,t

APPROVED BY:

/Systems Design

The

Allied

Bendix

Corporation

Guidance

Systems

Division

Teterboro

New Jersey

AUTONOMOUS

MOMENTUM

MANAGEMENT

FOR

SPACE STATION

07608

Final Report

15 October 1984

Prepared For:

George Co Marshall

Space Flight Center

Huntsville, Alabama

NASA Contract No.

NAS8-35349

Exhibit A

October I, 1983

to

August 31, 1984

PREPARED BY:

_..,i-v_--v_._

E. Hahn

Tecbr,ical _an_ger

I

k!
TABLE OF CONTENTS

SECTION NO. DESCRIPTION PAGE

Foreword ....................................................... iv

Abstract ........................................................ I

Nomenclature .................................................... 2

1.0 Introduction .................................................... 5

2.0 Description of External Torques ................................. 7

3.0 Techniques Considered .......................................... 50

4.0 Gravity Gradient Desaturation Using the Core Body .............. 22

5.0 Aerodynamic Desaturation Using Flat Plate Appendages ........... 35

6.0 Conclusions and Recommendations ................................ 62

7.0 References ..................................................... 64
FIGURE NO.

I

2

3

4

5

6

7

8

9

I0

II

12

13

14

15

16

17

18

19

20

21

22

23

24

LIST OF ILLUSTRATIONS

DESCRIPTION PAGE

CDG PLANAR SPACE STATION oolm,olueooiioulo.eo, .ou ._iomao_moouom_e 6

AERO TORQUE • CORE BODY o.o_ioo ..... ueooeGoeue.mmo • .eo*ogmo,J._*o 9

GG TORQUE, CORE BODY ........................................... 10

ATMOSPHERIC DENSITY IN ORBIT ................................... 11

AERO TORQUE• SOLAR ARRAY #I.................................... 12

GG TORQUE• SOLAR ARRAY #1...................................... 13

AERO TORQUE• FOUR SOLAR ARRAYS ................................ 14

GG TORQUE, FOUR SOLAR ARRAYS .................................. 15

AERO TORQUE • TOTAL VEHICLE llegolo_o .eiiol,geu_.ooeol.oemggmm_o 16

GG TORQUE• TOTAL VEHICLE ....................................... 17

TOTAL TORQUE TOTAL VEHICLE .... 18

MOMENTUM WITH NO MOMENTUM MANAGEMENT ........................... 19

CDG PLANAR SPACE STATION WITH AERODYNAMIC

DESATURATION APPENDAGES ........................................ 21

CORE BODY ANGLE ABOUT POP AXIS ................................. 23

POP MOMENTUM WITH CONTINUOUS CORE BODY CONTROL ................. 24

lOP MOMENTUM WITH CONTINUOUS CORE BODY CONTROL ................. 27

TILT PROFILE FOR CONTINUOUS lOP DESATURATION ................... 28

TILT PROFILE FOR SINGLE ORBIT DESATURATION,

RATE LIMIT = .01 DEG/SEC ....................................... 30

lOP MOMENTUM WITH SINGLE ORBIT CORE BODY CONTROL ............... 32

POP MOMENTUM WITH SINGLE ORBIT CORE BODY CONTROL ............... 34

IOP DESATURATION APPENDAGES .................................... 38

lOP APPENDAGE TORQUE IN VEHICLE SPACE .......................... 41

DESIRED AND PERPENDICULAR lOP APPENDAGE TORQUE ................. 42

OI X AND Z APPENDAGE TORQUE .................................... 43

ii
FIGURE NO.

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

LIST OF ILLUSTRATIONS (CONTINUED)

DESCRIPTION PAGE

IOP MOMENTUM, NO APPENDAGE CONTROL, RHOAVG = 1.7E-12,

BETA = -45 ..................................................... 44

lOP MOMENTUM, NO APPENDAGE CONTROL, RHOAVG - .34E-12,

BETA - -45 ..................................................... 45

IOP MOMENTUM, NO APPENDAGE CONTROL, RHOAVG - 8.5E-12,

BETA = -45..................................................... 46

IOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 1.7E-12, RHOCNTRL = 1.TE-12,.BETA = -45 ............... 47

IOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = .34E-12, RHOCNTRL = .34E-12, BETA = -45 ............... 48

IOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 8.5E-12, RHOCNTRL = 8.5E-12, BETA = -45 ............... 49

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = .34-12, RHOCNTRL = .34-12, BETA = 0................... 51

IOP MOMENTUM, APPENDAGE CONTROL, RHOAVG = 8.5E-12,

RHOCNTRL = 1.7E-12, BETA = -45 ................................. 52

POP AXIS TORQUE EQUILIBRIUM ATTITUDE AS A FUNCTION

OF AVERAGE DENSITY ............................................. 53

POP APPENDAGE TORQUE ........................................... 54

POP MOMENTUM, NO APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 1.7E-12............................................... 55

POP MOMENTUM, NO APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = .34E-12, RHOCNTRL = .34E-12, BETA = -45 ............... 56

POP MOMENTUM, NO APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 1.7E-12 ............................................... 57

POP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 1.7E-12, RHOCNTRL = 1.7E-12, BETA - -45 ............... 58

POP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG - .34E-12 ............................................... 59

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 8.5E-12, RHOCNTRL - 8.5E-12, BETA = -45 ............... 60

POP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 8.5E-12, RHOCNTRL = 1.7E-12, BETA = -45 ............... 61

iii
Foreword

This final report is submitted in accordance with "Scope of Work, Exhibit

D" for contract NAS8-35349. This study was directed from the Guidance Systems Division (GSD) of the Allied Bendix Aerospace Corporation. Mr. Eric

Hahn was study manager and he appreciates the valuable guidance of Ms.

Miriam Hopkins of The Marshall Space Flight Center (MSFC) who was the

contracting officer. Additional important suggestions were made by Mr.

Harry Buchanan and Mr. Stan Carroll of MSFC, Mr. Dave Barrows of McDonnell

Douglas Astronautics Company and Mr. Ray Kaczynski of GSD.

iv
Abstract

The report discusses momentum management for the CDG Planar Space Platform. The external torques on the Space Station are assumed to be gravity

gradient and aerodynamic with both having bias and cyclic terms. The integrals of the cyclic torques are the cyclic momenti which will be stored in

the momentum storage actuator. Various techniques to counteract the bias

torques and center the cyclic momentum were investigated including gravity

gradient desaturation by adjusting vehicle attitude, aerodynamic desaturation using solar panels and radiators and the deployment of flat plates at

the end of long booms generating aerodynamic torques.
Ap

cm

Cd

CMG

cp

GG

Hm

I

lOP

Ip

Ipcm

Iv

Kdb

LV

mp

N-m

N-m-s

OI

POP

n

S

TBy

!D

To

Nomenclature

transformation from p-th appendage principal axis to core body frame

center of mass

drag coefficient (assumed = 2)

control moment gyro

center of pressure

gravity gradient

instantaneous stored momentum

estimated average angular momentum

identity matrix

in-orbit plane

inertia of p-th appendage

inertia of p-th appendage in cm coordinates

total vehicle inertia tensor

atmosphere diurnal bulge

local vertical

mass of p-th appendage

unit normal to p-th appendage

newton meters

newton meters seconds

orbit inertial

perpendicular to orbit plane

unit radius vector from center of earth to center of mass

surface area of p-th flat plate

desired y-axis torque For bias counteraction

desired control torque

average value of lOP torque in desired direction
Nomenclature (Continued)

