Limited accuracy reference free angular position determination.
Brucas, D. ; Suziedelyte-Vysockiene, J.
1. Introduction
Testing and calibration of angle measuring instruments was
important ever since angle measurements were started to implement. These
tasks are still of an extreme importance today since lots of angle
measurement devices are being implemented in many branches of industry,
such as machine engineering, construction works, geodesy, etc.
Generally, there are several groups of plane angle measurement methods
which could be implemented for the mentioned task [1].
1. Solid angular gauge method:
* polygons (multiangular 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 scales, etc.);
* nonfull circle (sector scales).
Usually the calibration of angle measuring instruments (calibration
of the entire device) is performed by means of the comparison of the
tested device with the reference one. Several technical decisions for
the angle determination can be implemented in order to create a
reference measure. The most significant means used for this purpose are
[2].
1. Polygon/autocollimator.
2. Moore's Precision Index.
3. Circular scale/microscope(s).
4. Angular encoders.
5. Ring laser (laser gyro).
6. Interferometric angle generator.
In case of the implementation of the mentioned devices for the
comparation of angle measurements there is a need of expensive and
complicated device creating reference measures itself (one of the
mentioned), also the precise alignment (centering and leveling) of the
reference measure is needed [2]. Additionally, in some cases (as with
polygon/autocollimator) only a limited number of angular positions could
be tested or (as with Moore's Precision Index) the entire process
of calibration can hardly be automated, or the calibration devices (as
Interferometric angle generator) can not be implemented in industrial
conditions.
In case of the calibration (or testing) of the low accuracy angle
measuring instruments (turn tables in mechanical industry, construction
site geodetic instruments, etc.) some robust, uncomplicated (in terms of
adjusting) and less sensible for the environmental conditions system is
needed [3, 4].
2. Principle of the new angle testing/calibration method
As was mentioned before adjusting of the calibrating device to the
tested instrument is one of the most complicated and time consuming
procedures of the entire calibration process. Precise alignment of the
axis (centering and leveling) of the angle measuring instrument and
reference mean is needed.
The proposed here method of angle testing or calibration is based
on photogrammetric determination of the 3D points. Photogrammetry is a
method of determination of the coordinates spatial points by means of
two (or more) overlapping images of the object. Therefore by means of
the two properly calibrated and positioned cameras (or single camera
from different positions) and a special photogrammetric software
coordinates of the selected point(s) (visible on both images) can be
determined.
Principal geometry for coordinate determination of the
photogrammetric system with two cameras is shown in Fig. 1 [5].
[FIGURE 1 OMITTED]
In Fig. 1:
* the line connecting optical centers C and C' of the camera
is called the baseline - t;
* scene point X observed by the two cameras and the two
corresponding rays from optical centers C, C' define an epipolar
plane. This plane intersects the image planes in epipolar lines l,
l'. When object point X moves in space, all epipolar lines pass
through epipoles e, e' - epipoles are the intersections of the
baseline with the respective image planes;
* u, u' are projections of the scene point X in the left and
right images respectively;
* the ray CX represents all possible projections of the point X to
the left image, and is also projected into the epipolar line l' in
the right image;
* the point u in the right image that corresponds to the projected
point u in the left image must lie on the epipolar line l' in the
right image;
* K and K' are rotation of the camera, with R being the flip
of the cameras position.
The projections of the scene point X in fundamental matrix of
geometry of the system with two cameras are [5]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
Relative 2D coordinates of certain point in the images are obtained
and since u, u' provides a strong epipolar constraint that reduces
the dimensionality of the search space for a correspondence between u
and u' in the right image from 2D to ID; general 3D coordinates of
the point can be calculated.
Therefore, using the photogrammetric principles it is possible to
determine the point cloud of some kind of a flat plane attached to the
tested instrument and rotating together with it. Some photos of the
plate at different (calibrating) positions of the angle measurement
instrument can be made. Using these photo images the point clouds of the
plates at different angular positions can be obtained. Having these
point clouds flat surfaces can be drawn through these clouds (Fig. 2).
