NASA Contractor Report 178219

SPACECRAFT ATTITUDE CONTROL

MOMENTUM REQUIREMENTS ANALYSIS

Brent P. Robertson PRC Kentron Hampton, VA 23666 Contract NAS1-18000

V

Michael L. Heck Analytical Mechanics Associates, Inc. Hampton, VA 23666 Contract NAS1-17958 (NASA-CE-1782 19) SPACECRA€? A ' I ' X I ' I U D E Nd7-16864

CCNZtflCL C C M E N l U M i i E ~ U I R E f i E K P 5 AhALYSIS ( P R C K e n t i c n , Inc-) 9 2 p CSCL 228

U n c l a s G3/lS 43726

January 1987

Y

National Aeronautics and Space Administration

Langley Resesrch Center Hampton, Virginia 23665

https://ntrs.nasa.gov/search.jsp?R=19870007431 2020-04-21T20:12:10+00:00Z

TABLE OF CONTENTS

Title Page

ABSTRACT.. . . . . . . . . . . . . . . . . . . . . . . . . . . . iii GLOSSARY OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . iv

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. DERIVATION OF THE CONTROL MOMENTUM REQUIREMENTS

2.0 The General Equation (Inertial Coordinates) . . . . . . . . 2

2.1 Rotation From Local Vertical Coordinates . . . . . . . . . . 3

3 . DERIVATION OF THE CONTROL MOMENTUM COMPONENTS

3.0 Gyroscopic Effect . . . . . . . . . . . . . . . . . . . . . 5

3.1 Environmental Effects . . . . . . . . . . . . . . . . . . . 5

3.2 The General Equation (Local Vertical Components) . . . . . . 9

4 . DISCUSSION OF CONTROL MOMENTUM REQUIREMENT COMPONENTS

4.0 General Comments Regarding Secular Momentum Build-Up . . . . 13

4.1 Gyroscopic Component . . . . . . . . . . . . . . . . . . . . 13

4.2 Gravity Gradient Component . . . . . . . . . . . . . . . . . 13

4.3 Aerodynamic Component . . . . . . . . . . . . . . . . . . . 15

4 .4 Solar Component . . . . . . . . . . . . . . . . . . . . . . 19

5. RESULTS OF TEST CASES

5.1 Case 1 - Non-Articulating; Constant Density, Non-Rotating Atmosphere . . . . . . . . . . . . . . . . . . . . 20

5.2 Case 2 - Non-Articulating; Variable Density, Non-Rotating Atmosphere . . . . . . . . . . . . . . . . . . . . 20

5.3 Case 3 - Non-Articulating; Constant Density, Rotating Atmosphere . . . . . . . . . . . . . . . . . . . . 31

5.4 Case 4 - Non-Articulating; Variable Density, Rotating Atmosphere . . . . . . . . . . . . . . . . . . . . 46

i

TABLE OF CONTENTS (cont'd.)

. Title Page

5.5 Case 5 . Non-Articulating; Principal Axis Attitudes . . . . 46

5.6 Case 6 . Articulating; Gravity Gradient and Gyroscopic TorquesOnly . . . . . . . . . . . . . . . . . . . 60

5.7 Case 7 . Articulating; Full Environment . . . . . . . . . . 60 6 . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8 . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

APPENDIX A . Space Station Physical Properties . . . . . . . . . 84

ii

ABSTRACT

. The relationship between attitude and angular momentum control

requirements is derived for a fixed attitude, Earth orbiting spacecraft with large area articulating appendages. gradient, solar radiation pressure, and aerodynamic forces arising from a dynamic, rotating atmosphere are examined. It is shown that, in general, each environmental effect contributes to both cyclic and secular momentum requirements both within and perpendicular to the orbit plane. gyroscopic contribution to the angular momentum control requirements resulting from the rotating, Earth oriented spacecraft is also discussed. Special conditions are described where one or more components of the angular momentum can be made to vanish, or become purely cyclic. Computer generated plots for a candidate Space Station configuration are presented to supplement the analytically derived results.

Environmental effects such as gravity

The

iii

GLOSSARY OF SYMBOLS

- a

Ai

AMCD Angular Momentum Control Device

( 3 x 1 ) aerodynamic torque acting on spacecraft (LVLH coordinates)

body X, Y, and Z spacecraft projected areas

spacecraft coefficient of drag cD

( 3 x 1 ) aerodynamic forces acting on the body YZ(i=l), XZ(i=2), and XY( i=3 ) planes

( 3 x 1 ) solar forces acting on the body YZ(i=l), XZ(i=2), and XY(i=3) planes

- g ( 3 x 1 ) gravity gradient torque acting on spacecraft (LVLH

coordinates )

-A g ( 3 x 1 ) gravity gradient torque due to articular part acting on

spacecraft (LVLH coordinates)

- H ( 3 x 1 ) spacecraft angular momentum (inertial coordinates)

- ( 3 x 1 ) aerodynamic angular momentum contribution (inertial

coordinates ) HA

- ( 3 x 1 ) control device angular momentum ( inertial coordinates) HC

I ( 3 x 3 ) identity matrix

IOP in the orbit plane

II ( 3 x 3 ) spacecraft inertia tensor ( inertial coordinates)

EA ( 3 x 3 ) articular part inertia tensor (articular coordinates)

II ( 3 x 3 ) spacecraft inertia tensor (body Coordinates) b

spacecraft inertia tensor element (body coordinates ni j

LVLH local vertical local horizontal

iv

(GLOSSARY OF SYMBOLS cont ’ d. )

“A articular part mass

I

- P

POP

r

- r i

- S

t

(3x1) position vector from spacecraft cg to articular part cg (LVLH coordinates

perpendicular to the orbit plane

(3x1) unit vector down (LVLH coordinates)

(3x1) cp-cg offsets on the body YZ(i=l), XZ(i=2), and XY(i=3) planes

(3x1 ) solar torque acting on spacecraft (LVLH coordinates)

time

- (3x1) aerodynamic torque acting on spacecraft (inertial coordinates)

TA

- 3 (3x1) control torque applied to spacecraft (inertial coordinates) IC

- (3x1) environmental torque acting on spacecraft (inertial coordinates)

TE

- (3x1) gravity gradient torque acting on spacecraft (inertial

coordinates TG

(3x1) solar torque acting on spacecraft (inertial Coordinates) TS

Tb a (3x3) transformation matrix from articular part to body coordinates

(3x3) transformation matrix from body to LVLH coordinates Ti

(3x3) transformation matrix from LVLH to inertial coordinates

X, Y, and Z components of spacecraft unit velocity vector (body

coordinates)

1

V i

V

GLOSSARY OF SYMBOLS (cont'd)

V

vi

P atmospheric density

J,

0

0

magnitude of spacecraft velocity vector

X, Y, and Z components of spacecraft velocity (LVLH coo

ordered Euler angle about body Z (yaw)

ordered Euler angle about body Y (pitch)

ordered Euler angle about body X (roll)

dinat

- w

w 0 orbit rate

( 3 x 1 ) spacecraft angular velocity (inertial coordinates)

vi

CHAPTER 1

INTRODUCTION

In response to a variety of proposed nominal attitude profiles to be flown by the Space Station, the NASA Langley Research Center (LaRC) Space Station Office ( S S O ) has undertaken an analysis of Space Station attitude equations of motion, environmental effects, and control requirements and implications.

