Attitude Control of Mariner Jupiter-Saturn Spacecraft

K. Bahrami ATTITUDE CONTROL OF MARINER JUPITER-SATURN SPACECRAFT Khosrow A. Bahrami Member of Technical Staff J et Propulsion Laboratory Pasadena, California SUMMARY The Mariner Jupiter-Saturn is a sophisticated Mariner-type spacecraft with stringent attitude control requirements. This paper discusses the attitude control of the mission module, which contains the scientific payload. A software program residing in the onboard computer performs the necessary steps for attitude control. The algorithm within this implementation is based on a discrete stochastic single-step predictor (Kalman filtering). It acts upon the inertial or celestial measurements on ·the space craft to perform the estimation process. Torques to control the spacecraft are obtained by the actuation of small hydrazine thrusters whenever necessary, using estimated spacecraft position and rate values. The expected performance of the attitude control of Mariner Jupite rSaturn spacecraft was simulated on a digital computer, and the results are presented herein.

INTRODUCTION The Mariner Jupiter-Saturn is an unmanned spacecraft intended for scientific and planetary exploration. Scientific and planetary data will be gathered by the spacecraft and transmitted to Earth via a digital communication link. MJS scientific experiments include cosmic ray, imaging science, infrared spectroscopy, charged particles, astronomy, and others. One aim of the mission is to send to Earth high-r e solution photographs of Jupiter and Saturn. Thus it is required that the onboard cameras be pointed toward the appropriate location on the planet and the spacecraft antennas be pointed toward Earth. Hence precise attitude control of MJS s pacecraft is imperative. The MJS spacecraft consists of a propulsion module and a mission module. Early in the mission the propulsion module unit is used to put the spacecraft in the proper transfer orbit to Jupiter; then the module separates from the mission module. The mission· module contains the scientific payload. It consists of a stru c ture to which the magnetometer booms, high- and low-gain

41

K. Bahrami

antennas, radio isotope thermoelectric generators, TV cameras, and other members are attached (se e figure 1). The Mariner Jupiter-Saturn spacecraft will be the first US planetary vehicle to employ an onboard digital computer for the purpose of performing attitude control functions. The addition of this onboard digital computer has provided substantially greater flexibility in terms of the selection of a control algorithm based on estimation and prediction theory. For missipns of reasonable cost, the requirement for very high booster velocity increments to inject a spacecraft into planetary transf e r orbits places very severe weight constraints on the spacecraft and payload. Planetary distances create ground communication times on the order of hours for outer plan e t mis sions. The ev olution of controls for planetary missions must adhere to a minimum weight and autonomous control requirement. In contrast to Earth orbite rs, all computational ability must be onboard and must operate at minimum power and weight. Allavail a bl e computers meeting the MJS'77 control r e quirem e nts violated the weight and power constraints. The design of the MJS'77 onboard computer and control algorithms dictated the simplest computer design that could execute the best suboptimal control algorithms to meet the most string ent operational rr,ode of this unmaned sci e ntifi c flight. This was the special challenge of MJS'77, and the results of that design are presented for the control filter and estimator.

ATTITUDE CONTROL OF MISSION MODULE Attitude control of MJS spacecraft is achieved by an onboard digital computer (Hybrid Programmable Attitude Control Electronics, HYPACE), a set of sun sensors and star tracker, a s et of r ea ction control hydrazine thrusters, and an in e rtial s e nsor unit. T h e inertial s ensor unit consists of three identical two-degrees -of -freedom, tuned dry gyros. Attitude control of the mission module is based on either the inerti a l data from gyros (ine rtial) or the data from sun sensors and star tracker (c e l est ial). For an exce lle nt . d e scription of HYPACE, s ee 11 I. Inertial Cruise Details of the mission module controller during the inertial operation are discussed as follows : let k = 0, 1, 2, •.• be th e discr ete points in time denoting the start of each HYPACE frame time . At time k, the beginning of a new frame, th e spacecraft position in all three axes is obtained simultaneously . This is done by forming a running sum of the increme ntal gyro position outputs. The controller consists. of thre e similar single-axis controllers, one ea ch for 'pitch, yaw and roll axes. Th e HYPACE executes th e algorithms for these controllers sequentially within a 60-msec sampling time (s ee figur e 2). Owing to

42

K. Bahratn i PLASMA DETECTOR

HIGH GAIN ANTENNA 3.7 m DIAMETER LOW GAIN ANTENNA

SUBREFLECTOR

FEEDHORN

CRUISE SUN SENSORS

~ B~ .

