Acta Astronautica Vol. 17, No. 1, pp. 61-67, 1988 0094-5765/88 $3.00+0.00 Printed in Great Britain. All rights reserved © 1988 Pergamon Journals Ltd

A SOLAR ATTITUDE CONTROLLER FOR EXTENDING OPERATIONAL LIFE-SPAN OF COMMUNICATIONS

SATELLITES

K. KUMARt Auburn University, Department of Aerospace Engineering, 162 Wilmore Laboratories,

AL 36849-3501, U.S.A.

(Received 1 July 1986; revised version received 16 March 1987)

Abstract--Proposed here is the idea of incorporating system redundancy through the provision of a solar attitude controller for possible use in contingencies and/ or during the "last leg" of an axisymmetric satellite engaged in a geosynchronous communications mission when fuel available on board for orbital plane control maneuvers and/or attitude control is depleted. The solar torques generated by suitably regulating the movements of the control surfaces provide for continual tilting of the satellite ensuring its alignment with the satellite-Earth line-of-sight and, hence, uninterrupted communications even after its orbit becomes eccentric and its plane gradually drifts away from the equatorial plane.

I. INTRODUCTION

The feasibility of using solar radiation pressure for attitude control of spinning as well as three-axis stabilized satellites has been explored by several investigators[I-4]. Two on-board satellite experi- ments have even demonstrated the successful per- formance of the solar pressure attitude control systems[5, 6]. Yet, the potentialities of this useful concept, especially for geosynchronous satellites, re- main far from realized. This, perhaps, may be attrib- uted to the complexity of the proposed controller models and the associated excessive hardware costs. These appear to offset the gains accruing from the nearly passive nature of the solar controller, particu- larly in the face of competition with the well proven and reliable conventional alternatives.

An important aspect of satellites launched in the geostationary orbit is the adverse effect of environ- mental perturbations which introduce significant changes in orbital elements such as inclination and eccentricity leading to periodic longitudinal/lateral satellite drift with respect to the ground terminal. The resulting maximum "satellite misalignment" from the line-of-sight continually grows with increasing eccen- tricity and/or inclination. The fuel is provided on board for periodic station-keeping maneuvers in or- der to ensure continual, uninterrupted commu- nications. Here, an attempt is made to focus attention on how the versatility of the solar controller can be exploited for continual satellite tilting, in order to compensate for the periodic drift, as well as damping of the attitude disturbances. This particular feature of the solar controller may substantially enhance its overall cost-effectiveness for various space missions including communications. In light of the rapid

tVisiting Associate Professor, Department of Aerospace Engineering; on leave from I.I.T, Kanpur, India.

61

world-wide growth in demands on communications capacity, extension of the satellite applications to new areas and the resulting problems of excessive over- crowding of the previous geostationary arc [7-10], this may gain added significance.

2. PROPOSED SATELLITE-CONTROLLER MODEL AND EQUATIONS OF MOTION

The formulation begins with the analysis of cou- pled librational motion of an axisymmetric satrellite in a general, elliptic, non-equatorial orbit around the Earth (Fig. 1). The satellite is assumed to be at perigee while vertically passing over the equatorial

lee

icl OX

Fig. 1. Geometry of satellite motion.

62 K. KUMAR

ground terminal. Denoted by x 0, Y0, z0 are the prin- cipal body axes of the satellite, in the undisturbed nominal configuration, with the triad so chosen as to direct the z0-axis outward along the local vertical and the y0-axis parallel to the orbital angular momentum vector. An arbitrary satellite orientation is specified by a set of three successive rotations: ~b (pitch) about the y0-axis giving xl, Yl, z~, fl (roll) about the xl-axis resulting in x2, Y2, z2; and 2 (yaw) about the z2-axis leading to the final body coordinate axes x, y, z.

The controller model proposed here (Fig. 2) is similar to the one adopted earlier [11]. It consists of two sets of light, rigid and highly reflective control plates G~ - G~ and G 2 - G~. The Set G~ - G( is moun- ted on the main satellite body so that both the plates remain parallel to the y - z plane while undergoing displacements l ~ (along z-axis) and m~ (along y-axis). The plates of the other pair G 2 - G ~ also remain parallel. But these control surfaces are kept normal to xp-axis which lies in the x ~ plane and is inclined to the x-axis at ~/4 radians. This set undergoes displace- ment (/2) only along the z-axis. The translational motion of the two sets provides the variation of the overall center of pressure and hence the sign and magnitude of the solar torque for the satellite attitude control.

