Optimal aerodynamic attitude stabilization of near earth satellites

'

7

:-

('1 ?

OPTIMAL AERODYNAMIC ATTITUDE STABILIZATION OF NEAR EARTH SATELLlTES

VUEGïUlC 'J v ..

by

Sf

UOh

è:~

Rangaswamy Ravindran

,

OPTIMAL AERODYNAMIC ATTITUDE STABILIZATION OF NEAR EARTH SATELLITES

by

Rangaswamy Ravindran

Submitted DecemberJ 1970,

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..

ACKNOWLEDGEMENT

The author is pleased to record his indebtedness to the members of his thesis committee, Prof. B. Etkin (Chairman), Dr. J. B. French and Dr. P. C. Hughes. In particular he wishes to express

his gratitude to Dr. P. C. Hughes, his supervisor, for suggesting the problem, constant guidance and unabated interest shown through-out the course of this work.

The computations w.ere carried out at the Insti tute of Computer Science, University of TorontQ.

Financial support for this work was received from the

USAF, Office of Scientific Research, under grant AFOSR-68-1490. The author held the Applied Science and Engineering Fellowship, Wallberg Research Fellowship and the University of Toronto Special Open Fellowship during 1967-68, 1968-69 and 1969-70 respectively.

SUMMARY

A near Earth satellite orbiting in the a1titude range of 150 km to 450 km encounters sma11 but non-neg1igib1e aerodynamic forces. It is possib1e to generate sufficient aerodynamic torques by providing 'all-moving' control surfaces of suitab1e size to

achieve active attitude control. This report is a preliminary study of such an active attitude control system.

The satellite configuration considered has four a11-moving control surfaces and the dominant gravity gradient torques and aero-dynamic torques are considered in the analysis. The resu1ting equa-tions of mot ion are 1inearized and modern optimal control theory concepts are app1ied to synthesize a feedback control system for contro11ing the satellite by rotating the control surfaces to obtain the necessary control torques. The numerical studies carried out indicate that it is possib1e to control a ne ar Earth satellite by using control surfaces of reasonable size. Damping times of the order of from a few orbits to a fraction of an orbit seem reasonab1e.

.

'

_1. 2.

3.

4.

5.

TABLE OF CONTENTS Acknowledgement Summary Nomenclature INTRODUCTION 1.1 Gener al

1.2 Attitude Stabilization and Control Techniques

1.3

Present Study and Contributions

PRELIMINARIES

2.1 Coordinate Systems and Transformations 2.2 Disturbing Torques

2.3

System Description and Control Synthesis EARTH POINT ING SATELLITES

3.1.

Configuration, Coordinate System and Transformation Matrices

3.2

Assumptions

3.3

Determination of Drag and Dominant Torques

3.4

Determination of Nominal Control Angles

3.5

Equation of Motion

3.6

Synthesis of Feedback Control

3.7

Single Axis Pointing: A Special Case

3.8

Feedback Synthesis as a Linear Inhomogeneous System

RESULTS

4.1

Computational Considerations

4.2

Discussion of Results

COMPARISON WITH OTHER SYSTEMS AND CONCLUDIOO REMARKS REFERENCES APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D: APPENDIX E: APPENDIX F: APPENDIX G: TABLES FIGURES Atmospheric Prcperties Aerodynamic Analysis Satellite Lifetimes

Function Minimization with Constraints Optimal Regulators Evaluation of Derivatives Program Listings 1 1 2

3

4

4

7

16

19

19

21 21 22

24

30

31

33

34

34

36

43

45

.,

- - - _ . _ - - - "

NOMENC LAT URE

Note: Symbols of limited use are defined where used

erf(x)

~7

r

vector from spacecraft center of mass to the center of pressure of the i th control surface in $7 coordinate

system

error function of argument x orbital inclination

unit vector along the coordinate axis xi

k

unit normal to the surface element in

a7

coordinate system

normal pressure in

~7

coordinate system

position vector with respect to the satellite center of mass

t

7

unit vector along the direction in which the shear acts in

$7 coordinate system

u control vector

~ unit vector along the relative velocity vector xi vector

!

in ~ coordinate system

i x. J x A

Ca

CD'C ,C _{p T} D i components of x j

=

1,2,3

state vector

area of one control surface "wetted" surface area

principal moments of inert ia of the satellite about the xi,

x~, and x~ axes, respectively

cose

drag, normal force, and shear force coefficients, respectively derivative of moment coefficient with respect to angle of attack

drag

aerodynamic force

gravity gradient torque aerodynamic torque

K L Q R R R c

s

s

ex

1) e~ J. ei j (e). J. p

(J',

(J w

matrix of feedback gains

characteristic length of center body

positive semidefinite weighting matrix for the state

variables in the quadratic performance index

orbi tal radius

positive definite weighting matrix for the control variables

in the quadratic performance index

circular orbi t radius

molecular speed ratio

solution of the matrix Riccati equation

cross-sectional area of the center-body sine

orbital speed of satellite

relative velocity of 'air' with respect to the satellite coordinate axes in the

~

coordinate system, j

=

1,2,3 angle between the reversed incoming flow direction and the normal to the surface

orbital angle

angles defining orientation of body reference frame with

respect to orbital reference frame, correspond to roll,

pitch, and yaw respectively

ith control panel angle

rotation about

X~

axis

J

transformation matrix for rotation

e

around X. axis

J.

earth's gravitational constant

density of the atmosphere

normal momentum and tangential momentum accommodation

coefficients, respectively

shear in

~7

coordinate system

argun:ent of perigee

w E A (')i

(.)1

(.)7

(:-)

angular speed of earth

angle of attack of center-body t . . t . th t 1 f

per a~n~ng 0 ~ con ro sur ace

quantity expressed

in~

coordinate system quantity expressed in

X

7 coordinate system tilde

terms

operator: for arbitrary vectors a and b expressed in of an arbitrary vector basis {e}~ (~ x-~)

=

(e}T(a ~)

o

-a 3 a2 a 3 0 -al "" _a

₌

where -a ₂

o

(:) differentiation with respect to time

(

.

'

) differentiation with respect to orbital angle

tij transformation matrix from coordinate system

~

j

to Xi.

9

inertia matrix $1 coordinate system

~5

orbital coordinate system (see Sec. 2.1)

~7

body fixed reference coordinate system (see Sec. 2.1) $Hi coordinate system fixed in the ith hinge

$si principal axes coordinate system fixed in the ith control surface

I. INTRODUCTION 1.1 General

A large number of satellites have been launched to date and many more will be launched in the future to serve scientific, commercial and military aims. Irrespective of the purpose of the satellite, to be useful and effective, the satellite must be either attitude control-lable or i ts attitude must be measurable. By the former we mean that it should be possible to orient the satellite in a preferred or speci-fied attitude and by the later we mean that if the satelli te is un-controlled it should be possible to obtain a time history of its atti-tude by suitable instrumentation and telemetry. In gener al the satel-lites and the environment in which they operate are far from isotropic.

