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NELECTRICAL
ENGINEERING EXPERIMENT STATION
AUBURN UNIVERSITYAUBURN, ALABAMA
ENG
NEER
ING
https://ntrs.nasa.gov/search.jsp?R=19730010132 2020-04-26T00:43:21+00:00Z
MULTIVARIABLE CONTROL THEORY APPLIED TO HIERARCHICAL
ATTITUDE CONTROL FOR PLANETARY SPACECRAFT
PREPARED BY
GUIDANCE AND CONTROL STUDY GROUP
J. S. BOLAND, III, CO-PROJECT LEADER
D. W. RUSSELL, CO-PROJECT LEADER
DECEMBER 8, 1972
CONTRACT NAS8-27827GEORGE C. MARSHALL SPACE FLIGHT CENTER
NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONHUNTSVILLE, ALABAMA
APPROVED BY: SUBMITTED BY:
'9h a'4i? M. A. HonnellProfessor and Head (Acting)Electrical Engineering
S. Boland, IIIAssociate ProfessorElectrical Engineering
D. W. RussellProfessorElectrical Engineering
TABLE OF CONTENTS
FOREWORD
SUMMARY
PERSONNEL
LIST OF FIGURES
LIST OF TABLES
LIST OF SYMBOLS
I. INTRODUCTION
II. THE REACTION CONTROL JET ATTITUDE CONTROL SYSTEM
III. THE CONTROL MOMENT GYRO ATTITUDE CONTROL SYSTEM
IV. COMBINED REACTION CONTROL JET AND CONTROL MOMENTGYRO ATTITUDE CONTROL SYSTEM
V. COMPUTATION OF DISTURBING TORQUES DUE TO MOTION (TELEVISION CAMERA ..
VI. CONCLUSIONS AND RECOMMENDATIONS
REFERENCES
APPENDIX
ii
iii
iv
V
vi
vi
viii
1
6
18
36
OF
......... 41.
52
54
55
FOREWORD
This final report documents and summarizes the work accomplished
by the Electrical Engineering Department, Auburn University, in the
performance of Contract NAS8-27827 granted to Auburn University, Auburn,
Alabama. The contract was awarded July 23, 1971, by the George C. Marshall
Space Flight Center, National Aeronautics and Space Administration,
Huntsville, Alabama.
iii
SUMMARY
This report applies multivariable control theory to the design of
a hierarchial attitude control system for the CARD space vehicle. The
system selected uses reaction control jets (RCJ) and control moment gyros
(CMG). The RCJ system uses linear signal mixing and a no-fire region
similar to that used on the Skylab program; the y-axis and z-axis systems
which are coupled use a sum and difference feedback scheme. The CMG system
uses the optimum steering law [4] and the same feedback signals as the
RCJ system. When both systems are active the design is such that the
torques from each system are never in opposition.
A state-space analysis was made of the CMG system to determine
the general structure of the input matrices (steering law) and feedback
matrices that will decouple the axes. It is shown that the optimum
steering law and proportional-plus-rate feedback are special cases.
A third part of the report is a derivation of the disturbing
torques on the space vehicle due to the motion of the on-board
television camera. A simple procedure for computing an upper bound
on these torques (given the system parameters) is included.
iv
PERSONNEL
The following staff members of Auburn University have actively partici-
pated in the work of this contract:
J. S. Boland, III - Associate Professor of Electrical Engineering
D. W. Russell - Professor of Electrical Engineering
W. G. Legg - Graduate Research Assistant in Electrical Engineering
v
LIST OF FIGURES
I-1. Baseline CARD Planetary Vehicle Concept . . . . . . . . . . . . 3
I-2. Baseline CARD Planetary Vehicle Concept . . . . . . . . . . . . 4
I-3. Baseline CARD Planetary Vehicle Concept . . . . . . . . . . . 5
II-1. RCJ Attitude Control Thrusters ................ 7
II-2. X-axis RCJ Attitude Control System . . . . . . . . . . . . . . 8
II-3. Y- and Z-axis RCJ Attitude Control System Block Diagram . . . 11
iI-4. Phase-plane Trajectories of RCJ System . .... .. . . . . . 12
II-5. Phase-plane Trajectories of RCJ System . . . . . . . . . . . 13
II-6. Y-axis and Z-axis, RCJ System Using Sum and DifferenceSaturation Signal ......................... 14
II-7. Boundaries of No-Fire Region for RCJ System Using Sum andDifference Saturation Signals . . . . . . . . . . . . . . . . .15
II-8. Phase-plane Trajectories of RCJ System - Sum andDifference Saturation Signals . . . . . . . . . . . . . . . . .16
III-1. CMG Cluster ......................... 19
III-2. CMG System Block Diagram . . . . . . . . . . . . . . . . . . .20
III-3. Block Diagram of Open-loop CMG System; Clamped Mode andTorque Motor Dynamics Neglected . . . . . . . . . . . . . . . .23
III-4. Block Diagram of Open-loop CMG System; Clamped Mode andTorque Motor Dynamics Included . ..... . . . . . . . . . 29
III-5. CMG Closed-loop System; Clamped Mode . . . . . . . . . . . . .34
III-6. CMG Closed-loop System with Proportional-plus-rate Feedback;Clamped Mode . . . . . . . . . . . . . . . . . . . . . . . . .34
111-7. Closed-loop System with Proportional-plus-rate Feedback;Clamped Mode ......................... 35
111-8. Closed-loop System with Proportional-plus-rate Feedback;Clamped Mode ......................... 35
vi
CMG System with Sum and Difference Saturation Signals . .
