Single Gimbal Momentum Wheel Control Systems for Geosynchronous Satellites

SINGLE GIMBAL MOMENTUM WHEEL CONTROL SYSTEMS FOR GEOSYNCHRONOUS SATELLITES N. Swift, M.J. Hammond. British Aircraft Corporation, Bristol, England.

INTRODUCTION A number of control systems have been proposed for the three axis stabilization of geosynchronous satellites. In the absence of yaw attitude sensing a momentum bias along the pitch axis is necessary and either a double gimballed (DGMW), single gimballed (SGMW) or fixed (FMW) momentum wheel may be used. The use of a DGMW allows full freedom in roll and yaw steering of the satellite body. The freedom of the wheel within the gimbls allows large amounts of angular momentum to be absorbed and long periods can exist between de saturation manoeuvres. Cyclic disturbance torques are absorbed and do not in general cause unnecessary momentum dumping operations. The control system however, is complex and requires digital implementation by either special purpose digital computer or by microprocessor. Control systems using FMW's are relatively simple, but have no satellite roll/yaw steering capability without incurring a fuel penalty. Roll/Yaw control is performed directly by thrusters which are used continuously with an additional fuel requirement to damp nutation when compared to a DGMW system. The SGMW offers the possibility of a control system which has- the simplicity of a FMW system whilst retaining the features of a DGMW system. A satellite with such a control system has an arbitrary body steering capability about the axis containing the gimbal with sinusoidal steering about the other transverse axis. Implementation of the control system is straight forward and can be realised using simple analogue techniques. Desaturation is carried out continuously in order to maintain the required momentum vector orientation but the frequency may be reduced relative to that for an FMW system or the accuracy increased for similar thruster usage. This paper describes a number of ways in which control systems using a SGMW may be implemented. NOTATION ~,

~

roll and yaw attitude

<5 . Y

roll and yaw gimbal angles

h

wheel momentum

w

orbit rate

HI' H3

roll and yaw components of angular momentum

0

1~

M , M3 l Mo' My

roll and yaw components of external torque

111,122,133

roll, pitch and yaw moments of inertia Laplace transform variable

K, a,

control system parameters

roll and yaw gimbal actuator torques

T

Suffix 'd'

indicates demanded values to satisfy steering requirements

Suffix 'c'

indicates commanded values which are derived from a number of inputs.

GENERAL APPROACH The attitude control system is required to carry out four functions. These are: Alignment of the angular momentum vector Primary attitude control Nutation damping Steering. These functions can be carried out by either the thrusters or gimbal servo loop. Alignment of the momentum vector can only be accomplished using the thrusters and conversely steering requires the use of the gimbal servo loops. The remaining functions can be provided by either actuator and where the primar.y attitude control function is provided by using the gimbal servo loop then the nutation damping function is automatically achieved. A number of control systems based on the possible combinations of actuators and control functions is presented in the following table together with the designations by which they are referred to in the text. ACTUATORS

FUNCTION Primary Attitude Control

Thrusters

Thrusters

Gimbal Servo Loop

Nutation Damping

Thrusters

Gimbal Servo Loop

Gimbal Servo Loop

RI, YI

R2, Y4

System Designation

R3, R4, Y2

(R indicates systems using a gimbal whose axis is aligned with the satellite roll axis and Y systems using a gimbal aligned to the yaw axis). Systems R4 and Y2 use the gimbal servo in closed loop operation while R3 uses the gimbal actuator to generate torque directly. No realisable yaw system corr.esponding to R3 exists. EQUATIONS OF MOTION The equation of motion for a SGMW controlled satellite can readily be derived from the equations for a satellite with a DGMW. The linearised equations in this case are: 1~

Ills

2

+ hs(ljI+y) +

~

2

I33 s ljI -

hs(~+o)

(~+o)

M1

(1 )

+ W h (ljI+y)

M3

(2 )

W

0

h

0

where it is assumed that: Ill'

Wo

1 22 ,

Wo

Wo

133 «

h

and the contribution of the gimba1 inertia and transverse wheel inertia have been ignored. These equations may be factorised to give: sH sH

1

-

H o 3

W

3 + woH1

III

s~

=

+ h (ljI+y)

133 sljl + h

(~+o)

z

~

(3)

M3

(4)

H1

(5)

H3

(6)

The equations of motion for a single gimba11ed wheel satellite are obtained by setting: sy = y or so

0

o

as appropriate. STEERING COMMANDS Steering of the communication beam may be used to perform two functions: a)

To reduce pointing errors which have been determined by ground measurements.

b)

To reduce or eliminate the need for North/South Stationkeeping and thus make significant mass savings.

