Magnetic Torquing Scheme for Attitude Control of a Wheel-Stabilized Astronomy Satellite

Copyright © IFAC 91h Trienni .. 1 Wllrld Congn.:"'i Budapest. Hung.. r~. 19K4

MAGNETIC TORQUING SCHEME FOR ATTITUDE CONTROL OF A WHEEL-STABILIZED ASTRONOMY SATELLITE K. Ninomiya*, I. Nakatani*, T. Takahashi* and K. Maeda** ' I nstitute of Space and Astronautical Science, 6-1, Komaba , 4-chome, /vleg.-o-ku, Tok)'o 153. japall "NEe Space Development Division, 4035, Ikebe-cllO, Midori-ku, Yokohama 226. japall

Abstract. A new pointing control law by magnetic torquing is proposed for a wheelstabilized astronomy satellite. This law is based on a linear feed - back of attitude error and rate, and achieves both angular momentum control and nutation damping. Actuators are magnetic torquers along three axes plus one bias-momentum wheel. Sensors are rate-integrating-gyros and magnetometers . Stability analysis is carried out by root - locus method. A large angle maneuver scheme is also proposed, which utilizes the same pointing control law. The maneuver is carried out around the eigen axis of rotation, and the advantage is a small amount of processing required for the on-board computer. Computer simulation for the C2se of ASTRO - C, a Japanese X-ray-observation satellite, shows that the proposed system works satisfactorily . Keywords. Attitude control ; satellites, artificial; bias - momentum; magnetic logic; control system analysis. INTRODUCT iON

stably keep the extragalatic X-ray source within the field of view of LAPC for few days (pointing control mode). Large angle reorientations between different X-ray sources are also required. In this latter maneuver, the requirement for the reorientation time is moderate with the typical maneuver rate of 10 deg/hour or less, For such missions, the proposed control system has advantages inherent to magnetic torquing systems.

Attitude stabilization by momentum wheel has been widely used for earth- oriented satellites (Terasaki, 1~67; Dougherty and others, 1560; Powell and others, 1~71) , and various magnetic torquing systems have been employed for the unloading of the accumulated angular momentum or for the precession of angular momentum direction (Paiken and Fleisig, 1963; Br own, 1%5; Lindorfer and Huhlfelder, 1966; Tsuchiya, 1932). But few papers have dealt with the nutation damping of wheel- stabilized spacecraft by magnetic torquing (Mobley and Tossman, 1971; ~ticKler and Alfriend, 1976; Ninomiya, 19d1), mainly because other actuators (e.g . reaction jets or wheels) are usually employed whose torques are large in comparison with the magnetic torque and partly because magnetic torquers can not always produce the desired torque. However, for a class of satellites whose attitude is to be controlled with reference to the inertial frame with comparatively high precision , bias-momentum system combined with magnetic torquing can be advantageously applied as is shown in this paper. Here, a new magnetic torquing law is proposed for attitude pointing of this class of satellites. Also, proposed is a new scheme for the large angle maneuver which is based on the same magnetic torquing principle.

POINTING CONTROL LAW System Description The definition of coordinate system is shown in Fig. 1. The tar get attitude is inertially fixed and is represented by O-XYL or 0 coordinate system. The body-fixed system OB is depicted as O-xyz, where each axis coins ides with the principal moment of inertia axis. The relation between 0 and OB is expressed by Euler angles ex , e y and e z which correspond to error angles of the satellite. As shown in Fig . 2, the satellite has a momentum wheel whose axis is aligened with z-axis, Three magnetic

The proposed pointing control law achives the control of angular momentum together with nutation damping using a simple linear feedback system. The large angle reorientation- maneuver scheme tries quasi- statically to shift the attitude reference so as to cause the attitude change around the eigen axis of rotation. Assumed on- board attitude sensors are rate-integrating gyros (RIG ' s) and magnetometers for three axes. An on- board

y

J..::::=====------'''-''-'''-- Y

microprocessor is also assumed .

