Kalman Estimation and Control of Dual-Spin Satellites

1. lntroduction

Kalman Estimation and

Control of Dual-Spin Satellites

P.H.W. CHENGTelesat-Canada

N.D. GEORGANASUniversity of Ottawa

J.A.D. HOLBROOKTreasury Board Secretariat

Abstract

lt is necessary to maintain the spin axis of dual-spin geostationary

communications satellites with nontracking ground antennas to

within 0.1 degree of the orbit normal by periodic attitude corrections.

Normally, the data for attitude estimation are determined from the

analog sensor waveforms telemetered to the ground station. This

information is supplied to the attitude determination program, which

processes the data and outputs the right ascension and declination

of the spin axis.

An application of the extended Kalman/filter in estimat-

ing the attitude of dual-spin geostationary satellites is presented.

The precession of the angular momentum vector by the solar radia-

tion torque is considered to be the only natural attitude perturba-

tion. The orbital dynamics are considered to be known and are de-

coupled from the attitude dynamics. A periodic attitude control

policy is then derived.

The dual-spin spacecraft consists of two bodies spinningabout a common axis. The spin-stabilized geostationarycommunications satellite was first proposed by the HughesAircraft Company in the fall of 1959. The Anik satellite isa particular case of a dual-spin spacecraft. lt is composed oftwo parts: a symmetrical rotor (spinning 100 rev/min),and a platform which includes the dish antenna and whichfor all practical purposes is nonrotating. lt has a favorablerotor moment of inertia [1] 1S ><\(I112), where I is the spinmoment of inertia of the spacecraft in the z direction andI1, I2 are the moments of inertia in the x, y directions.

The control of Anik 1, II, and III (F-1, F-2, and F-3)satellites is achieved through a cyclic sequence of orbit andattitude corrections using four 1 -lb hydrazine thrusters [1].The data required for attitude corrections are obtainedfrom sun and earth sensor measurements.

This paper presents a periodic attitude control model forthe dual-spin stabilized geostationary satellites. An extendedKalman filter is used for estimating the spacecraft attitudeby processing the sensor data. A model for the precessionof the satellite spin vector by considering the solar radiationtorque as the only natural disturbing torque is used. Usingthe data of solar radiation force from [2] and by choosingthe appropriate initial conditions, a periodic attitude controlmodel is developed. In this analysis it is assumed that orbitaldata are provided.

The Kalman filter in our model could be used for provid-ing the attitude information of the spacecraft and also inestimating the new starting positions in each periodic con-trol cycle. The results obtained in this paper would be use-

ful in the design of an on-board attitude control system fordual-spin geostationary satellites using orbital informationtransmitted to the spacecraft from the ground stations (semi-active control system).

11. Coordinate Systems and State Equations

A. Coordinate Systems

Manuscript received April 27, 1976. Copyright © 1977 by The In-stitute of Electrical and Electronics Engineers, Inc.

This work was supported in part by the National Research Councilof Canada under Operating Grant A-8450.

Authors' present addresses: P.H.W. Cheng, Telesat-Canada, Ottawa,Ont., Canada K1L 8B9; J.A.D. Holbrook, Treasury Board Secretariat,Government of Canada, Ottawa, Ont., Canada K1A 0R5; N.D. Georg-anas, Department of Electrical Engineering, University of Ottawa,Ottawa, Ont., Canada K1N 6N5.

In order to deseribe the motion of a space vehicle, a co-

ordinate system is required. This coordinate system is chosento fit best the type of motion desired. Three reference co-

ordinate frames are used to define the geometrical relation-ships involved in the system model. These frames shall bereferred to as basic inertial (1), orbital (0), and body fixed(B). Transformations between these frames are shown belowand their geometrical relations are depicted in Figs. 1 and 2.

The basic inertial is derived from the more generalequatorial coordinate system (ECS). The origin coincideswith the center of the earth; the x-y plane coincides with theequatorial plane, with the x axis pointing to the direction ofthe first point of Aries and the z axis pointing to the Northpole. By neglecting the precession of the first point ofAries, this frame is considered to be inertial.