When using angular momentum control devices (AMCD's) such as Control Moment Gyros (CMG'S) to control attitude, it is necessary to examine the time history of the angular momentum requirements. Secular trends indicate a need for periodic desaturation or momentum "dumps." components impact the CMG sizing requirements. Therefore, both terms are critical when specifying "optimal" attitudes from a controls point of view.

Peak values of cyclic

A variety of attitude profiles, most of which are labeled torque

These have included principal to body axis equilibrium attitude (TEA), have been presented to the Level B Systems Integration Board (SIB). rotations, 2-axis (pitch, roll) angular attitudes, and 3-axis (pitch, roll, yaw) attitudes, (continuous, discrete, periodic, gravity gradient assisted, etc. One purpose of this study was to ascertain if constant TEA solutions could be utilized to eliminate or minimize frequent angular momentum dumping requirements, and what associated penalties, if any, are incurred. In order to accomplish this, an understanding of the contributions of gyroscopic, gravity gradient, aerodynamic, and solar effects on cyclic and secular angular momentum requirements was necessary. These effects were derived analytically, and demonstrated using the Articulated Rigid Body Control Dynamics (ref. 1) (ARCD) module of the IDEAS' Software (ref. 2) to simuluate Space Station control requirements for different TEA solutions.

all with associated momentum dumping strategies

-1-

CHAPTER 2

DERIVATION OF THE CONTROL MOMENTUM REQUIREMENT

2.0 General Equation (Inertial Coordinates)

The rotational equation of motion for a spacecraft in Earth orbit is given by

k = 1 TE + Tc

where ?i = ( 3 x 1 ) spacecraft angular momentum,

= ( 3 x 3 ) spacecraft inertia tensor,

- w = ( 3 x 1 ) spacecraft angular velocity,

TE = ( 3 x 1 ) sum of environmental torques acting on the spacecraft,

TC = ( 3 x 1 ) control torque applied to the spacecraft (e.g., to maintain a specified attitude).

For Earth oriented, fixed attitude spacecraft, the angular velocity is constant in both magnitude (equal to the orbit rate w,) and direction (perpendicular to the orbit plane), so the time rate of change of the angular momentum with respect to inertial space is given by

With rr(t> and w specified, once the environmental torques are computed over the orbit, Eq. (1) can be solved for the control torque fC which must be provided by the spacecraft attitude control system.

-2-

The momentum equation governing an AMCD is given by

- where HC = ( 3 x 1 ) AMCD angular momentum.

Substituting Eqs. ( 1 ) and ( 2 ) into (31, defining the gyroscopic torque

to be II 6, and assuming that the environmental torques consist of gravity gradient torque T momentum requirement at time t is given by

aerodynamic torque f,, and solar torque fs, the angular G’

t

1 dT + T G + T A + T S ( 4 )

gyroscopic environment a 1

where Eq. ( 4 ) is expressed in inertial coordinates thus avoiding integration in a rotating coordinate frame.

2 . 1 Rotation From Local Vertical Coordinates

The terms on the right-hand side of Eq. ( 4 ) may be expressed in more meaningful coordinates by using transformations. If local vertical local horizontal (LVLH) coordinates are defined to have a Z-axis positive down toward the center of the Earth, Y-axis perpendicular to the orbit plane (positive in the minus orbit angular momentum direction), and X-axis completing the right-handed coordinate system (positive in the velocity

i L vector direction), then a transformation T

coordinates is given by

cos w,t 0 -sin w,t 0 1 0

sin w,t 0 cos w,t

from LVLH to inertial

(5)

Notice that at time zero, the LVLH coordinate frame coincides with the L A transformation Tb from body to LVLH inertial coordinate frame.

coordinates using a Z(yaw)-Y(pitch)-X(roll) ordered Euler tramformtim ($,e,$) is given by

-3-

where ce = cos0

so = sine, etc.

At ($,e,$) = (O,O,O) the body axis coincides with LVLH axis. The i L The transformation Tb L transformation T is a sinusoidal function of time.

is constant f o r a fixed attitude spacecraft.

-4-

CHAPTER 3

DERIVATION OF THE CONTROL MOMENTUM COMPONENTS

3 . 0 Gyroscopic Effect

Using the transformations developed in Section 2.1, the gyroscopic torque in inertial coordinates may be expressed as

T Ti T~ T ~ T i '' = d it [ L b b b TL ] [-:o]

(7)

= ( 3 x 3 ) spacecraft inertia tensor in body coordinates (which b where varies with time for an articulating spacecraft),

I 1 1 XY YY YZ

3.1 Environmental Effects

As was the case with the gyroscopic effects, the environmental effects are more easily derived using non-inertial coordinates. torques in inertial coordinates may be expressed as

The environmental

-5-

- - - where g, a, and s, are gravity gradient, aerodynamic, and solar torques in LVLH coordinates acting on the spacecraft, respectively. (Notice that, no gravity gradient torque can be generated about the LVLH Z-axis. and Y components, g and g

spacecraft since is constant). In LVLH coordinates, b

Also, the X are time independent for non-articulating

X Y’

where r = unit vector down in LVLH coordinates,

The aerodynamic torque exerted on the vehicle is dependent upon the aerodynamic forces, coefficient of pressure - center of gravity (cp-cg) offsets, and the attitude. In LVLH coordinates,

- - - where rl, rz, r3 = ( 3 x 1 ) cp-cg offsets looking at the body YZ, XZ, and

XY planes, respectively,

-6 -

2, 8, = (3x1) time-dependent aerodynamic forces acting on the body YZ, X Z , and XY planes of spacecraft, respectively .

The aerodynamic forces acting on the spacecraft are given by

= -4pV 2 CDAi 1

V i (13)

where p = atmospheric density, V = magnitude of velocity vector, 5 = spacecraft coefficient of drag,

AI,AP, A3, body X, Y, and Z spacecraft projected areas, respectively, and - - - v1,v2,v3 = X, Y, and Z components of unit velocity vector in body

coordinates, respectively.

The LVLH aerodynamic torque's dependence on time is due to three effects: "bulge", time-dependent projected areas caused by spacecraft articulation, and the orbital inclination with respect to the rotating atmosphere.

the change in atmospheric density due to the solar induced diurnal

For a non-rotating atmosphere, the time dependent aerodynamic force acting on a spacecraft is always along the LVLH X axis for circular orbits and hence time-dependent aerodynamic torques can be generated about the L W Y and 2 axes only. aerodynamic force to oscillate slightly in the LVLH XY plane if the spacecraft orbit has a non-zero inclination. time-dependent ax.

In real life, a rotating atmosphere causes the

This gives rise to a small

A similar expression for the solar torque may be derived as follows

-s -s -s PI, Pz, Po = (3x1) time-dependent solar forces acting on YZ, XZ, where

and XY planes of spacecraft, respectively.

-7-

The solar torques acting on the Space Station are relatively small compared to gyroscopic, gravity gradient, and aerodynamic torques at near- Earth orbiting altitudes.