The required steering can be achieved either by steering the antenna relative to the satellite or by steering the complete satellite with the antennas fixed or a combination of both. Body steering is considered here as either an alternative or back-up to antenna steering. In the absence of external disturbance torques, the equations of motion, (1) and (2) reduce to: (~+ o )

o

(7)

+ woh (ljI+y)

o

(8)

hs (ljI+y) + woh -hs

158

(~+o)

These equations have the solution:

.p + 15

A Sin (w

lj! + y

A Cos (wo t + E)

o

t

+ E)

(9) (10)

A and E are constant and are determined by the orientation of the wheel momentum vector. If the gimbal axis is aligned with the satellite roll axis, y = 0 and the yaw attitude can only be steered sinusoidally. The wheel is oriented to give the phase and amplitude approximating most closely to the required steering. The roll attitude may be steered to follow an arbitrary command by steering the gimbal angle so that: (11)

Conversely, if the gimbal axis is aligned with the satellite yaw axis, the roll attitude may be steered sinusoidally while the yaw attitude steering is arbitrary. THE MOMENTUM PLANE

If nutation is damped, equations (5) and (6) become: h (lj! + y)

(12 )

-h ( + c )

(13)

The long term motion of the satellite is described by equations (3) and (4) and after nutational motion has been damped out, the attitude may be computed using equations (12) and (13) if the gimbal angles are known. As the long term motion is described by a second order system of equations, the solution can be represented graphically by plotting trajectories in a plane in which the coordinate axes are the roll and yaw components of angular momentum. This will be referred to as the momentum plane and is analogous to the phase plane which provides a powerful analytic tool for simple dynamic problems and control systems. The undisturbed motion of the satellite gives trajectories on the momentum plane which are circles centred on the origin which are described at orbit rate. If external torques act on the satellite due to thrusters operating for a short time, the motion may be regarded as moving instantaneously from one point on the momentum plane to another. The change in angular momentum is readily calculated from the impulse imparted to the satellite by the thrusters. The operation of the controllers may be understood by considering a simple case in which the steering commands are zero and assuming that 15

=y =

O. 159

Roll attitude is controlled by firing the yaw thrusters whenever the roll attitude error exceeds a given limit. A yaw thruster firing of fixed length will give a change in H3 which, when nutation has damped out, gives a resultant change in roll attitude according to equation (13). The roll attitude deadband may be expressed in terms of H3 using equation (13) and plotted on the momentum plane. The motion of the satellite therefore consists in a drift at orbit rate, until the deadband is exceeded. A thruster firing then returns the trajectory to inside the deadband after which further orbit rate drift follows giving an overall motion as tepresented in figure la. It can be seen that although roll attitude is well controlled, yaw attitude is relatively unaffected except in the region of the origin. If the opposite sign roll thrusters are fired at the instant that the yaw thrusters are fired, changes in both roll and yaw attitude occur and the motion is as shown in figure lb. It is evident that yaw attitude is now controlled more rapidly back to zero. The speed at which yaw is controlled, is determined by the relative magnitudes of the torques generated by the roll and yaw thrusters. The minimum roll deadband setting and hence roll accuracy is determined primarily by sensor noise and the thrust magnitude and pulse length which define the change in body attitude. To provide the greatest tolerance to noise the yaw thruster impulse should result in a zero roll error. For simplicity the roll thruster, which changes yaw attitude, is commanded on for the same fixed period as the yaw thruster so that the roll and yaw impulses are approximately equal. This restricts the speed of response in yaw. The momentum plane has been used to synthesise all the controllers considered in this study. In some of the control systems considered, roll attitude is controlled to zero by the gimbal servo loop and the yaw component of angular momentum is due to a roll gimbal angle or yaw attitude rate. The signal used to drive the thruster logic is different in these cases, but the principle of operating roll and yaw thrusters together is the same. In the presence of steering commands, the gimbal angles are not zero and the angles used in the momentum plane must be replaced by the errors between the actual and demanded gimbal or attitude angles. Control Systems using a Wheel with a Roll Gimbal Roll-Gimbal-System-RI. This system uses thrusters for both attitude and nutation control. This is therefore essentially a fixed wheel controller in which body steering can be carried out using the gimbal servo according to the limitations described above. An attitude control system such as that described in Reference /1/ is envisaged. This system appears to have no advantages and suffers from high RCS fuel consumption when compared with other controllers that have been considered. No simulation results are presented for this system. A schematic diagram is shown in figure 2.