Consequently in the case of scientific satellites, since the measure-ment of physical parameters made by satellite-borne instruments will be

strongly dep ende nt on the orientation of the instruments, it is neces-sary to either control the attitude of the satellite precisely or pro-vide information regarding the attitude of the satellite to the experi-menter to enable him to interpret his results realistically. In the case of communications satellites of active relay type the effective power transmitted from the satellite depends on the satellite

trans-mitter power, transmission efficiency and antenna gaine If the antenna gain is increased the beam width decreases and consequently the satel-lite will have to be pointed more accurately towards the earth station. The need for the attitude control of military satellites is quite ob-vious.

Attitude control is one of the most critical areas of space technology. Due to the absence of significant natural sources of

damping in the outer space, theslightest internal motion or the smallest external torque is bound to accelerate a satelli te into unwanted and undamped rotation. A satellite in orbit may experience disturbing

torques due to unbalanced aerodynamic forces, unbalanced solar radiation pressure, gravity gradients, magnetic fields, propulsion units, micro-meteoroid impacts, ejected mass, emitted radiation, internal motion etc. The relative magnitude of these torques depends on the orbital radius, the configuration and other aspects of the satellite. Wiggins

(Ref.68) studied the environment al torques experienced by a satellite of cylindrical shape, 5 ft. in diameter and 30 ft. in length, in the altitude range of 100 to 1000 nautical miles. Similar studies have been carried out by others on individual satellites. In general the aerodynamic torques are important below 500 km. altitude and become domi-nant below 300 km. altitude for many satellites. Solar radiation torques become important at very high altitudes. The gravity gradient torques depend upon the radial dependence of the gravitational field, and these are important for near-Earth satellites. Micrometeoroid impacts with a non-zero moment arm to the center of mass can infrequently cause sig-nificant angular accelerations but the average effect should be smalle

The other torques mentioned above are under the control of the designer and their magnitudes can be made negligible by suitable design.

The accuracy requirements of the attitude control system de-pend on the specific application for which the satellite is intended. Many control concepts have been analyzed for application in satellite attitude control and accuracy, cost, weight, power requirement, avail-ability, reliavail-ability, and simplicity are some of the major considerations

in the selection of a suitable control scheme. 1.2 Attitude Stabilization and Control Techniques

A multitude offascinating schemes have been devised for attitude control of satellites. These can be broadly classified into

the f~llowing categories (Refs. 21, 50).

a) Passive control systems: Depend on the environment to generate the necessary control torques and require no onboard sensing, logic or power. In general useful for maintaining an a priori ideal attitude. Spin stabilization, passive gravity gradient systems, stabi-lization systems which exploit Earth's magnetic field, solar radiation pressure or aerodynamic pressure to generate torques fall under this category.

b) Semipassive Systems: Differ from the above systems to the extènt that these require some onboard momentum storage. Gravity gradient stabilization with passive damping by control moment gyros is a typical example in this category.

c) Semiactive Systems: These require partial attitude sensing, minimal onboard logic and momentum control by mass expulsion and/ or environmental torque sources. Spin configurations, dual spin configu-rations and certain types of gravity gradient systems come under this class.

d) Active Systems: Stabilization in an arbitrary ideal atti-tude is possible. Attiatti-tude sensing and control carried out in all degrees of freedom. Momentum management generally by mass expulsion, but environment al torque sources may be used. For internal momentum storage, reaction wheels or control moment gyros are generally used.

\

e) Hybrid Systems: Involve control of more than three degrees of freedom, the additional degrees of freedom being related to

gim-balled auxiliary bodies, which are controlled relative to the main body by internal torquing. These systems allow for stabilization in an arbi-trary ideal attitude.

As one moves up the hierarchy from passive to hybrid systems, the accuracy, maneuverability, acquisition capability, power and/or fuel requirements, weight, complexity and cost of the system increases. Sabroff (Ref.50) in his excellent survey article has discussed in con-siderable detail the classification, historical progress, capabilities and limitations of various types of satellite control techniques.

Several studies* have been carried out on the possible utili-zation of environmental torques for satellite attitude control. A near Earth satellite experiences forces and torques due to the presence of Earth's atmosphere. The aerodynamic forces form a major source of per-turbation on the orbit of the satellite. A number of studies (e.g. * These are too numerous for individual citation in the text. The

list of references in this report and in Sabroff's article provide an exhaustive listing.

Refs. 29,62) have been carried out on the effect of aerodynamic forces on the orbits of near Earth satellites. The comparison between the measured and calculated orbital parameter changes has provided a means

of determining the density of the atmosphere at various orbital alti-tudes. On the other hand the torque due to the aerodynamic forces forms one of the major perturbation on the rotational motion of the satellite. The problem of stability of satellites influenced by gravitational tor-ques and aerodynamic tortor-ques has been studied in the literature (Refs. 16, 32,42, 57, 58). The possibility of passive stabilization by aero-dynamic torques has been discussed by DeBra (Ref.9), DeBra and Stearns

(Ref.69). Frye and Stearns (Ref.17) and Wall (Ref.65). Schrello (Refs. 57, 58) conducted an extensive study of the passive aerodynamic attitude stabilization of near Earth satellites. Sarychev (Ref. 54) studied the stability of a satellite with a aerogyroscopic stabilization system. The performance of such a satellite (Cosmos-149) equipped with such a system

is discussed by Sarychev (Ref. 53). 1.3 Present Study and Contributions:

In the present work, a specific configuration (see Sec. 3.1)

which utilizes aerodynamic forces to effectively control a satellite in circular orbits in an earth pointing mode is analyzed. The

configu-ration studied is gravitationally unstable. In the analysis only the

gravity gradient and aerodynamic torques are considered, the other tor-ques being assumed to be smalle Even though the other tortor-ques are not included in the analysis, the general formalism necessary to enable the inclusion of these torques as perturbations is provided. A linearized analysis has been utilized, but the essentials of a gener al nonlinear formulation necessary for a nonlinear simulation of the system is pre-sented.

The system equations are set up in state variables and the control system synthesis is carried out by means of time domain

tech-niques. The nominal control defined as the set of control angles which

minimize the drag due to the control surfaces and maintain the net

gravitational and aerodynamic torques acting on the satellite to be zero

when the satellite is in zero pointing error attitude are determined. The nominal control angles are in general functions of the orbital angle

~'. It is found that the nominal control angles are identically zero

for equatorial orbits, and for nonequatorial orbits the horizontal con-trol panels are to be set at zero nominal concon-trol angle and the vertical control panels have periodic control angles due to the cross-wind arising from the earth's rotation. Further for nonequatorial orbits, the nominal control angles are independent of the local density but are weakly de-pendent on the orbital altitude through orbital velocity. Numerical

re-sults are presented which indicate that the response of the' satellite to

initial perturbations is unstable with nominal control only.