TDIOP desired control torque magnitude in orbit plane

TGG

Tly

Tnom

T o

Ipa

U

Udb

Uon

UT

V

9

Xp

Z

aIi

a¢2

B

Y

¢p

_0

gravity gradient torque

desired y-axis centering torque

nominal torque from one appendage if p = _nom and air flow is perpendicular to surface

orbital period

aerodynamic torque due to p-th appendage

average value of lOP torque perpendicular to desired direction

vehicle angle from orbital noon

peak atmospheric density phase lag

half angle that lOP aerodynamic desaturation is "ON"

angle of desired torque lOP with orbital noon

velocity magnitude

unit vector along relative wind

moment arm of p-th appendage

z-transform variable

inertia difference (ali=Ik-lj, i, j, k in even permutation)

a-rotation of vehicle about POP axis to remove bias

A-rotation of vehicle about POP axis to center momentum

angle at panel "ON" makes with air flow For POP desaturation

angle of sunline to orbit plane

angle of "ON" POP desaturation panel with air Flow

angle of vehicle z-axis to local vertical about POP axis

angle of vehicle principal z-axis to local vertical about POP axis

vehicle angular rate in orbit

3.1416
p

Pavg

Pest

Pnom

@

@max

Nomenclature (Continued)

instantaneous atmospheric density

average atmospheric density

estimated average atmospheric density

nominal atmospheric density

vehicle x-axis rotation for GG desaturatlon

maximum @ adjusted For finite maneuvering rate

max rate for maneuvering vehicle about its x-axls
1.0 Introduction

Figure 1 represents the CDG planar space station configuration whose basic

attitude has the Z-axis local vertical (LV), the Y-axis perpendicular to

the orbit plane (POP) and the X-axis along the velocity vector. The solar

arrays are maintained nominally perpendicular to the sunline requiring two

degrees of freedom, one for the vehicle rotation and the second for the

sunline with respect to the orbital plane (B-angle). The radiators are

maintained POP and only edgewise illuminated by the sunline. However, unlike the solar arrays which have unlimited rotation capability, due to the

necessity of fluid transfer between the space station and the radiators and

the limited freedom of the plumbing interface, the radiators must be unwound during the dark side of the orbit.

If the balanced vehicle defined in Figure 1 were ideal, the principal axes

of the core body would be perpendicular to the orbit plane and along local

vertical and no gravity gradient torques due to core body would exist. The

center of mass would be at the geometric center and no aerodynamic torques

would exist. The only remaining torques would be a bias torque about the

X-axis due to the non-zero B-angle of the solar arrays and a cyclic torque

about the POP axis due to the solar arrays. The radiators only contribute

torque during the unwind cycle and were determined not to be significant

for sizing momentum storage and desaturation requirements. For the actual

Inertias and center of mass used For the study, significant core body gravity gradient biases and aerodynamic torques do exist although the balanced

design insures their being of small magnitude For the size of the vehicle

investigated. The momentum storage system assumed For this study was four

Skylab type double gimbal control moment gyros (CMGs) having an angular

momentum of 3000 N-m-s each.

The main effort of this study was to develop momentum management techniques

to counteract the bias torques and also to center the stored cyclic momentum along three inertial axes. Prior to developing momentum management

techniques, it was necessary to define disturbance torques for the basic

vehicle. Two types of disturbance torques were admitted for this study;

gravity gradient and aerodynamic. Gravity gradient torques were defined by
i

v

I

X

Z

I--

I.'._,

u._

CJ,_+

Z

FIGURE I CDG PLANAR SPACE STATION
the standard equations while aerodynamic torques were defined by a combination of flat-plate areas, a simple orbital curve fit to the Jacchla density

model and inelastic collision of particles with the flat plates. The definition of torques is complicated by the relative motion of the appendages

with respect to the core body leading to the selection of seven independent

bodies to represent the total inertia. These bodies are the core, four

solar arrays and two radiator sets. Nominally, the four solar arrays and

two radiator sets each move as units but separate motions were allowed in

order to investigate their use as desaturation techniques.

Z.O Description of External Torques

The inertia of the entire vehicle is defined as that of the core body plus

the inertias of the appendages reflected to the center of mass of the system. Due to their symmetric nature, the center of mass of the solar arrays

and the radiators are at the center of symmetry of the core body. However

the center of mass (cm) of the core body is slightly offset from the center

of symmetry. The inertia of an appendage relative to the cm of the core

body consists of the sum of the inertia due to its own inertia matrix about

its cm and the inertia moment of a point mass at the appendage cm acting

about the core body cm. The appendage inertia must be transformed via a

similarity transform from its principal axes to the axes Fixed in the core

body. The transformation varies with appendage attitude. The variable

inertia due to an appendage is defined by equation I while the gravity gradient torque is defined by equation 2. The gravity gradient torque on the

total vehicle can be defined two ways of which the First is the gravity

gradient torque due to the total inertia of the core body and appendages.

The second is to define the inertia of each appendage, determine the gravity gradient torque and add the individual torques to generate the total

gravity gradient torque. The latter approach allows one to see the contribution of the individual appendages for insight into possible desaturation

techniques. The aerodynamic torque due to the core body and each appendage

is simply the force on each body and a vector cross product of that force

with the vector between the cm and the center of pressure (cp). This

torque is defined by equation 3.
Ipcm = ApIpApT + mp(xpTXpl - IXpXpT) (t)

T_G = 3_o2 (_ , IVY) (2)

_pa = _p * (_ovZ)CoSpIO " _plO (3)

The flat plate area of the core is defined as the projection of a cylinder

onto a plane. Figure 2 shows the aero torques due to the core body. The

torque perpendicular to the orbit plane (POP) is in the same direction and

only varies in magnitude due to the diurnal bulge. The torques in the orbit plane (lOP) are essentially cyclic but have bias components due to the

diurnal bulge. The gravity gradient (GG) POP torque shown in Figure 3 is a

bias while the IOP torques are cyclic in Orbit Inertial (Of) space. All

lOP torques and momenta will be displayed in OI rather than body axis as

that is the frame where momentum buildup truly occurs and in which it must

be desaturated. The density variation in an orbit is given by equation 4,

a simple curve fit to the Jacchia model of the nominal atmosphere.

P = _avg (i + KDB cos(U - UDB)) (4)

Figure 4 shows a typical density variation over an orbit with the peak density occurring 30 degrees after noon and the minimum density occurring 30

degrees after midnight. Figure 5 shows the aero torque due to a single solar array with significant torques on all axes which is the same condition

observed for the GG torques of Figure 6. Figure 7 shows that the aerodynamic torque due to four solar arrays is much less as the majority of the

moment arms cancel. If the cm of the core body were at the center of symmetry, no solar array aero torques would exist. The cyclic gravity gradient torques due to the individual panels cancel lOP but add POP. Figures

8, 9, 10 and 11 show that the total torques on the vehicle are dominated by

the solar arrays. However the dominant bias component POP is the GG torque
I

I I

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FIGURE 2 AERO TORQUE, CORE BODY
FIGURE 3 GG TORQUE, CORE BOOY

10
FIGURE 4 ATMOSPHERIC DENSITY IN ORBIT

11
B

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o_'fIFIGURE 5 AERO TORQUE, SOLAR ARRAY #1

12
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FIGURE 6 GG TORQUE, SOLAR ARRAY #1

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FIGURE 7 AERO TORQUE, FOUR SOLAR ARRAYS

14
8

i=,.i

I ! !_ T T I I _ i I ! it iT

.I_ ' _ I ' I I _

I I _ I., i i I

.....",....(----i....I_---l' ,'....!

