SSOl14 1569839857
Rotation Angle Measurement Device: Principle of Operation
and Initial Calibration Results
Yu.L. Avanesov, K.S. Gorokhovsky, V.A. Granovskii, M.D.
Kudryavtsev, N.K. Kulachenkov
State Scientific Center Central Research Institute
Elektropribor
st. Petersburg, Russia valgr3 [email protected]
Abstract-A new device is developed for measurement of current
rotation angle of rocking platform. The device is based on the new
multivalued measure of plane angle - holographic prism [I]. A row
of linear CCD arrays is used as a reading unit. The device realizes
the method of comparison with measure, which excludes the effect of
platform orbital motion and provides highly accurate measurement.
The first calibration results are given.
Index Terms-Holographic prism, linear CCD arrays, rocking
platform, orbital motion, rotation angle measurement.
I. INTRODUCTION
Various types of angular transducers are used to measure the
platfonn rotation about horizontal axis, which are usually mounted
directly on the axis. For instance, encoders produced by DR.
JOHANNES Heidenhain GmbH and Renishaw pic can be mentioned [2,3].
So it is impossible to measure an angle at an arbitrary point of
platfonn, which is actually a pressing problem because of drive
imperfection and bending oscillations of the platform. The
presentation aim is solution of the problem.
II. THE DEVICE STRUCTURE
The problem can be solved with a new device consisting of
multivalued measure of plane angle and optoelectronic registration
unit (Fig.l, 2).
ceo arm s
Hot agraphIC measure
Rotary table
Figure I. The device - schematic picture.
978-1-4799-3866-7/14/$31.00 2014 IEEE
A.E. Angervaks, A.I. Ryskin, A.S. Shcheulin National Research
University TTMO
St. Petersburg, Russia
Figure 2. The device - general view.
The measure (Fig.3) comprises two holographic prism, i. e., two
crystal fluorite (3) samples fixed with respect to each other. Six
holograms are recorded in each sample (Fig. 4). Exposure to
reference laser beam (2) induces a response -several diffracted
beams (5) [4]. Diffracted beams form a fan with nearly plane
surface. The fan plane is perpendicular to the reference beam.
Beams of different fans lie in pairs in the same plane. Angles
between beams are functionally analogous to angles between the
nonnals to the faces of a quartz prism (polygon). So the sample
with holograms can be referred to as holographic prism. Crystal
grating of the sample is highly stable, so under nonnal conditions
of use and storage recorded holograms are very stable, and so are
the angles between diffracted beams [5].
4
Figure 4. Holographic prism - general view. 1 - reference laser;
2 - laser beam; 3 - crystal sample; 4 - rotary table;
5 - diffracted beam; 6 - the registration unit.
Generally, there are two modifications of a holographic prism.
In the first modification (I), a reference beam and diffracted
beams, which occur successively during prism rotation, are located
in the same plane oriented perpendicularly to the axis of rotation
(Fig. 5). In this case, each diffracted beam propagates in the same
fixed direction.
Figure 5. Holographic prism in modification I. Designations are
the same as in Fig.4.
The other modification (II) of holographic prism is used in the
device (Fig. 4).
The registration unit (6) comprises two rows of linear CCD
arrays, each row containing 10 arrays (Fig. 6). Arrays of both rows
are placed and fixed on the same plate in chessboard fashion. It
excludes blind zones with no sensitive elements (pixels) (each
array is packaged). The plate is fixed on moving foundation to
enable its placement at a point convenient for data readout. The
array height can be regulated. Data are readout from the
registration unit by PC with special software.
2
Figure 6. The registration unit - general view.
III. MEASUREMENT METHOD
The device functioning principle is comparing a measurand with
the angles reproduced by the measure. This comparison is
complicated by the orbital motion of fan centres during the
platform rocking. Coordinates of the centres are calculated using
the coordinates of the diffracted beam traces, that is, centres of
flashes on linear CCD arrays. Coordinates of minimum three traces
are needed to determine the fan centre coordinates. Using two beam
fans prevents the situation when there are no traces on all cells.
With any angular position of the measure relative to the
registration unit, only one beam from the pair of corresponding
beams can be out of sensitivity zone. As the platform moves, the
beams move along the registration unit, and fan centre coordinates
are changed. The centre trajectory is calculated using the centre
coordinates, and then the measure angular displacement with respect
to the fixed registration unit - that is, the angle under
measurement - is determined using the trajectory and the known
measure angles.
IV. THE DEVICE CALIBRATION PROGRAMME
Because of the design peculiarities and high required precision
of the device, its calibration is very complicated and cannot be
done as single procedure concerning the device as a single whole.
So it is necessary to carry out the definite programme embracing a
number of procedures, which is fulfilled step-by-step. Thus the
calibration programme includes the following generalized
procedures:
l . Calibration of holographic measure, 2. Calibration of CCD
registration unit, 3. Stability test of device as a whole, 4.
Calibration of device as a whole. Each of the generalized
procedures consists of a number of
relatively simple and functionally complete procedures.