The well known concepts from modern optimal control theory (Refs. 4, 5, 25, 26,51) are used to determine the feedback gain matrices which stabilize the satellite in the nominal attitude. The feedback gains are found to be constant for equatorial orbits but are periodic functions of the orbital angle for nonequatorial orbits. Further the feedback gains are functions of altitude. Numerical results are

pre-sented shöwing the response of the satellite to initial perturbations

with optimal feedback control at orbital altitudes of 200 km. and 300 km.

vari-ation in density due to the atmospheric bulge caused by solar heating,

and the variation in the sur~ace accommodation coef~icients. The effect

of using the equatorial orbit ~eedback gains when the satellite is in

nonequatorial orbits and, the effect of using the gain matrix computed

for 300 km. altitude orbit when the satellite is in 200 km. altitude

orbit is numerically investigated. A second satellite configuration

larger than the first is studied numerically. The possibility of

stabilizing the satellite in a single axis pointing mode in which one

of the satellite axis will point towards the center o~ earth but with

the satellite being allowed to 'weather-vane' around that axis is

investigated. An alternative method of synthesizing the feedback

con-trol treating the system as a linear inhomogeneous system is suggested.

The e~~ect of having the panels for aerodynamic

stabiliza-tion o~ the satellite on the satellite's li~etime is evaluated (Appendix

C). It is found that the presence of the control panels reduce the

li~etime of the satellite. But the results indicate that it is possible

to have reasonable lifetimes in the altitude range o~ 200 to 400 km.

with suf~iciently small damping times.

The problem of implementing the aerodynamic control system

is discussed. Further it is pointed out that a simple control law

can be devised to maintain a similarity in the performance of the

satellite for orbits of di~~erent altitudes by varying the control

panel areas as a function o~ al ti tude. If variable area control panels

are not utilized, the implementation involves a suitable gain

schedu-ling scheme, the gains being altered at discrete alt{tudes with

sub-optimal per~ormance in the intermediate altitude ranges. In such a

case, the per~ormance will be poor at the higher altitudes and good

in low altitude orbits.

11. PRELDITNARIES

In general one is interested in various types of

stabili-zation depending on the function of the satellite such as, 1) Inertial

stabilization where, the satellite has to maintain a fixed orientation

over very long periods of time unless reoriented on purpose (star pointing satellite). 2) Quasi-inertial stabilization where, the

satellite can be considered to have an inertial orientation over short periods of time but, over long periods the orientation is non-inertial

(Sun pointing satellite) and 3) Non-inertial stabilization where, the

satellite is "driven" at a certain angular rate about a fixed direction

in inertial space (Earth pointing satellite). The study of all of the

above types of stabilization involve the general steps of proper choice

of coordinate systems, derivation of expressions for the disturbing

torques and the determination of the equations of motion. In this

chapter a general formulation is established which encompasses all the

above types under a single ~ramework.

2.1 Coordinate Sytems and Transformations :

A judicious choice of coordinate systems is very important

and valuable in determining the disturbing torques and equations of

moti~n with ease. Roberson (Re~. 45) has discussed this problem

qualitatively at considerable length. In this section several

co-ordinate systems are defined which will be ·of value in determining

:t

i represents different coordinate systems, i = 1, 2, ••• , A, B, C ••• etc.

x~

J i x. J

e~

J ~ij _

represent the coordinate axes in the Xi coordinate

system, j

=

1,2,3

represent unit vectors along the coordinate axes X;

represent a vector

!

in

~

coordinate system

i

represents the components of

!

represents a rotation about

X~

J

,

j = 1, 2, 3

represents the transformation matrix from coordinate

system Xj to !ti

! i = l,ij xj (2.1)

' . "T

Note that ~1J

i

1J = I (Identity matrix) for orthonormality

and det[~j ] = + 1 for pure rotation.

The (e) IE 1 (e). 1

c

_e

~

s

_e

~

represents transformation matrix for rotation around axis X.

1

cose sine

~~Ij

-

implies that the transformation matrix relating the

two coordinate systems is the identity matrix.

fundament al transformation matrices are given by,

[:

0

:.]

Ce 0 - S _{e Ce}

Se

0

Ce

(9)2 -

0 1 0

(9) • -Se

_{Ce 0}

3

-Se

Ce

Se

0 _Ce 0 0 1

For a sequence of rotations the resultant transformation

matrix is obtained by compounding the individual transformation matrices

by multiplication from right to left.

With the above notation the following right handed,

ortho-gonal coordinate systems can be conveniently defined.

$1 _ Inertial coordinate system. Earth centered, with

X~ along Earth's axis of rotation. Xl and Xl lie in

equatorial plane with X

₃

pointing in the

dir~ction

of vernal equinox.

$2 _ Inertial coordinate system. Earth centered, with

X~

pointing towards the star of interest.

~2

can be

obtained from~ by two rotations. If a is the right

in the usual astronomical sense,

a

being positive

counter clockwise from vernal equinox, 5 being

positive towards north pole) the two rotations are,

8~

=

(a -90) about

x~

2

about X 3

$3 _ Defines orientation of orbit plane in space. Earth

centered. Let the no dal longitude of the satellite

orbit be

n

(same as right ascension §f nodal line) ,

and the orbital inclination be i. X

3 points towards

perig

3

e,

xi

in the direction of motion at perigee

and x~ normal to the orbi tal plane.

X3

can be ob-taine

3

from ~l by tUree rotations, 8~

=

n,

8

3

=

i,

and 82 = CA) where,

X

is an intermediate frame of

reference obtained by a rotation 8~ =

n

and w is the

argument of perigee.

~5

_ Orbital coordinate system with orlgln at the vehicle

mass center,

x5

pointing towards Earth's center,

x5

pointing along

3

the negative direction of the orbit&l

angular mom.entum vector and

xi

is such as to form a

right handed orthogonal coordinate system. :1:

5

can be

obtained from ~l by the three rotations 8~ =

n,

8~

= -(180-i) and

8~

=

180-(w+~),

where

~

is the true

anamoly.

~6

_

6

Body centered coordinate system with Xl pointing

to-wards the star. Since the stars ar~far away in

comparison with the orbital radius ,1)'-'<)0(>$2.

$7 _ Body fixed reference coordinate system (for simplicity

assumed to be principal axis system). ~1 can be

ob-tained by rotations of 8

1,82 and 8₃ about the Xl' X2

and X

3 axis respectively from, 1) ~5 - if the satellite

is to be earth pointing 2) !E6~l:2_if the satellite

is to be star pointing and 3)

:t

8<x>XS - if the satellite

is to be sun po~nting.

For small angles it can be shown that the equations of motion

uncouple such that the order in which the rotations are performed is

immaterial.

~8_ 8

~ Body centered coordinate system with Xl pointing towards

the sun. In general

i3

~l:S.

~S _ Earth centered solar reference frame,

X~ points towards

the sun.

x~ lies in the ecliptic plane. At any time t,

8i

=

_90

0

and

8~

=

(~S-900),

where

1

9

is an inter-mediate coordinate system obtained from ~l by a rotation

8~

=

%' %

=

23

0

. 26' 59",

~s

=

~Sl

+

21T / t=to

WS(t-to) and W

s

=

365.2563825

rad mean solar day. IE _ Geocentric Earth reference frame,

X~ along Earth's

axis of rotation, x~ and X~ lying in the equatorial plane with X~ passing through the intersection of the equator and 00 longitude reference meridian.