It may be emphasized that for the construction of control surfaces, the use of ultrathin and appropri- ately stiffened sheets of light material such as alumi- nized mylar is assumed. This is highly desirable to limit the associated weight penalties to a minimum. Furthermore, under these assumed design conditions, the changes in the location of the mass-center as well as the moments of inertia would be relatively insignificant. These variations are therefore ignored in the following analysis.

The Lagrangian approach is adopted for formu- lation. The satellite kinetic and potential energies are computed leading to the following expression for the Lagrangian:t

L = (½) m (~2 + rZ0Z) + (½) i[fl2 + ((~ + q~)2 cos 2 fl]

+ (½)I,:[~ - (0 + ~ ) sin fl]2 + i~rn/r

-(½)(t~/r3)( l -- 1=)(1-3 cos 2 ~, cos 2/3). (1)

The Lagrangian formulation in conjunction with virtual work analysis for evaluating the generalized forces contributed by the solar pressure on control surfaces leads to the following equations of motion:

et ~O" - (e2 + 2e, fl' tan f l ) f

+ 1.5 Ki sin 2qJ - K1 [(e~ pS. - e2p~)

sin/3/cos 2 fl + el pa/3'/cos fl] = Q, ,

el/3" - e2/3' + 0.5 ( e l f 2 + 3 Kicos 2 if)sin 2/3

+ KI e l p a f cos /3 = Qe,

Kl (el p~. - e2Pa) = Q;, (2)

tNomenclature is given in Appendix at end of paper.

X •

\ ~ To Ground stotion

Fig. 2. Geometry of proposed solar controller configuration.

where the generalized forces, evaluated for specular reflection assumed in the analysis, are given by:

Q~, = [(1 + e)/et]3[lUll{fq,(ulcos)~ cos/3

- 2 p vl sin 2 cos/3) + C;. (ul sin fl

+ 2 p wt sin 2 cos/3)} + lu2)Ca

x {u 2 cos a cos/3 - 2 v2P sin cr cos/3}],

Q# = [(I + e)/el ]3[luj I { -C~(u~ sin 2

+ 2 r i p cos 2) + C A w Ip cos)~}

- l u2lC~ {u2 sin o - 2v 2P cos a }],

Qa = - C~ ul lul 1[(1 + e)/el ]3 (3)

with

ul = cos 2 sin v - sin/3 sin )~ cos v

+ ~l{cos 2 cos u + sin 2 sin/3 sin u}

+ ~2 sin 2 cos/3,

u2 = cos a sin v - sin/3 sin a cos v

+ ~1 {cos a cos u + sin a sin/3 sin u}

+ ~2 sin a cos/3,

v~ = - sin 2 sin v - cos )~ sin/3 cos v

+ ~1 {cos 2 sin fl sin u - sin 2 cos u}

+ ~2 cos/3 cos 2,

w I = - cos/3 cos v + ~1 cos fl sin u - ~2 sin/3. (4)

A simple geometric analysis is carried out to obtain the expression for relative "satellite misalignment", ~,

Solar attitude controller 63

4 i= lO" ,

o I~l°mP2~ ° I ,

.005

6

o

I~10mp 3

a=.Ol

/

I 5 i o I0

Fig. 3. Steady state relative satellite misalignment as affected by "i" and "e".

in terms of three librational angles.

= cos- 1 [(r. cos fl cos ~k -- cos M cos fl cos u

- sin M cos fl cos i sin u - sin M sin i sin f l ) /

[r 2 + 1 -- 2r.(cos M cos 0

+ sin M cos i sin 0)]1/2]. (5)

This relation enables us to examine the adverse effect of the drifts due to increasing orbital inclination and eccentricity on the apparent "satellite misalignment" as viewed from the ground-station (Fig. 3).

3. AUGMENTATION OF USEFUL SATELLITE LIFE-SPAN

The three important factors restricting useful life of the satellite are:

--Reduction in solar power as a result of de- gradation in efficiency of the solar cells with time.

--Limit on the amount of fuel that can be carried aboard for the purpose of attitude control.