160

Roll-Gimbal-System-Rl. The additional fuel consumption which arises when the thrusters are used for nutation damping can be avoided if this function is carried out using the gimbal torquer. The thrusters are then used for attitude control only. The thrusters are driven by the roll attitude error through a deadband whose width is determined by the required roll position accuracy. This concept of nutation damping also results in an improved attitude accuracy when compared to system RI. The roll gimbal is controlled so that: O = K _s_ , s + a

(14)

The equations describing the nutation motion (5) and (6) become:

(15) -h (1 + ~) , + I s~ s + a 33

(16)

The characteristic equation is: 2 2 III 133 s3 + alII 133 s2 + (1 + K) h s + h a

=

0

(17)

Choosing K

6.5

a

0.2 sec

-1

gives roots at: - 0.032 and -0.084 ± .13 j This represents a compromise between the degree of nutation damping and the amount of random gimbal motion induced by sensor noise. Roll steering is performed by driving the roll and yaw thrusters with the error in roll attitude and the roll gimbal angle command becomes:

oc

+

(18)

A schematic diagram is shown in figure 3. Roll-Gimbal-S~stem-R3. In the previous system controlled us~ng the thrusters. A deadband is

roll attitude was incorporated to avoid excessive thruster firing and thus contributes to the overall roll attitude error. If roll attitude is controlled by the roll gimbal, the deadband is eliminated and potentially more accurate control achieved in roll.

The simplest system in concept is to use the gimbal torquer directly to generate roll control torques according to: (19)

161

The roll equation becomes: K (1 + TS) (1 + 0.1 TS) (~ - ~d)

+

o

(20)

The characteristic equation has roots at: - 0.12, -0.34 ± 0.23 j for K

=

10/Nm/rad and T

=

12.5 secs

The error between the actual and commanded gimbal angle gives a measure of the misalignment of the angular momentum vector and this is used to drive the roll and yaw thrusters through a deadband.

A schematic diagram of the control system is shown in figure 4. Roll-Gimbal-System-R4. Roll control torques may be generated indirectly by commanding the gimbal angle to follow a suitable law. If the yaw attitude is eliminated between equations (5) and (6) the roll dynamics have the form: 2

2

III 122 s ~ + h (~ + 0 ) (21)

If the roll gimbal is steered so that: o

=

K (1 + TS) (1 + 0.1 TS)

4> -

(22 )

K "'d + od

The characteristic roll equation is: O.lT III 133 s choosing

K

3

+ III 133 s

2.5

T

- .65,

2 + h2

T

2 (0.1 + K)s + h (K + 1)

o

(23)

12.5 sec gives roots at

=

- .068 ± .12 j

In the steady state, the roll equation reduces to: (24) since

H3

=

-h

(~ +

0)

(25 )

Equation (24) reduces to: (26)

162

The difference between demanded and actual gimbal angle is used to control the roll and yaw thrusters through a deadband and so align the momentum vector as described previously. The maximum roll attitude error is then determined by the external disturbance torques acting on the satellite, the width of the deadband and the loop gain. In practice, sensor noise limits the value of gain that can be used without excessive thruster operation and an integral term is added to the gimbal command. The roll attitude error is then independent of the external torques and the width of the deadband. The complete control system is shown schematically in figure 5. Systems using a Yaw Gimbal Yaw-Gimbal-System-Yl. This system is directly analogous to the roll gimbal system RI. The Control System is essentially a fixed wheel control system with sinusoidal roll steering given by the alignment of the momentum vector and arbitrary yaw steering obtained by steering the yaw gimbal. A schematic diagram of this system is shown in figure 6. Yaw-Gimbal-System-Y4. This system is analogous to roll gimbal system R2 in which attitude is controlled by the thruster system and nutation is damped by the gimbal servo loop. Setting: (27)

and substituting into equation (5) gives: (28) Eliminating 2

III 133 s

~

~

between equations (28) and equation (6) gives:

+ K 133

hs~

+

h2~

= 133 SRI - hR3 + I33 h

KS~d

(29)

- h I 33 SY d

The characteristic equation is: 111 133 s2 + K 133 hs + h Choosing

2

=

K =

(30)

0 this equation becomes:

o

(31)

which has two real roots at:

163

K is approximately equal to 2 for critical damping. of this system is shown in figure 7.

