Sub-Optimal Satellite Attitude Control Using Ionic Propulsion

SUB-OPTIMAL SATELLITE ATTITUDE CONTROL USING IONIC

C.DURANTE Laboratoire d'Automatique et de ses applications spatiales du C.N.R.S.

PROPULSION

J. C. LAPRIE Laboratoire d'Automatique et de ses applications spatiales du C. N. R. S.

A. COSTES Laboratoire d'Automatique e t d e ses applications spatiales du C. N. R. S.

INTRODUCTION

angles, L, M and N the control torques about the trihedral axes and w the orbital angular velocity.

In current Space Research programs the life-span imposed on earth satellites increases, and for the success of a mission the control systems must be more and more complex. It is therefore necessary to design orbit correction and attitude control systems using the possibilities offered by thrust systems that at present are in the testing or design phase.

Upon examination, the sy stem of equations (I) reveals that the equation related to the pitch motion is independent of the equations describing the yaw and roll motions. Equations (I-a) and (I-b) can be put into form[l]:

Within the scope of this study, the thrust systems utilised are ionic thrusters with cesium contact ionisation and electro-static deflection of the beam.

(Z -a)

L A

'I'

_ M

(Z-b) ~

The object of the present paper is to prove that: -micro-thrusters operating by large modulatErlamplitude impulses caused by the electro-static deflection of the beam make control laws possible that utilise a rotating vector whose amplitude is modulable from zero to a maximum value imposed by the type of micro-thrusters used. -execution of these control laws requires a much simpler logic system than the systems needed at present; reliability is thus improved. 1- EQUATIONS OF THE SYSTEM

C-B --A _4

B

w

C-A B

Z

C-B·

+ (I-A)

'I'

B

The transformation of the system of two second-order equations into a system of four fir st -order equations, is written in matrix form, by changing the variable 't; = wt. (3)

0

0

'I'

'1'0

(3

<1> 0



- Kl

0

0 0

0

0 0

(KZ_I)

0 1 -K

0 -4 KZ

1 -c

Pitch : C

0+

0

'1' 0 'I'

l

1

<1>0

0



C- B ---A

L

l-b Roll : B ( 4i + w~) + w (A -C)(q, x 3 wZt(C - A) = M

_ (1- C-A) wt

w Z

+

lL 0

~

with

Various authors have proved that the attitude variations of a satellite in a circular orbit round the earth, in the ca se of small deviations, can be written as : I-a yaw : A (;P-w~ )+w(C-B)( ~ +w'l' ) = L (1)

w

o 'I'

w~)

-d-'I'- -1 d~

w

d 'l' dt

1 .

0

- -w' I ',

d

C -A K =--

=--=d-c w

j]

Z

B

d

dt

Linear transformation [T i allows th e movements to be decoupled, only the control vector acts on the two sub-systems thus defined, this yields :

3 ...,2 (B - A) 0 = N

Where A, Band C represent the main moments of inertia about the yaw, roll and pitch axes , '1', and 0 the yaw, roll and pitch.

538

T ra jectory equations

0

o

Xl (4)

X

z

X3 X

W

0

0

0

Xl

0

0

X

- w I 0

0

0

W

0

0

-w Z

0

4

z

0

z

Cl 0

+

X3

C

z0

X

0

I

4

[:J

By integrating system (4) between instants Z. 1 and ~ I between which the control vector is 1+ constant, the following expressions are obtained: _ U L]cos (w

wl

sin

'2

where w and I

xl'>~)+

1

(w 1 xl'> 1> )

are the roots of the equation.

p4+ [(l-K ) (l-K )+ KI + 4 KJ p Z+4K KZ = 0 I Z I

J

The transformation matrix L:i and C are given by : J

and terms Cl

z

Kl w 1

0

2 wl - Kl

0

Wj(K -1) 2 2 -4K (WI-K ) 2 I

0

0

Wj (K2-1)

[TI~"

0

2 W2 -4 KZ

4KZ

w2 (I-K I )

w2

2 -K (w2 - 4K ) 1 2

0

0

0

"'~ (I-KO

W2 _ KI I w (K -1) l 2

Cl =

Matrix

[ Ci } j

C = Z

These are the equations of two circles plane (Xl XZ) centered at Pcint(~~ ;

2 Wz - 4 K2

wZ (I - K ) I

C~ ~R)('uthe other ~n ~lan)e (X 3

at point

effects the inverse transform:

z-

---..B:... Wz

-

1

Wz

L

one in

X 4 ) centered

.

