adaptive compensation for an optical tracking telescope

Aatomatica, eel. 10, pp. 125-131. Pergamon Press, 1974. Printed in Great Britain.

Adaptive Compensation for an Optical Tracking Telescope" Compensation Adaptive pour un T61escope Moniteur Optique

Adaptive Kompensation fur ein optisehes Naehlauf-Teleskop

/~anTrmHas roMneHcaRHa ~,rla Te.aeerona c orrrr~eerofi cae~amefi CHCTeMOfi

JAMES W. GILBARTt and GEORGE C. WINSTON,

A Model Referenced Adaptive Control System considerably improves the performance of an optical tracking telescope.

Summary--The application of model referenced adaptive control theory to an optical tracking telescope is discussed. The capability of the adaptive technique to compensate for mount irregularities such as inertial variations and bearing friction is demonstrated via field test lesults on a large tracking telescope. Results are presented which show a 6 to I improvement in tracking accuracy for a worst case satellite trajectory.

1. INTRODUCTION

THIs article describes the application ofa Liapunov- based parameter adaptive control technique to a satellite tracking telescope located at the Optical Research Facility of NASA's Goddard Space Flight Center. The overall goal of this effort was to improve the telescope's dynamic pointing accuracy by compensating for inertial variations, static friction, and other anomalies in the telescope mount. The model referenced adaptive compensation of plants with time varying parameters is based primarily on the work of Refs. [1 and 2]. For other references in this area as well as a detailed summary and comparisons of contributions of these and other authors, the reader is referred to a recently published article by HUASG and P~KS [3].

The technique was implemented on an EAI TR-48 analog computer which was linked directly to the telescope mount's drive motors and various sensors. In this manner, field tests were conducted to establish improved performance.

In Section 2, which follows, the parameter

* Received 22 May 1973; revised 28 September 1973. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by Associate Editor P. Parks.

I" Present address: The Johns Hopkins University Applied Physics Laboratory, Silver Spring, Maryland 20910.

~:/h-esent address: Goddard Space Flight Center, Greenbelt, Maryland 20771.

adaptive equations are derived for the telescope's configuration. Section 3 presents a brief description of the telescope and its various modes of operation, while in Section 4, strip chart recordings taken during field testing with and without the implementation of adaptive compensation are presented.

2. THE ADAPTIVE CONTROL TECHNIQUE

The configuration in Fig. 2 represents a functional diagram of one axis of the tachometer loop of the telescope mount. The command rate input has been denoted by ~ while the preamplifier output is

! MOT0g LOAD POWER | a TACH

FIG. 1. Analog tach loop including bearing friction and control input.

given by z. The preamplifier is actually a lag-lead network designed to boost the low frequency forward loop gain. The signal, u is the adaptive control input to be defined, ~p is the tachometer output, and f(0p) represents bearing friction, as shown in Fig. 2. The differential equation relating 0z to z is given by

0n-I- an0n = 140K~-I-f(0p) + 140K~. (1)

It is assumed that the plant parameters, Kp and ap, as well as the coulomb friction level, ~, are slowly time-varying parameters which are essentially

125

126 JAMES: W, GILBART and GEORGE C. WINSTON

/~ STATIC FRICTION LEVEL.

a COULOMB FRICTION LEVEL

Fro. 2. Bearing friction characteristic.

where the at and the fis are arbitrary positive constants. The inclusion of the t e r m s (fliegi), i----1, 3 in (6) was discussed in Ref. [2]. It was shown ithat these additional terms yield a "proportional" term in the definition of each adjustable parameter as shown below. The effect is a more rapidly convergent adaptive system. The time derivative of (6), utilizing (5), is

3

l)'= --24d 2 +2d ~ xsgl +2 - (x I i = 1 i= l \O~i

constant over the adaption period of the adaptive mechanism. An ideal model of the load, and tachometer, based on nominal parameter values, is chosen to be

Om+ 120m =0"25Z (2)

where 0., is the model or ideal velocity. Subtracting (1) from (2) yields the error equation

e + 12d = (ap - 1 2 ) 0 p - (140 K v - 0"25)z -f(Op)

- 140Kpu. (3)

In order to provide adaptive parameters to mini- mize the differences appearing on the R.H.S. of equation (3), the control input u is defined as

u=KIOF+ K2z+ K a sgn 0p (4)

where K1, K2 and Ka are adjustable parameters to be defined on the basis of Liapunov's direct method, and sgn 0p = + 1 for 0p > 0 and sgn 0p = - 1 for Op<0. The third term is utilized to compensate for axis bearing friction. Substituting (4) into (3), and rewriting (3) yields