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..... r .... r .... I"-'_..:._- TI_"- - - "T -T.... -I

I I I _ I I I

I _ i , II_ _ i I

I , , i kd , _ I I

>- i i i o I i -_ i

I(_ , , 1 , l,,_ ' _ '

I_I I

1_ I-.... _-.... _--_+, . .... i-..._- -_i_ -''_- 4_- - . .... -4.

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I I

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Io _-.... _ .................

I !

In o i

I - i I l I -_..1= ' i I

II_.II i o i _ , i i

................ ....

I0 I I I I I I 0 I _I I

I_- L_.., L I ', -=. I _ ' .... i _£ ......... J ........ J

, ---, f -f ..... _---,---_ I

_ , i i i ' i,'' I

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_._, ®._, oO'_ oo_, _f_"_""_._-' _o._' oo.,,L

FIGURE 8 GG TORQUE, FOUR SOLAR ARRAYS

15
8

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liiiliiii!

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LFI....... - -

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FIGURE 9 AERO TORQUE, TOTAL VEHICLE

16
I I I I I I

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t I I I ' I I

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FIGURE l.O GG TORQUE, TOTAL VEHICLE

17
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FIGURE II TOTAL TORQUE, TOTAL VEHICLE

18
due to the core body. The dominant lOP bias contribution is the solar arrays aero torque at a B-angle of zero while solar array GG torque dominates

at B-angles of magnitude close to 45 degrees. Figure 12 shows the stored

momentum profile if no momentum management were active and the initial momenta were -1000 N-m-s in Orbit Inertial (Of) X and Z respectively and

-3600 N-m-s along the Y-axis which is perpendicular to the orbit plane

(POP). The OI X-axis is along the velocity vector at noon and the OI Z-axis points up at noon. The start of all runs is orbital noon with the core

body approximately trimmed to remove the POP bias torque.

FIGURE 12 MOMENTUM WITH NO MOMENTUM MANAGEMENT

19
3.0 Techniques Cons|dered

A known technique for bias counteraction is the use of magnetic torquing

using four Space Telescope magnetic torquer bars. These would have the

capability of generating approximately .13 N-m torque per axis when averaged over an orbit and are independent of vehicle configuration, attitude

and atmospheric density. As their capabilities and control laws are known,

including the disadvantages of weight, power and magnetic contamination,

their use was not specifically investigated in this study but they remained

the benchmark for innovative techniques. Several innovative techniques

were investigated with various degrees of success. The emphasis of the

study was to investigate in depth those techniques that appear promising.

Among techniques that were investigated but eliminated were using the unwind profile of the radiator which did not generate sufficient torque.

Modifying the pointing of various solar arrays using both the B-angle gimbal and the local vertical gimbal did not produce adequate desaturation

without very large rotations during the dark side of the orbit which was

considered undesirable from pointing performance. Off-nominal pointing of

some of the solar arrays during daylight would produce undesirable power

profiles.

The promising techniques developed to be described are use of the core body

and the deployment of special aerodynamic appendages. All the techniques

developed would be autonomous in the Flight computer and simple to implement requiring only knowledge of inertias and approximate average atmospheric densities. Since the core body is maintained nominally local vertical (LV), its use for POP desaturation by means of gravity gradient torques

is well known. The control laws for this will be described. Phased motions of the core body about the vehicle X-axis can be used to desaturate

momentum lOP and will also be described. Initially, the control concept

was to make these desaturation maneuvers every orbit. Results showed that

after the initial transient decayed, the maneuvers were very small although

payload pointing interference was still possible. Therefore an alternate

control concept was to allow absolute momentum to approach a value inside

its capability sphere. Only at this point would desaturation maneuvers of

the core body be required although they would be much larger than those

20
performed each orbit. It appears that single orbit maneuversperformed

every tenth orbit might be adequateafter initial equilibrium is attained.

Using the appendagesof Figure 13, no core body motion is required. Closed

loop autonomouscontrol laws for the angle of attack of these appendages

have been developed. The appendagesalong the vehicle Y-axis desaturate

the lOP momentumwhile the appendagesalong the Z-axis control the POPmomentum. The hardwarerequired to deploy and control these appendagescould

be adaptedfrom solar array and antenna drives and does not represent new

technology. The attainment of a given torque at a certain atmospheric density dependson the product of area times momentarmwhich could be traded

off depending on considerations of weight, complexity and possible interference with payload pointing. The results of using these appendagescompares Favorably with magnetic desaturation in terms of weight, power, and

complexity of both hardwareand control laws. Whenatmospheric density is

high, the baseline sized appendagesproducegreater desaturation capability

than Four SpaceTelescopetorquer bars. During period of minimumdensity,

the torquer bars havemorecapability but the requirements are less.

1

III[IIOOST/AI*TITUO| CON_

PAO_*Ut.SJO_ IYSTI_ |TY_

T.

IOP D[SA'_UTIOW

APPENDAGE 1

FIGURE 13

CDG PLANAR SPACE STATION WITH AERODYNAMI(

POP DESATURATION

APPENDAGE I

"_/_I OP DESATURATION

DAGE 2

4_ LV* |lNTlAl_t. IVA ,I_ALGCKJ

POP DESATURATION

APPENDAGE 2

DESATURATION APPENDAGES

21
4.0 Gravity Gradient Desaturation Using the Core Body

Utilizing the core body, which maintains an attitude relatively close to

local vertical (LV) to desaturate the momentum perpendicular to the orbit

plane (POP), is a technique previously investigated (ref 1). Assuming one

principal axis is approximately POP and a second is approximately LV, the

gravity gradient torque along the POP axis is given by equation 5. The

total desired attitude change is the sum of that required to remove the

bias and that required to center the momentum along the POP axis. The attitude change to counteract the bias torque is given by equation 6 which

also utilizes the inverse of equation 5. The angle change required to center the momentum is given by equation 7. The new attitude about the POP

axis is given by equation 8 and includes an effect for increased momentum

due to the bias acting over an additional half orbit. This angle change is

accumulated In the flight computer and while the core body attitude about

the POP axis is controlled within a small range by the requirements of

momentum management, the flight computer will, at all times, have knowledge

of the actual attitude. Figure 14 shows a typical attitude about the POP

axis while Figure 15 shows the momentum. The large initial changes about

the POP axis occur due to the requirements of the centering logic.

TGGY : 3mo2A12_ p

(5)

(1-z-t)Hy

a¢1 = 3_o2Toal 2

(6)

a¢2 = Y

3mo2To •a12

(7)

= z-l_y - l.Sam I - ae 2 (8)

22
X

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C3

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I I I I

1 I I I

I I I I

,6 4- -.6 .... 4

I I I I

I I I I

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.... 4- ,6 4- .... -,!,

I I I I

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.... L .... -1. .... ..I, ....