Analytical stability study is carried out together with extensive computer simulations for the case of AS'fRO - C satellite which is due to be launched in Feburary , lS87. ASTRO-C is a Japanese scientific observation satellite having a Large Area Proportional Counter (LAPC) to be developed jointly by Japan and the United Kingdom . It is required to

x

Fig . 1 .

29 15

Coo rdinate s ys t e ms .

K. Ninomiya et al .

29 16

(3) does not provide nutation damping torque. As is shown in later sections, the control law given by Eq. (2) achives both the control of the angular momentum vector and nutation damping making a stable pointing control possible. The advantages of the law are the linearity of the control logic which makes designing easy and the simplicity of the required on- board calculation whic!1 can be handled by a micro-processor .

z

:~

I I

i 8I )-------

(ill[D

y

.------i:---

I,' "

x

L!-------Fi g. 2.

y

Spacecraft model.

torquers are provided one along each axis so as to produce a magnetic moment vector towards any desired direction . 8'x , 8 8 ~, 8 x , 8y and 8" are obtaind by rate-integrating- gyros, while the 3-axis components of the geomagnetic field are measured by on - board magnetometers.

y,

Assuming I 8'x I , I 8'y I , 18'" I , I 8x I, I 8y I , I 8z I , and I (h w- h o)/h ol« I, we have the following linearized equations of motion for the satellite alid for the wheel:

Ix Ei ~ + h o8y h o8'x ly~y lz 8z hw

Tcx + TDx, Tcy -+- TDy, Tcz + TD>. F Tw

The physical concept on which the proposed control law is based is explained below . In Fig . 3, a free nutation of a satellite is shown on a 8 X- 8 y plane. The point H is the center of the nutation and corresponds to the angular momentum vector direction. The point P (8 x , 8y ) denotes the z-axis direction . The target attitude is assumed to be the origin 0 and so, our objective is to provide a control torque making Hand P coincide with O. The satellite angular momentum vector is expressed by (ho( 8 y+8~/ wo ) ,-h o (8 x - 8 / wo ),hw) in L system . The torque for angular momentum direction control is along while the one for nutation damping is along~. The former one together with the unloading torque can be obtained by the contro l law of Eq. (3), if we set: 6 P.=~, = (h o ( 8 y+8 ~/wo ), -h o (8x 8X/~o ) ,hw-~ o ), whi:e the latter one by equatin g 6H=E 2=(h o8 x / wo ,h o8 y/wo ,O). The control law expressed by Eq. (2) is based on the assumption that Eq. (3) with 6H=E=E , +BE2 ( B> 0) will achieve angular momentum control and nutation dampin g simultaneously. Setting a= B+l, we have Eq. (2). In Fig. 3, ~ and HQ represent the torques f or momentum control and nutation damping respectively. The torque which is provided by the magnetic moment of Eq. (2) is ~.

HO,

hw, (1)

Th" notations are as follows: wheel angular momentum at

hw

each instant,

ho

nominal value of hw,

I x ,I y ,I2,.

moments of inertia about each axis,

ic

(Tcx,Tcy,Tcz)t

TD Tw

(TDx,TDy,TDZ)t

satellite control torque by magnetic torquers, disturbance torque, wheel control torque, wheel friction torque.

F

For the attitude pointing of the satellite, 8 x and 8 y are controlled by magnetic torques around x- and y- axes, while 8" is controlled by wheel reaction torque. The unloading of the wheel is carried out

o Fig . 3.

ex z- a x is mo t ion i n 8 x -8 y plane.