- -Limit on the amount of fuel that can be carried aboard for "north-south-correction"/station- keeping maneuvers.

We already have adequate experience whereby each one of these factors has been responsible for "premature death" of the spacecrafts. However, as the rapid developments in solar cell technology have led to considerable improvement in power generation over extended periods, I propose to focus attention on the last two factors and that how it may be

possible to do away with the need for additional fuel for ensuring longer satellite service.

Proposed here is the idea of incorporating redun- dancy through the provision of the solar attitude controller for possible use in contingencies and/or during the "last leg" of the satellite mission when the fuel available on board for orbital plane control maneuvers and/or attitude control is depleted. The device generating the solar torque by suitably regu- lating the movements of the control surfaces is likely to enable the continually controlled tilting of the satellite ensuring its alignment with the line-of-sight and hence uninterrupted communications even while the orbital plane may continue to gradually drift away from the equatorial plane. This approach of controlled satellite tilting to ensure the "satellite alignment" also seems feasible in the situation of the satellite's orbit becoming elliptic as a result of envi- ronmental perturbations.

4. SYNTHESIS OF CONTROL POLICY

With a view to develop a simple control law, several approximations are made. The pitch, roll and yaw angles are assumed to be small so that their second and higher order terms can be eliminated from the equations of liberational motion. This simplifies the equations to:

~b" + 3 K/~b - e 2 = Q ~ , ,

fl" + K2 fl =Qa,

KI p~ = Qa, (6)

where Q,, Qa, Qa represent the generalized solar pressure moments which we wish to control by moving the control surfaces. Georlaetry of satellite motion is analyzed to determine the required libra- tional variations which ensure "satellite alignment" with the line-of-sight at all times. The analysis shows that for this (i.e. achieving ( = 0), the desired satellite pitch and roll variations when approximated by their first harmonic terms can be written as:

= [2e/ (r , - 1)] sin 0, (7a)

fl = - [(sin i ) / ( r , - 1)] sin 0, (7b)

while, for the three-axis control considered here

2 = 0. (7c)

In order to achieve the desired sinusoidal pitch and roll variations in steady state, the harmonic terms such as (A sin 0 + B cos 0) are introduced in the "control expressions" for the generalized forces. The proportional + derivative control terms are added to provide for decay of the transients. Thus, the gener- alized forces are assumed to have the form:

Qj = - ( l t j j ' + vfl) + (Aj sin 0

+Bjcos O ) ; j = ~ , fl, 2.

On substituting these in conjunction with the expres-

64 K. KUMAR

sions for ~k, # and 2 as stated in (7) and solving the resulting identities for the constants A/and B~, we get the following control relations:

Q~ = -- (#~, t~' + v~ ~k ) + 2e [(3Ki + v¢,

- r,) sin 0 + #~ cos O ] / ( r , - 1),

Q~ = - (#~#' + va#) - (sin i)[(3K~

+ v¢) sin 0 + #~ cos O ] / ( r , - 1),

Q~ = - (u~2' + v ~ ) . (8)

For the control law thus obtained with the suitably chosen coefficients of sin 0 and cos 0, eqn (7) repre- sents the first order steady state approximate re- sponse to the three iibrational equations of motion. The presence of the proportional + derivative control terms restricts possible deviations from the desired satellite orientation. The parameters #~, v~ denote the control gains and can be chosen arbitrarily. On- board sensors are needed to measure the angles (~J, /~, ),) and the librational velocities (~ ', #' , ~.') whereas the true anomaly 0 and the sun-position angle ~b can be preprogrammed as functions of time.

The generalized forces Qj as dictated by the control law may not always be available, however. So, for a realistic simulation of the controller performance, a practical constraint is introduced based on maximum permissible design limits on movements of the geo- metric center of the sets of the control surfaces on either side of the satellite c.m. The constraint can be expressed in the form