Phase-plane Trajectory of Combined RCJ and CMG System . .
Phase-plane Trajectories of Combined RCJ and CMGSystems . .... . . . . . . . . . . . . . . . . . .
Phase-plane Trajectories of Combined RCJ and CMG Systems
Television Camera and Mount . . . . . . . . . . . . . . .
Television Camera Coordinates . . . .. . . . . . . . . .
1. Disturbing Torques Due to Motion of Television Camera
vii
IV-1.
IV-2.
IV-3.
IV-4.
V-1.
V-2.
Table
. . ..37
. . . 38
. . . 39
. . . 40
. . . 42
. . . 44
. . . 51
LIST OF SYMBOLS
DEFINITION
X-axis rate feedback gain (RCJ System)
Y-axis rate feedback gain (RCJ System)
Z-axis rate feedback gain (RCJ System)
X-axis feedback gain (see Fig. III-2)
Y-axis feedback gain (see Fig. III-2)
Z-axis feedback gain (see Fig. III-2)
X-axis rate feedback gain (see Fig. III-2)
Y-axis rate feedback gain (see Fig. III-2)
Z-axis rate feedback gain (see Fig. III-2)
Space vehicle X-axis moment-of-inertia
Space vehicle Y-axis moment-of-inertia
Space vehicle Z-zxis moment-of-inertia
X-axis reaction moment (see Fig. III-2)
Y-axis reaction moment (see Fig. III-2)
Z-axis reaction moment (see Fig. III-2)
X-axis acceleration
Y-axis acceleration
Z-axis acceleration
X-axis command moment
Y-axis command moment
Z-axis command moment
Gyro torque-motor time constant
viii
SYMBOL
Ax
Aly
Alz
FDPX
FDPY
FDPZ
FPX
FPY
FPZ
Jx
J
ix
y
Jz
MRY
MRZ
MXC
MYMy
MXCMND
MYCMND
MZCMND
Tm
SYMBOL. DEFINITION
T Disturbing torque about X-axisx
T Disturbing torque about Y-axis
Tz -: y~~~Disturbing torque about Z-axisTz Disturbing torque about Z-axis
Ui Input variables
Xi State variables
Zi State variables
6ij Gimbal angles (see Fig. III-1)
8o~~c.~~ ~Rotation of TV camera about the Y-axis
Of~~~~cx - X-axis command input
cy¢~~~~ ~Y-axis command inputOcy
Z-axis command inputOcz'
Odb Deadband
Saturation limit on feedback signals
Space vehicle rotation about the X-axis
oxby~~~ ~Space vehicle rotation about the Y-axis
Space vehicle rotation about the Z-axis
Rotation of TV camera about.the Zc-axis*C
ix
I. INTRODUCTION
The attitude control of a spacecraft visiting comets and asteroids
for scientific study requires a multivariable attitude control system.
During the journey from earth to rendezvous with the target (comet or
asteroid), the attitude control is provided by the thrust vector control
(TVC) system, the propulsion being supplied by a solar electric system
augmented by a chemical propulsion unit. When the spacecraft rendezvous
with the target, the attitude control for station keeping, docking, and
separation from the target requires more capability than the TVC system
can provide.
The objective of this study was to design a highly reliable attitude
control system for experiment pointing, station keeping, docking and
separation from the target for the Comet and Asteroid Rendezvous and
Docking (CARD) space vehicle [1]. Early in the study it was determined
that a combination of a reaction control jet (RCJ) control system and
a control moment gyro (CMG) control system properly mated would be a
good approach. Such systems have been successfully used on other space
vehicles although problem areas still exist [2].
The RCJ system is by nature an on-off linear control system and
for small errors will limit cycle; energy to generate torques is from
an ever-decreasing supply of pressurized gases. On the other hand the
CMG system is a continuous non-linear control system using solar.
energy to maintain the spinning of the gyros. For small disturbances
the CMG system can keep the attitude errors inside the RCJ system dead-
1
2
band. The RCJ system is available to desaturate the CMG system and to
correct for large disturbances.
The design objectives given primary consideration were:
(1) Minimize fuel-consumption when the RCJ system is the only
active system, and when both systems are active design
the system so that the torques produced are never in op-
position.