XE

can be obtained from Xl by one rotation, 8~ = ~E. where,

~E

=

~EI

+ WE(t-to) and WE =

7.292115

x

10-5

rad/mean

t=t o

solar sec. is the rate of Earth's rotation.

XM _

Geocentric magnetic reference frame;

x~

points to-wards t~ geomaMnetic North Pole (south magnetic pole); Xl and X

2 lie on the magnetic equator with

X~

lying at the intersection of the equatorial plane and the geomagnlftic equatorial plane.

X

M can be obtained from

X-

by two rGtations, 8~ = t3 and

M 80 0

8

3

= ~, where t3 = t3

M

+ ~E' t3

M

=

19.

and ~ =

17.5 •

The coordinate systems defined above are not the only ones necessary and additional coordinates systems will have to be defined as and when necessary to carry out the analysis of any problem on hand. For clarity all the above described coordinate systems are depicted in Fig.

2.1.

The trans format ion matrix for any pair of coordinate systems can be obtained by suitably compounding transformation matrices of

s~qcess~ve pai~s of intermedi~te coordinat~ sy~~ems. Thus one has

~l.J = .tl.~klt1J and knowing tl.k,~kl and ~lJ, ~l.J is easily determined. Thus all the necessary transformation matrices can be derived from the three fundamental trans format ion matrices,

(8)1,(8)2

and

(8)3

when necessary.

2.2 Disturbing TODques:

The various disturbing torques experienced by a satellite were listed in Section

1.1.

The disturbing torques have been the subject of intensive studies by various authors in recent years. In this section the torques acting on a satellite due to its environment are studied using a unified notation and suitable expressions are ob-tained. This section further provides a demonstration of the elegance of the notation and coordinate systems defined earlier in Section

2.1

for the purposes of the analysis on hand.

2.2.1 Gravity Gradient Torques:

Gravity gradient torques constitute one of the main factors influencing the rotational motion of space vehicles (Refs.

1, 19, 46,

58, 58,

etc). The gravitational force and torque acting on a finite

body situated in a simple inverse-square attracting force field are given by

(2.4)

where, r f _{- vector from spacecraft center of mass to dv}

p - density of volume element dv

•

R =R _-c + r'

craft.

R - vector from center of earth to center of mass of

space--c Ir

If the spacecraft consists of a set of n parts connected with each Other, let r'

=

_-1 c. + _{- 1 '} r. i

=

I _" 2 _{• • • • • • • ,} n _.

Hence,

and

c. - vector from spacecraft c.m to the c.m of the ith

-~

part

r. - vector from the c.m.of the part to dv.

-~ f ..

-1

E

R pdu--G_t )t3 - J 'IT~ (2.5)

GG

"" -

J (

r'

x p.

R)

P

du-- i. - R3 -"'L

(2.6)

Since only the torque expression is of interest here, we have

§Gi. = -

L}J- {

(~;,

+

rJ

x

(:~

c +

ft

+

!

t) }

Jt

3

P d\T

"

•

- I"

L

1

(~,

·

r,) •

R,}

R-'

pci

v

1-R-

3

can be approximated as follows:

s

{ ( Be

+

~\.

+

!

i.

) ·

(Re'"

ç;. ...

f

t)

r

2"

s

=

{g

c

.Be

+

2RC:·(~

i

+

.rt)

+ U~;,+r

i.1)2}

-

ï

R-3 {

.Be Re

2.

Be

(fi.

+ !'i. )

=

e _ . - + - - . + Re Re Re Re

R-

3 { 2 R e' + rt

}-.!

~ c 1 + --d.. -" - 2-Re Re

(2.7)

(2.8)

5

In

~ ~--j Re

.5

: -

[0]

0

R _3

e 1

The torque expres sion in X7 being of primary interest, in

X

7

3

Hence~

.

~

~ Noting that obtains

J!:.

R

3 C

-3

-3

2. 1'~75

5 (

)(~

_3{

3,975'5 )}

R ~ Re

{1 -

Re;t 13· ft + Ei.

J

~ Re

1

+ Re;t,

13·

(fi. + ti

{ -Re

~7S 1~

)(

[1

_tr~

~r

pd", +

1

Ui.

r~

pd\r]

-si, (

1."1: •

(f~·.r;

l)(

~

7S

j~

.

(f!+

r,'

l)

p

d~

J

and

r

pdu _ mi. ' the mass of part i one

Ju.

"

Hénce the total gravity gradient torque acting on the satellite is given by

since

~ ~~

,.

m.i. - 0 by defini tion of c. m of satelli te

For Earth pointing satellites

For Star pointing satellites

'l7S _

~76 ~Ól

:f1

S

~75 _

1

78;[811,15

where ;1.76 = (6_{3 )3}

(e,),~

(81 \

For Sun pointing satellites

For convenience define

K ..

J-J75 .5

13

For a satellite made up of a single body

wnere

rJ.,78

=

(e~\ (92,)2,

(9

1)1

[~]

and let consequently

J

where

~

K3

x; }

fdtr

~

35

c K₂K₃ (C-B) K~ Kl (A-C) K₁K2,(B-A)

1

x~

xj

f

dv '" 0 • \,

i-

j V'

1 ((

X:)2 + (

x~)2] peiu -

A ~ (2.12)

J

[(x;)1

+

(x~t]

pdu

=

B '11

II

(x:r~

+

(x;t]

pdU - C ~

The above analysis holds for an inverse-square attracting force field only, but it is a simple matter to extend the analysis to include the effects of anomalies in the force field.

2.2.2 Aerodynamic Torques:

For satellites orbiting below 700 km altitude the aerodynamic effects form the major source of perturbation. The orbital dynamics and the attitude dynamics of these satellites are profoundly affected by the aerodynamic effects, (Refs. 1,7,9,16,17,27,29,30,42,53,54,57,58, 62 and 65). Further a~ these altitudes free molecular flow conditions exist (see Appendix B).

Under free molecular flow conditions the normal, tangential and drag force coefficients are respectively given by

where ~I - Normal momentum accommodation coefficient ~ - Tangential momentum accommodation coefficient

a -

Angle between the reversed incoming flow direction

and the normal to the surface.

~I

•

S - molecular speed ratio, defined as the ratio of the speed of the satellite to the mean thermal speed of the molecules. Under infinite speed ratio assumption, (see Appendix B)

Cp = 2( 2-cr) Ca; ICerl

Ct; = 2cr SC{ ICal (2.14)

CD

=

2

{(2-ol)C;

+crs~}

ICa:I=

2(2-o-'-cr)C~

ICal+ 2/TIC_cr

l

The .above approximation is useful in computing the forces and moments acting on bluff bodies in free molecular flow.