[FIGURE 2 OMITTED]
After the planes 1 (Fig. 2) are obtained, it is possible to
determine the spatial angle between them (a). An important thing in that
case is that despite the position of the plate mounted on the tested
instrument (therefore the position of the point cloud 1 or 2) there will
always be a single intersection line of these planes 3, that is axis of
rotation of the tested/calibrated instrument. Rotation angle ([alpha])
is measured by comparing it with the reference one created in the
computer program used for that application. So, the measurements can be
considered as reference-free (with the virtual reference created inside
PC).
Therefore there is no difference how the plate (the point cloud of
which should be obtained) is positioned, the only limitation is
stability of the plate on the tested instrument and its visibility for
cameras. Consequently, there is no need in precise positioning neither
calibration plate nor the cameras (since the spatial angle is calculated
between two obtained planes), which makes the preparation for the
calibration considerably easier.
Some examples of application of such testing/calibration
arrangements are shown in Figs. 3 and 4.
[FIGURE 3 OMITTED]
In Fig. 3 an arrangement for testing or calibration of angle
measuring geodetic instrument (tacheometer) 1 is shown. Here the
calibration plate with marks 2 for better photogrammetric points
acquisition is attached to the spyglass of the geodetic instrument. The
arrangement of two cameras on a tripod 3 is placed at some distance (so
that the plate was visible at calibrated angular positions), there can
also be a single camera used but its position should be changed to
obtain the images of single angular position but from different camera
positions. Such arrangement allows calibrating both horizontal
([[alpha].sub.h]) (with the vertical movement of the spyglass being
fixed) and vertical ([[alpha].sub.v]) (with the horizontal movement
being fixed) angle positioning. After series of angular movements has
been done (and their images made) the results of angular positioning
from the geodetic instrument can be compared to the results obtained
from the photogrammetric software after processing of the images.
Obviously the calibration with single calibration plate can not be
performed at the entire range of tacheometer angular measurements
(360[degrees]) since the plate will not be visible on the images. To
perform the full circle calibration the plate or the instrument or the
cameras should be repositioned or there should be some special shape of
the calibration plate used (there can be used a three or four angle
object instead of the plate). The plate can also be positioned on any
part of the geodetic instrument where the suitable attachment points are
available. Obviously there is also no limitations in size of the plate
and its distance to the axis of rotation (the larger the plate or its
distance to the axis of rotation the larger linear movement, detected by
the photo grammetry, of the plate and its points will be obtained)
therefore the higher accuracy can be achieved with the only limitation
in visibility of the plates by cameras.
Another example of the application of testing/calibration method on
industrial milling machine is shown in Fig. 4.
[FIGURE 4 OMITTED]
In case shown in Fig. 4 angular positioning of the turn table (4th
and 5th coordinate) of five-coordinate milling centre is presented. The
calibration plate with marks 3 (same as used in Fig. 2) is firmly (for
example by magnetic holder) placed on the turn table of the milling
machine 2 at any position. The arrangement of two cameras 1 is
positioned next to the turn table (so that the plate is visible in the
images), the entire process of calibration is identical to the one of
the geodetic instrument. That way both the 4th and the 5th coordinate
rotations of the table can be calibrated using the mentioned instruments
in industrial conditions (workshop).
Similar angular position calibrations can be performed on many
kinds of machines as, for example, spindles of turning machines (in case
the turning was interpolated), turning tables of different machines,
etc. The measurements can be performed very fast with extremely short
time needed for the attachment of calibration instruments and high rate
of measurements (1-2 seconds to take the images of a single angular
position). The measurements can be performed at any indoor environment
(only avoiding direct sunlight, heat radiation and other sources of
optical distortions of the surrounding air), with further processing of
the images performed by computer software at any suitable facility. The
entire computer results processing can also be quite easily automated
with the automated marks recognition performed by the photogrammetric
software (which is widely implemented).