A schematic diagram

Yaw-Gimbal-System-Y2. This system uses yaw gimbal angle to control roll attitude, and hence nutation, directly. This has the advantage of giving better roll control than is possible using thrusters. The roll equation for a yaw gimbal system is: III S2$ + hs

(~

+ Y) + woh $ =

(32)

0

Setting: Sy

(33)

gives: 2

III s $ + hK2 S$ + h Kl$ with

sec

-1

=

K2

hKl $d - hs (Yd + =

~)

(34)

12.5

The characteristic equation has roots: - 0.39 ± 0.69 j The angular momentum vector is aligned using the error in the gimbal angle to actuate the roll and yaw thrusters through a deadband. A schematic diagram ·of the controller is shown in figure 8. However, it is readily shown, that the yaw stiffness is low and operation of the thrusters occurs continually as the gimbal angle repeatedly crosses the deadband. This system is therefore not viable due to the high fuel consumption. SIMULATION General ------The designs of the control systems have been initially carried out on the basis of linear analysis. A Hybrid computer simulation has been developed to confirm this analysis using an exact representation of the satellite and controller dynamics and including non-linear effects which at best can only be treated approximately by analytical techniques. The simulation uses a simplified model of the satellite in that the principal axes of inertia of the main body and components of the wheel are assumed to be aligned with geometric axes. The gimbal axes are also assumed to be aligned with the geometric axes and to pass through the centre of mass of the gimbals and wheel. The finite moments of inertia of the gimbals have been taken into account as has gimbal friction. The effects of array flexure have been represented by the lowest frequency assymetric bending mode and the lowest·frequency symetric torsion mode. lM

Two sets of gimbal dynamics have been used, corresponding to yaw and roll gimbal operation. The equations of motion have been programmed for the EAL 8945 Hybrid computer. The simulation can operate in either real time or at multiples of ten up to a thousand times faster than real time. This allows phenomena occurring at widely different timescales to be examined. Thruster operations are commanded on all the control systems simulated from simple deadband comparators. These commands are further processed by timing logic which produces a fixed length pulse (50 ms typical) followed by an off period. (lOs typical). After the off period, the timing logic resets and the sequence is repeated provided that a thruster command is present. This simple logic is necessary in order to prevent additional thruster firing due to the damped nutational motion holding the attitude on the deadband boundary. This also has the effect of increasing the transient response times, but avoids the possibility of unnecessary thruster operations. Parameters Typical simulation parameters are shown below: Satellite Body and Array Inertia Roll (Ill) Pitch (1 Yaw

(1

22 33

350 kgm

)

150 kgm

)

400 kgm

2 2 2

Inner Gimbal Inertia

0.15 kgm

Outer Gimbal Inertia

0.08 kgm

Thruster magnitude

0.5 N

Attitude Deadbands

±

Gimbal Angle Deadbands

±

Nominal Wheel Momentum

2 2

25 Nms

Simulation Results Simulation results for systems R2, R4 and Y4 are presented in this section. Systems RI and Yl will have an identical performance to their respective fixed wheel equivalents. The transient response of systems R2, R4 and Y4 are shown in figures 9, 10 and 11 respectively. The faster speed of response of system R4 to a-roll Initial condition is due to the tight roll control loop provided by the roll gimbal servo whilst systems R2 and Y4 rely on a thruster deadband loop whose response is restricted by the thruster timing logic. The response of Y4 is significantly modified by the nutation damping loop which commands the gimbal servo proportionally to the roll attitude error. 165

All three systems, exhibit a virtually identical yaw response since the alignment of the momentum vector is governed by orbit rate coupling of the axes and the deadbands set on roll attitude, systems R2 and Y4 or gimbal angle, system R4. The response of the systems to orbit rate sinusoidal steering commands are presented in figures 12, 13 and 14. A period of three orbits is shown which emphas1ses the capab1l1ty of the simulation to operate at 1,000 times real time thus enabling orbit rate effects to be examined. Initially the thrusters are operated at high frequency which gradually reduces to zero after approximately one half an orbit when the alignment of the momentum vector is complete thereafter no further thruster operations are required unless external disturbance torques cause momentum dumping to occur automatically when the attitude/gimbal deadband limits are reached. REFERENCES

11/

1~

Iwens, R.P., Fleming, A.W.and Spector, V.A.: Precision Attitude Control with a Single Body-Fixed Momentum Wheel. A1AA Paper No. 74-894 (1974).