3 . 2 The General Equation (Local Vertical Components) Substituting Eqs. (5) through (14 ) into ( 4 ) and performing matrix

multiplication gives the AMCD angular momentum control requirements for an articulating spacecraft flying at a fixed attitude in inertial coordinates as a function of time t.

+

4

Thus,

c

t t P

0

t

J 0

HC(t) = HC(0) - w g flcos w g ~ - f2 sin W,,T

f3

f,sin w o ~ + f2 cosow T

(gx+ a +s )cos w g ~ - (az+sz)sin wg?: x x g +a +s Y Y Y

(g x x x +a +s )sin W ~ T + (aZ+sZ)cos w g ~

-8-

d.c

+ I I + I I + (c+cqJ+s+ses+) I +I +I [ IxxIxy xy yy xz yz] [ XY YY YZ f 3 = ces+

" 1 A A A a X = cec$ 'lYFlZ - rlzF1 Y - rzzFzy + QYbZ

I -10-

S S S +(-c$s$+s$s9c$) rlZ Flx - rzXFzZ + rz F - z 2x

S S S +(s$s$+c$sec$) -rly Flx + rzxFzy + r3xF3y -

rlyFlz - rQ1 - rzZF2y + r3yF3Z S S S S

Y Y

and the f. represent gyroscopic contributions to the angular momentum requirements and where the I spacecraft.

1 are time-dependent for articulating ij

-12-

CHAPTER 4

DISCUSSION OF THE CONTROL MOMENTUM REQUIREMENT COMPONENTS

4.0 General Comments Regarding Secular Momentum Build-Up

The secular angular momentum build-up over one orbit is simply

fiC(2n/w0) - Gc(0) where GC(O> represents the initial angular momentum of the AMCD. Secular momentum build-up (and hence momentum dumping) can be avoided if a ($,€I,$) attitude can be found such that

One interesting phenomenon which can be observed from Eq. (15a) is that a torque which is constant when expressed in body or LVLH coordinates yields only cyclic angular momentum requirements in the inertial X and Z coordinates, i.e., in the orbit plane (IOP) (since the integral from t = 0 to 2n/wo of K cos wot and K sin w,t equals zero).

4.1 Gyroscopic Component

The second term on the right hand side of Eq. (15a) is the gyroscopic momentum which must be absorbed by the AMCD. whose motion equals orbit rate (a valid assumption for solar arrays and radiators), ~ ( ~ I T / U ~ ) = n(O), fi(2n/wo) = fi(0), and thus it can be seen that gyroscopic effects can never yield secular momentum build-up. articulating spacecraft in general has non-zero gyroscopic cyclic momentum both IOP and perpendicular to orbit plane (POP); however, for fixed spacecraft (no articulation) the f. are constant, and hence the gyroscopic momentum will be cyclic IOP and identically zero POP.

For articulating appendages

An

1

Cyclic IOP momentum for non-articulating spacecraft may be eliminated completely by choosing an appropriate ( $ , e , $ ) attitude. If the body axes are chosen to be aligned with the principal axes so that all off diagonal inertia tensor terms are zero, then it can be seen that fl and f, will be identically zero for any ($,8,4) = (O,O,O). articulating spacecraft has no gyroscopic momentum if flying at an attitude with any principal axis POP. Furthermore, a single degree of freedom (DOF) in pitch exists such that the gyroscopic momentum remains zero.

This indicates that a non-

4.2 Gravity Gradient Component

The gravity gradient momentum contributions in Eq. (15a) are in general secular both IOP and PO? due to the articulating parts which cause nb to

-13-

change with time. If no articular parts are present then g is constant and

the gravity gradient momentum will be cyclic IOP (about x local horizontal only) and secular POP.

X

- A gX

gY A

0 L -

If an articular part rotates about the articular part cg at orbital rate, the secular gravity gradient momentum contributed by the articular part may be derived as follows. spacecraft cg due to the articular part in LVLH coordinates is given by

The gravity gradient torque about the

T T T ~ T ~ b a A a T~ T~ b + mA(iTp I - p p

(17)

where T: = ( 3 x 3 ) transformation from articular part to body coordinates,

= ( 3 x 3 ) articular part inertia tensor in articular coordinates, A

m = articular part mass, A

= ( 3 x 1 ) position vector from spacecraft cg to articular part cg (LVLH Coordinates),

I = ( 3 x 3 ) identity matrix.

The transformation Tb is dependent on the motion of the articular part. a For example, if the articular part rotates about the body Y axis at orbital rate, as might be the case for a solar array, then the transformation from articular part to body coordinates is given by

Tb = a cos w o t 0 sin wot

0 1 0

-sin wot 0 cos wot

The secular gravity gradient momentum due to the articulating part in inertial coordinates may be evaluated by substituting Eq. (17) into Eq. (15a) and letting t = 271/w0. In general, gravity gradient induced

-14-

secular angular momentum due to part articulation will accumulate both IOP and POP. special case where all articular parts rotate at orbit rate about an axis coincident with an articular part principal axis and passing through the articular part cg.

Pure cyclic momentum requirements IOP can occur only for the

For non-articulating spacecraft, the gravity gradient momentum contribution may be eliminated with an appropriate ( $ , e , $ ) solution. body axes are aligned with principal axes then the following properties may be observed:

If

1) gx = 0 for ($,e,$) = (0,0,0> (see Eq. 15e)

Property 1) indicates that IOP gravity gradient induced momentum requirement may be completely eliminated if any principal axis is aligned POP, with a DOF in pitch existing such that this momentum remains zero. Property 2) indicates that POP gravity gradient momentum may be completely eliminated if any principal axis is aligned with the X direction in LVLH coordinates, and a DOF in roll exists such that this momentum remains zero. Property 3) indicates that both IOP and POP gravity gradient momentum may be completely eliminated if any principal axis is aligned with the local vertical and a DOF in yaw exists such that the momentum is zero.

A

4 . 3 Aerodynamic Component

The aerodynamic momentum contribution is in general secular both IOP and POP since the LVLH aerodynamic torques change with time. momentum contribution can be made cyclic if a ($,e,$) solution can be found such that

The IOP

2 7 1 I W o I (ax cos W ~ T - a sin W ~ T ) d.r = 0

0

Z

271/WrJ I (ax sin w 0 ~ + aZ cos q , ~ ) d.r = 0

0

(from Eq. 15a).

-15-

The aerodynamic momentum contribution for a spacecraft in a circular in body coordinates orbit may be derived as follows.

for a spacecraft in a circular orbit is given by The velocity vector

where V and V X Y

Substituting Eq. (20) into Eq. (13) gives the aerodynamic forces acting on the spacecraft .

are the X and Y components of velocity in LVLH coordinates.

-16-

2 =: -$pCDA3 I Vx(s$s$ + c$sec$) + V Y (-s$c$ + c$sBs$) I

X

vxcecJl + v cesll,

Vx(-c$s* + s$secJI) + v (c$c$ + S$SBCJI) Y

Y

VX(S$S$ + c$sec$) + v Y (-s$c$ + c$sesJI)

Substituting Eqs. (21a-c) into Eqs. (15f) through (15h) gives the aerodynamic torque in LVLH coordinates acting on a spacecraft in a circular orbit.

