular measurements, the same difficulty was experience to measure the distances from the theodolite trunnion axis and the line scale. Also, a line scale of 3,0 m in length was used which is long enough to get distance H2 as indicated in figure 6.
- The procedure follows in method 3 was no further investigated after it was found that, due to the weight of the clinometer, no repeatability in any measurement could be reached.
- This leaves us with method 4 which is the most reliable method till now with an calculated measurement uncertainty of ± 4,30 arc sec.
The next step, prior adopting this method as a traceable method for vertical angle calibrations, will be to participate into a comparison with at least two local laboratories.
References
1.ISO/IEC 17025:2005 – General requirements for the competence of testing and calibration laboratories.
2.Evaluation of measurement data- Guide to the expression of uncertainty in measurement (GUM)
3. Web site name -
4.Web site name -
5.Web site name -
6.New approach to vertical angle calibration by Lauryna Siaudinyte1, Vytautas Giniotis from Vilnius Gediminas Technical University.
2013 Test and Measurement Conference