3

e + 12~ = Y'. x#~ (5) 1=1

where xl = a p - 12-140KpKI, x 2 = - 140Kp+0.25 -140KpK2, xa=ot-14OKvKa, gl=Ov, g2=z, and ga=sgn Op. Furthermore, it has been assumed, from Fig. 2 that f ( 0 p ) = - = sgn 0p. The adjustable parameters, K1, K2, and Ka will now be defined in order that d = 0 is an asymptotically stable equili- brium point. Following the development in Ref. [2], a potential Liapunov function is chosen as

3 V=d=+ y' l (xs+fl~dgs)a (6)

f = 1 tZt

The adjustable parameters are now defined in order to make (7) negative definite; i.e.

d 5q = -- 140KpK i = - asdg i -- fli .---(eol).

d t (8)

Substituting (8) into (7) yields

3

1 2 = - 2 4 d z - 2 ~ fls(dg~) 2 (9) i = 1

which is negative definite for all d. Hence, the adaptive scheme utilizing (4) and (10) as the control input to the motor will drive i ~ 0 and Op~Om, forcing the motor, load and tachometer to act like its ideal model. If one divides both sides of (8) by 140Kp and integrates, the result is the definition of each Ks as "integral plus proportional" adaptive laws; i.e.

Kl(t)=nlf' ° oOp dt+CtdOp

K2(t) = B2 & dt + C2&

I t d sgn 0p dt +Cad sgn Op (10) Ka(t)=Baj o

where the Bs and C , i----1, 3 are arbitrary positive gains, since as and fls are arbitrary positive quantities.

The implementation of the adaptive design is shown in Fig. 3 with the Ks, i : 1 , 3 defined in (10). In the figure, the power amplifier, motor, load, and tachometer have been combined as the "plant" and the model system is represented by the transfer function for (2). The adjustable parameters, K1, /(2 and K a are defined as in (10). In the field test results which follow, values of Bi----10, i----1, 3 were chosen. The damping gains, Ci were set as large as possible without summing excessive noise

Adaptive compensation for an optical tracking telescope 127

MOOEL ¸

~ Z m, + u+Z

FIG. 3. The adaptive system. -

into the drive motor. The Bi were selected on the basis of a 10 to 1 ratio with the Ci for rapid adaptation. This is discussed in Ref. [2].

3. DESCRIPTION OF OPTICAL TRACKING TELESCOPE

A 24 in. telescope located at the Goddard Optical Research Facility was utilized as an experimental testbed for obtaining the data presented in Section 4. The telescope lenses are housed in a cylinder, approximately 7 ft long, 2 ft in dia, and weighing 1400 lb, which is gimballed about two axes (X, Y) in a cartesian coordinate configuration. Motion of the X and Y axes is along the East-West and North-South meridians, respectively.

The telescope is utilized for various laser ranging and optical communication experiments. Due to the proposed usage in future experiments of lasers having narrow beamwidths, imposed dynamic tracking accuracy requirements a re as severe as 0" 16 see of arc. Dynamic tracking accuracy, as it is used herein refers, to the telescope's ability to follow instantaneous command angles.

There are three principal modes of operation, each of which is investigated in Section 4. These are a search or slew mode, an autotrack mode, and a program mode. The search mode is utilized to acquire satellites within the laser beam-width, the autotrack mode is used to "lock on" to celestial bodies by means of their reflected or self-generated light, and the program mode is used to command the telescope to point along predetermined satellite trajectories. Each of these modes is discussed in Section 4.

4. FIELD TEST RESULTS

A. Rate step response, the search mode The telescope mount's response to a rate step

input is important when attempting to acquire a satellite within the field of view. If the mount does

not drive smoothly it becomes difficult to "lock on" to a low altitude satellite. This process is discussed more fully in Section 4-C Autotrack Mode. In what follows, the mount's rate step response will be examined with and without adaptive compensation.

I n Fig. 4, without Adaptive Compensation, the steady-state tachometer output is shown when f(t) was a constant input of 13 mV to the tach loop. This input rate was selected so that the

0

0'00"3 Y.O"

__ X: +80"

Y: o" L ~ " Y=O° x , o " - - ~ Y:O" X=*aO" r Y , o • x = - 8 0 •

' " ' I x = -4O"

I 0"005 deg/~e¢ ~ IVING ~AST TO WEST

I **..o X:+40 o

O'OC - - y : + 8 0 .