Vxc8c$+V ces$ Y a = BpCDVy

X [r ly [ceca( -s$c$+c$ses$)-(s~s$+c$*ec$ )ces$ I

+( - C$SJI+S$S9C$ ) ce s$]]

-1 7-

I + r3Y[c +( -s$c$+c$s9s$) ( -c$s$+s$sec$)

-se(vx(s$s$+c$sec$)+v (-s$c$+c$ses$)) a Z = 1pcDVX { A l IVxc9c$+V Y ces$ I Y

r and r are in ix’ iy’ iz For an articulating spacecraft, p, V Ai, r Y’ general time-dependent (in the case of a non-articulating rigid spacecraft, only p and V

inertial coordinates, fiA(t), is given by

change with time). The aerodynamic momentum contribution in Y

HJt) =

J

c

a (T) cos w0.r - a (T) sin w0‘c X Z

ay(.r>

a (T) sin w o ~ + a (TI cos w0‘c X z

d.r

where ax, a and a are defined in Eq. (22). Techniques and special

conditions resulting in zero secular momentum aerodynamic contributions are discussed in Section 5.

Y’ 2’

4 . 4 Solar Component

The solar momentum contribution is relatively small coinpared to the gyroscopic, gravity gradient, and aerodynamic momentum contributions for Space Station altitudes. Since L W solar torques change with time, the solar momentum contribution is secular both IOP and POP.

-19-

CHAPTER 5

RESULTS OF TEST CASES

In order to better understand the concept of momentum management, various attitude solutions for hypothetical test cases were considered using the ARCD program. Station Configuration (ref. 31, as shown in Fig. 1, was chosen as the subject spacecraft. attitude, with the dual keel aligned close to local vertical. Five comparatively massive modules (a U.S. habitation module, a U.S. laboratory module, a logistic module, a Japanese Experiment module, and a European Space Agency module) are located on a porch at the center of the station. hybrid power supply of solar arrays and solar dynamic dishes articulate to track the sun, and radiators articulate to anti-track the sun. Appendix A contains mass properties for the Space Station Configuration studied. All test cases were flown at an altitude of 250 Nm.

The reference Initial Operational Capability (IOC) Space

The station is designed to operate in an Earth-oriented

A

5.1 - Case 1 - Non-Articulating; Constant Density, Non-Rotating Atmosphere

The first case simulated was that of a rigid Space Station flying through a non-rotating atmosphere with constant atmospheric density (4.84E-12 KG/M**3) and no solar pressure.’ the control momentum which must be supplied by the AMCD in order to maintain an arbitrary attitude of ($ ,€ I , c$ ) = (-7,-1,3) in degrees (Case 1A). for this case, gyroscopic, gravity gradient, and aerodynamic torques in LVLH coordinates are constant, no IOP secular momentum requirements are generated. general be secular; however, a single DOF may be used to eliminate this secular momentum. maintain an attitude of ($,€I,$) = (0,-.36,0) (Case 1B). This solution balances the POP gravity gradient momentum shown in Fig. 4 with the POP aerodynamic momentum shown in Fig. 5 to give zero POP control momentum requirements. eliminate the cyclic momentum requirements arising from the gravity gradient and gyroscopic effects.

Figure 2 shows a time history of

Since

For an arbitrary attitude the POP control momentum will in

Figure 3 shows a plot of the control momentum required to

Furthermore, the Y principal axis could be aligned POP to

5.2 Case 2 - Non-Articulating; Variable Density, Non-RotatinR Atmosphere For a second case, a rigid Space Station was simulated flying through a

Figure 6 shows the density profile over the orbit

Figure 7 shows a time history of the control momentum

non-rotating atmosphere with a Jachia 1970 (570) atmospheric density model and no solar pressure.

(f=230, A =140).

required to maintain the attitude ($,€I,$) = (0,-0.36,O) from Case 1B. case is denoted as Case 2A. Both the POP and IOP control momentum requirements are now secular. This attitude no longer balances the POP gravity gradient with the aerodynamic momentum. The aerodynamic momentum contribution, shown in Fig. 8, is secular IOP due to the presence of a time- varying LVLH aerodynamic torque a shown in Fig. 9. It follows that the

P This

2’

-20-

. .

-21-

0, CU P cu 0, m

..

..

Lo 03 I a 1

I m a

v

u S -4

3 3 0 d

a -3 3 a cl 3 0

cu ln O s U A 0 C

.I

CL * * u- LL c a

+

c a c U \

U

z

a a

vt- I

u w

3 0 0 n

0 n W H K

a T

W v,

v 3

a

v 0 n

0 n c3 Y K

a c

w m a mlk : 0 3 v 0 n

0 n c3 U K

0, W a

0

A

u W 0 Y

W A > 'U c 2 3

I-

a

H

(Y

0

m

c v, c W c

I ffl

-22-

(33 ‘VI I

a v

CD CD m

c.

W W 0 Y

m QJ d

61 61 c3 c3 c

h 3 v u o w

Y

0

Y

W Q ) rn .y a u v 3

u t3 h

o w H,P

01

W vl U V 3 v 0 W

0

W U

U . . E 8 0 3 c 3 r u 43

c3

H

5 L 0

m

h

(3 W 0 v

W J>

L Z I

I-

a

U

-23-

. .

W

1 I I I I I I

m 6) CD 0

Q

m W m

-> t

h 3 VCJ o w 0 H

0 CI

(3 clc

W Ln a U 3

U

0 H

0

W 0 v

U

M .I4 Fr

..

-24-

I I I I

J

-25-

n

Q

3

-26 -

Ix) t " I IU

bl

5 m W W - I IU

a m W m

m W W 4. m

3 V 0 H

0 H W c(

H

E 0

m

A

W W 0 v

W J X W G z x

I-

a

U

Iy

0

m

I

-27-

. D . I

s i t

w a -

a a W m

m 43 W m

I I _ I - w m ,” ”,

-28-

. .

h

a

I-!! " 1

W =- Ln U V

Q, a I

W 0)

c

(D

(D co 0

W

- w m a a v l l x

0 3 u w o w

n

-29-

determination of an attitude such that a = 0 can be used to eliminate

secular momentum build-up IOP. Z

For a non-rotating atmosphere, V = V and V = 0, and from Eq. ( 2 2 c ) X Y

the aerodynamic torque a is given by z

One possible means of eliminating a is to roll the spacecraft so that z

the cp-cg of pressure offset looking down the LVLH x axis is in the orbit

-30-

plane (which results in a zero moment arm). in Eq. (24) it can be seen that

By setting J, = 8 = 0

lY

lZ

tan 9 = - r

results in a = 0, and thus, zero secular momentum IOP. For the Space

Station under study a I$ = -35" was calculated using Eq. (25). pitch angle 8 can be utilized to balance the POP gravity gradient momentum with the POP aerodynamic momentum. different values of (I$,$) such that the aZ = 0. plots of the control momentum required, aerodynamic momentum, and the aerodynamic torque, respectively, for an attitude of ($,e,$) = (0,-.92,-34.79) [Case 2Bl. Observe that the secular momentum requirements have been driven to zero.