Wheel Control System

oy magnetic torque around z - axis. The control of 8 z is performed by a conventional method using the wheel as an actuator. The magnetic torque Tcz which results fr om the control law of Eq. (2) g ives disturbance a gainst 8z as well as the unloadin g torque. However, this di s turbance

Hagnetic Torquing Law for Pointing Control According to the control law presented in the present paper for reducing 8 x and 8 y and for unloadinb' the spacecraft magnetic moment is cor.trolled as follows :

M = g(~xi)

,

g>O

( h O( 8y+a 8~/ wo ) ) E = -h o (8 x - a8y / wo ) hw-h o A

, a> 1

(2) A

t

where g and a are constants, B=(Bx,By,E~) is the geomagnetic flux-density vector, and wo =h o/ I IxIyiS the nutation angular velocity.

e',

~--------------~----~ -si e,

In the past, the control law, (3)

was proposed for the unloading of zero-momentum wheels, where 6H is accumulated angular momentum to be unloaded. However, for a bias - momentum system with the satellite angular momentum error 6H, Eq.

I"" 6n

""heel moment of in e rtia the devi at ion of whee l rotation speed from nominal value

F i ~.

4.

Bloc k diagr am fo r t he control system a r ound z- ax i s.

29 17

Magnetic Torqu ing Sch eme for At titude Con tro l 1S samll enouLh compared with the control torque provided by the reaction of the momentum wheel. The block diagram of the wheel control system is shown in Fig. 4 .

(de g) 180 00

<= .,.; UJ

150

"

UJ

lAP.GE PNGIE I'.ANEUVER

.,.;

x

<= 0

... .... ...

IU"';

Large angle maneuver scheme using magnetic torque for a wheel stabilized satellite is proposed. The significant feature of the scheme is that the maneuver is carried out around the eigen axis of

rotation such that a feed-back loop suspends the maneuver under unfavourable geomagnetic field conditions.

The control strategy is as follows . A satellite is to be rotated around the eigen axis of ratation at

1

N

0

o ...

Cl) UJ ..-i"';

OOX <=

IU

120

IU

90

60

IU

...

... Cl) Cl)..-i

30

> "111 Cl) <=

"

IU

120

S

a predetermined constant angular rate wo , hence w ~,

150

mi nimum maneuv er angle of z-axis

w~ and w ~ , which are respectively the x, y and z

component of wo, are to be kept constant throughout the maneuver . In the actual control scheme , the rotation around each axis is controlled so that Ei =JS(e·i-wVdt be ~ero at each moment, where suffix i stands for x, y or z . The integration is carried out by a simple digital counter with inputs e·i ar.d

w1·

Fig. 5 .

Comparison of z-axis maneUver angle.

by J~exdt - Lol ' J~eydt-202 and J~ezdt - 203 around x, y and z axis respectively. He r e, ei indicates the dev i ation of e~ from w ~, and 0j(j =1 , 2 , 3 , 4) is given by the following equation, under initial condition OJ =0.

o

As magnetic torque is used, we cannot always expect

that sufficient torque is available for keeping e·i=W ~. Therefore , i f anyone of ! Ei(t) ! ' s exceeds a specified threshold, the input w ~, a target angular rate, is turned off for all 3 axes to wait for the reduction of ! Ei (t) ! . When all of !Ei(t) ! ' s are reduced to less than the predetermined threshold, the input w1 to the inte~ral counter is initiated again . The basic concept of the scheme explained thus far is that the time integral of (e·i-WV, which corresponds to the instantaneous attitude error , is used as a

feed - back signal to keep the 3-axis components of wO constant. This can also be said as setting the 1nstantaneous target attitude JS w ~dt so that ei is a specified constant value for each axis .

1

"7

0, 02 03 04

=J>

the on-board computer, because an instantaneous

"

wO wO y (t - T) x wO qe - q ~ qO2 z ex 0 qg qO4 - qf e ciT y - qg qO1 qO e 4 z - qf - qg - q g

°g y -w

In the equaton, q~ is the Euler parameter of LB on condition that e~Jis always kept to be wl. The numerical evalution of cross - couplin~ error is presented in the later section. STABILITY M1ALYSIS OF POINTING CONTROL Here, the stability analysis of the pointing control law in the previous section is presented . The magnetic control torque T=~xB can be calculated using Eq. (2), while the equation for the wheel torque is easily obtained from Fig . 4. Substituting Tc and Tw into Eq . (1), we obtain Ixe·~ +hoey