X :+eO" I I

Y:+BO° x : o ° L .. L

I ~ Y: 4-Bc," Y=+8O" " X: - t ,o* X= -8o "

y:o • X: -8o o

0 COS I

¥:0 • y=o • ' Y=O o Y:O° X=÷40" X '÷BO"

X : - 4 0 " x~o"

DRIVING WEST TO EAST

o 1 J I . ] V:~BO. Y=+ 80" Y=480" x= -t 40 " x :+4o "

i Y" + e ° " x : o ° x = - 4 0 "

0005 y=+60.

x : - 80"

DRIVING WEST TO EAST

I , I I 0 50 150 200 ~ (~ecs)'

Fro. 4. X-axis tachometer output without adaptive scheme (f = 13 mV).

X-axis of the mount would move at 0.005 deg/sec when the Y-axis was fixed at 0 degrees and the X-axis was initially positioned at 0 deg, driving East to West. The X-axis was then positioned at

_ 80 o, _40 o, +40o, + 80 ° and the output rate recorded for 30 sec. The Y-axis was then fixed at +80 ° and the various X-axis rates were again recorded. The last two plots in Fig. 4 show the output rates when the same procedure was followed with the mount driving in the opposite direction (West to East). Hence, it is observed that, depending upon the positioning of the Y-axis and the initial positioning of the X-axis, a step input of 13 mV will drive the mount at rates ranging from 0 to 0.0094 deg/sec. It is apparent, then, that variations in X-axis inertia due to changing Y- axis position, mount imbalance, and static friction cannot be neglected if the mount is to move at precise rates over the entire field of view. It is here that the adaptive technique can be of use.

Consider now the steady-state response of the adaptive system in Fig. 3 to the same step input. Figure 5 illustrates the tachometer output go(t) and the adaptive control input u(t) for various positions of the X- and Y-axis. In Fig. 5, the mount was moving West to East with the Y-axis fixed at 0 °. This corresponds to the third plot in Fig.~4,

128 JAMES W. GILBART and GEOI~OE C. WINSTON

Ol u{t}

{Volt~)

de~ ooos ~T~

, ! i

_'~ ~ L L

DRIVING WEST TO EASY

. . . . ~ ...... I I -I - Y=O" Y-O" Y=O o v:o* Y=O o

X;-80 o x=-40 o X:O* X:+4O o X:+80"

I I I 5O ioo 15o tlsecsl

FIG. 5. Tachometer output and associated input using adaptive scheme.

which was obtained without utilizing the adaptive system. It is seen that when Y is fixed at 0 ° and X is initially positioned at - 8 0 °, the mount drives at nearly 0.006 deg/sec without the adaptive scheme. Hence, the control input, u(t) in Fig. 5, is negative in order to slow it down. However, for the next four runs ( - 4 0 °, 0 °, +40°, +80°), the mount will drive too slow or not at all without the adaptive technique. In Fig. 5, it is seen that the control input gets increasingly more positive as the X-axis is moved further East, overcoming the static friction present. Table 1 illustrating the steady state values of K1, K2 and/(3 for the 35 see runs in Fig. 5 is given below.

TABLE 1. DRIVING WEST TO EAST (Y-AX~ AT 0 °)

Initial X-axis location KI K2 K3

--80 ° 0"65 --0"13 --0"33

- -40 ° --0-32 +0"02 +0 .88

0 ° --0"39 +0"31 +1"31

+400 --0"65 +0"85 + 0 ' 8 8

+ 8 0 ° --0.89 + 1"33 + 1 "54

The table shows the different configurations which the system in Fig. 3 assumes as the position of the mount changes.

To further illustrate the ability of the adaptive scheme to drive the mount smoothly at low rates, a comparison of dynamic position errors with and without adaptive compensation is showa in Fig. 6. These position errors are obtained by integrating d(t) in Fig. 3 and removing the fixed bias. The peak to peak error over a typical 30 see run without the adaptive scheme was 1.4 arc sec or 0.5 arc sec rms, whereas with the adaptive technique the peak to peak error was only 0.11 arc see or 0"039 arc s e e s r m .

The position error responses for both the X- and Y-axes while being driven simultaneously at 0.005 deg/sec are shown in Fig. 7 on an expanded

~1'0.

i 0.0

.~-1"0

1:4 ~C-~3~IIIS p ~ - ~ DEVIATI0I/

~ 1"0

i 0'0

~-1'0

20 t (SECS) 3 0

WIIH AII~TI~

FIG. 6. Position errors when driving at 0-005 dog/see (X-axls only).