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L .... L,.=. _ _ __L .... J _ ____.___ _ _ _

|

I I I I I I

I I I , I I

I I I I I

I I i I I

o9' o

OB"

"In

"_nl__I

B

"tl'l

g-m

FIGURE 14 CORE BODY ANGLE ABOUT POP AXIS

23
...J

0

n--

Z

0

l_.-

r F

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t

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t

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oo' o911-

FIGURE 15 POP MOMENTUM WITH CONTINUOUS CORE BODY CONTROL

24
If the vehicle Z-principal axis is moved out of the orbit plane by a rotation about the vehicle X-axis, a body fixed gravity gradient torque about

the X-axis will be produced. However this torque is cyclic about two axes

fixed in the orbit plane. In order to generate a gravity gradient bias

torque in the orbit plane, it is necessary to generate a phased rotation

about the vehicle X-axis. If the orbit inertial (01) axes have average

momenta over the previous orbit given respectively by Hx and Hz the total

bias torque to be generated about the 0! X and Z axes are given by equations g and 10. The magnitude of the total 10P torque is given by equation

11 and the direction of the desired torque is given by equation 12.

X

TDx = T

0

(9)

TDz = To

(10)

TDIOP : /TDx2+ TDz 2

(11)

rTDx_ _ (12)

UT : tan-I lDl_zJ +

After the desired magnitude and direction of the 10P torque to be generated

is calculated, the second part of the calculation is the definition of the

vehicle attitude profile with respect to the orbital plane. Instantaneously, the gravity gradient torque in body space is given by equation 5 with

alI replacing al2. If that attitude is maintained over one-half orbit centered about a given direction, the average torque in that direction is given by equation 13 while the average torque in the perpendicular direction

(equation 14) is zero. Reversing the rotation about the X axis during the

other half of the orbit produces a rectified torque in the desired direc25
tion and still zero in the orthogonal direction. The angle required to

desaturate the lOP bias is given by equation 15 which shows an efficiency

loss of 2/_ compared with the angle required For POP gravity gradient

desaturation. Examining Figure 16, which shows the lOP momentum with core

body desaturation active, indicates that the average values of OI X and Z

momenta impose only small momentum storage requirements on the CMOs and

that the bias torques are removed. Figure 17 shows a typical core body

rotation profile about its X-axis For lOP desaturation which shows that

after the transients have decayed, the required angles are quite small and

that even maneuvering at .01 deg/sec, the times to attain the new attitudes

are negligible.

3=o2eal UT+ _ 3=2 2eal I

7D = _ 1 _ cos(U - UT)dU = _ (13)

UT -_

3=o2eAI1 UT+

T i = f _ sin(U - UT_dU = 0 (14)

x UT --2

e = sin-i[ _ToToP ) (15)

3_o22-AT1

It should be noted that since torque and momentum are not in the same

direction, it is difficult to remove both bias and average values of the

momentum in two OI axis. However the average momenta are at values close

to zero and in maintaining those values with the control law, bias torques

are also counteracted. Examining the POP rotations of Figure 14 and the

lOP profile of Figure 17 shows that after the momentum along POP is centered and the bias is removed, both the POP corrections required and the

lOP profiles represent small angles which, nevertheless, could disturb payload pointing. Due to the balanced structure of the vehicle, the momentum

26
J

"1

I

I

I

I

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I

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I

I

I

I

I

I

[

I

I

I

I

I

I

I

I

I

I

I

a _m m ._

I

I

P

I

I

I

!

FIGURE 16 fOP HCNENTUNWITH CONTINUOUS CORE BODY CONTROL

27
I

....

I

1

I

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t

I

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1

I

I

I

I

t

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I

I

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[

I

t

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1

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-1

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J

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!

1'I

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0_' l'-

FIGURE 17 TILT PROFILE FOR CONTINUOUS lOP DESATURATION

28
accumulation, on a per orbit basis, is very low once POP bias has been removed by the appropriate rotation of the core body. This led to the development of an alternate method of gravity gradient desaturation using the

core body. Instead of performing an adjustment every orbit, the vehicle

was held Inertially fixed until the absolute momentum attained a software

limit. The maximum momentum storage capability of four Skylab type Control

Moment Gyro (CMG) is 12000 newton-meter-seconds (N-m-s) leading to the selection of a software limit set at 9000 N-m-s. The logic was set so that

if the stored absolute momentum attained this value, corrective action for

both POP and IOP momentum would be performed.

The amount of momentum left for maneuvering is quite low so that a maximum

core body maneuvering rate of .01 deg/sec about any vehicle axis was assumed. Since the angles required for desaturation will be much larger when

momentum is allowed to go to the inside of the 9000 N-m-s momentum sphere,

the assumption of square wave attitude angle profiles of Figure 16 for lOP

desaturation can no longer be made. Therefore with the assumed maximum

core angle with infinite rate capability calculated, the actual angle is

adjusted for finite rate according to equation r6. This maintains the same

integral of area x time as if the new attitude were attained instantaneously leading to the trapezodial profile of Figure 18.

e' = 6max(To/2 ) [i_¢i 8o )

4 6max(To/2)

(16)

These considerations are less critical for the new angle about the POP axis. Desaturation about the POP axis is not dependent on angle in orbit for

the adjustment of the momentum states. Therefore Finite maneuvering times

will compensate at the beginning and end of the desaturation profile. Due

to the significant time required for the completion of the single orbit lOP

desaturation, it was decided to perform a maneuver in only two steps as

shown in Figure 18 rather than the three steps of Figure 17. It should be

noted that Figure 18 shows the desaturation profile lOP from time of start,

which could be any time in orbit, and assuming the calculations are performed at a fixed position in orbit, the end of the desaturation could take

29
ri

i

i

I

ti

C3 i

i

I-- '

nh

rr" _,

C_ W*

....

U3 C3'

,F

n-" I

0 III

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.... _

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t- tI _--- I

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I 1 I

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B

'_--- - -I' r "I

i i I

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.... "T ! ....

I I I

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I_ I I I

.... 1 .... 1 .... 1

I I I

I I I

1=5• I I 1

._.-- -, .... -+ .... -4

I I I

I I I

I I I

'_ I I I

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_ J --'r-----__.

N,,i,,, I 1 I

• 1

-_--- 4 ....

I

, I

B i

I

B

I I

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i I

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I 1

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-I .... .I

I I

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._.__ _ .L.... ._....

t i ,J

I I ___II

0,.j_

L

I

oJ

FIGURE 18

TILT PROFILE FOR SINGLE ORBIT OESATURATION, RATE LIMIT = .01 DEG/SEC

30
two orbits from the time that the calculations are made. The other change

in lOP desaturation when no longer performed at orbital frequency is, rather than drive the lOP momentum to zero when desaturation is required, the

momentum will be driven to the opposite point on the lOP momentum circle

given by equations 17 and 18. Since the bias should drive the momentum In

the same direction as previously, the maximum time allowed before the next

desaturation is required should be extended. Figure 19 shows the momentum

IOP about two inertial axis with single orbit desaturation.

-21_x (17)

TDx = To

(18)

The single orblt POP desaturation technique utilizes only minor modifications to the POP desaturation technique at orbital frequency. The angle to

remove the blas is given by equation 6 and that remains as the new vehicle

attitude until the next desaturation is required. The angle to center the

momentum is given by equation 7 and that angle is removed at the conclusion

of the single orbit desaturation. The new angle at the beginning of a desaturation is given by equation 19 and that at the end of one desaturation

period Is given by equation 20. The period of the single orbit POP desaturation is concurrent with that of lOP desaturation which generally is not

the position (orbit noon) where the calculations are performed.