MOTION DUE TO JET FIRING ORBIT RATE DRIFT

••

a ) MOTION WITH YAW THRUSTER FIRING ONLY

b ) MOTION WITH ROLL AND YAW. THRUSTER FIRIrlGS

TRAJECTORIES ON THE MOMENTUM PLANE

FIG.l

167

+VE YAW -YE ROLL

TIMING LOGIC

-YE YAW +VE ROLL

(OEAOBANO) (THRUSTERS )

ROLL GIHBAL SYSTEM R1

FJG.2 +VE YPJ/4 -YE ROLL

TIMING LOGIC

-YE YAW +VE ROLL

(THRUSTERS) }--_ _ _ _ _--"-~c=---+ (GIMBAL SERVO)

~+ ~.

ROLL GHIIAL SYSTEM R2

FIG.3

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

!I-d_ _-t

~

+VE YPJ/4 TIMING LOGIC

'-----.1........,

-YE ROLL -YE YAW +VE ROLL

(DEA~ANO)

(THRUSTERS ) ROLL GIIIIAL SYSTEM R3

~I-d_~~>--t ~

FJG.4

(""'"

sr.",

+VE YAW TIMING LOGIC

-YE ROLL -YE YAII +VE RQLL

(DEAOBANO) (THRUSTERS) ROLL GIIllAL SYSTEM R4 168

FIG.S

+VE YAW -VE ROLL

TIMING LOGIC

-VE

YA\~

+VE ROLL (OEAOBANO)

(THRUSTERS)

FIG.6

YAW GIMBAL SYSTEM Yl

+VE YAW -VE ROLL

TIMING LOGIC

-VE YAW +VE ROLL ~c

(GIMBAL SERVO) FIG.7

YAW GIMBAL SYSTEM Y4

+

~c }------- .::.--(G IMBAL SERVO)

-VE YAW TIMING LOGIC

+VE ROLL -VE YAW +\"E ROLL

YAW GIMBAL SYSTEM Y2

FIG.8 169

t

40 SEC 0.0005 ROLL RATE (RAD/SEC) ~--========-~=-----0.0005

ROLL ANGLE (OEG)

1.0~

-1.0

t..::========"'-_"""="...-___

0.0005 YAW RATE I (RAD/SEC) tor -0.0005

....

• ......

YAW 2.5 ~ ANGLE f---~============(DEG)

- 2. 5

I

+VE fHRUSTERI IIIIIIIIIIIIIIIII1 COMMANDS -YE

ROLL INITIAL CONDITION

1.00 '-----'

4000 SEC

0.0005

~~~6/~~~ t co. -o. ooos~

""""'"

""""

. 0.1 ~

~~~tE· lr."lnl!MlIII~~WrWrWvWMWNvVM1MMM/lc1~--= ( DEG)

-0 . 1

s.Op

O.OOOSt YAW RIITE 11ottll-+111... ( RAD/ SEC ) 1-i__..""".. WllltltllltttllltttIll1l
.....,..-----t---

- 0.0005

YAW

MIGLE

(DEG ) -5 . 0 THRUSTER...VE COMIWIOS -YE

I.1111111111111111111111 1111111111 III I I I I I I 0 YAW HllTIAL CONDITION 4. 0

INITIAL CCNDITION RESPONSE ROLL GIMBAL SYSTEM R2

170

FIG.9

>-------< 10 SEC O.OOlf ROLL RATE ~ "(RAD/SEC) I-..L-_ _~~~_,......-r.-==--=-=---====-~---

A

__

___

-0 . 001 nO.LL 0 . 5~ ANGLE W-------~~~~--~==~-------------------(DEG1-0.5 -O'OOSl YAW AA.TE (RAD/SEC)

-0.005

~

V

1.0[

YAW ANGLE (DEG)

~""\7:----'-7""""'--------"===:::::"-------"'==========-\..../

-1.0 +VE\

t~~~~6~

1---L_--,_-,-_...L_L----lL-_ _ _-1_ _ _ _ _ __

-YE ROLL INITIAL CONDITION 0.4

0.0005 ROLL RATE (RAD/SEC) .. 0.0005

0

>------<

~ . .·. ..._-,...,...,...,........,...,. . . .,. .,. . . ,. . ... Ir---. ,,-.. . . . . . . . . . .---------4000 SEC

t

ROLL ·0.05 ANGLE "IIIIIIIIIIIIIIII!I!lIII!l1I11I11I11111111 1 I I I . (DEG) -0.05

0 . 0005

(Y:D~~~~)

t

·"'.'1111111111111111111111 !I Ill!