CHOICE AND OPERATION POSSIBILITES OF THE THRUST SYSTEM

As was pointed out in the introduction, the increase of satellite mission life-span and complexity require thrust devices that will operate with the smallest possible thrusting mass. We have studied the possibilities offered by the electro-static deviation of ion beams provided by cesiurn contact ionisation thrusting For this technique, differential voltages are applied between the various elements making up the accelerator electrode [3J (fig. 1):

539

A

Single burl-on rhrusrer

Single srrip I-hrusrer _Figure 1_

,+, = 3n:

2

F i gu re 2

540

- 4 circular sectors, in single-button thru ster s, that allow the deflecti On of the thrust beam in two p e rpendicular planes; - Z parallel bars in single-strip thrusters, that allow the deflection of the thrust beam in one plane . In this second type of thruster, if Vo is the voltage applied to the accelerator electrode in normal operation, the deflection is obtained by bringing the Z parallel bars to voltages Vo +I1Vand Vo-tJ.V.

Figure Z

\.f> =)( It should angle of va lueoof

!fie

be pointed out that the deflecti on has a linear variation that is a function a de v iation of 100 corresponds to a DV equal to 5'10 .

(8)

(0)

+ U~ ('C)

U

max

For the v alue of the maximum deflection angle chosen~, equation (6) takes the form Cm x = x GA x sin 0( N' which for a deflection angfe ofD( <~N sin 0< C C max sin o(N

Vo

In the general case the torque is

(6)

Ju~

C=FxGAxsino<

with F : thrust vector GA: distance between the center of gravity of the body considered and the thrust system 0( : defl ect ion angle.

Applying this to equalities (7-a) and (7-b), the angle
The yaw and roll control torques L ('2:,) and M ('l:.) can be supplied, e . g. by four single-strip micro- thrusters placed in pairs on j"he pitch and yaw axes; appropriate control signals create e qual and opposed deflection angles for both micro- thrusters of a pair, thus there is no r esu lting forc e during the entire duration of the control.

(9-a)

yaw

sino(=

sino(N

cos\}'

(9-b) Roll

sin 0( =

sinflA.' N

sin Ij!

Note: With current thrusting, to ensure the dependa bility of the sy stern, a deflection angle C( of + 100 is not exceeded. Therefore for ad'~ val~es ofC( ~O( (error E being less than 5.10-3) sine( = 0( N

Within th e pre sent study, both pos si bilitie s offered b y this micro- thrusters system will be used: - amplitude variatio n - phase va riation

Ther e for e, fo r the yaw torque: L'.V 6. V L K N cos lfl K Vo Vo

Th us the control law will be such that the components of t he control vecto r satisfy the equalities: (a) U

(7)

L =R A

C max A

C'max (b) U = = B L B M

with

C .rnax A

C'max B

for the roll torque cos 'j.I

sin

(ID-b)

If

!J.

V

R

=

6.v'

N

sinljl

3 - STUDY OF THE CONTROL LAWS

U max

3.1. Study of the phase angle of the constant amplitude control vector.

Th ese e qualities, illustrated in figure Z, involve at every moment:

The quality criterion chosen for this study is such that the position of the control vector must maxirniz e the action of the control vector

541

in the two planes (Xl' X z interval of applicatior•.

X ' X ) for each 3 4

The condition that t = tg order equation in t:

Therefore the theoretical problem is the search for control laws that will give the values of the components of the control vector U and U that minimize, from initial krfown dat~ Xl (Gi), X (<=i) , X3 (Ci), X (~i) 4 z the quantity

\f gives a fourth Z

The numerical solution of this equation by BAIRSTOW's method[5]gives angle It' which, to be a solution of the problem, must also satisfy:

Approximate Law:

where

When quantity (wlll'C) is small, equations (11) and (13) can b e redu ced an can be put into the form:

b. 'G: time during which the contr 01 Vf'ctor remains constant in phase and in amplitude.

(14-a)

3.1.1. - Study in one plane- Minimisation

;[----2-----------------------

ofQ .. = x. + X. ___ }J ___ l;. ____ J

(4)

with (l4-b) -Cl Xl (L.) sin 1

We have chosen plane Xl' XZ. The treatment of the problem is identical for plane X3' X • 4

if

(E:)- xZcn cos lji (t) 1

A

cOS,¥ + B sin '¥ = 0

with the condition - A sin

""1

J

sin CU 1 m~xl cos (wILl'C)

+[ :xz(r/+

'f~

l)+sin

Xz(tl

~ -cos(wl ~'t) +Cl~ (Z)sin(wl A"t)]

13. (-Cl ~ -cos (u.ll tl't)