= z-l¢ _ l.Sam I - as 2 (19)

¢ :, z-l_, + a(b2 + .Sa_,I (2O)

31
d

0

Am

LT--

"1

I

I

I

I

"1

I

I

I

I

4

I

I

I

I

"t

I

I

I

I

J

I

I

I

I

1

I

I

I

.J

I

I

I

I

1

I

I

I

00' 013_-

FIGURE 19 lOP MOMENTUM WITH SINGLE ORBIT CORE BODY CONTROL

32
The POP attitude maneuver at the beginning of the single orbit desaturation

is the sum of the bias removal and momentum centering angles while that at

the conclusion is the negative of centering angle. Using the momentum desaturation technique of allowing absolute momentum to reach the maximum allowable value of 9000 N-m-s, the POP momentum with single orbit desaturation is shown in Figure 20. A subsequent desaturation was not required for

30 orbits which is probably too optimistic for a real mission as simulation

did not include varying B-angle, atmosphere density and mass properties

between successive orbits. The real trimming of torque and momentum will

probably not be as accurate as shown in the simulation. However even the

attainment of ten orbits of pure Inertial Hold before momentum management

trlm maneuvers might represent adequate viewing time for many payloads.

Another desaturation technique, magnetic desaturation using the control

laws developed during the Space Platform study was considered but not actually investigated in detail For Space Station. These control laws would be

directly applicable to Space Station and the magnitudes of the lOP bias

torques (.05 N-m) on OI X and Z are well within the capability of four

Space Telescope torquer bars. However the .5 N-m bias torque about the POP

axis is beyond the capability of 4 bars unless the vehicle Is first trimmed

by a closed loop POP axis rotation. Magnetic desaturation also has undesirable properties such as the weight and power required and the potential

for magnetic contamination of payloads.

Two alternate desaturation techniques would be mass shifting for gravity

gradient desaturation and appendage control for aerodynamic desaturation

with the latter considered more promising. The appendages considered for

this study are two flat plates along ± vehicle Y-axis for IOP desaturation

and two flat plates along ± vehicle Z-axis for POP momentum trimming. It

was decided that the magnitude of the POP bias was too large for complete

aerodynamic desaturation and the flat plates would act as trim tabs after

several orbits of gravity gradient maneuver desaturation about the POP axis. This report will discuss sizing, control laws and show closed loop results of aerodynamic desaturation using flat plate appendages. In addition,

results will be shown For those cases where the atmosphere is different

from the nominal density used to size the appendages and control law gains.

33
_I

CD

A_

l,x..

0

(..)

i

n_

n-"

0

=I

. I

0

i

I

I

I

00'0_I

I I

I I

I t

t t

@

i

I

I

I

I

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I

I

I,,-

I

I

I

I

I,,.-

I

I

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II

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I

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I..

I I

I I

I I

"- I I

®.& oo.&

I

I

I

.I..

oo

I I

I I

I I

I I

"T .... "T .... "_

I

I

I

I

I i

I I

i I

I I

I I I

I I I

I I I

I I I

4 .... 4.---- H

I I I

I I I

I I I

i t I

•4 .... .4 .... q

I I I

I I I

I I I

I I I

4 .... 4 .... q

I I I

, t I

I i I

I I I

I I I I

FIGURE 20 POP MOMENTUM WITH SINGLE ORBIT CORE BODY CONTROL

34
5.0 Aerodynamic Oesaturatton Using Flat Plate Appendages

Simulation results have indicated that for the baseline vehicle inertlas,

baseline offsets of the center of mass from the center of symmetry and the

baseline atmosphere model (Pavg = 1.73 x 10-12 kg/m 3, Kdb = .5, Udb = 30

degrees), the OI X and Z bias torques are about .0135 N-m each and the magnitude of the total lOP bias torque which is the vector sum of the components is about .02 N-m at a zero degree B-angle. However at a non-zero Bangle, although the aerodynamic forces and torques are less, the gravity

gradient bias lOP is greater. The worst case lOP bias torque occurred at a

B-angle of -45 degrees about the X-axis. This can be realized intuitively

that as the solar arrays are rotated from POP for the B-angle, a gravity

gradient bias torque along the OI X-axis is generated having its maximum

value at 45 degree magnitude B-angle. For the inertias and cm used, the

magnitudes of the aerodynamic and gravity gradient biases add about the OI

X-axis at the negative B-angle producing the worst case torque of .045 N-m

consisting of 0.035 N-m gravity gradient and 0.01 N-m aerodynamic torque.

Using the efficiency factor of 2/_ (to be shown later) which is identical

to that for lOP desaturatlon using the core body, the required torque from

the panels would be about .075 N-m. To allow margin, the panels for lOP

desaturation were sized to generate Tnom of 0.1 N-m. Assuming an orbital

velocity squared of 5.8 x 107 meters2/sec 2 and a drag coefficient (Cd) oF

two, the product of area times moment arm must equal 1000 meters cubed to

generate Tnom. Typically, this would represent an area of 40 square meters

with a 25 meter moment arm or 20 square meters with a 50 meter moment arm.

The sizing requirements for the POP axis trim tabs is less clear-cut as requirements depend on the degree of trim attained by gravity gradient desaturation using the core body. In order to simplify hardware considerations,

it was decided to use the same requirements as for lOP desaturation, namely

Tnom torque capability at an atmospheric density of Pnom" In the POP axis

35
aerodynamic torque, the unfavorable factor of 2/_ for the theoretical

torque achleved in the desired direction does not exist so the torque attained could be Tnom. Referring to Figure 13, panel 2 generates a positive

torque about the POPaxis. The general requirements for deploymentand rotation of the panels are consistent with solar array and antenna mechanisms. Candidates for actuators include stepping motors and torquers.

Since the vehicle is rotating at orbital rate, in order to maintain a fixed

orientation to the air Flow, the panels must only be rotated whenreorientation is required. This could Favor stepper motors as actuators since the

devices would be off whennot actually rotating. An estimate of the weight

is 15kgfor each actuator/boom/panel or 60 kg for the system.

The control law for lOP desaturation was based on assuminga constant atmosphericdensity and sampling the momentumof the two lOPaxis at orbital

frequency with the sampling performed at orbital noon. The choice of noon

for momentumsampling is arbitrary for this simulation only. Perhapsmidnight which wasused in Skylab for Ephemerisupdating would be operationally preferable but results should be independentof samplerplacement in orbit. The requirement of knowledgeof averagedensity makesthe aerodynamic

desaturation somewhatless desirable than gravity gradient desaturation.

Currently, the inertia matrix of a spacecraft is assumedto be well known

while atmospheric data at the SpaceStation altitude is knownbest from its

long term averages. Therefore the study involving aerodynamicdesaturation

with flat appendageswill consider the case where the control law estimate

of Pavgis different from the "real" atmospheric density. It will be shown

that momentumdesaturation is still viable even with a certain amountof

mismatch. The density variations within the orbit, due to the diurnal

bulge are less critical than the average value of the density for aerodynamicdesaturation and becauseof the uncertainty of the exact density distribution during an orbit, are not included in the control law.

36
The momentumsampling schemefor the generation of the required torques

follows along the samegeneral lines as those used for the core body desaturation with the lOPbias torques to be counteracted given by equations 9

and 10, the magnitudeof the total bias torque to be generated is given by

equation 11 while the direction of the desired torque is given by equation

12.