÷0.15 ~- 0"07 ~'e¢ P-P

k . . . . . . . . . --~ . . . . . o,,~,=, m oF L . . . . . . . . . . . . . . c~Tec)

- O q 5 × - AXiS

CROSS- ODUPLING EFFECTS

+0.25 ~- / ~ p.p . . . . LJ, L ita " - - ~ . . . . . . . . . . r . . . . , . . . . . . . . . . .

I l ~ { i / 0 ~ ~ i l " [ l ~ " , | l r " 1 1 1 1 1 " l i p 1 I I I I " I I T 1 i I I i -

o / 2 < 7 >

FiG. 7. Simultaneous X and Y axes position errols utilizing adaptive scheme.

ordinate axis to illustrate detailed variations while using the adaptive technique. The peak to peak and rms enors during this run were 0.07 arc see and 0.025 arc see, respectively, for the X-axis which drives with a smaller position error than the Y-axis. It is evident from Fig. 7 that the two most significant remaining factors contributing to the mount's position errors are the natural resonant modes of each axis and the cross-coupling which exists between the axes. For instance, the 5-6 Hz natural frequency of the X-axis is clearly evident, while a high degree of correllation between errors is seen at the indicated points.

B. Sinusoidal response In order to illustrate the effectiveness of the

adaptive technique to compensate for bearing friction, a sinusoidal rate command signal given by

#(t)=2.5 sin 2n(O.O1)t (deg) (11)

was chosen. The frequency in equation (11) was chosen since it embodies the position and velocity characteristics of a worst case satellite in a 100 mile


orbit. A plot of equation (11) is shown in the first graph of Fig. 8. The second graph is a plot of the mount velocity response to this input without the adaptive compensation. The effect of increasing retarding torque caused by increasing bearing friction is clearly evident when the mount velocity passes through zero. The third plot, on the other hand, illustrates the mount's velocity when the adaptive compensation is employed. It is seen that the 5 sec dead zone region has been virtually eliminated and in addition, the velocity response is completely symmetrical about zero. The compensation afforded by utilizing the additional adjustable parameter, K3(t), is highlighted in Fig. 9. When comparing the first plot in the figure with the mount's velocity response, the second plot in the figure, it is evident that K3(t) is tracking the changing friction level as drawn in Fig. 2. Since K3(t) is multiplied by sgn Op, as shown in Fig. 3, the adaptive mechanism automatically provides a control signal of the proper sign, depending upon the sign of Op(t). The third plot in Fig. 9 is the time

; ( H '~

(0eg/lec) 0

-2-5

STATIC FRK;TION

--2 SCHEME

~,l,I o ~ (~e~/se¢)

-2 '5 WITH ADAPTIVE SCHEME

F[o. 8. Comparison of tachometer responses to sinusoidal rate command.

response of d(t), the error between the model reference and the plant. The peaks in this curve are due to the discontinuity of the friction non- linearity at the origin.

C. Autotrack mode In the autotrack mode, a "star-tracker" system

consisting of light sensors provides two continuous voltages which are proportional to the mount X- and Y-axis position errors, referenced to the origin of the coordinate axes of the optical system in the telescope. These error signals are then injected into the respective compensation network in each axis.

Figure 10 illustrates the mount's X- and Y-axes position errors with and without the adaptive compensation. The voltages from the autotrack system were calibrated in arc-seconds by allowing a star to drift through the field of view with the mount's

I

e x,._,( I ) 0

(see) - I

I

e i , (H O

(s~) - ,

WITH ADAPTIVE SCHEME -- 0 , 7 ~ c p l p

J

-- 0.9~c P-p

" f l ~ r l l l l r l q l q r ' ' ' ~ i ' l r ' l " l r L ~ . " " n " ~ ' " " " " 1 r ,

IO l (sees) 20

ex{i) 0

- I

ey(t) 0

(~J _,

WITHOUT ADAPTIVE SCHEME > 1"4 ~ c P-P

I -

I r ~ w ~ ~ - " U - W " - ' " l ' l l l l l ' " l f b I / DISTURBAt~CE 1.6 s~'c P-P / EFFECT

, J ] L

/ y I ~ VF-' I l,"ll,,"rl i -~ fFr !llr l r r ,- i , ,frl~ n r, lr i t t-- t I0 f (sees} 20

0.5

0.5

°1

-IO

- g ~ ( t ) (volts)

~tplll (Vi l t l )

r ;<,, a I

L , i ; 'P v ' j I~o. 9. --K3, ~ and ~ utilizin8 adaptive mechanism

with sinusoidal input.

