We seek the X~ by analogy with Section 3 by successive approximation, if necessary with ex3 ponential smoothing of the ABji from (15) in accordance with (14). The solution for the previous example in this case with ~ = 0.5 is obtained in six iterations: X, = 0.741, X2 = 0.259, X3 = 0.287, X, = 0.036, X5 = - 0 . 0 4 6 , EX 2 = 0.702. Incorporating both errors thus produces a larger reduction in the test (2) for minimal adjustment. The unit positions for the regulators must be chosen correctly in determining the Bij. In principle, the unit position can be any value up to the maximum one corresponding to full scale. However, it is not difficult to show that there is an optimal unit position that minimizes the relative error in determining Bit. If the regulator deflection in the unit position is small, Bji will be small, and the ~elative error DBij/B~j is determined by the error of the instrumen~ for constant DBij and may become arbitrarily large. If the regulator deviation in the unit position is large, the nonlinearity error will increase, which is not incorporated in the linear approximation of (I). If the regulators show weak nonlinearity, the optimal unit position X for a regulator is given by [X] = V / 4DYo/(Yo)~

,

if this does not exceed the scale limit. Here Y(X) is the dependence of the instrument reading on the position X of the regulator for some arbitrary linear scale, and Y~ is the second derivative of Y(X) for the zero regulator position. In the general case, it is difficult to establish Y(X), and the more so Y"(X), so the unit positions for the regulators should be taken as large as possible within the linearity ranges of the scales with respect to the nominal zeros.

DESIGN PRINCIPLES FOR A CLASS OF SPATIAL-POSITION ADC B. I. Bogatyrenko

UDC 531.74.087.92.082.1.082.5

Very precise distance measurements became possible when lasers became available as coherent monochromatic sources. Commercial one-coordinate interferometers have relative errors of 1-5.10 -7 in the range up to 60 m [i]. Interferometers can be combined to measure several coordinates and as systems for measuring angular and linear displacements, velocities, velocity nonuniformities, and so on [2]. Such systems have the disadvantage of complexity. Also, interferometers enable one to make simple measurements only on relative displacements, while there are serious difficulties in measuring spatial positions. This is particularly so when the mobile component has several degrees of freedom, e.g., moves in space. Such problems often arise in research in automating equipment having adjustable components (mirrors, lenses, stops, and so on) [3], in installing the operating accelerators [4], in assembling aircraft [5], and so on. Here as a rule one does not require high accuracy, and it is sufficient to make measurements with a relative error of I-i0"I0 -~ in ranges up to 0.i m. Interferometers are particularly inconvenient in such circumstances. Laser radiation is also highly collimated, which has facilitated progress in optoelectronic scanners [6], particularly displacement ADC [7, 8]. Until recently, the use of such ADC was restricted to measuring coordinates in a plane, as for example in graphics input to computers, systems for marking-out sheet materials, copying devices, and so on. It has been suggested [9] that an angle can be converted to a time interval in a scanner for measuring spatial coordinates. Here we consider the design principles for optoelectronic scanning ADC for spatial positions. A major component in such a scanning ADC is the opticomechanical converter (OMC), which cgnverts angles to proportional time intervals. Various forms of OMC have been examined for t~wo-coordinate ADC, which has shown that the highest accuracy is obtained when the circular scan is provided for the mobile object, while the immobile photocells lie on the coordinate axes [i0]. We consider the scope for using such a two-coordinate OMC in a spatial-position ADC. The two-coordinate OMC contains a circular scanner and four photocells PCI-PC4 (Fig. i). The beam from the highly collimated source such as the laser L falls on an inclined mirror Translated from Izmeritel'naya Tekhnika,

118

0543-1972/87/3002-0118

$12.50

No. 2, pp. 8-10, February,

1987.