STAR TRACKERS

MAGNETOMETER

RAD IOISOTOPIC THERMOE lECTRIC GENERATORS ASTRONOMY

MONOPOLE (2)

HIGH · GA IN ANTENNA ~

('

------------- / :; ---/'

RADIOISOTOPE

\" \ \

GENERATOR

Figure 1: Two views of the tnission tnodule , showing radioiso tope thertno electric generat ors, radio antenna s, scan platform , and other tnetnbe rs 43

K.Bahrami

x

- y

I

I I I IX

(k + 1;\) '--_ _ _ _ _--'

C _ __ ___

---..J

-..2!.RO~<>-1.......

iJ ~ llelay MOT = Minimum On Time DRLRU = Dry I nertial Reference Unit

L_

Fi g ur e 2 ; B lock diag r a m o f mission m odule in e r t i a l cont rolle r

44

K. Bahrarni the similarity between the three axes, only the pitch axis controller will be discussed in detail. The spacecraft position measurement data at time k is used to compute the best single-step predicts of spacecraft position, rate, and acceleration in all three axes at time (k + 1). The actual time required for real-time estimation and prediction done in HYPACE will be well within the 60-msec frame time. The actual estimation process is based on the following assumptions and procedure. For the purposes of estimation a reasonable model for the dynamics of the spacecraft is needed. Because of HYPACE speed and memory size limitation, a simple model of spacecraft dynamics (a 3-axes rigid body) is selected. It is assumed that the three spacecraft axes are decoupled. dynamics of the spacecraft are given as follows:

x (k -p M

+

P

1)

(k+l,k)x

-p

(k)+ r

-p

(k+l,k)[T

(k) = H x (k) + '1 (k) - --p . p

~=(l

0

p

The

(k)+Wp(k)]

(1 )

(2)

0)

where ~p is the state of the system, Mp is the spacecraft pitch position measurement, is the state transition matrix, Tp is the control torque, r is the control translhon m 2 tnx. H- IS the observatIon vector -p and Wp and '1p are the measurement noise and torque disturbance processes, respectively. Denoting the pitch spacecraft position, rate, and acceleration by ap ' w , and" , respectively, we have ~T = (a , w p ' "p)' Since the spacet'raft dyrR:mics are considered time-invatlant, tlien

(k + 1, k) =

1 6T i6T2) 6T (o o

0

45

K. Bahrami

l'p

(k

+

1, k)

r

( 3)

-p

where ll.T is th e sample time and Jp is the spacecraft pitch InOInent of inertia. Th e sequential state predictor i s outlined in the appendix. The results fro In the appendix (A-IS and A-16) are repeat. here for conv enience.

X (k/k) -p X

-p

(k

+

X (k/k -p

Ilk)

et>

1)

+

x

K [M (k) - H (k/k -p p --p

(k + 1, k)

x

-p

(k/k)

+ -p r

(k

+ 1,

k)

1)]

TP

(k)

(4)

(5)

where ~p (k/k) is the best estitnate of spacecraft pitch states at k giv en IneasureInent Mp (k), ~p (k + Ilk) is the best one-step predict of spacecraft pitch ~tat e s based on IneasureInent Mp (k), !5 p is the Kalman gain vector, and Tp is an estiInate of the torque developed by the pitch gas thruste r s . During the frame time and after the best single-step predicts of spacecraft pitch position, rate, and acceleration are computed, a decision will be made to see if either of the pitch thrusters must be turne d on at time k + I . The decision whether to turn the appropriate thruster on at k+l is based on the weighted e rror, d e noted by E (k+l). namely p

E

p

(k

+ 1)

x

(1 K O ) (k rp - p

+

Ilk)

( 6)

where Kr is rate to position gain. When tiIne (k + 1) arrives, the appropriafe thrusters will be turned on; they will be turned off 20 Insec later. This leads to a Ininitnuzn on-tiIne of 20 Insec. Again at tiIne (k + 1) sensors indicating spacecraft position are read. A new cycle of estitnation to arrive at best predicts for the spacecraft position, rate, and acceleration at titne (k + 2) is initiated, and the dec is ion is Inade as to which thrus ter is to be turned on a t (k + 2). This process is sequentially repeated in reaI-tiIne.