The orbital frame differs from the basic inertial frameby two Euler angles, the right ascension Q2 and the inclina-

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-13, NO. 3 MAY 1977236

NORTH POLE

Fig. 1. Basic inertial and orbital coordinate systems. Fig. 2. Orbital and body-fixed coordinate systems.

tion i, which define the orbital plane of the satellite. Con-sequently, only two rotations are sufficient to bring the twoframes into coincidence. The transformation matrix, de-noted as TO1, can be derived as in [4]:

cosQ2 sinQ OTO 1

= -cosisinQ2 cosicosQ2 sini . (1)sin i sin 2 -sin i cos Q cos i

The differential equations relating the state rates (4,, 0, 4)to the body rates (w, wy>, wc), as measured about the x,y, z body axes, are [7]:

_Q_ O sin 4> cos 1_= (cos 0)- 0 cos 0 -sin 0 xY .

4>L 1 sin4>sin0 cos4>sinO JLzJ(4)

The body frame differs from the orbital frame by threeEuler angles 4,, 0, 0. Under nominal attitude conditions,the orientation of the body-fixed frame is such that XB iS inthe orbital plane, YB is along the local zenith, and ZB com-pletes the right-hand triad. There are six possible ways oftransforming the orbital frame to the body frame. The se-quence of rotations we have selected is developed in [5]and the matrix TBO is given by

cos 0 cos 4

TBO =Lsin (ksin 0 cos ,-cos 0 sin 4,cos= sin 0 cos 4 + sino0 sin 4,

cos 0 sin 4,sinm sin 0 sin 4, + cos t cos 4cos 4> sin 0 sin 4, - sin 4 cos 4,

-sin 0sin cos 0cos cos 0j

The general state equation can thus be written as

x = G(x) W (5)

where (W)T = (Cx woy, CoZ) is the vector of the body rates.

111. Measurement Equations

To formulate the Kalman filter, it is essential to know themeasurement equations. Two different measuring devicesare used, namely, sun sensors and earth sensors. Fromthem, the declination, the right ascension of the spin axis,and the angle between the spin axis and the sun are obtained.Thus, three measurement equations are required. We arealways assuming that orbital data are provided continuously.

(2) A. Earth-senor measurement equations

B. State Equations

The attitude of a spacecraft is defmed as the orientationof the body axes (orbital frame). Thus the state x of themodel is chosen as the three component vector

(x)T = (4,, 0,) (3)

where 4,, 0, and 0 are Euler angles defining the body attitudeas shown in Fig. 2.

The earth-sensor measurement equations can be derived bytransforming the unit spin vector from the orbital to thebody frame. Consider the unit spin vectorA in the orbitalframe as

sin 8 cos eAo = sin sin e

Lcos

(6)

where 8 and e are the declination and right ascension of the

CHENG ET AL.: KALMAN ESTIMATION AND CONTROL OF DUAL-SPIN SATELLITES 237

zo

UNIT SPIN VECOR ( A0) UNIT SPIN VECTOR (AO)

Yo

xo

Fig. 3. Unit spin vector in orbital frame.

spin vector, respectively (Fig. 3). The unit spin vector in thebody frame is AB (Fig. 4), where

1AB = L° j (7)

Since the transformation matrix TBO is known, we have

Fig. 4. Unit spin vector in body frame.

sin Q cos P

SI = sin Q sin Pcos Q J

(10)

where Q is the equation describing the annual movement ofthe sun with time zero as the vernal equinox,

Q = 25.00 sin (2irDwst/365.0)

AB TBO AO. P = wst, ws = 1 degree/day

After carrying out the matrix multiplication, simplification,and equalization, the following equations are obtained:

sin 6 cos e = (-sin2 Vi sin 0 cos 4 + sin IPsinI cos /

-sin 0 cos ik)/ cos k

sin 6 sin e = sin 0 cos «sin &-sin p cos k

cos 6 = cos 0 cos b.

For simplicity of the linearization procedure, the twoearth-sensor measurement equations are related as

Yi 4 sin 6 sin e = sin 0 cosq sin -sin4 cos it

Y2 -A cos 6 = cos 0 cos q.

B. Sun-Senror Measurement Equations

The sun-senor measurement equation can be similarlyderived by considering the unit sun vector in the inertialcoordinate system and transforming it to the body frame.The unit sun vector in the inertial frame is SI (Fig. 5):

(8)

t = 0, 1, 2, *- , 365. (11)

The sun vector in the body frame can be represented by thevector SB (Fig. 6):

sin p cos a

S = sin p sin a .