I y 8Y- h oe·x

Ize~

One of the problems of this scheme to be investigated is that the maneuver of z- axis which LS colinear with the wheel axis does not follow along the shortest path. The comparison of the nominal path with the optimum one is shown in Fig. 5, where the z- axis maneuver angle around the eigen axis is plotted v . s. the optimum one. The z- axis

°

wO -w ~ Y - wO - W

(4)

The advar.~age of the above mentioned scheme is that only a small amount of processing is required for attitude needs not be calculated. Although i~( e·Cw Vdt is used as a feed - back signal, the att1tude maneuver is carried out by an open- loop sequential control whose algorithm is as proposed 1n second section for a pointing control. An example or a block diagram of the attitude control system using the maneuver scheme mentioned above is shown in Fig . 8 . In Fig . 8, the portion which is added to the pointing control system to achieve the large angle maneuver is enclosed by dashed lines .

w ~ - w~

-w ~

6h

g( 6hB x Bz -ho(ey+ae~/wo) (B~ +B~) -ho(ex- aey/wo)BxB y ) ~TDx ' g( 6hB y Bz +h o(ey +ae~/wo)BxBy +ho(ex- aey/wo) (B~+B~)) +TDy ' g( - 6h(B~+B~);ho(ey+ae~/w o)~xBz ) - 6h+TDz' -h.o (ex-ae y / wo) ByB z KwKrez +KwKpez - KwKf6h/lw

(5)

seen from Fig. 5, the deviation from the optimum path is small and causes no practical problem for such a satellite as ASTRO - C whose typical attitude 0 maneuver is between 100 to 30 with moderate

where 6h=hw- ho. We will study the nut at on stability in a short period, where the variation of the earth magnetic field can be neglected . To verify the stability of the system , we have to show that the roots of the characteristic equation of Laplace transformation of Eq. (5) exist in the left half plane . If the onboard microprocessor of the control system can ' t calculate the control law quickly enough to guarantee the continuous system, the stability of the system can be checked by confirming that all of the roots of the characteristic equation of Z-t ramsformaton are

maneuver time requirement.

within unit circle .

Another shortcoming of the scheme is that a control error is caused due to the fact that e ~, e ~ and e ~ car.'t be completely kept to be the target rates, w ~, w~ and w ~. We tentatively define to call this

the seven roots of the characteristic equation of Laplace transformation is always at the origin , and one of the roots of Z- transformation is always at (1.0 , 0 . 0). But these can be tolerated as stated below.

maneuver angle around the eigen axis lies in a

hached region in Fig. 5. For the satellite with angualr momentum along z- aixs, the maneuver angle of the z-axis should be as small as possible to make the maneuver time small .

error as cross-coupling error.

l-!owever, as can be

This error is given

In our case, however , one of

K. Ninomiya et aL.

2918

When controlling a satellite attitude by magnetic torquers, there is always the restriction that the torque along the magnetic field can't be obtained. Hence, the disturbance torque component along the magnetic field can't be suppressed, causing the accumulation of the error angular momentum along this direction. The root at the origin or at l+jO in the case of Z-transformation of the characteristic equation corresponds to this fact. Neverthless, we can use magnetic torquers in practical applications, because the geomagnetic field vector direction changes considerably as the satellite orbits round the earth except for near equatorial case. An example of the root loci of Laplace transformation with respect to a is shown in Fig. 6, and the root loci of Z-transformation with respect to the sampling time in Fig. 7. Fig. 6 (a) shows the loci of all the roots, where a pair of the seven roots are near -0.48+Oj, another pair which correspond to nutation are near O±j wo and the other three are at or near the origin. An enlarged locus of the nutaion pole is shown in Fig. 6 (b), from which the nutation stability is concluded for a>l. In Fig. 6 (c), the pair of poles near the origin determinds the speed of the precession control. In Fig. 7, one of the roots is at (1.0,0.0) as explained already, but it is seen that all other roots are in the unit circle if the sampling time is shorter than about 10.6 seconds. For Fi~. 6 and Fig.7, g=6x106 and Bx=O T, By=-1.S6x10- T, Bz=1.Sx10-s T are chosen and for Fig.7, a=10.0. Spacecraft parameters used are those for ASTRO-C.