Ki=,9 - -Cont ro l -on ' ~o:45", p=O, Cjmax=2

jo=O ~]=5 , v j :5

jo=.l - - - Control-ofl

e i t

) .002 .004 .006..094 .096 .098 .I I l I~ I l I ?I

i= 0 [ '~ rr~ I

5 - ,~ ~ I " I ' , '~, ~ ,

L.~--",~.--~ I 2 3 4 7 4 8 4 9 5 0 I

iO q

O

0

~-Orbits

0 .2 .4 .6 ,9.4 9.6 91.8 I0 I0{,, , , , "~ ~ /~,

[ " i~i~ . - o i ~ I 1

, [ ^ ~ ¢ , . . v . . . . . ' , , / l ~ - i I V

0 ~ ~ ~ t~-~ f-.~ 0 I 2 3 "47 4 8 4 9 50

~-Orbiis

Fig. 4. Typical satellite response demonstrating the effectiveness of the controller for uniformly growing "e" or

Ki=8 - -Con l ro l -on ' ~b o :45"~ Cjmax:2

io:O ~tj=5 , v]:5

J'o=.l . . . . Control-off

e !

0 0 0 2 . 0 0 4 . 0 0 6 ~094 , 0 9 6 . 0 9 8 .I

I ~ - U r b l s , i I I I n L

I i ! J ~ 1 1 ' , , ' \ P ~q 51 , ~ . I . , , I t ,

O0 .I .2 .3 --4.7 4.8 4.9 5 r

i0 C .002 .004 .006 t~094 .096 .098 .I

o ° = / ' . ~ I I~1 r e O r b , , , . ~ ,* /',

2 3 4z) , ; ',', 49 50) s ,x I ~ 1 , i%1 Iii I ! ' , t ~ , l t ' , ; I

• 0 ~ " ~ - - 1 t c / " , ~ ~ . . ' ~ . ~ .I .2 .3 ' 4 . 7 4 . 8 4 . 9 5

i o

Fig. 5. Typical satellite response demonstrating the effectiveness of the controller for simultaneously increasing

"e" as well as "i ' .

This design constraint may marginally bring down the attitude accuracy as compared to that achieved with the constraint Iajl-< Iajm.,I used in an earlier study [11], yet, it is favored here due to significantly reduced on-board computation needed for control implementation.

5. R E S U L T S A N D D I S C U S S I O N

The system dynamic behavior is studied through numerical integration of the exact nonlinear equa- tions of motion (i.e., eqn (2)) in conjunction with the proposed control law subject to practical constraint imposed on the controller design. The orbital inclina- tion and/or eccentricity are assumed to have small, arbitrary but constant rates of growth while com- puting the results• The numerical simulation also accounts for the uniform rate of change of the Sun-position-angle ¢k due to Earth's revolution around the Sun.

Figure 4 compares the typical response results for continually growing "e" and "i" with and without the controller in the presence of initial impulsive attitude disturbances (i.e. ~J6, #6, i-0 4= 0). The results show the controller to be quite effective in realizing a much closer synchronization of the satellite orientation with the line-of-sight. Even with a marginal fall in the control surface reflectivity resulting in an increase in the value of the parameter "p" from 0 to 0.1, the controller continues to perform satisfactorily with only a slight reduction in attitude accuracies (Fig. 5).

Solar attitude controller 65

Ki-.8 ~ C o n t r o l - o n , ~o=lO'T p=O

jo=O t~j = v j = 5 , Cjmax-2

j~=O . . . . Control-oft

21 f

° t e•'°l 1~4o~p . . . . . . . . . . . . . . . . . . . . . . . . . . . I

2.5 5

0

Zl;

°it It4or.p

0

i* 2,5 5

i i

. 05 .I e

Fig. 6. System plots showing the effects of 'T ' and "e" on steady satellite misalignment.

i = I% e • .001 - - C o n t r o l - o n ' p=O

Ki=.5 pj= vi=5,Cjmax = 2

0=0 . . . . Control-off

4 [- . . . . . . . . . . . . . . r . . . . . . . . . . . . . . 1

I~l~mp2 t i~=o

o / ~ , ~ , 0 rr 4o 2rr

5 i o ~o=10° I

I / / " / j 1 1° 5 1-/j//- J

01, - ' - ~ - I I 0 . 0 5 . , .1

Io

Fig. 8. System plots showing the effects of Sun-position angle and initial impulsive disturbance on steady state

relative satellite misalignment.