Z A non-zero

Then, Eq. (24) can be solved for Figures 10, 11, and 12 show

Note that aZ can also be eliminated with solutions involving $ f 0 by

contributions to a . For the case of a pure yaw, by balancing the ;.x

setting 8 = I$ = 0 in Eq. (241, it can be seen that 1 1 Z

tan2$ = - Al lY (26)

results in a = 0, and hence, the IOP momentum is zero. For the mass

properties in Appendix A, a $ = 58' was calculated using Eq. (261, and an attitude of ( $ , e , $ ) = (54.0,5.15,0) was determined using the program ARCD. Figures 13, 14, and 15 show plots of the control momentum required, aerodynamic momentum, and the aerodynamic torque, respectively, for this attitude (Case 2C). Again note that secular momentum build-up is zero since

Z

a " O Z

5.3 Case 3 - Non-Articulating; Constant Density, Rotatinv Atmosphere

For a third case, a rigid Space Station was simulated flying through a Figure 16 rotating a'aosphere with constant density and 110 solir pressure.

shows a time history of the control momentum required to maintain the attitude ($,e,@) = (0,-0.36,O) [Case 3A] from Case 1B. Although this plot seems to show only cyclic momentum requirements (due to the large scale), the IOP momentum requirements are in fact secular and result from a secular IOP aerodpamic momentum contribution of approximately 250 Nms for this

-3 1-

6) W Q) Q)

a m ' w w

a ? + +

-32-

. .

A

Q,

b)

6) 6) 6) V

I

6)

W W W 03

1

I '

i

n 3 0 0 o w H C 3 I -

I i- :

W 3 v 0 i H

-I

U

m L1! 0

A

W W 0 w

W J> 'U u ZI a I- H m

0

h

t3 W 0 Y

W J X w u LI

I- a

H

a 0

m

-33-

Ln m W m u-

..

.. 7

M I c3 3 Q I

m -r-

u S

3 3 0

-4

4

a 3 3 a -3 3 0

a a s U L: c c

./

n * * U

cc c a

c

L [1: C U \

U

z

a a

-5

n

t

1

I- U m OL 0

L U

Q)a . s Q) --) W J -

u 1 w

V a o

h 3

I \ I .. ."

-34 -

A

Q

In - - L

\ I-- m a

H I" giP L

CUO U

W

t

W

3 V 0 H

0 n (3 H D:

V CU

w m m a v 3

v w o w n o 0

a 0

n

Y $: H

v l r x u o V 3

w m

A

v w o w - 0 I -

m

t9 W 0 Y

W J J N '(3 v L I a I- U

-35-

d

tD d

cu

..

.. - c1) 03

1 a 3

I a d 7

m S

3 AJ 0

-d

-.I

a 3 3 a 3 3 0

W m U Q U 3 0 n

.. N * 4

v)

W 0

a

H

V ai 0 Ln \

v, a z

a

a V

6)

3 0 0 H

0 H

W

m

c3 W 0 .d

W JN u u LI a I-

m ry 0

U

A

t3 W

-36-

n 3 v w o w n o I -

a W a tr:

1 6,

W 6,

CI,!

I

"I Nl-

C U I - "I -

W VI U V 3

W v IT Tu

c W I VI

U J - - C a l U L U W D L O C O l Y f 3 3 W - L I E - 0

-37-

H

(1: 0

m

h

W W 0

w x Jl- (3N za ai-

v

I- Y

m (Y 0

v, 0

Ln e w

..

.. - W 03

I a 3 CT

I d- - U S --. 3 J 0

a 4

-3 3 a L, 3 0

al ln U n U L)

U n

.. (u * * v) a W 0 H

U

c L': \

Li:

z

a a

I-

L- W J na

\ "'t 3J

0 Q U Y

I l l 1

6)

6) m ~

m N

m m h3

I - U

I -3a-

vehicle, as show in Fig. 17. An inspection of Fig. 18 shows that the secular IOP aerodynamic momentum contribution results primarily from the time-dependent torque ax due to the effect of the rotating atmosphere, since

a is nearly constant for this case. 2

An attitude was determined using ARCD which produced cyclic control

In the case of pure roll ( 4 ) motion only, the momentum requirements by eliminating a

secular aerodynamic momentum. IOP aerodynamic torques in LVLH coordinates may be obtained from Eq. (22a) and Eq. (2212).

and not allowing a to contribute X 2

+A3 V S$ I Y I[ -r,YS@ll

Since Vx )) V the V2 terms may be neglected in Eqs. (27 ) to give Y’ Y

a X N Ipc D v All ( -rlys$ - r,2c@)

a = $pCDVx {AllVxl( -rl c@+rlzs$) + A2 I Y V c$ I r,2sc$ - A3 z Y

By setting ax = 0 in Eq. (28a), it can be seen that

-39-

. .

3 ‘3 0 H

0 U

‘3 U

- 0 L a

n

‘3 W 0 Y

W A N ‘a u L X a I- Y

m Ili 0

A

‘3 W

- w Ln U v 3

a m u

U

m Iy 0

A

c3 W n Y

W A X w u L I a I- Y

aL 0

m

-40-

.r

I

3 0 0 U

0 U

c3 U u U m W v)

v 3

a

v o w H O

IY

W w m a o m u

-41-

lZ

lY

tan 4 - r -

An examination of Eq. (28b) reveals that the only time-dependent term . affecting az

has

It can be shown that a spacecraft in a circular orbit is IVYI

so a can be written as Z

where C, and C, are constants.

contributed by aZ is given by

For this case, the secular IOP momentum

2.rr/w,

(c,+ c,( cos w o ~ l ) cos w0-c d.c HA Z (271/w0) = s 0

Since the integrals in Eqs. (32) are identically zero for constant C, and

C2, the (O,O,O) solution given by Eq. (29) will result in purely cyclic IOP control momentum requirements. For the Space Station under study a 0 = 55' was calculated using Eq. (291, while an attitude for Case 3B of ($,e,@) = (0,0.81,53.0) was determined using the program ARCD. control momentum requirements, aerodynamic momentum, and aerodynamic torques are shown in Figs. 19, 20, and 21, respectively, for this case. Note that although aZ is not constant, the IOP aerodynamic induced momentum contribution is zero.

Plots of

-42 -

W 6) W W I h Y ,

3 V 0 H

0

(3 I4

H

h

c3 W 0 Y

W A h IW L L I a I- rn U

u 0

A

W W

-43-

La m W d

m

..

.. Y

W 03

I c3 1 a

I d- 7

m c -d

3

0

[1 4

3 S Q -3 3 0

a, In 5 II a -3 u Q

.. N * * Ln cf W 0 H

0 cx Q Ln \ a m a L

/

C L L

C L u w I-E 0 0 W E

\ U 3 u

UQ O L u >

-

- E -

- c 63 U 63

43 ID

Y I I

sc

$b

sc

sc

ii 6r

W

W

43 W -4 0

h 3 v a o w H O I -

(Y 0

L u

? E 6) -- W-J

0

w $2 -I>

ci - w m r n W u o V 3

v w o w

h

63

Q) h; Q N 0)

-44-

zl- W E %- og

2 1 I

v u \ O L

c- 1

Q)

cu I

m

Q) W is m

m

W m

V 3

a

v 0 H

0

U H

U Iy

I 1 m r)

0 m

H (D

0

n

u W 0

w x JI- '0 n. z a ac

W

!- U

rY 0

m W ul

-45-

5.4 Case 4 Non-Articulating; Variable Density, Rotating Atmosphere

For a fourth case, a rigid Space Station was simulated flying through a rotating atmosphere with a Jachia 1970 (570) atmosphere density model and no solar pressure. required to maintain the attitude (+,e,$) = (0,-0.92,-34.79) from Case 2B [Case 4A]. aerodynamic momentum contribution as shown in Fig. 23. the secular IOP aerodynamic momentum contribution results from a

which are now both time-dependent in LVLH coordinates.