The aerodynamic forces and moments acting on the satellite

will have to be determined in

!7.

where

~ - relative velocity of "air" with respect to satellite

k -

a coefficient which depends on altitude and accounts for any relative rotation of the atmosphere wi th respect to earth; 0

<

k

<

1. WE - angular speed of earth

V - orbital speed of satellite

(2.15)

[ J

l

C ,]

S,

c".~

]

0

-vc rit

+ kw! Re Ci.

5 5 Si 0 5 Y

~R.

- - Y.. +

k~

CA)E

~

1t

B:e -

V 0 + kWE -ei. x 0 _{kU)E Re Si. CU>+lJ}

Sy.

_{-St SIAl} + '1

-Re

_VS

y

*

where

y*

is the inclination of the orbital flight path to horizontal and tan Y*

=

flr*

=

ES

7/P.+EC?). For near circular orbits

y*

is small since the eccentricity E-.O • Further for low altitude orbits k.l -V + W R C. E c l. Hence _wERcS i

qÜi-?

(2.17)

o

The magnitude of ~ can be approximated as follows:

(2.18)

For the altitudes of interest R wE

~0(7

x 102) m/sec, V

=

0{7 x 103 )

m/sec. and hence wERc/V is small compared to unity.

Inverting (2.18), the following expression is obtained

V VR

=

t

1 - 2 ( !.IJ

~

Re)

ei

Hence 1. (2.20) 1. + wE Re

c.

V " and wher~ V \ WE Re } V -_R. 1 - - -_V C. _\ where

1

75 is given by (2.11) (2.22)

If n

7

represents the unit surface area and t

7

the vector lying plane at the centre of the elemental and ~, we have

inward normal to the elemental

and

cos fr

=

"Q.7.

~

at the intersection of the tangent area and the plane containing n 7

t 7 __ cota n 7 + cosec a: ~ (2.23)

t

7

may also be expressed as t

7

=

(n 7

• ~~ ) X "!J:.7

I

t

~7 X Y'~) x ~

I

The normal pressure and shear acting on the elemental area are given by

( 2,24)

The net aerodynamic force on the satellite is given by

cl.

=

J

(~7

+

~7)

d.A

Aw

If r

7

is the position vector of the center of the elemental

surface area, then net aerodynamic torque on the satellite is given by

iJ,.

=

1

r

7 J( ( f1 + 'Ç. 7 ) ciA (2,26) Aw

where ~ is the "wetted" surface area. 2.2.3 Solar Radiation Torques:

For satellites orbiting at altitudes in excess of 1000 km the solar radiation torques form a major source of perturbation on the rotational motien of the satellite (Refs. 1,6,19,23,37,40,44,59,60 and 67). In the case of satellites travelling in interplanetary space the solar radiation torques form the major source of perturbation apart from torques due to meteoroid impacts.

The incident energy fl~ from sun in the vicinity of earth ~ • Es /ItTrr'J. , wheri E

=

3.86 x 102 joules/sec is the solar constant and r

=

1.5 x lOl m~ters is the mean solar distance from the earth.

~ • me~ by virtue of Einstein's mass-energy equivalence relation.

C

=

velocity of light in vacuum. 2 7

Hence the incident momentum flux V

=

mc

=

tIc

=

E /4~r c

=

4.64 x 10-kg/m2• Let«, 'C' and cr' be the coefficients of

r~flecti

vity, trans-missivity and absorptivity respectively, characterizing the satellite surface element. For continui ty ct + t:' + a'

=

1 and in gener al t:'

=

O. Hence cr'

=

(I-a).

Consider an elemental surface area, the position vector of whose centroid is given by

r"(.

Let n7 be the unit vector along its outward normal. Further let vS _{be a-Unit vector directed from the} vehicle center of mass to the sun.

In

"_X

S , V

S

=

'1-8

=

[g]

and in

~l,

v7

=

~.1s'1-s

=

'1-

78 v8•

For earth pointing case

1,7

s

=

tJ..75

1.

51

~lS

. For star pointing case 1.7s

=

:1.

76 tJ..61

t-

s•

,~7s __ .,78.

For sun pointing case ~ ~

(2.27)

The net force on the satellite due to solar radiation pressure can be easily shown to be given by,

f~R

• -

v

{J

(1+c:.C)