F~o. 10. Comparison of autotrack error signals with and without adaptive scheme.

brakes energized. It is seen that the X-axis peak to peak position error is about 1.4 arc sec whereas the Y-axis deviation is less than 1.6 arc sec with the exception of a couple of transients which range to 2 arc sec peak to peak. With the adaptive system in effect, the peak to peak deviation of the X-axis error is less than 0.7 arc sec whereas the Y-axis error remains below 0.9 arc sec. In addition, it is seen that the first set of plots possess more high frequency content than the second set, indicating that more of the low frequency content of the error signal is being tracked with the adaptive compensation. Finally, from the data in Fig. 10 as well as from other runs which were performed, it appears as if the adaptive compensation renders the control system less susceptible to transient oscil- lations from signal shocks due to disturbance effects such as the one indicated in the figure.

130 JAMES W. GILBART and GEORGE C. WINSTON

At this point, it is:instructive to examine the nature of the error signal emitted by the star- tracker. It has been well established that incident light received from a celestial body undergoes random refractions as it passes through the atmosphere [4, 5]. When observed from the earth, the effect is one of apparent random image motion. Several studies have established the rms value of the motion to be in the range of 0.3-1 arc see rms. This is the reason that adapative compensation is less effective, only a 50 per cent improvement, than in the search mode which has over 10 to 1 improvement. Image motion is composed of frequencies well beyond the bandwidth of the system.

D. Program mode

In program mode, the telescope responds to instantaneous predicted satellite trajectory commands which are based on previous pass data. Since in this mode, the predicted satellite velocity is available, this information can be used to advantage to command the telescope [6] using feedforward compensation. However, several authors have shown, i.e. Ref. [7], that this approach has the severe disadvantage of being highly sen- sitive to plant parameter variations. As indicated in Sections 4-A-4-C, this drawback might be eliminated with adaptive compensation. In this vein, the system in Fig. 11 was implemented, incorporating feedforward compensation, closed loop compensation, and adaptive compensation. For a thorough treatment of the particular configuration in Fig. 11, the reader is referred to Ref . [8]. A comparison of the mount's position error for a worst case satellite trajectory with and without the adaptive compensation is shown in the first set of plots in Fig. 12. The rate command utilized in the second set of plots simulates a

typical pass. It is sgen that in.both cases the rms, tracking error is reduced by more than a factor of 6.

NOTE: Figure 11 does not depict the "sync" circuit which was utilized during actual field tests to circumvent the problem of preamplifier satur- ation for large step commands. When the error signal, e(t), exceeds a certain threshold, the adaptive compensation is frozen and the closed loop compensation integration is removed until e(t) settles down. Incorporating such a "sync" mechanism is common practice in systems of this type.

MODEL

+

+ + + O I

I

ADAPTIVE I

FIG. 11. Program mode configuration.

5. CONCLUSIONS

A Model Referenced Parameter Adaptive Con- trol System based on Liapunov's Direct Method was applied to an optical tracking mount. The ability of the adaptive technique to compensate for mount irregularities such as inertial variations and bearing friction was demonstrated. Utilizing the adaptive scheme, the mount would drive at a fixed rate to within 0.11 arc sec peak to peak position error, whereas without adaptive compensation the

(t)

3

(°tSEC)

0

e (t)

RATE COMMAND SIGNAL

1"8

~s~c~ 0

e I t ) I 1.al

(SE0Cl

1~0 " ~ ' ' ' ~ 0

t (SECS)

CLOSED LOOP

eR.M.S. = 1-27 SEC 0.75

eR.MS ' = 0"2 S~C COMBINED

0"75

= . ~ . . . . ( s ~ c }

SYSTEM ALONE eR Ms, = @53 S~C

OMPENSATION enM~. = 0 . 0 9 S"EC

i i i II _ - = ~ l~ I

FiG. 12. Comparison of position errors in program mode with and without the adaptive scheme for two different command signals. The center plots are taken from the system not using adaptiveconffol or

feedforward compensation,


same rate command yielded an error of 1.4 arc sec peak to peak, an improvement greater than 12:1. Furthermore, when the velocity command signal passed through the null point, the adaptive compensation virtually eliminated the dead zone region caused by static friction. In the autotrack mode, a moderate 50 per cent improvement resulted when the adaptive system was employed. For example, the peak to peak pointing error was reduced from 1.4 to 0.7 arc sec for the X-axis and from 1.6 to 0"9 arc sec for the Y-axis. Greater improvement could not be achieved due to high frequency image motion. Finally in program mode, an improvement in tracking accuracy of 6 to 1 was achieved with adaptive compensation.