INAe;

/'

a-50

I.'

"'a-O ftEAL

,1.

(a)

0.2

.'.<4 1.25

t

a-JO

\~a-o a-I

e. BI

• •.• 2

(b)

1.22

~a-50

-'.I£-Il

/' a-O

-2.1[-1l

.... a-50

Fig. 6.

l.t(-1S

1.It-U

(c)

-1.1[-11

Root loci with respect to a.

lHAG

The stability of the large angle maneuver system presented in the previous section is also assured from the above discussion on a pointing control system, because the large angle maneuver itself can be viewed as a pointing control with target angular velocity input. DESIGN CONSIDERATION Value of nutation momentum a proper

REAL

a. Large a value results in quick damping and slow precession of angular vector and vice versa. We have to choose a value by simulation.

Value of g. Large g value achieves quick control, while too large g saturates magnetic torquer rods causing the degradation of control performance. Adequate value of g depends on orbit and target attitude. It will be possible to form an adaptive system in which the g value is changed such that one of the three torquers is excited at its maximum producible magnetic moment. Simulation results for ASTRO-C case indicate a positive answer. Gyro drift. To calibrate gyro drift, sensors are necessary (e.g. star sensors). However, calibration sensors can't be used during large angle maneuvers, so low drift gyros need be selected for the present system. Terminal error of large angle maneuver. When the large angle maneuver is finished, there remains an attitude error due to the gyro drift and croSscoupling error described in the third secton. To achieve the final attitude, a pointing control is to be carried out after up-dating the gyro data by, say, a star tracker. This means that the final error of the large angle maneuver should be small enough for the star tracker to acquire the target star. Extensive simulations have demonstrated that this requirement is fulfilled for ASTRO-C case. SIMULATION Figure 8 shows an example of the attitude control

sampling time (sec)

Fig. 7.

Root loci of characteristic equation of Z-transformation with respect to the sampling time.

system of ASTRO-C which has been designed based upon the previous discussions. Simulation results for ASTRO-C are discussed below. The assumed parameters of ASTRO-C are shown in Table 1. Common conditions of the simulation are as follows: 1. To represent the target attitude with respect to an inertial frame, Euler angles shown in Fig. 1 are used. Here, X and Z axes are chosen in the direction of the ascending node and celestial north pole, respectively. 2. Uncanted dipole model is tentatively adopted for the earth magnetic field. 3. Fricton torque of the wheel is assumed as 1.0x10- 2 Nm. 4. Gyro drift is neglected(specification for the stability of the actual non-g drift rate of ASTRO-C gyro is 0.008 deg/hour). Supression of External Disturbance. To investigate the effect of the external disturbances against the pointing control system, the torque due to gravity gradient, residual magnetic moment and step constat torque are taken into account. Step

2919

Magnetic Torquing Scheme for Attitude Control

MAGNETOMETER's STAR TRACKER X

-XAXIS

-

"i'

~""1;··'····-I··'

STAR TRACKER Y

Y AXIS

'-'--1 s i'

9yo

r--r-r._-..l ...... _,

'"'T

i;~ B~: ' -__

910

9 xo

:

r---rt .... ...!.....

.J

"

L----L-J~·-·,··-·~-··-i-··! 9 10 .