_cosp(12)

The angle p is the angle between the sun vector and thespin axis and is provided by the sun-sensor measurements.Equating the two vectors, we obtain:

SB=TBO TOI SI

Since we are considering a known orbit, the values of Q and9) P can be precalculated. Thus TO,-SI is assumed to be

known. Let us define

VlTO I .SI- V2

L-V3_where vl, v2, and V3 are constants. After carrying out thematrix multiplication, we select the sun-senor measurementequation as

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS MAY 1977

Z 8

238

UNIT SUN VECTOR ( S,)

QIYl

UNIT SUN VECTOR (SB )

/~~~~~P /pX

xI

Fig. 5. Unit sun vector in basic inertial frame.

Y3 cos p = vl (cos 0 sin 0 cos / + sin 0 sin P)

+ v2(cos sin 0 sin i - sin 4 sin P)

+ V3(cos 4 cos 0).

.XB

Fig. 6. Unit sun vector in body frame.

U= F * öx

where

(13) (6X)T = (X -X*)T = (6X1, 6X2, SX3)

The general measurement equation for the system canthus be written as

y = h(x) (14)

where y= [Y1, Y2 y3IT.

IV. Kalman Estimation

and

F = (a/öx) [G(x)W] IW=W *, X =X*

[O O O= O 0-w* .

LO z* O

The state and measurement equations derived earlierappear in terms of quantities that have a nonlinear algebraicrelation with the state variables. The nonlinearity of thedynamical and output equations must be removed for theapplication of Kalman estimation techniques.

As described earlier, the dynamical system evolves ac-

cording to the vector differential equation x = G(x) W,where x is the three-dimensional state vector. The nominalvalues of our model are selected by considering the dual-spin geostationary satellite in an ideal attitude position.This is the situation when the Euler angles 0 and 0 are bothequal to zero, the angle is moving at the orbital rate w0,

and the spacecraft is spinning about its body z axis at thenominal rate, WZ-

Thus the nominal values ofx are defined as x* where,(X*)T = (> 0*, 0 ) = (ot, 0, 0) and (W*)T = (W,*, WYs4*) = (0, 0, 4*).

Equation (15) represents the state linear perturbation equa-tion. This equation describes the motion of the actual sys-tem relative to the nominal.

The measurement equations must also be linearized. Themeasurement equations are related to the state by y = h(x).Using the same technique as that of the state linearization,we obtained the linearized measurement equations as

by = H8x (17)

where,

(8y)'= @ _ Y*)T = (6Y , bY2, Y3)

y* = h(x*)

H = (ahlax)lx=x*A. Linearization of State and Measurement Equations

To linearize the state equations, (5) is expanded in a

Taylor series about the nominal values. We neglect allterms except those of first order. Thus we obtain

= sin wo0t -cos WOt0 0 0

° V1 cosOOt+v2 sincot v1 sinco t-v2 cosWOt

The measurement noise {v(t), t > to} is considered to be an

CHENG ET AL.: KALMAN ESTIMATION AND CONTROL OF DUAL-SPIN SATELLITES

(15)

(16)

- ~ ~ ~ ~ ~ ~7- b

Z

IC-

,

239

additive zero mean Gaussian white noise with covariancematrix

E{v(t)vT (,)} = R(t) 6(t- r)

where R(t) is a known positive defmite matrix and 8(t) isthe impulse function. Thus the linearized measurementequation becomes

by=H *x +v.

rotational inertial of the satellite, and w is the angular ve-locity of the satellite.A torque is produced by the radiation force of the sun

acting through an effective center of mass displaced fromthe center of mass of the satellite. The torque direction isperpendicular to the angular momentum vector resulting inprecession of this vector.