With a view to assess the effects of the various orbital and design parameters, numerous response results were obtained. The resulting information is summarized in the form of system plots showing their influence on steady state amplitude of the relative "satellite misalignment" (Fig. 6). In the absence of the controller, this angle may become unacceptably large even for a moderate growth in orbital inclina- tion and/or eccentricity. However, the proposed con-

i = I',e=.O01 - - C o n t r o l - o n , ~o=10"

Jo=O pj= vj=5.t Cjrno x =2

Jo=O . . . . Conlrol-off

2

°it I~lomp

0

p=O \ \ I

\ , , \

I I .5 Ki I

IO I / i o I Ki=5 /

!i- -

O , ,

Fig. 7. System plots showing the effects of satellite mass distribution and control surface reflectivity on steady state

relative satellite misalignment.

trol operation appears to be successful in restricting the misalignments to well within acceptable limits for space communications.

Figure 7 shows the influences of satellite inertia parameter and control surface reflectivity. The con- troller continues to be effective for an arbitrary shape including normally unstable "stubby" satellite configurations, thus allowing freedom in the choice of the satellite mass distribution. The success is however subject to the control surfaces retaining a sufficiently high degree of specular reflection.

The effects of the initial Sun-position-angle and the attitude disturbances are indicated in Fig. 8. The relative "satellite misalignment" does seem to suffer some undesirable growth due to changes in the Sun-position; however, the steady state performance is practically independent of the level of the initial disturbances. The effect of various other design pa- rameters such as control gains, the maximum permis- sible limits on ~ was also examined. It was found that it is possible to improve the controller per- formance to some extent through a judicious choice of the parameters. A comprehensive optimization of the control strategy is likely to be counterproductive, however, as it would involve a heavy penalty by way of considerably enhanced complexities of hardware required for implementation while providing only a marginal reduction in "settling times".

A comment concerning the Earth-shadow which would render the controller ineffective would be appropriate here. The periods of the shadow encoun- tered by a geostationary satellite are fortunately limited to ~1.25% of its life-span. The results showed the controller performance to remain virtu- ally unaffected.

66 K. KUMAR

Success of the proposed controller presents an interesting possibility for station-keeping of commu- nications satellites. The servo-power required is rather modest and so is the size of the control plates. A preliminary estimate using data for the INTEL- SAT IV series of satellites shows that the peak power levels can be limited to ~ 5 W for a controller with typical plate sizes of 0.5 m area and permissible movements of 0.1 m providing for the required C ~ x values of 2. The associated weight penalty can be minimized by selecting a material that would be light, rigid and also capable of retaining a high degree of specular reflection over extended periods even in the hostile space environment. In the past, aluminized mylar membrane has been suggested as a possible choice for the control surfaces. Availability of "bet- ter" materials may further reduce the associated weight penalties. Perhaps, adapting of even tiny portions of the solar panels themselves as the control surfaces, particularly at the tips, is a feasible alterna- tive worthy of serious consideration.

5. J. R. Scull, Mariner IV--Revisited--or the Tail of the Ancient Mariner. Presented at the XX Congress of the International Astronautical Federation, Argentina, Oct. 1969, pp. 747 758.

6. U. Renner, Attitude control by solar sailing--a promis- ing experiment with OTS~2. European Space Agency. Vol. 3, pp. 35-40 (1979).

7. G. E. Lavean and E. J. Martin, Communications satellites, the second decade. Astronaut. Aeronaut. 12, 54-61 (1974).

8. K. Kumar, Some aspects related to the satellite applica- tions in non-stationary 24-hour orbits. Acta Astro- nautica 9, 147-154 (1982).

9. D. L. Akin, Some applications of non-stationary geo- synchronous orbits. AIAA /AAS Astrodynamics Confer- ence, Palo Alto, Calif. Aug. (1978).

10. O. L. Dial and J. L. Cooley, Mission design implications of an inclined elliptical geosynchronous orbit (Inter- national Ultraviolet-Explorer). J. Spacecrafts Rockets 14, 401-408 (1977).

11. K. Kumar and A. K. Padhi, Lateral drift compensation for satellites in non-equatorial synchronous orbits through attitude control. Proceedings of the Symposium on Engineering Science and Mechanics, American Astro- nautical Society/N.C.K. University, Taiwan, Dec. (1981).