46

K. Bahrami' For yaw and roll axes, the subscript pin (3), (4), (5), and (6) is changed to y or r. Celestial Cruise The operation of the mission module controller during celestial cruise is similar to the inertial cruise case except that the spacecraft position information is obtained from the sun sensors and the star tracker rather than the gyros. The measurements obtained from the sun sensors and the star tracker are corrupted by noise, thus leading to a new set of Kalman gain constants.

COMPUTER SIMULATION OF CONTROLLER The expected performance of the spacecraft mis sion module during cruise is analyzed with the aid of a digital computer. Both inertial and celestial cruise cases are considered. Some of the details of these simulations are discussed in t\:>.e following paragraphs. To achieve a good representation, the spacecraft dynamics, shown as one block in figure 2, are modeled as a collection of hinge-connected flexible bodies (see /2./ and /3/). The behavior of hydrazine thrusters is also simulated with a computer subroutine which 1s based on thruster dynamics. The results of the simulation are presented in the following paragraphs. Inertial Cruise Without Disturbance In this case it is assumed that the disturbance torques on the mission module are small and negligible. This case is simulated on computer. The results of simulation are shown in figure 3. The spacecraft was kept within the dead-band with low rates. Inertial Cruise With Disturbance The most significant disturbances are encountered during the trajectory correction maneuver. Four thrusters provide the velocity increment and are identical to the attitude control thrusters. They are located symmetrically about the center of mass of the spacecraft. Alignment errors and individual thruster tolerance variations create the major disturbances to be counteracted by the attitude control system. For the study of attitude control during the trajectory correction maneuver, disturbance torques of 0.07 N-m for pitch and 0.028 N-m for yaw are assumed · This case is simul a ted The results shown in figure 4 indicate that even with disturbance torques of such large magnitude, the attitude of the spacecraft was maintained. The disturbance torques on the spacecraft are in the pitch and yaw axes. As expected, the spacecraft maintained its pitch and yaw positions near one edge of the deadband.

47

...

~

Cl>

0 .8 ~ (b)

(0)

."

eE

."

-O.~

V

(~c.. Q.

'"

I' -0.8

'\

~ r?

~

I"\"

'i

."

e

E

,d," -0.4

<~>..

"\..

'""

d,"

50

100

(e) u

0

{

0

..

u 0

ll!I! 1.1::.1

..

e

I

~'"

3" -20 -20

'3"

3'" -10

50

TIME , .ec

100

100

~

Cl

;a.

50

10' (f)

20

e

'a. <3

",

'""

k-'"

TIME, .ec

TIME, .ec

(d)

V

/

-0.8 I'\.

100

10

o

~

~

-0. 8

u

tXl

'" 8.'"

:>"'

e

i.:? 50

"e..

'\

E

TIME, .ec

."

(c)

50

TIME , sec

100

I IJ I I I I I I I I I 0

50 TIME, sec

Figure 3: Inertial cruise without external disturbance. (a), (b) and (c) are plots of spacecraft position and estimated position for pitch, yaw and roll, respectively; (d), (e) and (f) are plots of spacecraft rate and estimated rate for pitch, yaw and roll, respectively

100

(c)

0.8~

(a)

0

0.8 ."

."

2

."

e

2

1;

E

E

<~Q. Q.

(~

0

01 11

I I I 1 1 1 1 1 I

CD

-0.6

-0.81

o

50


J-

->-

CD

,

o

100

TIME, sec

I IIIIIIII 50

-0 . 8

;,.~

200

."

...

;a-

(I) 10

0

~

e...