I

~~~11 (g~

n.

7

)!!;1

ciA +

f

(1-«) \

~~

n

1

\ (n

7

x

y,7),q{

dA }

AW AW

The net torque on the satellite due to solar radiation pressure is given by

(2.28)

~:R.

.. -

V

{f

(i +

0:)

I

!~

!l71

r

7

x

(~~!7)

!!:7

dA

+

J

(1_a:)\~?~71/x [(~1xi)

X!l]dA }

(2.29)

Aw Aw

sunlight and zero when the satellite is in the umbral region of the earth's shadow. V varies from

4.64

x 10-

7

kg/m2 to zero in the pen-umbral region of the earth's shadow and the nature of this variation is not treated in this study. Further, the earth reflected solar radiation and the earth emitted radiation has not been included in the analysis, their effect being smal 1 in comparison with direct solar radiation.

2.2.4

Torques Due to Earth's Magnetic Field:

A satellite in orbit experiences disturbing torques due to the interaction of the satellite's magnetic dipole with the earth's magnetic field. The satellite dipole moment is due to either perma-nent magnets or closed current loops present in the satellite. The magnetic torques acting on earth satellites have been extensively studied in recent years. (e.g. Refs. 1,

37, 59,

etc).

The earth's magnetic field may be approximated as the field due to a simple magnetic dipole at the center of the earth with the dipole axis inclined at 17.50 to the spin axis of the earth. The potential function for the magnetic dipole in spherical polar coordinates is given by where m sin+

t = -

=-..;;=..!. 2 r m - dipole strength

r - magnitude of the radius vector from dipole to the vehicle e.m.

~ - latitudinal position of vehicle c.m. relative to the magnetic equator.

The magnetic i'ield B is gi ven by

[-

2.5.

~+]

(2.30 )

(2.31)

where $"is a geomagnetic reference spherical polar co-ordinate sytem. e x where

c"

0 (2.32) -St SA Cf C.5). S+

Hence,

[3S.S,c. ]

BM

=

m 3S~ - 1 (2.33) - r3 3S.C.CA. ~

=

!f51!flM xM

=

!5M x M

~

_(tij) x M (2.34) since

~~

=

-J;

comparing

-~

and

~5M,

the fo11owing are obtained.

Hence

""

l31 .. - C~ S). t3~

_ -

S •

.t

13 - - C. CA.

- - t:i

since

t.

t'

_k

-

t·.

tiC

~J " Jlo I. Ci Cn. Cw.~ - SA SI.t)t'1 ~S1 _ Si. CA -ei. CA SW+'1 - S4 CIAl+, consequently B5 _ m - r3 3t». t_l1

t

st + 3

t'2

t~- ~'2

+ 3t'3

t

32. tal 3!u

t

,n

t

Zl +

~t2.2t;2.-t22+ 3tu.t~2t&3

3

tu

l~

+ 3

t:

_{z - tu}

+

~

in,

l~s

•

8

_jk Sl.CW+? - Cl. SA CW+')- CA SW"l - ei. - $" S.n.

- Si. Su.>+'1 Ci, SA SW+') -CA CU)+~

!t

iM

CIoI)+, ( S"C)'M - SAM C"C~_A) - S~_A s",.'1 SAM -(S;,S)'M C~_A + C).M ci.) ~ { Sw., ( S~ C)'M -

ei

S),M C,_

A)

+ CIIU" S AM S

,-A.}

B 7

=

1.

75 B 5, where !,75 is gi ven by (2 .11) (2.35) (2.36) CpC~M = _S>'M

-spe)'M

(2.38)

In general the torque on a satellite due to the interaction of the earth's magnetic field with magnetic moments fixed or generated within the satellite is given by (expressed in $7 coordinate system)

(2.40 ) where

M7 is the magnetic moment of the satellite composed of:

-C~ SAM S~ C"", 0 St SAM

Ct

due to permanent magnets in instruments and current earrying deviees

~=

due to magnetization of the satellite huIl in the

geo-magnetic field.

In general

Mi

=(v/4~)i~7

where v - volume of the satellite huIl.

Hence G

7

=

(M

7

+

~)

x B

7

-=M ~

=r

-Torques Due to Eddy Currents:

! -

is a matrix representing a

symmetric tensor.

(2.41)

When a satellite has an angular veloeity relative to the magnetic field, eddy eurrents are indueed and the torques produced by the eddy currents (Ref.l) tend to reduce the component of the angular velocity perpendicular to the external magnetic field.

The torque due to the eddy eurrents is given by

where

w7 _ angular velocity of the satellite

ks - a constant ealled the dissipation eoefficient, which can be approximated for the particular satellite in question by

k

=

ct J tiR

s

where

a -

a non-dimensional eoefficient if Band Rare

expressed in eleetromagnetie units.

J - longitudinal or transverse moment of inertia

of the satellite huIl.

t - thiekness of the satellite huIl.

R - specific volume resistance of huIl material. 2.3 System Description and Control Synthesis:

(2.42)

Having established the formalism necessary to determine the torque acting on the satellite, in this section the equations of motion will be obtained and the problem of synthesizing a suitable control will be posed. There are several alternate formulations available for the derivation of the differential equations of rotational motion. Hughes (Ref.22) diseusses the relative merits of some of these formula-tions. In the present study the Euler's equations will be employed to obtain the differential equations for the dynamies of the system.

The rotational dynamics of the satellite ean be represented

where d dt 7 U) )( _G7 ₊ G7 _-c -D

3

7 -

Inertia matrix of the satellite

- - - -- ----.

w

7 -

angular velocity vector with respect to inertial coordinates.

~

- disturbance torque vector G7 - control torque vector

~

The expressions for

w.7

for various pointing schemes can be obtained as follows:

-For the earth pointing satellite

For star pointing satellites

For sun pointing satellites

consequently

'E7 '"'

A;

é

₃ + (e₃

\1:

ê~

..

(9a)3

(e2)2.~~

é

₁ +

(e~)3

(91\ (91\ 43 JÎ.s .. (ea \ (92)1 (91\ (JL_{s - 90), (-}90 )113

As

(2.46) In gener al the angular rates other than

ë

l '

á

2,

è

3 and ~ are small and hence can be neglected. Depending on the orientation scheme under study the respective equation in (2.44)-(2.46) can be inverted to obtain

ê7

in terms of 9

7

and w7 which being the kinematic differential equation of the system. Alternatively the expressions for w

7

given by ( 2 . 44) -( 2 .46) can be substi tuted in (2 '.43) to obtain a set of three second order nonlinear ordinary differential equations.

Thus the following set of first order differential equations describing the kinematics and dynamics of the system are obtained.

·7

W~ where,

77(S-C)

Wz. ₀₀₃ -,;;-= iD(w~,e~.u.j.t) = w;

W;

(C~A) (2.48) OOi w~ ( A~ B )

u =

[Uj]

is the control vector.

[~

0

~]

3

7 = B

0

Equations (2.47) and (2.48) form a set of six first order nonlinear ordinary differential equations, These can be expressed as

x

=

!.

(x, ~, t)

where

xT

~

(9l,92,e3,êl,è2,ê3)

fT

~ (~, ~)

The above system which is to have x

=

0 as its equilibrium state is to

be controlled to remain in the neighbourhood of x

=

0 in the presence

of small perturbations. The control necessary

to

ensure that

x

=

0

when the satelli te is at x = 0 s.tate in the absence of all

perturba-tions is given by f (XO, uO, t)

=

O. UO is called the nominal control

and xO

=

0 is called the nominal state~ Various methods of determining

~o are discussed in Appendix D. In the present study, the aerodynamic

forces on the control panels used to generate control torques also

contribute to the drag on the satellite which causes orbital decay.

Of the several possible ~o, the present study considers the particular

uO which will also minimize the drag on the satellite in order to

ensure that the satellite lifetime does not get reduced drastically because of the aerodynamic control.

Since the perturbing forces and torques on the satellite are small, the excursions in the state of the system from the nominal

state due to the extern al disturbances can be assumed to be small,

con-sequently as an approximation, the system equations given by equation (2.49) can be linearized to obtain,

.

X

=

A( t)

x+

B(t) U (2.50 ) where U = u - u 0 X = x - x 0

=

x of A(t)= ~x x = x 0

=

0 0

~.

B(t)

=

of.

_ê)u

x

=

xO

=

0

o U

=

U

The problem of control synthesis consists of determining ~