6. C O N C L U D I N G REMARKS

It seems possible to utilize the proposed controller in order to handle simultaneously the twin problems a of attitude control and changing orbital shape or e inclination. This approach may prove to be particu- et lady effective for geosynchronous satellites in con- e2

i tingencies and/or during the "last leg" of the satellite mission when the fuel available onboard for orbital is plane control maneuvers and/or attitude control is depleted. Some suitable modifications in the ground J segment equipment would of course be needed to I, m I ensure alignment of the ground antenna-main lobe- axis with the line-of-sight in all such situations. 12 However, with the recent related technological ad- vances, these may be possible at reasonable costs with

m little adverse influence on the overall economics. The p feasibility of carrying out arbitrary attitude maneu- vers, already established, may also prove to be quite handy, adding further to the versatility of the solar controller.

Acknowledgement--The author would like to express his appreciation to Mrs Marjorie McGee for typing the manu- script of this paper.

REFERENCES

f P~

r

re

r n l u

u

x, y, z 1. V. J. Modi and R. C. Flanagan, Librational damping of gravity oriented system using solar radiation pressure. Aeronaut. J., Royal Aeronautical Society, 75, 560-564 (1971). x0, Y0, z0

2. V. J. Modi and K. Kumar, Attitude control of satellites Xl, y~, z~ using the solar radiation pressure. J. Spacecraft Rockets x2, Y2, z2 9, 711-713 (1972). Xp, yp, Zp

3. V. K. Joshi and K. Kumar, New solar attitude control approach for satellites in elliptic orbits. J. Guidance Ap Control 3, 42-47 (1980). Cj

4. K. Kumar, Satellite librational damping using a quasi- C~, feedback solar controller. IEEE Trans. on Aerospace C# and Electronic Systems, 18, 591-597 (1982). Ca

APPENDIX

Nomenclature

Semi-major axis of the satellite orbit Orbital eccentricity (I + e cos 0) (2e sin 0) Inclination of orbital plane with respect to equator Inclination of ecliptic plane with respect to equator ~,, ~, ~ Distance of geometric center of first set of controller plates (G~, G't) from satellite mass center along z and y axes, respectively (Fig. 2) Distance of geometric center of second set of controller plates (Gz, G~) from satellite mass center along z axis (Fig. 2) Satellite mass i ( 1 - p - - z ) / ( 1 + p - ~ ) ; 0 for totally 2 reflective plates and 0.5 for totally absorptive plates (1 +q, ') Generalized momentum corresponding to :o. D- ' - (1 + ¢ ') sin fl] Length of the satellite radius vector, a(1 -e2)/ ( l + e cos 0) Earth-radius Dimensionless radius of satellite orbit, r/r, Time (0 + ~ ) ( 0 + ~ - ~ ) Principal body coordinates with origin at the mass center and z-axis along the satellite axis of symmetry (Fig. 1) Intermediate body coordinates preceding the final modified Eulerian rotation about the satellite axis (Fig. 1) Rectangular coordinate axes with xp normal to the plates (G2, G~) (Fig. 2) Total area of each set of the controller plates Solar parameters C~, C#, C~ R p 3 S A p l l (I + p - r ) / ( # C J )

R,,3S A?I 2 (I + p --z)/(pC~,l) R,~S Apm I (1 + p -- ~)/(pC,.l)

Solar attitude controller 67

Cjmax

C I

KI

L M

S

Ct 2

6

Physical limit on Cj as imposed by satellite 0 design; j = ~,, fl, 2 ,u Velocity of light /~j l~x= Iyy vj Mass moment of inertia about k-axis; k = x, p, y, z (Fig. l) Mass distribution parameter, [1 --(17ill)] a (1 - K,) ( (I + 31(,) Lagrangian I ~ l o... Mean anomaly Generalized forces Q¢,, Q~, Q~ ~b Solar constant ~, fl, 2 [I - c o s (i s - i ) ] sin q~ (°) - s i n ( i s - i) sin 4~ ( )'

Angle denoting the position of perigee as ( )0 measured from the vernal equinox (Fig. 1); taken as zero in the analysis

True anomaly (Fig. 1) Gravitational field constant Control gains; j = ¢,, fl, 2 Control gains; j = ~b, fi, 2 Reflectivity and transmissivity of control sur- face material, respectively (2 - n / 4 )

"Relative" misalignment of satellite axis (Fig. 2) Maximum "relative" steady-state misal- ignment Sun-position angle (Fig. I) Pitch, roll and yaw angles, respectively (Fig. 1) d( )/dt d( )/dO The value of (0) at 0 = 0