9.1 9r1

:

1 .

f-----' 1 9 ! Ir===~

Wal

IIOIIEIoITUII WHEEL

Q. WHEEL SPEED BIAS

~

Fig. 8. TABLE 1

ASTRO-C block diagram. TABLE 2

System Parameter of ASTRO-C

orbit

0 500Km, i = 30

Maximum Error in 2 Days 0

e = 0.0

,

moment of inertia

2 Ix = 161.9 Kgm 2 Iy = 161.9 Kgm I z = 170.0 Kgm 2

attitude error allowance

ex ey ez

< < <

additional circuites for large angle maneuver

0

0.5 0 0.5 0.1 0 (for 2 days)

0.131 0 0.078 0 0.017 0.086 Nms residual magnetic moments in LB coordinate target attitude g C}

actuator parameters wheel magnetic torquer constants of wheel control block constants of large angle maneuver

TD Nms ho = 40 2 Iw = 0.106 Kgm ATm2 100 KwKp = 40.0 KwKr = 161.7 3 KwKf = 2x10Wo I £il

'y

Nm/rad Nms/rad Nms/rad

,

~

0.2

0.4

o-l.S

I/'

'x

(1", I", 1) oATm2

(0 ,0 ,90 ) 6 x10 6 10 (0 ,5X10-s, 0) Nm

.-5 /

ey

( (do.)

V

ex 0.2

0 . 4 (d ••)

= 1. 3x10- 4 rad/s 0 OFF threshold 0.6 0 ON threshold 0.5

'y constant torque was adopted to tentatively aSsess the effect of airdrag and solar radiation pressure torques and it's amplitude value was chosen as the sum of maximum airdrag and maximum solar radiation torques. Maximum values of le xl, le y l , lezl and 16hl during 48 hour period are shown in Table 2. The Dependency on a of Pointing Control Characteristics. A step response and a nutation damping feature of the pointing control system for several a 's are shown in Fig. 9. As is expected from the disccusion in the fourth section, the larger the a ,the faster the nutation damping. Large Angle Maneuver Results. An example of the large angle maneuver l oci are shown in Fig. 10 (a) to (c). Figure 10 (a) shows the locus of the x-axis on the celestial sphere. Units are in degrees for both absissa and ordinate. The absissa represents the ideal locus which could be obtained when e'i =wi (i=x,y,z) holds through the maneuver. (b) and (c) are y and z-axis loci respectively.

0-10

'y

/

/'

(:--' ~

0 .2

'x

0 . 4 (do.)

initial condition target attitude g

Fig. 9,

Q-20

( '-'

ex 0.2

0.4 (de.)

0

ex • ey a 0.5 ex • 1. 0x10- 4 rad/s e'y • eZ • s'z = 6h = 0 (0 0 ,0 0 ,0 0 ) 6x10 6

Pointing control performance.

K. Ni nomiya et a l.

2920

Cross - Coupling Er r ors. An example of c r oss coupl i ng error discussed in the third sec t ion du r ing the large angle maneuver is shown i n Fig . 11. The hor izontal axis is time . The parameters are the same as those for Fig . 10 case . Cr oss co upling e rror s are small and lie well wi thin the acqui si tion range of ASTRO - C star tracke r s. The Dependency on a of Large Angle Mane uve r Characte r is t ics . The req uired t ime for t he la r ge angle mane uver is summarized in Table 3 fo r several a ' s . The assumed parameters are the same wi t h those fo r Fig . 10 case . As is expected, i t i s seen that too la r ge value of a results i n longe r

maneuver scheme which is based on the same point ing control law is also proposed . The significant feature of the scheme is the simplicity of the on- board hardware and software implementation . Extensive simulations are performed for the case of ASTRO - C satellite . The results show that the stab ility of t he system is assured while the requi r ed poin t ing accuracy can be achieved by a relatively simple on- board equipment. The simulation resul t s also indicate that the proposed large angle maneuver scheme works satisfactorily and is worthy of the application to the wheel-stabilized satellite .

maneuve r time .