Figure 22 shows a time history of the control momentum

Secular IOP control momentum requirements are due to the Figure 24 shows that

2’ and a

X

The program ARCD was used to determine a unique 3 axis attitude solution of ( $ , e , $ ) = (-39.2,8.09,-27.2) for Case 4B such that control momentum requirements were purely cyclic. secular aerodynamic momentum contributed by ax with the aerodynamic momentum

contributed by a to achieve zero secular IOP control momentum requirements. In addition, the POP gravity gradient momentum balances the POP aerodynamic momentum to achieve zero secular POP control momentum requirements. Figures 25 through 30 show the control momentum requirements, control torque requirements, aerodynamic momentum, aerodynamic torques, gravity gradient momentum, and gyroscopic momentum, respectively, for Case 4B.

This attitude balances the

Z

In all of the previous cases, no attempt was made to reduce the control momentum cyclic peaks which influence the AMCD sizing. Case 4B that although large angle attitude solutions may give rise to zero secular control momentum requirements, they also tend to result in large IOP control momentum and control torque cyclic peaks. These huge peaks are caused by the gravity gradient and gyroscopic effects, which in general become large whenever the principal axes are not aligned with LVLH.

It can be seen from

5.5 Case 5 - Non-Articulating; Principal Axis Attitudes

For a fifth case, the Space Station was simulated flying through a rotating Jachia atmosphere at a principal axis attitude of (+,e,$) = (-0.77,0.33,-5.66) [Case 5Al. No gravity gradient or gyroscopic torques can be generated at this attitude, so the control momentum requirements shown in Fig. 31 are due solely to the aerodynamic momentum contribution. For this attitude, relatively small peak control momentum requirements exist; however, a large secular control momentum build-up must be incurred in order to maintain this orientation.

The program ARCD was used to determine an attitude of ($,e,$) = (-0.77,-0.32,-5.66) [Case 5B] which minimized the secular momentum without substantially increasing the peak momentum. In section 3.1 it was shown that a DOF in pitch about the principal axis solution existed such that the IOP gravity gradient momentum and the gyroscopic momentum remained zero. This DOF was used by ARCD to balance the POP gravity gradient momentum shown in Fig. 32 with the POP aerodynamic momentum shown in Fig. 33, to give zero secular POP control momentum requirements, as shown in Fig. 34. It was not possible to reduce the secular IOP control momentum requirements resulting from aerodynamics with small yaw and roll angles.

-46-

n

ul r- P m

I

N ~r

L

bk

W

-p n

c3 W 0 Y

W J h (3c; LI a + m U

CY 0

A

W

-4 l-

/

m

Q m Q * I

6)

6) W 0 73 m

I ’

1

n 3 u a o w H O I -

i -1- GI

CI

13 W 0 Y

W -I> w a LII a I- H

Ix 0

m

I I

I

-48-

. .

D t -4

3 J 0 d

a 3 3 Q r: 3 0

e, m 0 3 0 3 u c3

.. cu * * Lo

W 0

a

H

n

wl P

Q- m

1

n 3 vc3 o w

Y

W h

U

w m m c r a 0 v 3

v u o w

A

n w m t n ( Y a 0 U 3 v w o w

h

I d I -

- I

N 1 1 a 1

m 0

I 4 w m

-49-

W

W a 0

W

6) W 4 m

n 3 v u o w H O I -

I - - U

A w m m u u o u 3

v u o w H O

0

CI

Y 9' - w

Ln U 0 3 v 0 H

H a c1: 0

h

U W 0 Y

W J > -23 c 2 3 a I- U

-50-

h

cu b cu I

W U

E \O

m - OVI E3 c -

t-m -

a m

a a +-- C-(C.I I =I- L a w UE

7 z - H

W ” z - U W i-3 ao +E

-

t

6)

1 1: I:

h

W W 0

n 3 0 0 o w H O I “ 01

- w m a 0 f J Y W

0 3

A

u w o w ~n i l-

H W Q , u J ( Y

UJ a v

a o J-- c P) u L uozl-E0-I +ouo3w -L ’ E- o

-51-

. .

m r m 0 "

W 6 ) w m L n - a , .

6)

CJ 43 In I

63

W W Ln - l

n 3 w u o w n o I -

-52-

I I I

3 0 0 H

0 H a c(

n

(3 W 0

W N JL '0 N za a+

v

I-

m Y

[1! 0

n

- w c n W E u o V 3

n

Y n o I -

-53-

m 6) 6) a 6) V

l

3 0 0 H

0 H W CI

1

- w m m n : u o V 3

n

u w 1 E::

0 m N d m 03 W

I v, a

I- m U

CY 0

-54-

Q) In

N d

W m

.. .*

W co l a 3 a

I v N

U S

3 3 0

Q

-d

d

3 S

3 3 0

rl, ro U II 0 3 0 0

.. N * * Ln

W 0

a

H

0 E 0 Ln \

L7

z

a a

Q)

W

I I I

a a to 0

03 W ~. m

n 3 0 0 o w

n

I w

i- :

U

m CL 0

A

W W 0 v

c13 W M I U - J I Y

u o

-55-

D 9

n W

rn U

0 0 o w H P I - n

Y

w m m u u o v a - v w o w

Y

H

W M

a 0 a

VIE

V

v w o w - 0 I -

-56-

. .

W W

W W 0

m 6, W m

-57-

v, 6)

N N

W

.I

.I

7

Lo 00

I W 3

1 a 6) cu

u c 3 3 0

-d

4

a 3 3 D 3 3 0

a fi c c c i c c

* * U

LL c a

I-

C. 0: C v \

U

z

a a

UU II I- a K-- w w cz

a> - 0 u - W a 1 I

m a

1

a a to 0

- W M w r y c o 0

.- ut3 o w - 0 I -

-58-

i

7 W

W CD 0 -la W

. m

o w o w

0

-59-

In summary, Case 4B had zero secular momentum requirements, but large attitude angles with corresponding large cyclic momentum requirements, while Case 5A (principal axis attitude) has small cyclic momentum requirements, but non-zero secular momentum requirements. Case 5B maintained the small cyclic momentum requirements utilizing the degree of freedom to reduce secular momentum requirements. momentum and torque requirements are relatively small at an attitude close to the principal axis attitude for a rigid spacecraft.