Prediction of the natural precession of the satellite canbe achieved by solving the following equations [2]:

(18)

V. Kalman Filter Equations

From the linearized state and measurement equations(15) and (18), respectively, we obtained the flter equations[8]:

ö'..(t) = F(t) Sx(t) + K(t) [by(t) -(t)bx(t)

t > to, with 6xA(to) = O

K(t) =P(t)HT(t)R'-(t)

P(t) = F(t) P(t) + P(t) FT(t)-P(t) HT(t) R-1 (t) H(t) P(t)

(19)where öx(t) is the optimal estimate of bx(t), K(t) is the Kal-man gain matrix, and P(t) is the error covariance matrixwith initial condition given by

P(to) = E{bx(to) bxT(to)}

which is assumed to be known.Thus from (15), (16), (18), (19) and knowing the values

of w0 and w* (as an example, in the Anik satellites co0 = 1degree/day, w* = 100 rev/min), the attitude of the space-craft can be deterrnined at each instant of time. With thisattitude information, it is then possible to decide the atti-tude control cycle.

Vl. Periodic Attitude Control Policy

lt is necessary to maintain the declination of spin axisfrom the orbit normal to within 0.1 degree for communica-tions with nontracking ground antennas. The periodic atti-tude control model derived below is such that the spin axisis kept within its 0.1 -degree limit cone.

A. Natural Precession of Spacecraft

The precession of the spacecraft angular momentum vec-tor H by the solar radiation torque is considered to be theonly natural attitude perturbation of the dual-spin geosta-tionary satellite:

H=I* w (20)

where H is the angular momentum of the satellite, I is the

l = - (TI) COS0 t

2 = - (rIH) sin w.t

(21)

(22)

where x1, x2 are orthogonal components of the spin vectorprojected onto a plane passing from the center of the satel-lite and parallel to the inertial x-y plane, w, is the angularrate of the projeetion of the sun vector on the same plane,H is the angular momentum of the satellite, and r is the solarradiation torque magnitude.

To solve (21) and (22), it is necessary to know the solarradiation torque magnitude. The data for solar radiationforce and the natural precession rate of a typical dual-spingeostationary satellite (Anik 1) with an augular momentumof 900 ft4lb-s are obtained from [2].

preeession rate = (F X LA)/H = r/H (23)

where F is the solar radiation force and LA is the lever ormoment arm length. and knowing the value ofH, we can findthe solar radiation torque magnitude. The natural precessionrate curve and the solar radiation curve are plotted in Figs.7 and 8, respectively, as functions of time. Time zero isconsidered to be the tirne of earth-sun-Aries coincidence, i.e.,March 21. At this time the sun is in the equatorial planeand moreover in the inertial x axis.

B. Attitude Control Policy

As mentioned earlier, the attitude constraint of thismodel is that of limiting the spin-axis declination to within0.1 degree. That is, the projection of the unit-spin vector

should not be permitted to move outside the limit circle ofradius equal to 1.74533 X 10-3 (equal to sin 0.1 degree)units in the x1 -x2 plane. No control is necessary as long as

the spin-vector projection is within the limit circle.Attitude corrections using single axial jet in pulsed mode

are performed periodically within a cycle. There are two

cases [2]: 1) attitude corrections that precede the inclina-tion corrections are such that the spin axis would be normalto the equatorial plane at the time of the inclination maneuver,and 2) if an inclination maneuver does not follow the atti-tude correction, then the attitude is targeted to coincidewith the orbit normal midway between attitude corrections.Thus if

a = (7r/2 -8) exp j (a + fr/2) (24)

is a representation of the spin vector with 8 and a as the


50 100 150 200NUMBER OF DAYS FROM 21st MARCH

250 300 350

Fig. 7. Natural spin-axis precession rate.

Fig. 8. Variation of Solar radiation torque.

15 _

0>>[

06,.: 10 5

I°5

300 350

declination and right ascension, respectively, then corre-

sponding to cases 1) and 2) we have the targeting strategies[2]:

1) a, + at = 0

2) a, + A a. = itwhere a, is the target spin-axis representation in the attitudemaneuver, Aat is the precession (due principally to solartorque) over half the attitude correction cycle and it is the

inclination vector [2] at the midpoint of the attitude correc-

tion cycle. The attitude control policy selected in this modelcorresponds to that of case 1). Since the orbit normal pre-cesses approximately symmetrically about the pole duringan inclination maneuver, this strategy minimizes the cross

coupling of the inclination velocity increment into the orbitplane. The drift rate and eccentricity are therefore disturbedas little as possible [2].