Acknowledgements--The authors would like to thank Mr. D. Griffin and Mr. W. H. Schaefer of the Advanced Data S~stems Division for their valuable assistance in this project. This work was supported by NASA, in part under Grant NGL-22-010-018, and by the National Academy of Sciences.

REFERENCES

[1] P. C. PARKS: Liapunov redesign of model reference adaptive control systems. 1EEE Trans Aut. Control AC-II, 362-367 (1966).

[2] J. W. GILBART, R. V. MONOVOIZ and C. F. PmCE: Improved Convergence and Increased Flexibility in the Design o f Model Reference Adaptive Control Systems. Proceedings of the Ninth Symposium on Adaptive Processes (1970).

[3] C. C. HANG and P. C. PARKS: Comparative Studies o f Model Reference Adaptive Control Systems, 1973 Pre- print Vol. Joint Automatic Control Conference (JACC), IEEE, pp. 12-22. Also IEEE Trans. Aut. Control AC-18, 419--428 (1973).

[4] J. L. BUI~TON: A System to Monitor Stellar Image Qnafity. NASA X-document X524-69-101 (1969).

[5] V. I. TATARSKI: Wave Propagation in a Turbulent Metffum ('translated by R. H. Sn.VnMAN). McGraw- Hill, New York (1961).

[6] J. C. LOZIER, J. A. MORTON and M. IWAMA: The servo system for Telstar antenna positioning. Automatica 2, 129-149 (1965).

[7] BOWER and SCHULTHE~S: Introduction to the Design of Servomechanisms. Wiley, New York (1958).

[8] J.W. GILBART, G.C.WnqsTON, and R.V. MONOPOLI: Combination Open Loop, Closed Loop, and Adaptive Compensation: Its Application to an Optical Tracking Telescope, Proceedings of 1972 International Conference on Cybernetus and Society, Washington, D.C. (IEEE) (1972).

R6sum6----L'application de la th~orie de contr61e adaptif par module de r6f6rence /t un t61escope moniteur optique est discut6e. La capacit6 de la technique adaptive A compenser pour les irr6gularit6s de monts tels que des variations iner- tielles et le frottement de paliers est d6montr6e par des r6sultats protiques sur un grand t61escope moniteur. Des r~sultats sont pr6sent~s qui montrent une am61ioration de 6 A 1 dans la pr6cision de la surveillance pour la trajectoire d'un satellite du pire cos.

Zusammenfassung--Die Anwendung der Theorie der adaptiven Steuerung mit Bezugsmodell auf ein optisches Nach- lauf-Teleskop wird diskutiert. Die Eignung tier adaptiven Technik zur Kompensation von Unregelma fligkeiten in der Halterung wie Schwankungen in der Trligbeit und Lager- reihung wird gezeigt und zwar an Hand von Feld-Test- Ergesbnisscn bei einem gro Ben Nachlauf-Teleskop. Angege- ben werden Ergehnisse, die eine 6 zu 1 Verbesserung in der Folgegenauigkeit f'tir den schlechtesten Fall einer Satell- itenhahn ergeben.

Pe3ioMe--O6cyxaae'rcg npRMeneHHe Mo~e.m~, c~3aHHofl C reopneg a~anTmsHoro ynpanaeana ~-m ~aecxona c onra~ec~og cae~ameg cgC'l'eMog. BO3MO~gHOCTH Me'rO~OB a~anTaw~H ~ KOMneaoau~u~ HeperyJDIpnocTH yCTaHOBKH H "i'pORH~ B no~unuemKaX ,aeMocTpRpylOTC~ C noMon~lo pe3yJIbTaTOB HaTypn'hlX XCnWramatt 6oabmoro cae~amero Toraecxolra. IIpno~tCTa pe~ymbTaTH, noga3bma]onlHe OT 6 ~o I yayqmem~ B To,mocTX c~exe~s ~m Hamty~m¢l~ TpaeKTopma CnyTamm.

adaptive compensation for an optical tracking telescope

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