Figure 21 showsthe two appendagesFor lOPdesaturation with 2 "on" and i

feathered producing a negative torque about the vehicle Z-axls. WhenappendageI, along vehicle +Y-axis is rotated perpendicular to the air flow,

a positive torque about vehicle Z is generated. The appendageswhich develop torque along the vehicle Z-axis lOPare along the two ends of the vehicle Y-axis with their axis POP. The control law developed for lOPdesaturation admits only three states for the two appendages:both Feathered,

appendage1 perpendicular and 2 feathered and 1 feathered and 2 perpendicular th the velocity vector. Therefore only three torque states about Z-vehicle axis are possible. Theseare zero torque if both or noneof the appendageswere feathered, and maximummagnitude if only one appendageis

feathered. The zero torque condition will be assumed"by both appendages

feathered, if the sameappendagewere perpendicular to the air flow for

the entire orbit, the torque would be a bias in vehicle space but cyclic in

lOPaxes. Therefore it will be necessary to develop a control law involving feathering each panel part of the orbit and possibly both panels part

of the orbit to generate a torque with desired magnitudeand direction.

It is assumedin this study that the separation of the cm from the cp is

negligible comparedto the momentarm of the individual appendageand given

identical densities, the torques due to each appendagewould have the same

magnitude. It was also assumedFor this study that the appendagesdo not

affect the gravity gradient torques. A typical case that is also easy to

visualize is a B-angle of zero where the bias torques are -.014 N-mabout

both OI X and Z axes. This also represents a torque of magnitude .02 N-m

about an axis 315 degrees from noon. In order to counteract this external

bias, the appendagesshould generate a positive torque along an axis 45

degrees from noonwithout any componentalong a perpendicular axis.

37
This Is done by using panel I symmetrically about 315 degrees and panel 2

about 45 degrees. All subsequent calculations are based only on density

equal to Pnomwith closed loop calculations taking care oF actual atmospheric density. The theoretical torques generated are symmetrical in 4

quadrants about the torque direction so the equation for turning panels on

is given only for two quadrants symmetric about the desired torque direction.

_PZw_

I

JUPPI_DAGS

2

FIGURE 21 lOP DESATURATION APPENDAGES

38
Theaverage torque In the desired direction over one half orbit Is given by

equation 21. If a certain magnitudeis desired, the inverse of equation 21

is solved for the angle Uon which represents the "on time" of the appendage

torque IOP. This angle is given by equation 22. The maximum torque that

can be generated In the desired direction if the angle is ninety degrees is

Tnom _ 2/x. The torque in the direction perpendicular to the desired

torque is given be equation 23 and can be seen to be zero if o Is actually

constant.

TD Tn°m {°est) _UT+ U°ncos( U _ UT)dU

_nom T- Uon

2TnomPest

_Pnom

sin(Uon ) (21)

Uon - sln-lr_'PnomTDIOP_

_2-OestTnom J

(22)

UT+ Uon

Tn°m (°est) _ sin(U - UT)dU = 0

Ti = T Pnom UT- Uon

(23)

The POP axis control contains an integral path so the desired setting is

the sum of a bias and integral path. The bias contribution to the desired

POP torque is given by equation 24 while the desired centering torque is

given by equation 25. The total torque desired is the sum of the bias and

centering torques. The POP axis trim tabs control law utilizes a different

approach from the lOP control law. Either panel I or panel 2 is "on" the

entire orbit with the same angle based on the POP momentum sample. Unlike

the lOP control law, the panel that is "on" can be set at any angle between

zero and ninety degrees. The magnitude of the torque is determined by equation 26 while the angle that the normal to the panel makes with the airflow

to generate the desired torque is given by equation 27.

39
rl-z-l_u

TBy=-ITo jny + z-lTiy (24)

.5(l-z-1)H

TIy = - _o - To (25)

TDy = TBy + Tiy (26)

= sin-l( Pest'TDy )

Y Pnom" Tnom

(27)

The sign of the desired torque determines which panel is "on". Figure 22

shows the lOP appendage torque in vehicle space for a typical orbit, Figure

23 shows the torque in the desired and perpendicular direction and Figure

24 shows the torque in OI axes. The torques shown are not perfect as they

represent the torques generated by the control law using the first momentum

samples. The closed loop nature of the calculations will assure that after

every several orbits, the profile converges to produce the torque in the

desired direction despite the non uniformity of the atmosphere. This non

uniformity can be seen in Figure 22 where a constant p would produce a

square wave profile.

Figures 25, 26 and 27 show lOP momentum buildup for nominal, 20% nominal

and 5 times nominal average density over 9 orbits without bias counteraction. Figures 28, 29 and 30 show lOP momentum when the average density,

nominal, 20% nominal and 5 times nominal, used in the control law exactly

40
a_miimni_n_

I I

I I

I I

'1- 'r

I I

"I "1

i i I

t I t

B , ,

.... "1' I'

I I I

I I

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i I

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..... l- - -'- i- 4-

i I I

i I I

I I I

i I I

..... t- .... 4- 4-

I I I

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I I I

! I I

..... l- .I- .II I I

I I I

I I I

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---L. .... J. ....

It

FIGURE 22 [0? APPENDAGE TORQUE IN VEHICLE SPACE

41
8

T ] 7 ]

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r--- 1"

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i I I "t... / i tS i

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I- .... r - I- - I- -£- _I- - -_ - - - -_ - "I

I I I I l I _ I I

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I__ _1 ' ' _ ' _"h_ ' I

I- - _'--- _" "_---_-_" .... "_---'_" -_

1

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I- .... _- ---4- -+ .... + .... ÷_--,-

I I I I I ill _I I I

, I I 1 I l_ t I

I " I 1 I I -_ I /

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I I I I I _"_ I I

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1,.-.... 1- .... "t"---1. 1 --i_ -I--_-x-l_----l, . 1

I

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FIGURE 23 DESIRED AND PERPENDICULAR lOP APPENDAGE TORQUE

42
I I I I I _ I I

I t I I I iO. I

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F- F : ., ., _........

! i ' '

I I I

..... I T ---

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I I I I I

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Fl--F r ........

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', ,I _+ .. _--_ - - - i

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I I I I I

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FIGURE 24 OI X AND Z APPENDAGE TORQUE

43
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.I.

I

I

I

I

I

I

I

I

I

I

I

°I

,-a

I I

I t

I t

4II .... !

I I

I I

4 .... 4-;-..J

I I I

I I I

I I !

I 1 I

.L__.__JIII III.... ]

I 1 t

I I I_ __ I /

®._, ®._ oo.I _'_I;-_ oo'_-®.o_-

m i[ [ •

i

FIGURE 25

lOP MOMENTUM, NO APPENDAGE CONTROL, RHOAVG = 1.7E-12, BETA = -45

44
I I I I I I

I

8

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

W I

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I

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I

@0' _ 00'

I I

I I

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r -_

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÷

I

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llIl_llII

I I

I I

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I I

____ _ ....

I I

i I

I I

I I

I

liII_

FIGURE 26

lOP MOMENTUM, NO APPENDAGE CONTROL, RHOAVG = .34E-12, BETA = -45

45
rii_iii

I

I

I

I

_ i iii fill iii

I

!iiii

m

I I

I I

I I

I I

I

I

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I "1

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I

iilI_

I I

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t I

1 !

l_Ilii

!