->(J

0

0

r'I

". '41lt

100

I' ~

,

11

l

1\ '\: ""'\J

b

t

>-

l

I

" 1

(3"

3

,

-"to.

u

~

...

Q.

<3

50

TIME, sec

e

2

........

~

TIME, sec

1001~ ..("

"- f....-'

o

100

-20

-100

o

-400 50 TIME, sec

100

0

50 TIME. sec

100

o

50 TIME, sec

Figure 4: Inertial cruise with constant acceleration disturbance. (a), (b) and (c) are plots of spacecraft position and estimated position for pitch, yaw and roll, respectively; (d), (e) and (f) are plots of spacecraft rate and estimated ratefor pitch, yaw and roll, respectively ~

100 ~

t:tI

~ '1 i»

g,

K. Bahrami Celestial Cruise This is the case in which noisy star tracker and sun sensor data are used for spacecraft position measurements. The Kalman gains must be reduced as dictated by the covariance matrix of these processes. The results are shown in figure 5. Again the spacecraft attitude was maintained and the rates were low.

CONCLUSION A major challenge of MJS'77 was devising suboptimal attitude controller that could meet the demanding mission requirements . This challenge was met by implementing a discrete stochastic controller for a specially designed computer onboard the spacecraft.

APPENDIX.

SINGLE-STEP PREDICTOR

Zero-Mean Systems Let the dynamics of the system be given the linear difference (A-I) and those of the observation by (A-2):

cl> (k

+

(k) = Hy (k)

+

y.. (k + 1) M

-y

-

I, k)

1.

(k)

Tl{k)

k

-

+ [' (k +

(A-I)

I, k) W (k)

(A-2)

0, 1,2, • • •

where y is the state n-vector, wand '1 are zero mean disturbance and noise processes, cl> and!:, are the-state and disturbance transition matrices, H is the observation matrix, and My is the measurement. Furthermore assume that (1)

Processes {y{k), k = 0, 1,2, . . . } and {M

-y

(k), k = 0, 1,2, . . . }

are Gaussian, zero mean.

(2)

E[y" (j) w' (k)]

(3)

E[My

50

=Q

for all k -; j; j, k

(j) w' (k)] = 0 for all k'! j; j

= 0, 1,2,

...

0,1,2, . . .

(b)

(0)

0.8

'c

e

-0

e

E

E

(~o..

tr"...oj

-0.4

'"

p

<3

'",

1,1'

Q.

~

-400 50

100

...

o

!fl

0

.

~

..,

IR ~

V'

JT\j

v

..

~.

IP J

50

100

TIME. sec

40

. u

(I)

11

20

~ -0

0

'"

<3'

'"

3'

,

-;;;

'\.

.

e

<3

3

V

"I

o

100

e ,

'I

~

jJ T

0

~

~

r..

W'

"Y n

"

-40 -20

-20

o

50

TIME. sec

~

~

- 0 .8

M

~

n 1\

[,

,r

.;,'

u

1\

-0

50

- 0 .4

TIME , sec

(d )

20

"e..

'",-

~

(CJ:>~

V'

TIME , sec

<3 Q. 3

e

E

0

~ ~

o

(c )

-0

~

f'

-0.8'

u

.,.. ~~.

3

,/

.

0

100

o

50

TIME. sec

100

o

50

TIME , sec

Figure 5: Celestial cruise. (a ), (b) and (c) are plots of spacecraft position and estimated position for pitch, yaw and roll, respectively; (d), (e) and (f) are plots of spacecraft rate and estimated rate for pitch, yaw and roll, r e spectively

100

~

Ol

P>

p-

"

P>

8.

K. BahraIni (4)

E

Lr (j) 21'

(5)

E

[My

(6)

E [w (j) w' (k)] = Q (k)6 jk; j, k = 0,1,2, . . .

(7)

E [.2(j + l)~' (k + 1)]

(k)]

=Q

= 0,

for all j, k

I, 2, . . .