-:XZ(t:"/sin (W AL)] l

t.:~

(1 -cos (£0 60 ))] 1

+sin If cos If

(l-q)

+ B cos '¥

>

0

In the approximation given by equation (14), if the Cl term is equal or close to 1, the results are: Xl (ei ) tg IjI (Z)

The search for an extreme leads to the solution of the equation: cos

IjI

The results obtained show that the componerts of the control vector in plane X X z are such that the corresponding veclor is positioned _Vi behind the bisector of the e rror vector rotation angle (6)

U cosjo.m(Wl 6,.7:.) X(e: 1)= C1 Umaxsinlfl- E X (<:'.) - max

-11

>0

In equation (lZ) if the term Cl is equal or close to 1, the equation to be solved is of the form:

z

<.Ul

1

Observations

The results of equations (4) and equalities (7 -a) and (7 -b) yield, in plane Xl' X :

21+

1

with

(<:;.) > 1

o

0

These results show that angle ljl must be chosen so that the components of the control vector in plane (X XZ) cause the control vector to be positilonea 71/ 2 behind the error vector.

This means that to find the value of ~ that minimizes the quantity (!::i+l) Z . Z . QIZ = X, (e;I+l) +:xz (e;. 1 \' an equatIOn of the following form mus~+De solved:

The preceeding results are directly applicable to plane (X X ). 4 3

542

3.1.2. Study in both planes - Minimization

r equire previous calculation of the control vecto r amplitude as a function of the l evel to be attained (presence of a limit-cycle) and incr ease the response time of the system.

--2---i----i----2---------------

of Q = XI

+X2 +X3

+ X

4

Th e results obtained in paragraph 3.1.1. show that to find the value of 't' that will minimize the criterion chosen Q, an equation of the following typ e must be solved (15-a) a

- variable amplitude and application interval control laws. In thi s study, the second pos si bility ha s been used, and a control law has be e n found which is a function:

cos '!'+ b sin'!' + c s in '!' cos '!' = 0

with

X2 (I5-b) a=C x I

- of the properties of the trajectories in both pla nes X l X and X3 X · 2 4

2) CC) cos( 11'lL)J+C ~ 1 l

(Z) W1

[1-

- of the results indicat e d in paragraph 3 .1 according to which for small intervals of application the values of the control vector pha se and amplitude ar e independ e nt on~ from each other.

sin (w{"l) -

~(~)

~

(uJ26\lJ+~.)sin

[I-COS

~

(9

2

X2

2) C1)

(w1 1'l1)J-~(~)sin

(15-c) b=W'l"" [I-COS X4 -C ~ 2

(W I'lL)

1

[

X3

I-co s (w I'lt)]-C 2 2

Calculation of the control vector amplitude and duration of application, thus determined, is indicat ed in schematic diagram I, broken down into two part s :

(w 1'l'(;) 1 _

--wz (Gi )sin (ut'b)

(15-d) c=-2u max[...L (l-C l(l-cos ~ I'lb ) 2 1 1

- Part 1. A. calcula t es the amplitude of the control vector

W 1

- 2-

w 2

2 (1 - C ) +(l-cos (wi G 2

)]

- Part 1. B. calcu lat es the duration of its application 3 .3 - Gen e ral Diagram

the angle '!' found must satisfy inequality (15-e) :

2 . 2 ) 15 -e) -a sin'!' + b cos '!' + c (cos'!' - SIn '!'

>

Diagram 2 r e pres e nts the general scheme of the digital comp ensator and of the periphericals that execute the control law.

0

Approximate Law

Part A of thi s diagram puts the quantities delivered by the sensors (paragraph I) into state v ec tor form.

If quantities (w xl'lG) and (w xl'lt:;) are small, 2 1 the result is :

Cl Xl ( t:;) + X X

4

(

Part B ca lculates the three quantities u, 'P and d Z (Paragraphs 3. 1 and 3.2 .)

'G)

("0) + C 2 X3 ( G i) 2

Parts C and C are the interfaces between · . I 2 t h e d Igltal compensator and the analog piloting loop systems.

with the condition to eliminate the indetermination due to the tangent

4. RESULTS AND PERFORMANCES (16-b) - sin 'f [C -co s

X ( I I

'G)

+ X

4

((;)J -

[X2(l::)+C2X3(l)J >

The proc ess chosen as an example for digita l simulation is a satellite whose featur es are:

o

KI 3.2. Study of the amplitud e of the control vector and of th e duration of application

= 0,4 0,46

0,23

'" 1

From the results obtained in Paragraph 3. I , two solutions can be con side red: - constant amplitude control laws