! I

I I

I

I t

I!! lil_lllI

I I

I I I

I I I

I I

I I I

I I I

I I

I I

.... 4

I I

I I

I I

I I

I I I I

I I I I

I I I I

I I I I

_ .... J____J

I I I I

! I I I

I I I I

I I I I

_T IIII

i I I |

I

I

I I

I

---I' .... "I

I I

I I

'

! I

I I

I I

I I

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I. I

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I !

I I

I !

1 I

4II.... 1

I I

I 1

FIGURE 27

lOP MOMENTUM, NO APPENDAGE CONTROL, RHOAVG = 8.5E-12, BETA = -45

46
B

I

I

I

......... _- 4------

I

I

FIGURE 28

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 1.7E-12, RHOCNTRL = 1.7E-12, BETA = -45

47
I

I I I

1 I I

4 .... 4 .... 4

I I I /

I I I

I I I I

.... •4 .... 4 .... J

I I I I

I I I I

I I I

I I I I

|

I I I

I I I

I I I I

oo'o_ I- oo'o_1-

FIGURE 29

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = .34E-12, RHOCNTRL = .34E-12, BETA = -45

48
E]

I

i

I

I

"1-

t_- _ -N ) I,,,,11"1J N _ I,_I_W

FIGURE 30

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = 8.5E-[2, RHOCNTRL = 8.5E-12, BETA = -45

4g
matches the average atmospheric density. At the B-angle of -45 degrees,

Figure 29 shows that regardless of the control law, the appendages can not

counteract the lOP bias about the OI X-axis bias torque when the atmospheric density is only 20% nominal. The only solutions are to increase the panel size or raise the solar arrays For 0 degree B-angle as shown in Figure

31. This would reduce the available power if the actual B-angle were -45

degrees. Figure 32 shows lOP momentum when the Oest assumed by the control

law is different from the true Oavg. The true density is 5 times that used

by the control law and makes the lOP bias desaturation marginally stable.

The performance of the panels for POP desaturation is heavily dependent on

the approximate balance of the vehicle gravity gradient and aerodynamic bias torques. Figure 33 shows the approximate POP axis trim as a function of

Pavg where vehicle trim is the angle between the vehicle Z-axis and the local vertical. Figure 34 shows the POP appendage torque when one is perpendicular to the air flow and with the nominal atmospheric model. The appendages are capable of generating .1 N-m torque at the nominal Oavg which

represents trim capability over one degree. At 5 times the nominal density, the trim capability is 5 degrees while at 20% the trim capability is

only .2 degrees of vehicle attitude. Therefore reaching an approximately

balanced attitude about the POP axis using gravity gradient desaturation is

most critical during p_riods low atmospheric density. Figures 35, 36 and

37 show the POP momentum for an approximately trimmed vehicle at nominal,

20% nominal and 5 times nominal density without additional appendage desaturation. Approximately trimmed means the the POP bias Is not perfectly balanced but within the capability of the aerodynamic desaturation panels when

the density in the control law is exactly matched to the average density.

Figures 38, 39 and 40 show typical POP momentum where the vehicle is

trimmed and the true density and that assumed in the control law are

matched. Figure 41 shows the POP momentum when the true Oavg is 5 times

that assumed by the control law. Only a slight oscillation can be seen so

that POP desaturation is quite insensitive to a Pavg greater than that assumed by the control law. However if the Pavg is less than that assumed by

the control law, the degree of core body trim might be inadequate and the

POP momentum would diverge. The divergence is not caused by lack of robustness in the controller but by insufficient torque capability.

50
B

I= " --'l"='t = .... I" ...... "t .... "t=--"- - "I

I I I

I I

t I

•4,.... ,4 .... -,l

I I I

I I I

I I

I i

4 .... 4 .... .-i

I I I

t i I

t I

I I

I I I

I I I

I I

I I

.... ..I .... -I

I I

I I I

I

, t

oo.dFIGURE 31

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = .34-12, RHOCNTRL = .34-12, BETA = 0

51
I

II

B

I I I I I

r- I I I I

t I I I

I I I

------'1" ? "T ....

I I I

I I I I

FIGURE 32

lOP MOMENTUM, APPENDAGE CONTROL,

RHOAVG = 8.5E-12, RHOCNTRL = 1.7E-12, BETA = -45

52
FIGURE 33

POP AXIS TORQUE EQUILIBRIUM ATTITUDE AS A FUNCTION OF AVERAGE DENSITY

53
I I I I 1 I I

I I I I I I t

I I I I I I

II I I I I I l

II"-- " " - _" .... I" 4- ÷

' I I I I, 'I -

i-- , , I I I I

-- --_- 4- 'I- ÷ -_ -4

I I I I I I

I I I I I I

I I 1 I

I I __ I

i_ I. .... L_ L .I. 4. .4 .J,....

I I I I I I I

I I I I I I I

I I I I I I I

I I I I I I I

I ! 1 1 1_ I I

ol "o al 'b

,,._ _o __ .o.'o _'o

8

g

i

g

_o'?

FIGURE 34 POP APPENDAGE TORQUE

54
Xm

I I

IS

I I I I

I

I

I.

!

I

4

i

I

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l I

I I

..I.... 4

I I

I I

I I

I I

l II .... 1

m I

I ;

I i

oo'_- oo'o_-

FIGURE 35

POP MOMENTUM, NO APPENDAGE CONTROL, TRIMMED CORE, RHOAVG : 1.7E-12

55
I

I

I

I

I

XN I

I

II

I

Bo L

I

I

I I

I I

| i

I

I I

I I

I I

I I

•4 -4 ....

I I

I I

I I

I I

.I .4 ....

I I

I I

I I

I I

® ,o,.°°i ,._o°_ °°'_"

FIGURE 36

POP MOMENTUM, NO APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = .34E-12, RHOCNTRL = .34E-12, BETA _ -45

56
iliiu

I

i

t

I

I

I

mi_Ii

I

XN I

I I

I

I

7

I I

I

I

i

( _-W-N )NFIIN_WOW

7

I

I

I

I

.... "I' .... "I .... "I

I I I

I I I

I ! I

i i I

I I I

I I I

I I I

I I I

+ _ .... .4 .... -'t

t I i

I I I

t i I I

I I I

4. 4 .... -_.... -4

I I i

I I I I

I I I I

I I I i

4. .... 4 J, .... -4

I I I I

I I I I

I I I I

I I i i

.... J. -i -i.... J

I I I I

I I I I

I I I I

I I I 1

J. .L .... .L .... -I

I I I

I I I I

i l I i

i I I I

oo'oe I' oo'o -

FIGURE 37

POP MOMENTUM, NO APPENDAGE CONTROL, TRIMMED CORE, RHOAVG : 1.7E-12

57
I

I

I

I

I

I

I

i

I

I

I

I

I

I

-I

I

I

I

I

-I

I

I

I

I

I

I

i

I

.J

FIGURE 38

POP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = L.7E-12, RHOCNTRL = 1.7E-12, BETA = -45

58
"I

I _,

I

I I

I I

4

I

4

i

I

I

I

-I

I

I

I

I

I

I

I

I

" -t

I

I

I

I

4

I

I

I

I

I

I

I

.... -L .... J

I t

I I

I I

I I

I

I

i I

I I

oo.d: oo, -

FIGURE 39

POP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE, RHOAVG : .34E-12

59
B

I_ r"----Ir "=" - - _- -" -_t-,- - "r -r

0 i _ _ ' __

'_'I I l-- _ I _ I

l='_II I I _, 1, t

r (If i i _

r I'-I i,=,___ I ' - i i

_,,,,,tL.Lit I i ' L I

•-(NI I I '" __

l ' i !