Complication of the calibration is conditioned on the
following influencing factors. According to a holographic prism,
there are the facts that the diffracted beams lay not in a plane
but in nearly conical surface, and they, coming from
which are equal to readings difference of two scales; A - plan
matrix; X - vector of desired parameters. L SM-solution of the
system
x = (ATArlATy; D(X) = (ATArl 0"2; &2 = ; (Y_AX)T (Y-AX),
(2)
where &2 - dispersion estimate for separate reg.ging Yi;
D(X)covariation matrix of measurement results of X which are
get
owing to primary data superfluity p>O. Then, discrepancies of
conditional equations E = V-AX are summed up what permit to get
generalized estimates of precision for each series.
The calibration results of the samples of a holographic prism in
modification I are shown in Tab.I.
TABLE I. CALIBRATION RESULT FOR HOLOGRAPHIC PRISM IN
MODIFICATION I
Calibration model and parameters for two Limits of the prism
sample of the prism error
Sample I (five holograms, nominal angle between 3,5" adjacent
beams equal 15)
Sample 2 (six holograms, nominal angle between 3" adj acent
beams equal 5)
As to the holographic prism in modification I, it is necessary,
for its calibration, to develop previously a number of geometric
models which describe systematic distortions caused by imperfection
of mutual location of a diffracted beams fan, and the crystal, and
its pivot pin, and the reference beam. Besides of the
above-mentioned factors of imperfection (see section IV) there is
no fan centre as a point of crossing all beam axes. All those
models are required for estimating corrections for above-mentioned
distortions for each crystal being a part of the device.
VI. CALIBRATION OF REGISTRATION UNIT
Calibration of registration unit includes measurements of mutual
orientation of linear CCD arrays in each row and between the rows.
Then correspondence between the numbers of sensitive elements
(pixels) in adjacent cells is set. As a result, the common scale of
the registration unit is formed, which consists of consecutively
numbered pixels.
VII. TEST OF THE DEVICE STABILITY
Device stability is tested by measuring the dispersion in its
readings as the measure is multiply returned to the same position
fixed with respect to the registration unit.
Two experiments have been performed in which five and four beams
have been used (Fig. 9). Corresponding data are shown at Tab. II.
Standard deviation of data is less than 0" = 0,9 pixel. Taking into
account the distance equal 1150 mm between the holographic measure
and the registration unit, we get the stability parameter estimate
0" = 2". It is important to note that this estimate includes a
constituent which is caused by dynamical instability of the
platform in use (random value).
N
I
2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43
5
Figure 9. Beam traces on the CCD arrays.
TABLE II. RESULTS OF THE STABlLlTYTEST
Pixel number on the following CCD array I 5 9 10 14
1040,16 917,22 1337,58 376,11 1208,01 1041,23 918,77 1338,91
377,36 1209,03 1040,96 918,09 1338,80 377,07 1208,56 1042,01 918,80
1339,01 377,89 1209,14 1041,38 918,63 1338,99 377,06 1208,74
1041,89 919,27 1339,32 377,46 1209,21 1042,24 920,59 1340,27 378,34
1210,03 1041,51 918,37 1338,26 376,57 1208,47 1040,41 917,57
1337,30 375,51 1207,27 1039,74 918,62 1338,50 376,45 1208,14
1041,01 918,44 1337,95 376,13 1207,83 1040,60 917,72 1337,30 375,43
1206,86 1043,30 920,26 1339,51 377,91 1209,48 1044,31 920,93
1340,55 378,81 1210,61 1041,20 918,31 1338,03 376,08 1207,64
1042,24 919,06 1338,82 376,85 1208,35 1040,75 917,69 1337,86 376,06
1207,83 1041,70 918,15 1338,34 376,75 1208,36 1040,82 917,82
1337,98 376,38 1207,97 1042,12 918,39 1338,60 377,07 1208,44
1041,58 918,07 1338,49 376,94 1208,71 1042,99 919,27 1339,37 378,00
1209,65 1042,16 919,10 1338,95 377,14 1208,61 1042,13 919,11
1339,14 377,18 1208,97 1043,09 919,47 1339,27 377,11 1209,23
1042,03 918,81 1338,49 376,43 1208,02 1043,07 920,19 1340,03 377,73
1209,47 1041,43 918,75 1338,19 376,08 1207,55 1043,02 919,94
1339,65 377,20 1209,23 1041,91 918,97 1338,89 376,68 1208,20
1041,48 918,05 1337,91 375,91 1207,70 1042,61 919,37 1339,08 377,11
1208,86 1042,37 919,43 1338,99 376,91 1208,12 1042,14 918,91
1338,59 376,81 1208,00 1041,76 918,86 1338,54 376,63 1208,06
1042,73 920,00 1339,70 377,43 1208,99 1042,32 919,04 1338,50 376,58
1208,39 1041,65 918,47 1338,36 376,06 1207,54 1042,60 920,06
1339,71 377,50 1208,97 1042,39 919,16 1339,12 377,16 1208,63
1042,62 919,82 1339,33 377,36 1208,93 1043,28 920,24 1339,90 377,77
1209,54 1042,13 919,11 1338,66 376,91 1208,79