3. Testing of the accuracy of calibration
To test the suggested calibration method practically a special
measurement arrangement was composed (Fig. 5).
The calibration plate with the marks attached to it 1 (Fig. 5) was
placed on a precise turn table 2 produced by Wild Heerbrugg company (its
angular accuracy of 0.3" [6, 7]). Large number of marks was used to
decrease the random errors of the photogrammetric coordinates
determination of a single point by obtaining the number of points and
calculating the average best fit plane. A professional digital camera
Canon EOS 350D (8 mega pixel) calibrated at the University of Technology
of Bonn by TCC software was used for the experiment (with camera
position changes for each measurement step). Special tie points 3 were
used for later camera photogrammetric orientation.
The turn table (with the calibration plate attached) was turned
with a step of 15[degrees] starting and ending at the positions where
the marks were no longer visible by the camera and the measurement cycle
repeated after that. Several photos (from different points) were taken
at each measurement step and coordinates of the calibration plate marks
for each step were calculated using PhotoMod photogrammetric software in
manual mode (since no special software was prepared for the initial
preliminary test).
[FIGURE 6 OMITTED]
After obtaining the point clouds (for each step) the best fit plane
for each measurement step was created using Imageware software. Angles
between the plates were later calculated by the Unigraphics CAD software
(Fig. 6).The results of tests comparation of angular position generated
by the turn table, precisely determined by its encoder and considered as
reference are shown in Table. The deviations of angular position
determination by the suggested calibration method are shown in Fig. 7.
[FIGURE 7 OMITTED]
As can be seen according to the results of experiment the maximal deviation of angular position determination by means of the suggested
photogrammetrical method does not exceed 2.53' (test Nr. 6).
Standard deviation of the tests performed is 0.89' (with the
standard deviation of the rotary table positioning neglected due to its
small value).
[FIGURE 8 OMITTED]
It should be noted that the results are quite good for initial
tests with high potential for further improvement of the measurement
accuracy.
4. Possible sources of errors and higher accuracy achievement
Since the described experiment was only the initial trial to adopt
the described method of calibration of angle measuring devices, there
are plenty possible sources of errors with huge possibilities of the
accuracy increase. Among the many possible the main sources of errors in
case of that particular experiment could be named [8, 9]:
* errors of camera orientation;
* errors of camera calibration (lens and CCD matrix distortions);
* errors of photogrammetric point position determination;
* optical distortions of the air;
* limited resolution of the camera.
Since the measurements were performed using a single camera, some
tie points had to be used to determine camera position in several images
(perform camera orientation inside the photogrammetric software).
Accuracy of such orientation is absolutely essential for further point
position determination. The results of deviation of angular position
determination in comparison with the root mean square (RMS) of
photogrammetric camera orientation are shown in Fig. 8.
As can be seen from Fig 8, the largest deviations of angular
position determination (tests 3, 6, 11) correspond well to the largest
RMS of the cameras orientation. It should be possible to decrease
mentioned errors by using an arrangement of two cameras (Figs. 3 and 4),
since in that case the position of the cameras would be predetermined in
all images.
Digital Canon EOS 350D camera used in the experiment was calibrated
(at Technical University of Bonn by Tcc software) with focusing to the
eternity, unfortunately due to quite short distance from the camera to
the rotating plate (1-2 meters) in case of the experiment described most
of the images were quite blurry (with focusing to the eternity), and
refocusing of the camera would lead to loss of the accuracy of
measurement due to undetermined optical distortion of the lenses.
Therefore it is necessary to perform the calibration with focusing at
the shorter (work) distances suitable for the described task. Such
camera calibration should increase the accuracy of measurements
considerably.
Some errors of the photogrammetric points (on the plate) position
determination could occur due to manual nature of the point collection
in the experiment. Such influence was decreased by the use of a large
number of points but errors could still influence the measurements.