as a function of X and t. Since the system has been linearized a linear feedback control of the form U

=

-K(t) X can be obtained (see Appendix E). Such a linear feedback-control is advantageous from the viewpoint of both synthesis and implementation of the control.

lIl. EARTH POINTING SATELLITES

Earth pointing satellites are a class of satellites which are intended to have one of their axis pointing towards the center of the earth continuously. Earth pointing satellites find promising appli-cations in present day space programs in such varied areas as communi-cation, navigation, geodesy, meteorology, earth resources surveys and reconnaissance. Several control schemes have been applied for the

attitude control of such satellites. In this chapter a specific configu-ration is studied for possible application of the aerodynamic forces for attitude control.

3.1 Configuration, Coordinate System and Transformation Matrices: The configuration under study is shown in Fig. 3.1. It essentially consists of a center-body with a long axis of symmetry with cruciform control surfaces mounted as shown. These are 'all-movable' con trol surfaces which are capable of being rotated about their cent-roidal axes lying normal to the center-body' s axis of symmetry. The control surfaces are to be of light weight construction and of suitable area so as to provide sufficient aerodynamic torque but producing neg-ligible inertial torques due to their angular accelerations.

Apart from the coordinate systems defined in Sec. 2.1 the following coordinate systems will be of use in subsequent analysis.

$Hi _ coordinate system fixed in the ith hinge,

~Hl

_ is obtained from

~7

by one rotation:

e~

=

_900

X

H2 _ is obtained from

~7

by one rotation:

e~

=

+900 IH3 _ is obtained from

$7

by two rotations:

e

7

= 90

0 ,

2

e

H3

_{= 90}

0

1

Alternative choice:

e

7

=

900

e

H3

=

_900

1 ' 3

~4

is obtained from

X

7

by two rotations:

e~

=

_900 ,

e~4

=

900

Alternative choice:

e

7

=

_900

e

H3

=

_900

1 ' 3

Hl H3 Note that X

2 and X2 along the negative

xI

7 H2 H4

point in Xl direction whereas X

2 and X2 point axis.

~S.i th

:"S - Principal axis Hoordinate system fixed in.i control surface • ~ J. is obtained from ~ J. by a rotation e~i= e~

b! - Vector from spacecraft c.m to the center of pressure

of the ith

~ntro1

surface.

Some of the usefu1 transformation matrices are given below:

~'5 = d15 \-%QncQ ;L • 1 0

o

~.,,7.

[: -: :]

~.'.7

_ [_: :

~]

s\ Cw.-'1 - Cl.

-s,

SLo"? Se Se Ce + Co Se 1 & a .... i .s Ce1 Ce) - Se1 Se₁ Se. - Se C _{1 e~} ~~i..S;' _ 7 141 ~.

...

!".,7. [: _:

:J [: : -:]. [: :, -:]

i 0

o

o o

(3.1)

(3.2)

(3.3)

(3.4) (3.6)

[

: : :] [: : :] _ [_°1 : :]

_o -1 0 -1 0 0 0 -1 0

7,1014 [0 -1 0]

'l ,.

0 0 -1 1 0 0 (3.8) 3.2 Assumptions:

Generally any assumptions made are constraints on the generali ty of the analysis and these can be easily relaxed at the cost of added complexity of the analysis. The following assumptions have been made since the present study is in a sense a feasibility study and as such it is advantageous to maintain a certain degree of simplici ty.

1) The vehicle is assumed to be in a circular orbit.

2) The rate of change of the orbi tal parameters.Cl ,i" w and Re due to the earth's oblateness and atmospheric effects are small in camparison with the orbital rate of the satellite. The order of magni-tude of the ratio of the rate of change in the orbital parameters to the orbital rate of the satellite can be estimated approximately (Refs. 1, 29, 58)as,

. =

-"1

-3 ~ 10 2

Re

P

,where CD - drag coefficient of satellite

Ae - effective cross-sectional area of satellite

m - mass of the satellite

f -

atmospheric density

For a typical satellite Rc/? varies approximately from a value of 10-5

m./rad. at 700 Km altitude to 10-lm./rad. at 150 Km altitude. Hence the

orbital parameters may be assumed to remain constant over several orbits. 3) The isodensity contours are geocentric circles and the

density at any orbital radius is taken from the 1962 U.S. standard

atmos-phere. But the effect of the diurnal density variations on acontrol

synthesized with this assumption is investigated numerically by

intro-ducing a periodic variation in density at any given altitude (see

Sec. 4.2).

4)

In the altitude range under study free molecular flow

conditions are assumed to exist (see Appendix B).

5) The control panels are assumed to be rigid and consequently the effects due to their flexibility are neglected.

6) The control panels are of light weight construction and hence the moment of inertia of the control panels are assumed to be

the satellite and the inertia effe cts arising from the rotation of the control panels,

For order of magnitude purposes the moment of inertia of a spacecraft may be approximated as being 0(mL2 ) where m is the mass of the spacecraft and L is characteristic dimension. Similarly the largest contribution to the satellite moment of inertia from a control panel of mass m may be approximated as O(m ,f2), where

t

is the distance of the contro! panel center of mass from that of the satellite. Hence ratio of the change in the satellite's moment of inertias due to control

panel rotation to the moment of inertia of the satellite is approximately of O(m /m). The present generation solar panels on spacecraft have a mass dênsi ty of about 5 Kg/m2 • Hence solar panels of reasonable si ze will lead to O(m /m) ~ 0.025. On the other hand it is possible to

con-c

struct panels using either metallic foils or MYlar sheets stretched over light weight frames to obtain mass densities of 0.25 Kg/m2 leading to O(mc/m) f::i 0.002.

7)

Only aerodynamic torques and gravity gradient torques are considered in the analysis and the other torques are assumed to be negligible. The expressionsfor the other disturbing torques obtained in Sec:. 2.2 may be utili zed to study the effect of these perturbations on the performance of the attitude control system.

8) Since the external disturbances acting on the satellite are small, the excursions in the state of the satellite from the

nominal state is small and consequently a linearized analysis is carried out. The essential details necessary for a nonlinear formulation is presented and i t may be used to carry out a nonlinear simulation to study the effect of nonlinearities on the system performance.

9) The atmosphere is assumed to be rotating at the same angular veloci ty as the earth.

10) The shadowing of the control panels from the oncoming flow when the center-body is in an off-nominal attitude is not con-sidered. The effect of such shadowing on the control can be eliminated by having the "all-movable" control panels a certain distance away from the center-body.

3.3 Determination of Drag and Dominant Torque: The control surfaces can be considered as negligible thickness. Let C ,Ct':and C

_n

represent the

Pi " i

flat plates of normal force,

tangential force and drag force coefficients for the ith control surface in free molecular flow. Further let C denote the moment coefficient

m

of the center-body alone. C is a function of A , the angle between the

m .

veloci ty

symmetry leads to

vector of the body relative to the atmosphere and the ax~s of of the body. The drag on the satellite due to the center-body orbit perturbation and is treated in Appendix C.

·Si Let

J

3 be

n!

=

~

the unit normal to the ith control surface. Then,

:l,H'

:t"

"j:'

=

:r".' :r""" [:]

•

7

Let t. be of unit magnitude and lie at the intersection of the control ~

aurface and the plane containing ~ and ~.

In the sequel the superscript

7