_-_1- .... _--_--- t- -'-' - -"=". - - ÷ ....

Z ,9 I i '

_ i_._" i .... f i

I_1- .... _- - - _- ..... -_i-_- - - + ....

cT' Z_ i .. i

m]r'_ t I

I_ -I I I 1 I I

_IIA I I_ --I J I I

ii_.... _.--,_--=':_F_--_r-- -- _-_- -- -- -- _-.... 4

..... _ _ J I I

_-, ,I I

n_ "" L___,.:_---, .-,-_---,i. .... _.

n ,t I _ : _l

I I _ I -- I I

_ _ _1 ...... __

i I _ ,_ ' I 1

_¢'_111 oo'o_ oo'O_zOI_ oo'o_-N-i'l _ oo'oZ- ) _nIN3_OLl oo'o_-

I

I

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I

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I

I

I

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I

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I

I

"I

I

t

I

i

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i

i

J

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i

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l

I

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I

J

I

I

I

I

I

I

FIGURE 40

lOP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RHOAVG = B.SE-12, RHOCNTRL = 8.5E-12, BETA = -45

60
i

I,,, B

------r ---- -.r- ----r -T_---_

1"" I

}rC_i ---r .... _=c_. _ .... _-m'--- _ --

w _ l i c 'I

,

I . I I II ---"....

" "

I_,_ ,' _' ' ' __" "---', ,

g!--- '

_'-- _--- __-- _"_- - _-- " I- I

_(_ I I I _ I

I ,._| ' '_" I '-' I

, , I I

_.__. .... _._-___..____÷+_--- ___. ' J

_,',_.... "--_- *---'.-_.... ',.... 1

2_.i -. ,

I I I

I ! .= i

_C_F .... _.-6--,., , ..... .,, ....

nC3_ "- ._.' ' , ,

oe ____ _. ,,-_.....

_IlI_ IIm_p" ' I '

I I '-'--- ' -

I I

I I

!

"-4"--- .1 ....

.I I

_ I _ _t " I -- t

oo'o'_, oo' dz oo"? co' oe

101 (9-N-N) Wl-lINqWOI4

I I

I I

I I

I I

I I

I I

I I

"_II.... 1

I I

oo'os,I- oo'_-

FIGURE 41

POP MOMENTUM, APPENDAGE CONTROL, TRIMMED CORE,

RNOAVG = B.5E-12, RHOCNTRL = 1.7E-12, BETA = -45

61
6.0 Conclusions and Recommendations

Two viable new techniques of momentum management for the the CDG Planar

Space Station configuration have been developed. These are the use of the

core body for gravity gradient desaturation and the implementation of four

flat plate appendages at the end of long booms which would be used for aerodynamic desaturation. The use of the core body, whose attitude is nominally local vertical, for gravity gradient desaturation about the POP direction is a technique previously considered. The control law developed in

this study shows that the core body can also be used to autonomously center

POP momentum therefore minimizing momentum storage requirements.

Using the core body to desaturate momentum lOP required a phased rotation

about the vehicle X-axis to generated torque lOP with the desired magnitude

and direction. All calculations using the core body for desaturation were

autonomous and required only knowledge of the core body inertia matrix.

With the exception of momentum sampling, calculations were only performed

at orbital frequency and should impose very little burden on a flight computer. Examination of the results showed that after equilibrium was attained, only small core body rotations at orbital frequency were required

to control the momentum.

As even these small rotations could interfere with payload pointing, an alternative method of momentum management using the core was developed. This

method allowed the absolute momentum stored by the CMGs for an approximately trimmed vehicle to slowly drift to a value inside the maximum storage

sphere of four Skylab type CMGs. At that time, the control law would initiate a maneuver, using the remaining CMG momentum, to remove the POP bias

and re-center the POP momentum. The law would drive the lOP momentum along

a diagonal to the opposite value of the lOP momentum circle. For these

larger maneuvers, the profile was adjusted for finite vesicle maneuvering

capability. This concept worked extremely well as maneuvers were required

only about every thirtieth orbit. However the simulation did not include

time varying external parameters such as B-angles and density Functions.

62
To allow a trimmed vehicle to retain an inertial attitude for even longer

periods of time, the aerodynamicdesaturation appendagesconsidered could

be used. They do represent additional hardwarerequirements but an examination of the results showsthat the methodis competitive with the benchmark, magnetic desaturation. The total weight would probably be less than

the 170kgof the SpaceTelescopemagnetic desaturation systemand the power

required much less than the 120 watts required at maximummagnetic moment. Howeverthe control law utilizing these appendagesdoes require at

least an approximate value of the average air density during an orbit.

Doubt exists as to the availability of this parameter but the control law

developedhas been shownto be relatively insensitive to atmospheric density uncertainties. Onepossibility is to autonomouslyobtain an estimate of

Pavgfrom the POPaxis trim angle.

During periods of high atmospheric density, appendagedesaturation is capable of muchlarger torques than Four SpaceTelescope torquer bars. During

periods of very low atmospheric density, a higher degree of attitude trim

might be required for the POPaxis. lOPdesaturation could be difficult

since the appendagesdo not generate sufficient torque to counteract solar

array GGbias torques at 45 degree B-angle. Appendagesdo not present the

problemsof magnetic contamination of payloads but potential problems could

exist with line-of-sight of payloads and attitude control position sensors. Due to the light weight and stiff construction proposed (Kevlar

Boom),no attitude control structural modeinduced control problemsare anticipated and the transmission of torques to the core body should be quite

rigid. By meansof transformations stored in the attitude control computer, magnetic desaturatlon performs independently of vehicle attitude since

a magnetic momentcan be defined in orbit inertial space. The appendage

control law defined in this study only operates at the CDGPlanar Space

Station baseline attitude. Should appendagedesaturation be desired at a

variety of attitudes, the modified concepts could still be feasible but the

appendageswould require at least two-degrees-of-freedom. Additional

studies could define the appendage/boom/actuatorsystems in greater detail

and also evaluate control law performanceover changing B-angles and atmospheric densities.

63
7.0 References

Wertz, James R. (Editor) Spacecraft Attitude Determination and Control, D.

Reldel Publishing Co., 1978.

Barrows, D., Bedell, H., Hahn, E., Kaczynski, R., Levinthal, J. Momentum

Management for the Space Platform, American Astronautical Society, AAS 82-

004, 1982.

Kaczynski, R. F., Momentum Management Utilizing Gravity Gradient Torquing,

Bendix GSD Report MMS-83-1, Oct 1983.

Kaczynski, R. F. Spacecraft Aerodynamic Torques Due to an Inertially Oriented Flat Plate Area Bendix GSD Document No. 5460476-MT, April 1982.

Levinthal J., Morata, L., and Powell L., Space Platform Attitude Control

System International Federation Automatic Control, Netherlands, 1982.

Hahn, E., Kaczynski, R., Barrows, D., Autonomous Momentum Management for

the Space Platform AIAA-84-2006; AIAA/AAS Astrodynamics Conference,

Seattle, 1984.

Nasa SP-8021, Models of Earth's Atmosphere, Revised March 1982.

64