Torquer nonlinearity and residual magnet i c moment . Since magne t ic torquers inevitably have nonlinearity and residual magnetic momen t, the influence of these factors has been checked by s i mula t ion using Pr eisach model for magne ti zation . For pr ac t ical torquer rod characteristics, no no t iciab l e degradat i on of cont r ol pe r fo r mance waS obse r ved. CONCLUSION A linear control law is proposed for the pointing control of a wheel - stablized astronomy satell i te by magne t ic t or quing. The stability analysis is carried out, which has guaranteed that the nu t ation angle as well as the error of the angular momentum vector direction are eliminated. A large angle

10

(a)

2u (dea)

y-ax is 20

10 ~

'I~:-""~A '" initial attitude target attitude g

a

Fig . 10.

(b)

(d e g)

,c::...,,20 (de,)

( 0

0 0

,

0

0 0

,

0

0

(c)

)

(15 ,15 , 15 0 6xl06 10

Loci of axes on the celestial sphere for the la r ge angle maneuver.

.

...!l,

10000

0.

o

u

.

~ o -0. 1 ~

u

Fig. 11.

Cross-coupling er ror in large angle maneuver.

TABLE 3

Required Time for the Large Angle Maneuver a 1.5 5.0 10.0 20.0 50 . 0

Time (minute) 184 1117 192 210 241

( .. c)

REFERENCES Brown , S . C. (1965). An Analytical Comparison of some Electromagnetic Systems for Removing Momentum Stored by a Satellite Attitude Con t rol System. NASA TN-D-2963 , Ma r ch. Dougher t y, H. J., Scott, E. D., and Rodden , J. J. (1968) . Analysis and design of WHECON - an a tt i tude cont r ol concept . AlAA Pape r No. 68 - 461, San Fr ancisco, calif. Lindorfer, W. , and Muhlfelder, L. (1966) . Attitude and Sp i n Control for TIROS Wheel . AlAA/JACC Guidance and Control Conerence Papers, Seattle, Wash., pp. 448 - 461 . Mobley, F . F. , and Tossman, B. E. (1971). Magnetic Attitude Control System for HEAO. NASA- CR- I03025. Ninomiya, K. (1981) . On the development of Attitude Stabilization and Control System of a Japanese Scientific Satellite . IFAC 8th t r iennial world congress, Kyoto, Japan, Aug 1981. Paiken, M., and Fleisig, R. (1963). Momentum Control of OAO Spacecraft Utilizing the Earth ' s Magnetic Field . XIV International Astronautical Congress, Paris, France,

Sept- Oc t. Powell , B. K. , Lang, G. E. , Lieberman, S . I, and Rybak , S. C. (1971) . Synthesis of Double Gimbal Control Moment Gyro Systems for Spacecraft Attitude Control. AlAA Paper No . 71 - 937 , Hofstra University. Renard, M. L. (1967) . Command Laws for a Magnetic Attitude Control of Spin-Stabilized Earth Satellite. Journl of Spacecraft and Rockets, Vol. 4 , Dec. 1967, pp. 1631 - 1637 . Schmidt, G. E. Jr., and Muhlfelder, L. (1981). The Application of Magnetic Torquing to Spacecraft At ti t ude Con t rol . AAS 81-002, Annual Rocky Mountain Guidance and Control Conference , Keystone, Colorado, 31 Jan - 4 Feb. Stickler, A. C. , and Alfriend, K. T. (1976). Elementary Magnetic Attitude Control System . Journal of Spacecraft and Rockets, vol. 13, No . 5, May 1976, pp. 282 287 . Terasaki, R. (1967). Dual Reaction Wheel Control of Spacecraft Pointing . presented at Symposium on Attitude Stabilization and Control of 'Dual- Spin Spacecraft. Tsuchiya, K. (1982). New Control Schemes for a Magnetic Attitude Control Systems . Joint IFAC/ESA Symposium on AUTOMATIC CONTROL IN SPACE , Noordwijkerhout, The Netherlands, 5- 9, July. Wertz, J. R. (Ed . ) (1978) . Spacecraft Attitude Determination and Control. D.Reidel Publishing Co . Whisnant , J. M. , Anand, D. K. , Pisacane, V. L., and Sturmanis, M. (1971) . Modeling of Magnetic Hysteresis. Scientific and Technical Reports.