I

1

I

11 1 1 1 1 1

-0 . 0005 YAW ANGLE

5'0~

~DEG)

_

-5 .0 THRUSTER+VE 'COIot1ANDS

I_10101111111111111111111 III " I 1I I I I

I I

-YE YAW INITIAL CONDITIml 5.0

INITIAL CorWITION RESPONSE ROLL GHlBAL SYSTEMR4

0

FIG.l0

171

0 . 002 ROLL RATE (RAD/SEC)

[~

40 SEC

f-LI\~=-------------'=""

-0.002 ROLL ANGLE (DEG)

LOthr--------=----~I~

V

-1.0 0.00' [ YAW RATE ( RAD/SEC)

-

~

-0.001

~t~~.::::::::===============-­

l:LE 2.5 (DEGt2.5r

I

+VE THRUSTER hrrrTTTrrnTTTrrrlrrr-l---COMMANDS IIIIIIIIIIIIII I II I I -YE ROLL INITIAL CONDITION 1.00 4000 SEC

0.0005

fg~g:~gl ~=...

hi"""""""""O"""O 0 0 0 0 0 0 0 0 0 0 0

0

0

~~t~·05~/1/1d~ (DEGl -0.05

r-

.

~ -_ _ _~~~~~~.........~~~__'_~~~_~_ _ YAW 0.0005 RATE t (RAD/SEC) ,-0.0005 YAW 5'0r===== ANGLE (DEG) -5.0

I

THRUSTER +VE c._"'lIIndIUJIIIWIIIJllIIIJllIIIWllllllwllIwllwlll1.lUllJJIII1.I.LILJIIUlululul-.JILILLI-L-.JL-.JL-L_-L-_-LCOt+lANDS ,-YE 0 YAW HlITIAL CONDlTIotl 4.0

INITIAL CONDITION RESPONSE YAW GIMBAL SYSTEM Y4

172

FIG.11

0.0005

~

40000 SEC

~

1--__ ___ ___ ___

RATE "'_................_ _ ROLL (RAD/SEC) -0.0005

1.0~/"\.

~~tE

/"\.

/"\.

(DEG)~ -1.0

0.0005 YAW RATE (RAD/SEC) -0.0005

~LE

5.0

r r

,._111"....' ....' ---- '--- ---- ---- --

1\

/l

1\

~\J

(DEG) -5.0

+VE ~ THRUSTER illlr.-ICOMMANDS .......... -YE

- - . - - - - ---- - -- - - ROLL GIMBAL SYSTEM R4

FIG.12

>-----I

0.0005 RATE ROLL (RAD/SEC) -0.0005

~~tE

40000 SEC

t

"'_M +-<- -+--- ------___ __

l'O~

~

-1.0 0.0005 YAW RATE (RAD/SEC) -0.0005

~LE5.0 (DEG) -5.0

/"\.

/"\.

~V

(DEG)

r,._III"+<',1-- -1-- ---- ---- ----

r

1\

1\

/l

\J\JV

I

+VE THRUSTER , _ " " ! " I I - - r - - - - - -COMMANDS .. -VE ROLL

GIt~BAL

-----SYSTEM R2

RESPONSE TO STEERING COMMANDS 0 (O.6 u ROLL 3.6 YAW)

FIG.13

173

----.

f

40000 SEC ROLL0.0005 RATE (RAD/SEC) 1--.........- - - - - - - - - - - - - -0.0005

:~~El.O ~ (DEG)

-1.0

~

~

/l

~""-/\.-.7

~:D;:!!:5 Ft'."'.,. '. . '._____"--__________ -0.0005

r

I

+VE mi ll I I THRUSTER COMMANDS F"ll.LL.-

-----'---

-VE YAW GIMBAL SYSTEM Y4 RESPONSE TO STEERING COMt'1 ANDS (0.6° ROLL 3.6

174

0

YAi~)

FIG.14