K2

they

543

0,565

W2

= 1.41

1----- ----

I

I I I I I I I I I I

FLOWCHART

~----~------------~

u

12 =

(X2 + X2) 2)_ W1 (- 1 2(X COS '!' - X Cl sin ,!, ) l 2

-1-,I I

tUz

I >

o

I I .------'-----, I

~~~---~~~~~~I

I I

I

I I

u

r------

L = U

lA - Calculation of th e Control V ec tor Amplitude

cos '!'

L...-u_R_=_U_s_irn_'!'___-_-_-" _

__ _ _ _

_

__

A=C

2

l

2

Z

U +U + w (X Cl U - Xl U ) L R L l Z R

I

I

I

I I I

I . Arc s In Arc

I I I

J I

~---------~--------~

I

II

~------I

I I I L ____ _

dt

. Sin

I

A +B ) ( ZCB ( CZ + D Z

I

I

<

lZ

dt

f - -:-z-:--2 ZAB ))

<

dt

max

e

1-------1

YES I. B-CalculatlOn of the Duration of application

I

I I I

- - --------~ 544

r

'!' '!' :

-

<1>.

An alo g -Di g i t al C o n ve rt e rs

I

I

<

--

--

I

--}----~

thr es hold

threshold >

E ?

--4. .

I =

(,V L =0

= dt

-- - - - - I

0

'!'

=

I

1

--

'!'

;j;

;

Cl!

= I w

.

A

I

~

I

~

I

I

Tran sfo rmation

I dt

Z

E = ( ,!,2 + <1>2 ) l / Z

,----

(,VR

F LOWCHA RT

r

Cl

I

L

dt

---- ------,

rr i j)

-

I

( X]

~-=-===-F-=-~ B~

m

I I I

I I

I

t g 'I' = Cl X?

Calculation of sin 'I'

X

4

X,

and cos '!'

~ I = -s i n 'I' [Cl

rx z

sin 'I' cos 'I'

1 - cos x 31

~+

+ Cz

<[(

I

X4

•

'I'

)-

?

0>

I

= - sin 'I' = - cos ,!,

1

I I I I

+ c?

~

I I I

Xl

+

1

lA -Calculation of the Control Vector Amplitude lB

I

-Calculation of the Duration of Application

r - - - ---1- - --~ C I z I Digital-Analog Converters I I 1 (, VR i (, VL ~ L _ _ _ _ _ _dt_ f __ J 545

_.. INITIAL CONDITIONS Components

FINAL CONDITIONS

Criterion Q

X =-0,41x10 1

Xl = 0,1 X

2

= 0,

20000 10- 6

X4 =

°

X =-0,28x10 4

Xl = 0,1 X

2

= 0, 20 000 10-

6

X3 = 0, X

4

= 0,1

Xl = -0,1 X

2

= 0 20 000 10-

6

X3 = 0,1 X X X

4 1

= 0, =-0,1

X

4

= 0,1

20 000 10-

6

-3 -3

0,884 10-

6

10,23

-3

1,000 10-

0,91610-

6

6

8,79

8,68

-3

-3 X =0,45x10 2 -3 X =-0,17x10 3 -3 X = 0,87x10 4

546

T

-4

-3 X = 0,29x10 1 -3 X = 0,45x10 2 -3 X =-0,18x10 3 -3 X = 0,87xlO 4 -3 X = 0,35x10 1 -3 X = 0,39x10 2 -3 X =-0,61x10 3 -3 X = 0,51x10 4 X =0,29 x1O 1

2 = 0,

X3 = 0,

X =-0,82x10 2 X = 0,14x10 3

X3 = 0,1

Criterion Q

Components

1,07910-

6

8,80

For a constant value of the torque corresponding to u max = 0,01, the laws given by equations (15) and (16) were simulated. In figure 3, which corresponds to application intervals of dZ of 0,1, both trajectories are identical. In figure 4 which corresponds to intervals of application d~ of 0,5, it will be remarked that the response times that bring the system to the fixed level are different from 0,5 at the most; this corresponds, in the case where the threshold is equal to 1 0_ 4 , to a time differenc e smaller than 1,3'10 of the total application time.

The sub-optimal law proposed reduces the complexity of the control logic organ very considerably in comparison with the complexity required for the optimal control law [8) calculated from Pontryagin's Principle of the Maximum; in this control law, the control vector cannot be connected to the state vector otherwise than by means of an auxiliary vector and this involves lengthy computations whereas for the law proposed, the control vector is calculated from the state vector by simple equations that only involve elementary arithmetic operations for the execution of the digital compensator.

Observation The application of the possibilities of ionic micro-thruster s to the problems involved in the attitude control and orbit correction of telecommunications satellites is at present under joint study by the L. A. A. S. and the C.N.E.S. [9]

The oscillations In figure 4 illustrate the drawbacks of constant amplitude control.

Taking a maximum application time of 0,1, the proposed control law was simulated on a digital computer.

NOTE:

The results indicated in Table 1 correspond to identical error standards and different phasedifferences.

Figure 5a,b,c, was not available at time of printing.

DISCUSSION

Q. If I understand correctly, control is achieved using differential voltages. Is there nothing provided to discharge the jets in the thruster?

Figures 5-a, 5-b, 5-c, show the variations of the error vector in planes Xl X and X3 X , 4 2 and the variation of the chosen criterion Q as a function of time (The trajectories of figures 5-a and 5-b, are gr
A. There is in fact a neutralizer at the exit of the thruster and it is essentially the presence of this neutralizer and the problems of erosion which limit the deflection angle to 10° at the moment.

The results indicated by Table 1 and by figures 5-a, 5-b, 5-c show:

Q. To what extent have you studied the reliability of the computer which would give us a control law similar to the one I have just outlined.

- small response time variations due to different initial phase angles - much improved performances with regard to those obtained by many authors [1] [7J using conventional onloff operation systems. - the possibility of limiting the error vector to a small value, as wa s indicated in paragraph 3.2, distinctly lower than the threshold imposed by the missions considered at present. The lowest limit attainable is imposed by technological constraints: minimum torque, minimum application duration, function of the response time of the entire piloting chain.

A. At the L.A.A.S. we are at present trying to build a specialized computer which would give us a control law similar to the one I have just outlined.

CONCLUSION Amplitude and phase modulation possibilities introduced by beam-deflection electrical propulsion systems give considerably improved results (simplicity of associated electronics as well as performances) in comparison with those obtained from conventional onl off thrust systems.

547

,o'~------------~~----------------~-----------'

Initial

10

Conditions x,(O),. 0,2 X2 (O)'" 0, X3(O)'O 0,2 X4 (O)" 0,

'~' ----'~----------~----------------4-~~~----'

,

10-

Log Q

t--- r---..

,

r.........

~

3

~

Initial Conditions

f---

'I----

X, (0) X, (0) X,(O) X, (0)

Approl(iml!lte law-

,0,' • 0, • 0,2 .0,

Vo\/'\

v:-~

Tog Q I r (?) 'Id? = 0jS

,

10- 0

10

20

_

'" Figure

~

4 __

"

"

O,02~

-,------r----,.-------,

(,;o" ,'''"--(,,2)- -r;;O,.)

_ Figure 5 b _1

Log Q

10

I

,~

Hr-

,

~"'"

I

Log Q = re?)

~

1\ , •

1\ 10

_ Figure 5e_

548

'2,'

?

-BIBLIOGRAPHY -

[1]

A. J. CRAIG and 1. FLUGGE - LOT Z Investigation of optimal control with a minimum fuel consumption criterion for a fourth - order plant with two control inputs; Synthesis of a efficient sub-optimal control. J.A.A. C. 1964 (Session 8 - Paper 1)

[2J

P. LUQUET Propulsion ionique. Rapport C.N.E.S. n° l846/ PR / ED - Septembre 1968.

(31

J. R. ANDERSON and G. A. WORK Ion beam deflection for thrust vector control - Journal of Spacecraft and Rockets. Vol. 3 - n° 12 - December 1966

(4J

A. COSTES - J. C. LAPRIE

a

Contribution la synthese de systemes de commande pour la stabilisation d'un satellite l'aide de micropropulseurs electriques.

a

(A paraltre dans "Aeronautique et Astronautique").

Cs]

E. DURAND Solutions numer iq ue s de s equations algebrique s. MASSON et Cie - 1961.

(6]

A. J. CRAIG and 1. FLUGGE - LOTZ The choice of time for zeroing a disturbance in a minimum fuel consumption control problems. Transactions of the A. S. M. E. 1963 (Paper 63 - WA 33)

[7]

G. PORCELLI and A. CONNOLLY Optimal Attitude of a spinning space body. A graphical approach. 1.E.E.E. Transactions on Automatic Control - June 1967.

[81

M. ATHANS and P.L. FALB Optimal Control. Mc - Graw hill book Company 1966

[9]

J. P. PUJES Les systemes de correction d'orbite et d'attitude des satellites synchrones. 3eme Symposium LF.A.C., Toulouse, 2- 6 Mars 1970.

549