9 1987 Plenum Publishing Corporation

Z

,:& i,m

'

;-<~x

iV-'

X

L'

L

a

PC 0

i

i/i b

Fig. i

Fig. 2

attached to the shaft of the motor M at 45~ at point P (the scan center), the ray is reflected, and when the motor turns, it rotates in the plane. This is linked to the Oxyz coordinate system, whose axes Ox and Oy bear the photocells at equal distances ~ from the origin O, those being fitted with slits. The purpose of these is to shape the flux passing to the photocell (photomultiplier, photodiode, etc). The slits are fairly extended and are perpendicular to the Oxy plane, which provides for reliable signal detection under possible uncontrolled displacement of the scanner along the Oz axis or tilt by an angle ~. The coordinates xp and yp of point P are given

[i0] by

where

to=tl(V'tg2~+tg2~+l+l), ~=2~t~lT is the angle between PC3 and PCI as seen at point P in the positive direction, while 8 = 2~tB/T is the similar angle between PC2 and PC4, to and t B are the time intervals between the pulses from PC3 and PCI or PC2 and PC4, respectively, which are proportional to ~ and B, and T is the beam rotation period. The spatial coordinate zp of point P remains undefined, since t~ and t B do not alter on account of the slits when the scanner is displaced along the Oz axis. Therefore, one naturally combines several two-coordinate OMC in a single multicoordinate one. We have examined possible forms of disposition for the two-coordinate OMC and the number of them, which revealed two features. In the first one, which is called a three-channel one, the scan centers for the three two-coordinate OMC form the triangle ABC, while the scan planes are perpendicular in pairs (Fig. 2a). In a four-channel OMC, there are four scan planes (Fig. 2b). The slits in each channel (not shown) are oriented along the corresponding coordinate axes in the immobile Oxyz system. This system is designed such that the axes Ox and Oy pass through the points A'-D' in the initial position of the object, where the scan centers for the instruments are at those p o i n t s . In the three-channel OMC, there are 12 photocells, while there are 16 in the four-channel one (the positions of them are numbered in Fig. 2). The photocells in a single channel are equidistance from one of the points A'-D' and are on the coordinate axes. We consider the three-channel OMC. Interval measurements on all three channels give directly only the coordinates x A and z A of point A, YB and z B of point B, and x C and z C of point C, while YA' XB, and YC are not determined. If these lacking coordinates were known, the spatial position would be known: the position of a solid is defined by the coordinates of three points on it not lying on a single straight line. 119

To calculate YA" XB, and YC, we express the distances between the scan centers in terms of the coordinates of A, B, and C to get

ACZ= (XA---Xc)ZJI-(EA--EC)2+(ZA--Zc)~; BC'2= (XB __Xc ) 2+ (YB --VC ) ~+ (ZB --Zc )~"; A B 2= (XA --X B ) z + (YA "--YB ) 2 _~_(ZA __ ZB ) Z. We solve

} (1)

(i) as an equation system to get YA, XB, and YC' for which we represent

(i) as

YC = ] f (XA --Xc )Z+(ZA --ZC )~--AC2+gA ; I xt3 =],/-BC~'"'--'(zB --Zc )~--(YB --Yc )'2 + x c ;

;/A System

=

]

~AB~'--(ZA --z~ ) L ( X a - - X ~ )~ +VB 9

(2) is readily solved by computer,

(2)

e.g., by simple iteration.

In the four-channel OMC (Fig. 2b), the input measurements in all four channels give directly the coordinates x A and ZA, YB and ZB, x C and Zc, and YD and z D. The coordinates YA, YB' YC' and x D remain undefined. However, here it xs possible to calculate all the coordinates of point P at once. In fact, let the scanners be placed in a way such that the quadrilateral ABCD is a square. Then xp =0,5(x A +Xc) ; yp =0,5(y~ ~go ): zp ==0,5(zA +z c ) =0,5(z~ +z o) =0,25(z A ~Z8 and a n y l a c k i n g e.g.,

coordinate

is

+ z c +zo~ ; found in a single step ~A = | / A V 2 - - ( x A - - x # ) ~ - - ( ~

w h e r e AP= AC/2 i s

a design

parameter

(3) f r o m an e x p r e s s i o n

analogous

to

(2),

--zP)~ 89

o f t h e OMC.