Figures 34 and 35 show that the peak control

5.6 Case 6 - Articulating; Gravity Gradient Gyroscopic Torques Only

In order to illustrate the effects of part articulation on the gyroscopic and gravity gradient momentum contributions, a sixth case was simulated where an articulating Space Station was flown with all aerodynamic and solar torques zero. with articular parts which rotate about their respective c.g's at orbit rate and have their principal axes aligned with the articular part axes was simulated. Since gyroscopic momentum is always cyclic if articular parts rotate at orbit rate (see Section 4.11, and for this special case gravity gradient momentum is cyclic IOP (see Section 4.2), a single DOF is sufficient to ensure that control momentum requirements are cyclic by eliminating the secular POP gravity gradient momentum contribution. 36 and 37 show time histories of the gravity gradient and gyroscopic momentum contribution, respectively for an attitude of ($,e,$) = (0,0.33,0) [Case 6Al. Note in Fig. 38 that the POP gravity gradient torque is not constant due to part articulation, even though the spacecraft maintains a constant attitude.

For Case 6A, the special case of a Space Station

Figures

For Case 6B, a Space Station with actual articular parts (which do not rotate about their c.g. and do not have their principal axes aligned with the articular part axes) was simulated. histories of the gravity gradient and gyroscopic momentum contributions, respectively, at the same attitude of ($,e,$) = (0,0.33,0) used in Case 6A. This attitude still eliminates the secular POP gravity gradient momentum; however, the IOP gravity gradient momentum is now secular and equal to approximately 250 Nms.

Figures 39 and 40 show time

5.7 Case 7 - Articulating; Full Environment For a final case, an articulating Space Station was simulated flying

through a "full" environment comprised of a rotating Jachia atmosphere and solar pressure. In Case 7A, the program ARCD was used to determine an attitude ($,e,$) = (-16.3,-2.34,15.8) which eliminated secular control momentum requirements, as shown in Fig. 41. This attitude balances the secular gravity gradient momentum contribution, shown in Fig. 42, with the secular aerodynamic and solar momentum contributions, shown in Figs. 43 and 44, respectively. Since the articular parts rotate at orbit rate, the gyroscopic momentum contribution shown in Fig. 45 is purely cyclic. Figure 46 shows that relatively large peak control torques (60Nm) and hence large peak control momentum (99,300 Nms) requirements are necessary to maintain attitude.

-60-

V 0 H

0 H

(3 U (Y

m Lo

W VI a 0

v 0

I - O

-61-

-62-

. .

.

ct) In

* m

6 l

..

.. Y

cn I

m c3 3 Q:

I N cu

m S -d

3 3 0

a 3 3 Q 3 2 0

al In 0 s D 3 U c3

-.I

.. N * * Ln a W 0 H

W

0 cn \ d Ul a L

Q

61

6)

6) (D m

c

W

6) W m

A

f.3 W 0 Y

I- U

m (Y

0

L a

W -

Y

wae, V ) u a 0

W VI

0

E > m \ 3

a

0

H

N - a W ct, W

1 W v)

-63-

-64-

. .

6)

6) a 0

ID

6) 01 m

I 1 I 63

I

a m * 0,

1

z: c -J 3 U H t- cr

H

a

a QI cr,

W m U U z H l-

A 3 v H t- (Y a m

a

(2)

W w U V z c U

a

h

C3 W 0 Y

W N

Lo ax

Id W

c m Iy 0

U

h

W W 0 Y

w >

a x

*= W

c H CD (1: 0

A

W

W M m r Y E 3

-05-

Q)

6 c3 6l d I

Q)

6 (D 0

63

c3 W m

- a c3

n

c3 W 0 Y

a x I- U

m rY 0

- W W

- w LD U 0 t

c U

a

0 U

M *d .La

,

-66-

a K w o I- T

-Iu I" L

(30 r x n I-J 'g ao L t ao

m 5 8 8 5 03

8

Q a Q Q d

-67-

. .

G3 v a, .. c .. IT, m

CD 00 I

W VI

1 G3 cu

a.

p: S

4 3 0

a 3 S Q -3 3 0

a, Y, 5 II a -3 a 0

-4

d

.. N Jt * Lo

W 0

a

H

0 DL

v, \

Ln

L

n

a a

<$-- ?a

-68-

W VI a U

- w VI a V r U I-

m m

Y

m 13! 0

n

W W 0 Y

W 1)- oa L X a I- W

(1: 0

m

h

W

w r n m w P O

-69-

a

CD 0

m

8 m J n m

z W l-

1 3 v H c

a h

a W 0 v

W Jh b U

a L I

I- I4

- w v) U 0 z u I- i a

a II: 0

h

W W 0 Y

W J: ’0 ( 2 :

c a

U

d

c rc N a

1

-70-

8

1

H

V 1 L

-71-

. .

d d

m N

tn PI

..

..

LD 03

I a. w m

1 43 cu

a S

& -4 0

a

-4

&

4 3 Q 3 3 0

a3 a 0 12 0 3 U 0

.. tu * * Ln

li C

a

* c; Q? c U \

LT

2

a a

(D

7 - 1 E L U I- J 3 v H I-

a

] -,%

-1 ( D d G - A

Q d

(D m * In I I

I - w a w

U

"I Fl- Y

-72-

In order to minimize the peak control torque and momentum requirements, the program ARCD was used to determine an attitude ($,€I,$) = (-0.1,0.29,-4.80) for Case 7B close to the principal axis solution (Case 5). Figures 47 through 52 show the control momentum requirements, gravity gradient momentum, aerodynamic momentum, solar momentum, gyroscopic momentum, and control torque requirements for Case 7B, respectively. The peak control torque (4Nm) and peak control momentum (4100 Nms) requirements have been reduced significantly for this attitude. In addition, the secular POP momentum has been eliminated with a small bias pitch angle offset; however, the secular IOP control momentum is non-zero and has a magnitude of 1600 Nms. The secular control momentum requirements result from the secular IOP aerodynamic momentum contribution, which cannot be eliminated or counterbalanced with secular IOP gravity gradient momentum resulting from part articulation at an attitude near the principal axis solution. indicates that periodic roll and/or yaw dumping would be required to reduce the accumulated secular momentum build-up.

This

-73-

7

c3

m .. c .. c- y

LO 03

1

W v,

I c3 (u

a

m S -4

L)

-3 0 4

a 3 3 n c, 2 0

QJ ln W a U r) a Q

.. (u * * m

W a n M

c: e c U \

U

z

a a

r- W VI

m W W 0

m

W W @Y

h

W W 0 Y

L I-

J 3 v I4 I- d a m

U

a

b

I-

co U

a 0

L U

.s W a

D

- W M VIE u a 0 L U - C W a w

6) la 6)

a m b

L:

I- a

F U I w m

-74-

, . .

d m W In m

..

.. F

a 03

I

W v) I c3 cu

e

m C -4

3 3 0

R

d S Q -3 3 0

al ro 0 II U 4J

U 0

d

.. Rl * * VJ

W 0

a

H

0

0 U J \

cn a L

a

a W W c

z l-

-I 3 V H I- o!

H

a

U

m F W v)

V z H I-

a

a

v

n W M

a a V l K

V r I - w a w

H -

I-! w c o

-75-

Y

m a! 0

n

W W n Y

W -I>

LI

!- W M DL 0

a a a

n

W W

-76-

r: F

J 3 U H I- oc a m

U

a

rc W m U L H c -I 3 U H I- (1!

a

a

a m r- W vl

0 z Y n I - w a w

a

U

U

m a w m

n

W W 0 Y

W J N bv, LI a c m U

Q! 0

A

W W a v

W -I> W v , T I

I-

a

H

K 0

m

-77-

. .