will be dropp~d and all vectors without any superscript are to be treated as having superscript

7.

where

(n .. ~) = cos,,"., t. = -cos!rin.+ coseca.~

~ ~ ~ ~ ~ ~~

t.

m~

also be represented as t.

=

(~X

YR)X

~

~ ~

P.

=

21

f~R

A.C n.

~ ~ Pi~

I (n.X!n)x n.1

~~ ~

~

=

~ rv~

Ai Cri.

~

=

~ PV~AiCti

{

cosecai~-cotai!!:i

J

A . - area 0 f th e ~ . th con ro sur ace. t 1 f

~

Due to the control surfaces alone:

(3.10)

4 4

~

=

L

Ei.

x

(~;. +~;.)

-

L

ir

V: Ai. bi.

(~"

Cp" i" cosec ctd!R Cti. - cotai,!li, Cr;)

c.s ;'.1 ".1

c.s - control surface

superscript ~ - represents tilde operator (see nomenclature

for defini tion) •

'OC

For the center-body, for small angles Cm

=

0:

A

=

cmA'

A.

C is a function of body shape and location of the center of mass.

m

A

~

_c.b

where, c.b - center-body

SA - characteristic cross-sectional area of the body. L - characteristic length of the body.

4

+ G

=

lrV:SAL{ClII.

A(-It_R

1

1 ) +

LcA\.&J~;.c"i.+CO$QC«'t~IlCt~-cot«;'~i.Cr;;.)

~

=

~c.s ~c.b

2 A

~::1

T.There d.

=

Ai

.g.

bi (3.11)

" c)'f~ SA' -1

=

L

D

=

(3.12)

The gravity gradient torques acting on the satellite can be approximated by: K2K3(Cl - BI)

~

~ 3j.L K₃K_l(A_l- Cl)

7

c K_{IK2(Bl - Al)}

~

- !,5

k

S

_

:lS

m .

[:~

::: :

::::::~.~

1

CeS. Cez.

J

Under steady state zero pointing error condi tien.

~75

=

I since

9

=

0 \)' \ = -R. ~. 0 5 ~R.. • (-V + we'Rc

C,,)

'IR.. ( we Rç:~

C

U)+')

o

3.4

Determination of Nominal Control Angles:

(3.13)

The notion of nominal control enunciated earlier in section 2.3 will be utilized here. The drag acting on the satellite due to the control surfaces, which is a direct consequence of using aerodynamic control leads to a reduction in satellite life. Under steady state zero pointing error condition in the absence of other perturbations it is necessary to maintain the combined aerodynamic and gravitational torques on the satellite at zero in order to maintain the satellite in that attitude. But it is also essential to ensure that in the process of doing so, very little additional drag due to such a nominal control is

imposed on the satellite. Thus the naminal control SC

(?)

has to be

--0

determined so as to minimize the total drag on the satellite under steady

state zero pointing error condition (9 = 0,

ë

=

0) subject to the

constraint that

XI

=

0 where, xT

=-(e

_{1 ,} 6₁

,e;,ë

₁,èz., êa).

x=O 211' Hence

!

Dd~

I!:o

(%+ ~)I =

o.

I!.=0

~I

leads to

(~+ ~)

Îx=O 9=0

=

0

will have to be minimized subject to the constraints

It is easy to show that the above calculus of variations

problem degenerates to a ordinary minimum problem for each value of 1J

because of the absence of derivatives of dependent variables in the

constraint

{~+ %}~=o

=

0 for various? values from 0 to 211'. The prob-lem of function minimization subject to constraints is treated in Appendix D.

The force coefficien~are given by (refer to Appendix B)

(3.14 )

Substituting equations (3.14) in (3.11) and (3.12) there results,

-(SCd.)2{(2(2-~'-~)

C ) .

2.~

}]J

+e ~

\ii

C( n. + - '\tR,

11" S ~ _ W

fifs-Drag due to control surfaces alone becomes,

(3.16)

The drag due to the center~bo~ is not included here since once a con~

figuration is selected for the center-body, under nominal attitude condition its drag is fixed and cannot be altered without altering the center-body configuration.

When

~

= 0, the following results since

~75

= I.

Hence

n., ..

_{!h •} ( ttn • s j1 . ) = { _ V + WE Re Ci. } -.. - V, R. "'s 'U'R. ~1

("bI

.(.2. Cec -~2. Cee l2. Cse

- t

2 Cee

!!.l)

=

1

...

2- ~ q.

t CSe ( b₁

n

2. )

..

.ti

Cee (bl!h) '"

-tz

Sec

(b_

n,J

a - t

2

s

gc

i. 1 _{2- ~ lt}

.t

z SeC ₁

t2,

Sec _2-

ti

Ce~

.ti Cee

4 o o o (~(~+w; Re

ei.) _

ti (WERe:i.CW+?) Il R. _ ~2.

(WE

Re Si. Cw ... ,) VR. iJ. ( wli Re Si. C"" ... , ) Vit _tz.(-'I"'WE.R.eCi. ) VR. .

-!z(-V ...

;E~e4)

(b4~~)

=

R.

_ ti (

we Re Si.

e

U>+7J ) VA.

_ ti (

WE. Re Si. Cw

+' )

VR. = S e (-V ... w~ Re ei. ) 92,

V

R.

For earth pointing satellites under nominal attitude con-dition

~=m

and hence substituting in (3.13), G

=

o.

Hence the constraint

rela--"=G 9=0

tion reduces to,

~I

=

o.

9=0

Referring to Appendix D on function minimization with constraints it becomes quite clear that most of the methods require the gradient of the function to be evaluated. The necessary derivatives are evaluated in Appendix F. The first derivatives are given by equations (F.14) and

(F.15) whereas for minimization methods involving the utilization of second derivatives the necessary derivatives are given by equations ( F .16), ( F .18), ( F .19) and (F. 20) •

Of the several computational schemes discussed in Appendix D, the Penalty Function approach using gradient method for minimization gave good results. A listing of the computer program for the above is given in Appendix F. From trials made using other methods it was found that even though these require fewer iterations, the computation time required was higher because of the increased number of computational steps involved.

For equatorial orbits all control angles are zero for all altitudes. For other orbits the nominal control angles are functions of orbital angle, orbital altitude and inclination. Figures

4.3

and

4.4

show the nominal control angles for a particular satellite configuration ( configuration

® :

see Chap. 4) in polar orbi ts at 200 Km and 300 Km

altitude respectively. It is clear from these figures that the altitude dependence is very slight and e~

(1)

and e~ (~) are alw~s zero for nominal control.

It should be pointed out here that the method enunciated above for the determination of nominal control is very general and powerful. Thus the above formal approach permits the determination of the nominal control when other deterministic but periodic disturbing torques are included and when the orbit is elliptical. In the case of circular orbits, if only aerodynamic and gravitational torques are considered, recognizing the quadratic nature of the drag, the nominal control can be easily obtained by setting the yaw torque under nominal attitude condition equal to zero and thus obtaining an algebraic equa-tion for the vertical control panel angles which are related by

9

₃

= -

9

_4,

Since the resulting algebraic equation will be nonlinear in the unknown, it can only be solved by some iterative scheme such as the gradient method. In the case of elliptic orbits both the yaw torque and the pitch torque will have to be set equal to zero (under nominal attitude conditions leading to two algebraic equations for the vertical and horizontal control panel angles respectively with the conditions

e~

= -

e~

, e;

= _

e~. These equations can again be solved by using any

one of the several variations of the gradient method. In the presence of other deterministic but periodic disturbances such an approach m~

not be useful if those torques were included in determining the nominal control angles. Thus the second method described above is not general, but the computational scheme in cases where it is applicable involves

one form or other of the gradient method employed in the approach used in the present study. Obviously the second method requires lesser

amount of computations than in the method utilized in this study for the circular orbit case with only aerodynamic and gravitational torques being considered.

3.5

Equations of M6tion:

In this section the equations of motion for the configuration under study are derived. These are then linearized around the nominal state and nominal control. Referring to section 2.3, for an earth point-ing satelli te,