Automation of this process is possible and should increase the accuracy
of point position determination and therefore the accuracy of
measurements. Automation should also allow collection of the larger
number of points (on the calibration plate) which should decrease random
errors of specific point coordinates determination due to final
approximation of the measurement results (creation of the single best
fit plane through the point cloud).
Optical distortion of the surrounding air is always a great problem
in case of optical measurements; therefore at the mentioned experiment
it was tried to decrease the fluctuation of air, changes of temperature,
etc., but still those influences definitely were present due to
movements of the operator controlling the camera (since a single camera
had to be moved from one place to another to obtain needed images). The
errors of measurement caused by the distortion of images by air
fluctuation should also be decreased by the implementation of the
previously mentioned camera arrangement (Figs. 3 and 4).
Additionally higher accuracy of measurements could be achieved by
the implementation of higher resolution (15-20 mega pixels) camera.
One of the possible sources of the errors of measurements was
considered to be the angle of the calibration plate to the camera but
according to the results of experiment (Fig. 7) there is no obviously
noticeable influence of the rotation angle on the deviations of its
determination. Nonetheless since the tests are only preliminary, the
mentioned hypothesis could not be completely rejected till some further
researches were done.
Generally in classical digital close range photogrammetry it is
considered that achieving the accuracy of point coordinate determination
of [+ or -] 0.001% (or even [+ or -] 0.0005%) from the distance to the
object is quite possible (depending on various factors) [10]. Therefore
theoretically it may be stated that in case of the described experiment
(plate size 240x240 mm and distance to the camera 1.3 m) the accuracy of
the points position determination could be around [+ or -] 0.013 mm,
which makes the angular accuracy of [+ or -] 0.37'. Therefore
accuracy of the measurements could be increased considerably even
theoretically.
5. Conclusions
1. A new method of the calibration of angle measuring instruments
allowing rapid settlement of simple calibration equipment and fast
calibration process at the instruments work environment was suggested by
the authors of the paper.
2. The experiment of implementation of the mentioned calibration
method showed accuracy of the measurements not exceeding [+ or -]
2.53' (standard deviation 0.89'), which allows implementation
of the method "as is" for the calibration of less accurate
angle measuring instruments.
3. A huge amount of the improvements could be made to the mentioned
method allowing increasing the accuracy of measurements considerably.
4. Further tests should be performed both in the field of accuracy
increasing of measurements and practical implementation of the method.
Acknowledgments
This work has been funded by the Lithuanian State Science and
Studies Foundation, Project No B 32/2008.
Received December 12, 2008
Accepted February 05, 2009
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D. Brucas *, J. Suziedelyte-Vysockiene **
* Vilnius Gediminas Technical University, Sauletekio al. 11, 10223
Vilnius, Lithuania, E-mail: domka@ktv.lt
** Vilnius Gediminas Technical University, Sauletekio al. 11, 10223
Vilnius, Lithuania, E-mail: j_visockiene@hotmail.com
Table
Results of the angle measurement tests
Angular Determined
Test set Test Nr. position, deg angle, deg
1 1 0.0000 0.0028
2 14.9909 15.0053
3 29.9840 29.9427
4 44.9789 44.9650
5 59.9767 59.9922
6 74.9773 74.9351
7 89.9809 89.9950
8 104.9866 104.9839
2 9 14.9912 14.9747
10 29.9848 29.9965
11 44.9790 45.0094
12 59.9766 59.9904
13 74.9770 74.9812
14 89.9813 89.9911
Deviation of angle RMS of camera
determination, orientation in
Test set Test Nr. arc min images, mm
1 1 0.17 2
2 0.87 2
3 -2.48 5
4 -0.84 2
5 0.93 1
6 -2.53 4
7 0.85 3
8 -0.16 1
2 9 -0.99 2
10 0.70 4
11 1.83 5
12 0.82 4
13 0.25 2
14 0.59 2