It is then sufficient to have a three-channel OMC t o d e t e r m i n e s p a t i a l positions. A four-channel OMC is more complicated, but the excess measurements greatly simplify the working formulas and accelerate the ADC operation. It is convenient to evaluate a spatial position not from the coordinates found for points A, B, and C but from the transformation parameters of the mobile coordinate system Oxyz linked to those points. The direction cosines for the Ox and Oy axes in that case are clearly

x~,, ..--xp t,~---))___

,

Y~ --Vp t~t=----~--- ;

"

/~'~=

x A -- xp

t;~:---while those for the O'z' PA a n d PB:

axis

AP

Gt=

gA - - Y P

can be calculated

;

ZA - - Zp

A------P---: as

zB --Zp A~

tm=

the vector

A~

;

product

of the directed

segments

tls=f~lt~2---tsl[2a; t~s=tsttl~-41t/s~; tss=tztt~---t~itl~, while the parallel-transfer

parameters are defined by (3).

When the transformation parameters are known, one can readily calculate the coordinates of any point on the mobile object, which provides a general solution for determining the spatial position. It is readily seen that any motion of the object will be perceived by at least one OMC channel: the angles ~ and B (or one of them) will alter by Aa and AB. We define the sensitivity of the OMC as the ratio of the larger of the two angle changes to the displacement causing it. Figure 2b shows that the sensitivities to linear and angular displacements are correspondingly

S~=SAy=SA~=21I; SA~x=$A~y=S~t=2Lll, where Z is the distance of the photocells

120

from the origin L = AP = BP = ... = DP.

(4)

Formulas (4) provide general recommendations on OMC design. In particular, the angular sensitivity may be raised by increasing L and reducing l, whereas L does not affect the linear sensitivity. There is an upper bound to L from the design of the converter, while Z cannot be made less than the possible scanner displacement. The displacement range is here limited by the lengths of the slits and cannot be more than 50-100 mm. The OMC scanners may be combined in a scan unit, which is mounted on a stand rigidly coupled to the mobile object and placed within an immobile framework with scope for free displacement within it within the required limits. The framework bears the photocells fitted with slits. Such a multicoordinate OMC is a basic component of an ADC for spatial position, which also includes the digital meters recording the time intervals and a computer, e.g., a microcomputer. These design principles for multicoordinate OMC extend the functions of existing scanning ADC and enable them to measure spatial position. The accuracy is determined by the errors in the angle digitizer based on beam scanning. Estimates [i0] show that the error in determining coordinates in a plane is not more than i0 ~m for a displacement range of 500 mm with 20-second angle measurement error. For the linear displacement range of i00 mm and an angular range of 3 ~ , it has been found by experiment that the spatial-position ADC has an error of not more than 5 ~m (for the coordinates of the scan centers, i.e., points A, B, and C). The error in the computed angular coordinates of the object is then determined by the errors in the linear coordinates and by the design parameter L. For example, the error does not exceed 5" for L ~ 0.2 mm. LITERATURE CITED I. 2. 3. 4. 5. 6. 7. 8. 9. i0.

V. P. Koronkevich et al., Laser Interferometry [in Russian], Nauka, Novosibirsk (1983). Yu. F. Zastrogin, Monitoring Motion Parameters by Laser: Methods and Means [in Russian], Mashinostroenie, Moscow (1981). Yu. V. Stupin, Computer Methods of Automating Physics Experiments and Systems [in Russian], Energoatomizdat, Moscow (1983). Kh. K. Yambaev, Precision Position Measurements [in Russian], Nedra, Moscow (1978). E. T. Vagner, Lasers in Aircraft Construction [in Russian], Mashinostroenie, Moscow (1982). B. S. Rozov (ed.), Scanning Measuring Instruments [in Russian], Mashinostroenie, Moscow (1980). E. P. Basov and M. K. Sulim, Inventor's Certificate No. 514311, Byull. Izobret., No. 18 (1976). K. I. Bogatyrenko et al., Inventor's Certificate No. 1078449, Byull. Izobret., No. 9 (1984). K. I. Bogatyrenko et al., in: Applications of Lasers in Science and Engineering, Abstracts for the First All-Union Conference [in Russian], Leningrad (1980). K. I. Bogatyrenko, Izmer. Tekh., No. 6, 10 (1984).

121