(j).2' (k)] = 0 for all k> j; j, k = 0, 1,2,

=

R (k + 1) 6jk; j, k

= 0,1,2,

where 6 ij is the Kronecker function. and (') denotes the transpose operation. Then, the optimal single-step predict Y(k + l/k) is given by the following recursive relationships (for the proof and also for the definition of an appropriate performance criteria see /4/)

I

(k/k)

I

(k+l/k)

K (k+l)

=

r

=

=P

P (k+l/k)

(k/k - 1)

+K

(k) [M (k) - H.r (k/k-l)]

(A-4)

(k+l/k) .r (k/k)

(k+l/k) H' [HP (k+l/k) H' + R(k+l)

=

(A-3)

~-l

(A-5)

(k+l, k) P (k/k) cb' (k+l, k)

+L P (k+l/k+l) = ,.1

(k+l, k) Q(k)

-

£.'

(k+l, k)

(A-6)

I

K (k+l) H (k+l) P (k+l/k);

P (0/0)

=P

(0); k

= 0,1,2,

..

(A-7)

This completes the procedure for obtaining the optimal single-step predicts of a system whose states are zero mean.. Now the results are extended to a system whose states are not zero mean. Non-Zero-Mean Systems Let the dynamics of the system be given by

~ (k+l) =

52



(k+l, k) ~ (k) + ~(k+l, k) [T (k) + w (k)]

(A-B)

K. Bahrami

Mx (k) = H ~ (k) + !l. (k)

(A-9)

where x is the state vector, cl> and [' are the state and control transitiOn matrices, T(k) is the actUal forcing function, wand '1 are zero-mean disturbance and noise processes, and Mx is the measurement vector. Furthermore, assumptions 1 throufh 7 for the zeromean case hold, with the exception that processes ~ (k); k=O, 1,2, . . . andtMx (k) ; k=O, 1,2, . . . } are not neces·sarily zero mean. Now the optimal predictor for this system is derived.

T

Define

(k) ~ E [T(k)]

Y. (HI) (:, ~ (k+l) -

f

(A-10)

£

(A-ll)

(k+l)

where k

~

(k+l) =2: cl> (k+l, k+l- j).£ (k+l- j, k- j)

T

(k- j)

(A-12)

j=O From the definition of£(k + 1) in Eq. (A-12) it follows that

£

(k+l) = cl> (k+l,k)

£

(k) + £(k+l,k)

T

(k)

(A-l3)

From Eqs. (A-9) and A-ll) it follows that

M

-y

(k)

= -x M

(k) - H C (k) --

(A-14)

Since ~(k) is zero mean, then (A-3), (A-4), (A-ll), and (A-14) can be used to obtain the optimal single-step predict,i (k + l/k):

i

(k/k) = i(k/k-l) + K (k)[M

i

(k+l/k) = cl> (k+l/k) ~ (k/k) +

x

(k) -

Hi (k/k-l)]

£ (k+l, k) T (k)

(A-IS)

(A-16)

Eqs. (A-IS) and (A-l6) together with (A-S), (A-6), and (A-7) provide a method for the iterative calculation of the optimal single-step state predicts, provided that the expected value of the input to the system is known.

53

K. Bahrami

REFERENCES /1/ Smith, L. S., and Kopf, E. H., Jr.: The Development and Demonstration of Hybrid Programmable Attitude Control Electronics. 27th Meeting of AGARD, May 27-31, 1974, Athens, Greece. /2/ Fleischer, G. E., and Likins, p. W.: Attitude Dynamics Simulation Subroutines for Systems of Hin e-Connected Ri id Bodies wit N on-Rigi Appendages, Technical Report 32-1598, J et Propulsion Laboratory, Pasadena, Calif., Aug. IS, 1975. /3/ Likins, P. W.: Dynamics and Control of Flexible Space Vehicles, Technical Report 32-1329, Rev. 1., Jet Propulsion Laboratory, Pasadena, CaliL, Jan. IS, 1975. /4/ Meditch, J. S.: Stochastic Optimal Linear Estimation and Control, McGraw-Hill Book Co., New York, 1969. ACKNOWLEDGMENT The author is indebted to H.K. Bouvier fo·r his invaluable help. This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract NAS 7-100, sponsored by the National Aeronautics and Space Administration.

54