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 9@mail.ru

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.

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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

common centre, are refracted on the interface between crystal body and air medium. Besides of the above-mentioned facts, there is non-parallelism of the crystals and, the hence, of the beam fans.

According to the registration unit, the factors are as follows. Firstly, there is non-planarity of the CCD rows and their non-parallelism. Secondly, the beams are refracted on the both interfaces between, correspondingly, air medium and external surface of the glass coating of each CCD array, internal surface of the coating and sensitive layer of each array.

According to the device as a whole, all above-mentioned factors become complicated because of their influence dependence on the current monitored (measured) angle.

Below the results are presented of initial steps of the calibration programme.

V. CALIBRATION OF HOLOGRAPHIC MEASURE

Calibration of measure includes measurements of plane angles between diffracted beams of every crystal [6] and of linear parameters of crystals mutual orientation.

Inasmuch as calibration is transmission of unit from a primary reference standard to another standard or measuring instrument, it is important that the specification of a primary standard of a plane angle [7] is based on the inseparably linked notions of the complete angle and procedure of its even division, which aims at obtaining a prescribed circular scale. This procedure forms a basis of nearly all angle-setting and angle-measuring devices of a revolution type, which have circular scales of the complete range 0-360 [8- 10]. The basic element of this procedure is a stepwise rotation of the carrier of the circular scale, and the cessation of rotation in a given number of steps is performed using an indicating device by juxtaposing one of the scale marks with the immobile index of the device. In this case, each scale mark forms a certain rigidly fixed to its carrier movable direction in the plane of measurements (which is perpendicular to the axis of rotation), while the index forms the initial immobile direction (from the center of rotation).

To reduce the required precision of a reference standard, we use the procedure of the simultaneous calibration of the two scales inherent in the standard and a calibrated device [7,8, 1 1]. These types of measurements are based on a pairwise comparison of the even marks of the two scales of the same discreteness n, which are alternately superposed with each other at several positions of scales with different mutual angular shifts. The results of the simultaneous calibration of the scales are estimates of their systematic errors, i.e., individual deviations {

The schematic picture of a device for the collation of scales according to which the calibration procedure [ 12] is performed is shown in Fig. 7. A holographic prism ( 1) is mounted on a rotary table A (2), which, in turn, is placed on a rotary table B (3). The table A can freely rotate with respect to the table B or can be fixed with respect to it with a device that is controlled by a PC (4). The rotation of the table B is also controlled by the PC (4) and is performed by a stepper motor (5) via a system of reduction gears (6). The angular coordinate of the position of the table B is specified by the number of steps with respect to the initial (zero) position. As reference lasers (7, 8), the radiation of which is collimated by telescopes ( 9, 10), act on the holographic prism, it emits diffracted light beams, which are focused by objectives ( 1 1, 12) onto photodetectors ( 13, 14). The intensities of signals of the photodetectors are fixed by the PC in relation to the angle of rotation of the prism, which is specified by the number of steps of the motor. The detected signal is maximal if the Bragg condition is satisfied; i.e., at a given position of the prism, the angle between the reference and diffracted beams is equal to the doubled Bragg angle. The corresponding angular position of the prism forms a mark of the (holographic) scale. The assignment of marks of the measuring (complete circular) scale is governed directly by the rotary mechanism through the number of the step of the motor with respect to the zero position. The values of the marks of this scale correspond to the nominal values of the angles recorded in the holographic prism.

The use of two lasers and two photodetectors in the calibration device is based on the following reasoning. Variable errors along the circular scale (irrespective of the nature of measured angular quantities) can be expanded into a Fourier series, where, as a rule, the largest contribution is made by the first harmonic, which is usually attributed to radial beatings of the axis of the measuring device. To suppress this harmonic, one can use the average procedure of repeated measurements of angular quantities performed upon shift of the recording scale by 180. It is expedient to calibrate the scale of the holographic prism by simultaneous reading holograms with two counter directed reference laser beams while recording diffracted beams by two photodetectors (from each hologram).

Unknown calibration parameters Readout in k" series, Ii" position

Figure 8. Two basic movements (within and between series) and relations between parameters upon simultaneous calibration of two circular scales in

planar geometry.

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As was noted in the preceding section, the calibration procedure consists of the collation of two scales, which are specified by the holographic prism (the scale to be calibrated) and by the table of the rotary mechanism, on which the prism is placed (the reference measuring scale), by means of numerous comparisons of angular positions of marks implemented by these scales. It should be noted that the two scales are equivalent in this procedure. More specifically, the calibration includes a number of measurements of angles of the holographic prism with respect to marks of the reference measuring scale, and these measurements are grouped into series. A characteristic feature of these series is that the angles of the prism are measured relative to a certain fixed mark specified by the reference measuring scale. The number of this mark (relative to the initial mark) is the number of the series (k). During each series of measurements, the positions of the number of marks specified by the holographic scale are collated with the positions of a number of marks of the measuring scale that are covered by the sector of the holographic scale (by the angle determined by the holograms of the prism that spectroscopy extreme angular positions).

The collation procedure of the two scales is illustrated by Fig. 8, which is based on the traditionally accepted twodimensional geometric model (in the projection onto the plane of measurements). In this figure, [i] is the nominal ith mark of each scale (calculated from the chosen initial mark [0]); (i) is the actual /h mark of each scale; Jll is the count in the kth series in the I/h position of the coupld scales, which is equal to the difference between the readings of the first and second scales;

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

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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

The graphs of data are shown in Fig. 10. The graphs are very closed in form, so it is expedient to make second differences of data. The latter lead to the estimate (J = 0,4 pixel that is equal to (J = I". This estimate evaluates instability of both the system for the platform fixation (autocollimator with quartz prism) and the program system for data processing.

ceo array I

1 J 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 Measurement number

ceo arrayS

1 3 5 7 ') 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 Measurement number

ceo array 9

I 3 5 7 ') 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 Measurement number

CCD array 10

3RO liI 1 I 3 5 7 9 11 13 15 17 19 21 2J 25 27 29 31 33 35 37 39 41 43

Measurement number

CCD array 14

I 3 5 7