A

m w .

E+ W

6)

m (D m 4 m

- w m a 0 VIE

v L U n I - - a w

-78-

6)

$$

-79-

CHAPTER 6

SUMMARY

This study represents a comprehensive analysis of the relationship

The sources of attitude control torque requirements were between spacecraft attitude and attitude control angular momentum requirements. examined, specifically, gyroscopic, gravity gradient, aerodynamic, and solar radiation pressure. attitude/momentum relationship, and were used to determine attitude solutions to eliminate secular angular momentum build-up and/or minimize cyclic angular momentum peaks for non-articulating spacecraft. of aerodynamic induced disturbance torques were examined by studying the individual contributions due to a variable density atmosphere, a rotating atmosphere, and spacecraft part articulation. Gravity gradient and gyroscopic contributions to the cyclic and secular angular momentum requirements were examined for both fixed and articulating spacecraft. Peak cyclic momentum values due to gyroscopic or gravity gradient effects were as large as 110,000 Nms for the Space Station configuration studied. The effects of the dynamic atmosphere contributed up to 1300 Nms of secular angular momentum in the orbit plane (IOP) after one orbit, while non- symmetric articulation gravity gradient induced secular momentum contributed up to 250 Nms IOP after one orbit. secular momentum perpendicular to the orbit plane (POP) could always utilize a pitch angle bias to offset the average aerodynamic and gravity gradient sources. Gyroscopic effects, even with part articulation, did not contribute to secular momentum accumulation.

Analytic equations were derived which showed the

The effects

For the configurations studied, the

A series of seven test case sets were designed and solved numerically to demonstrate the validity of the analytic results, complete with selected computer generated plots of the torque and momentum requirements and contributions.

For purposes of quick reference, Table 1 lists the sources of secular angular momentum build-up for both fixed and articulating near Earth orbiting spacecraft.

-80-

"

' I )

TABLE 1 - SOURCES OF SECULAR ANGULAR MOMENTUM BUILD-UP

IOP POP (in orbit plane) (perpendicular to orbit plane)

FIXED (Non-Articulating S/C)

Gravity Gradient NO

Gyroscopic NO

Aerodynamic

Solar

ARTICULATING

1.

2.

3.

4.

5.

6 .

Gravity Gradient

Gyroscopic

2 YES

4 YES

5 YES

6 NO

1 YES

NO

3 YES

4 YES

1 YES

6 NO

Aerodynamic YES YES

Solar 4

YES 4

YES

Can be eliminated with principal axis attitudes or pitch bias aero off sets

Due to atmospheric dynamics, can be eliminated with "large" angle attitudes (roll, yaw)

Can be eliminated using pitch bias gravity gradient offsets

Small compared to aerodynamics for Space Station altitudes

Zero if all articular parts rotate at orbit rate about an axis coincident with an articular part principal axis and passing through the articular part cg.

Assumes final and initial configurations are identical (e.g., articular parts rotate at orbit rate)

-81-

CHAPTER 7

CONCLUSION

The angular momentum control requirements for Earth orbiting spacecraft result from a variety of environmental effects. realistic, simulation of articulating spacecraft, three rotational degrees of freedom were utilized to determine attitudes which resulted in zero secular momentum build-up; however, this required large angle attitudes for the Space Station configuration studied and large cyclic angular momentum peak values. Three axis solutions were found, however, which resulted in acceptably small cyclic and (non-zero) secular momentum requirements. This, of course, implies the need for periodic desaturation of the angular momentum control devices. Deviations from this attitude resulted in large cyclic momentum penalties for relatively modest reductions in secular momentum build-up. Since the primary contributor of the secular momentum is aerodynamics, appropriate mass and area adjustments to the spacecraft configuration might be considered to reduce composite center of pressure - center of gravity offsets, thus minimizing torques resulting from the aerodynamic forces. Additionally, symmetric articular components designed to rotate about their center of gravity would eliminate gravity gradient induced secular momentum build-up due to part articulation.

For the most general, and

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CHAPTER 8

REFERENCES

1. Heck, M.L., Orr, Lynne H., DeRyder, L . J . , "Articulated Space Station Controllability Assessment," AIAA Paper No. 85-0024, Jan., 1985, Reno, Nevada.

2 2. "IDEAS Users Manual," Structural Dynamics Research Corporation, Dec.,

1985, San Diego, CA.

3 . "Space Station Reference Configuration Description," Systems Engineering and Integration Space Station Program Office, NASA Johnson Space Center, JSC-19989, Aug., 1984 (updated 1986).

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Appendix A - Space Station Physical Properties

total weight = 192,000 kg

location of cg

(origin located at center of transverse boom)

(x, y, z) = (-3.82, 0.91, -7.72) m

= 1.52 E8 kg-m2 IXX inertia tensor elements

I = 7.24 E7 kg-m2 YY

I = 1.01 E8 kg*m2 zz

I = 1.05 E6 kg*m2 XY

= 4.00 E5 kg*m2 IXZ

I = -2.86 E6 kg*m2 YZ

projected areas in body x : A , = 2110 m2

y : A 2 = 884 m2

z : A , = 995 m2

cp - cg offset looking in body x : r, = -1.82 m, r = 2.56 m

y : r = 1.67 m, r2 = -5.50 m

z : r = 2.55 m, r3 = 9.22 m

Y lZ

2X Z

3x Y

. L

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Standard Bibliographic Page

. Report No. NASA CR-178219

2. Government Accession No.

'. Author(s)

Brent P. Robertson and Michael L. Heck

17. Key Words (Suggested by Authors(s))

). Performing Organization Name and Address

PRC Kentron and Analytical Mechanics Associates, Hampton, VA 23666 Inc.

Hampton, VA 23666-1398 .2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, DC 20546

18. Distribution Statement

3. Recipient's Catalog No.

19. Security Classif.(of this report) Unclassified

5. Report Date

January 1987

20. Security Classif.(of this page) 21. No. of Pages 22. Price Unc las s if ied 91 A0 5

6. Performing Organization Code

8. Performing Organization Report No.

10. Work Unit No.

11. Contract or Grant No.

NAS1-18000, NAS1-17958 13. Type of Report and Period Covered

483-32-03-01 15. Supplementary Notes

Langley Research Center Technical Monitor: L.J. DeRyder

L6. Abstract

The relationship between attitude and angular momentum control requirements is derived for a fixed attitude, Earth orbiting spacecraft with large area articulating appendages. Environmental effects such as gravity gradient, solar radiation pressure, and aerodynanic forces arising from a dynamic, rotating atmosphere are examined. It is shown that, in general, each environmental effect contributes to both cyclic and secular momentum requirements both within and perpendicular to the orbit plane. The gyroscopic contribution to the angular momentum control requirements resulting from a rotating, Earth oriented spacecraft is also discussed. Special conditions are described where one or more components of the angular momentum can be made to vanish, U L briviiis ~ ; r c l y c y c l i c . Cnmpiiter generated plots for a candidate Space Station configuration are presented to supplement the analytically derived results.

Angular Momentum Requirements Attitude Control Requirements

Unclassified - Unlimited

Subject Category 18

For sale by the National Technical Information Service, Springfield, Virginia 22161 N A S A Langley Form 63 (June 1985)