The natural precession of the satellite is predicted by

CHENG ET AL.: KALMAN ESTIMATION AND CONTROL OF DUAL-SPIN SATELLITES

0It ,112

241

XI (t ):--- cOs ( Ws t )

2(t)- H SIN(WX2(t)-Z H( SIN( wst )

Fig. 9. Natural precession of satellite in a year.

solving (21) and (22) with an appropriate set of initial con-ditions. The spacecraft spin axis is allowed to precess ac-cording to its natural path until it reaches the boundary ofthe limit circle. Control is then applied by activating one ofthe axial jets in pulse mode during a particular segment ofthe satellite spin and thus producing the required torque tobring the spacecraft spin to a new location within the limitcircle. The satellite is permitted to move naturally againand the cycle is repeated. The control penod is dependenton the initial position of the satellite spin axis.

MEASUREMENTS ARESPACED TEN DAYS APARTRADIUS OF LIMIT CIRCLE : 174533x 10-3

VII. An Example

Consider that time zero is the time when the sun is in theequatorial plane and in particular on the inertial x axis(March 21). Equations (21) and (22) are solved with a digitalComputer with initial conditions selected as

x,(0) =1.6760X 0-3

x2(O)= 1.4545 X 10-4.

(25)

(26)


t

242

x tt - [SIN (W5t ) SIN ( 210 )-COS l Wstl COS (z21° )

X2(t)c 8- COS(Wst) SIN(21° i )-SIN( Wst) COS{21°- i)]

t: O,I,2,*----, 21 (FOREACH CYCLE)

INITIAL CONDITlONS FOR EACH CYCLE ARE:

xl (0) =x (0)COS(21°)-X-(0) SIN 210)2

141 i~x2 (0) ZX (0)SIN ( 21)+x (O)cos Z O)

CONTROL PERIOD * 25 DAYSRADIUS OF LIMIT CIRCLE 74533 x 10 3

Fig. 10. Periodic attitude control of dual-spin geostationary satellite.

The initial conditions were selected such that the sun beinginitially at the equatorial plane starts to "push" the spinaxis from its initial position causing its projection to movepast the center of the x1, x2 coordinate system on thetenth day and beyond the limit circle after 21 days. If nocontrol is applied after 21 days, the satellite projections willcontinue to move in a spiral fashion as shown in Fig. 9.

Since the spin-axis projection moves outside the constrain-ing circle after 21 days, it is possible to formulate an attitudecontrol model with a period of 21 days. That is, the satellitespin axis is moved to a new starting position within the limitcircle by applying controls. To determine the new starting

conSditions and the control magnitude, (21) and (22) haveto be modified.

The sun has travelled 21 degrees around the earth in thecourse of 21 days and the solar radiation torque is variedaccordingly (refer to Fig. 8). Thus the modified equationsare

il (t) = [T(t)/H] [sin (cos t) sin (210 -i)

(27)- cos (W.t)*cos (21°*i)]

x2(t) = [r(t)IH] [-cos (wst)sin (210°i)


(28) VIlI. Conclusions

where tO0, 1, 2, -, 21.The new starting positions after each 21 -day cycle are

found from the following recursive formulas:

x (0) = x' (0) * sin (210) +x' (0) - cos (21°). (29)

xi+1(0) = x'(0) * sin (21 0) + x' (0) cos (210) (30)

The periodic attitude control model is found by inte-grating (27) and (28) with initial conditions given by (29)and (30). The first starting conditions x4(0) and x'(0) areselected to be equal to (25) and (26), respectively. With iequal to 0 the equations are solved with t running from 0to 21 days. The solutions provide the natural precessionof the satellite spin axis. After each 21-day cycle, the satel-lite is brought to a new starting location as defined by (29)and (30) with i incremented by 1, by applying a correctiveforce.

Sample trajectories are plotted in Fig. 10. lt is observedthat the distance traveled is not uniform for each of the21 -day cycles. This is due to the fact that the solarradiation torque magnitude is not constant. Thus, theapplied corrective force will also vary. However, the 21 -daycontrol cycle is selected for the practical reason that thecontrollers will perform the attitude maneuver on the sameday after every three weeks. In addition, this choice mightalso help the personnel manager in scheduling the shift andvacation periods for the controllers!

A mathematical model for the estimation of the attitudeof the dual-spin geostationary satellite using the extendedKalman filter has been formulated. In addition, a periodicattitude control model has been established.

The state for the Kalman filter has been selected to be avector of the three Euler angles P, 0, and 0 defining theattitude of the spacecraft. This choice of state variables arosefrom the definition of the attitude of a spacecraft. The mea-surement equations were derived by transforming appro-priate unit vectors from one reference frame to another.Due to the inherent nonlinear characteristics of the state andmeasurement equations, linearization by means of a Taylorseries expansion had to be introduced. The nominal valueswere chosen by considering an ideal attitude situation.

In deriving the periodic attitude control model, the assump-tion that the spacecraft had achieved an almost perfect orbitand also that the orbital parameters were known at eachinstant of time was made. The solar radiation torque magni-tude and the moment arm length were found by combiningthe precession rate data and the solar radiation force datagiven in [2].

The targeting strategy selected was that the attitude cor-rections preceded the inclination maneuver. The inclinationcorrections were done in the middle of the attitude correc-tions cycle when the spin axis is normal to the equatorialplane. This strategy minimized the cross coupling of theinclination velocity increment into the orbit plane and alsodid not perturb the drift rate and eccentricity.

The results obtained in this paper could be useful forthe design of on-board attitude attitude control systems forthe dual-spin geostationary satellites, utilizing orbital infor-mation transmitted to the spacecraft from the groundstations.


sin (wst)-cos (21'- i)]

244

References

[1] Lenkurt Electric, "Developments in communications," Len-kurt Electric Co., Inc., Calif., 1972.

[2] W.H. Wright and B.M. Anzel, "Telesat station keeping meth-ods and performance," Proc. AIAA 5th CommunicationSatellite Systems Conf., Los Angeles, Apr. 1974.

[31 Lloyd Harrison et al., -Canadian satellite, A general descrip-tion," presented at the Int. Communications Conf.. Montreal,Canada, June 1971.

[4] R.L. White, M.B. Adams, E.G. Geisler, and F.D. Grant, "Atti-tude and orbit estimation using stars and landmarks," IEEETrans. Aerosp. Electron. Syst., vol. AES-11, pp. 195-203,Mar. 1975.

[51 P.C. Hughes, "Attitude dynamics of Canadian satellites,"Can. Aeronaut. Sp. J., vol. 20, May 1974.

[6] C.T. Leondes, Ed., Advances in Control System Theory andApplications, vol. 3. New York: Academic Press, 1966.

[7] D. McRuer, 1. Ashkenas, and D. Graham, Aircraft Dynamics andAutomatic Control. Princeton, N.J.: Princeton Univ. Press, 1973.

[8] J.S. Meditch, Stochastic Optimal Linear Estimation and Con-trol. New York: McGraw-Hill, 1969.

[91 W.R. Perkins and J.B. Cruz, Jr., Engineering ofDynamic Sys-tems. New York: Wiley, 1969.

[10] W.T. Thomson,Introduction to SpaceDynamics. New York:Wiley, 1961.

Ppter H.W. Cheng was born in Sabah, Malaysia, in 1950. He received the B.ASc. andM.A.Sc. degrees in electrical engineering from the University of Ottawa, Canada, in1974 and 1975, respectively.

Since 1975 he has been with the Satellite Control Systems of Telesat-Canada,Ottawa.

Nicholas D. Georganas was born in Athens, Greece, in 1943. He received the Dipl. Ing.degree in electrical and mechanical engineering from the National Technical Universityof Athens in 196e, and the Ph.D. degree (summa cum laude) from the University ofOttawa, Canada, in 1970.

Since 1970 he has been with the electrical engineering department, University ofOttawa, where he is currently an Associate Professor and Coordinator of the GraduateProgram. His current research interests are in the areas of large scale systems andcomputer communications.

J. Adam D. Holbrook was born in Birmingham, England, in 1946. He received the B.Sc.degree (hons.) from Dalhousie University in 1966, the M.Sc. degree from the Universityof Western Ontario in 1968, both in physics, and the B.A.Sc. degree in electrical en-gineering from the University of Ottawa in 1974.

He was with Telesat-Canada at its founding in 1969 and until 1976. Presently he iswith the Treasury Board Secretariat of the Government of Canada, Ottawa.

Mr. Holbrook is a member of the Association of Professional Engineers of Ontario.


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Kalman Estimation and Control of Dual-Spin Satellites