(2) Minimize coupling between the three axes of the attitude
control system.
The spacecraft's general appearance, size, configuration, inertias,
etc., were taken from the Northrup Services, Inc., final report of its
feasibility study [1]. Figures I-1, I-2, I-3 are taken from that report
An additional area of study included in this report concerns
the disturbances on the spacecraft due to the motion of the scanning
platform on which is mounted the television camera. A procedure to
compute the disturbing torques as a function of the television camera
inertias and scanning rates is included.
3
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II. THE REACTION CONTROL JET ATTITUDE CONTROL SYSTEM
The attitude control system controls the orientation of the space-
craft with respect to the XYZ axes; the outputs are the angles Ox, y,
%z. The reaction control jet (RCJ) attitude control system uses cold
gas thrusters to produce torques about the spacecraft center-of-gravity.
The location of the thrusters is taken from [1]; the numbering cf the
thrusters and the spacecraft axes are as shown in Figure II-1. The space-
craft center-of-gravity is at the origin of the XYZ axes and it is as-
sumed that the thrusters produce torques in the Y-Z plane.
To simplify the analysis the dead-time, rise-time and decay-time of
the thrusters are neglected;'in addition it is assumed that all thrusters
produce the same torque with no misalignment. Once an engine is acti-
vated it is assumed to continue to produce a constant torque for at least
fifty milliseconds. This is a commonly used value.
The X-axis control system uses thrusters which produce no torques
about the Y or Z axes. The'control system uses proportional-plus-rate
feedback with a saturation block in the position feedback path. This
produces a no-fire region similar to that used in the Skylab program [2].
Figure II-2 contains a block diagram of the X-axis control system
and a phase-plane showing the boundaries of the no-fire region. The con-
troller parameters were calculated assuming that the system deadband,
*db, and the saturation value, *R, were specified and that a trajectory
originating on the rate limit boundary would intersect the ~x axis at
Odb. The *db specification represents a compromise between fuel consump-
6
7
Thrusters
Z Thruster;
3~~ 2
6 X Axis out-of-page 7
NOTE: Attitude control thrusters are larger
to show numbering system.
Figure Il-I. RCJ Attitude Control Thrusters
9, 11, 13, 15 produceforces out of papers 10, 12, 14, 16, produceforces into paper
r than scale
A
ex < -Odb Thrusters\ 2,4,6,8
Mx (H1 $ R = Odb
R42te Limidb)Rate Limit
No Fire Region
-
Rate Ledge I
R
-*R, + (!, - *a) \
\/(R + d,2 )
1 ( \2 + x
=
db
r I ll2 t .I i A1%~ j~ R 4
Figure 11-2. X-axis RCJ Attitude Control System
8
>x ' Odb
Alx
I~Alx qm Ox \+ ox = db
x
I4cx = 0o
!pir
I i1
I
� i
4
II
9
tion and steady-state accuracy; the Tate ledge specification represents a
compromise between fuel consumption and speed of response.
Alx is computed in the following manner:
Assume *c= 0 and ex < - db so that
*x dt =-Mx (II-1)dt
Also d x (II-2)
dt x' i
Dividing (11-2) by (II-1) yields
d¢x = - ~xdo ~ H
or dox o x dfx
Mx
Integrating and requiring that the trajectory go through (OdbO) yields
2
1 / x\ 2~2 + o x fdb (II-3)+ x db
For - OR -< *x -< R ' the boundary is given by
ex =-db (Alx ox + {x) (II-4)
For the curves given by (II-3) and (II-4) to intersect at Ox =
-OR requires that
Aix @R Idb (II-5)12-M
10
The rate ledge is( hR- 'db)
¥ (~R + ~db)
Since the Y and Z axes share thrusters a linear-signal-mixing scheme
[3] is used to insure that opposing thrusters are not energized at the
sa-me time. A block diagram of the system is shown in Figure II-3. Ay
and AlZ are calculated in the same manner as AIx.
For the digital simulation the rate ledge is taken to be 0.172
degrees/second. From [1] ~db = 0.3° and the acceleration produced in
each axis, for error in that axis only, and the thrusters producing the
acceleration are:
X axis: Solar panels(pitch) Solar panels
+4x produced-x produced
Y axis: Solar panels(yaw) Solar panels
+fy produced-fy produced
Z axis: Solar panels(roll) Solar panels
+~z produced-~z produced
rolled: +.208 deg/sec 2extended: +.031 deg/secby thrusters 1,3,5,7by thrusters 2,4,6,8
rolled: +.0391 deg/sec2
extended: +.0214 deg/sec2
by thrusters 9, 11, 14, 16
by thrusters 10, 12, 13, 15
rolled: +.0378 deg/sec2
extended: +.0126 deg/sec2by thrusters 10, 11, 13, 16by thrusters 9, 12, 14, 15
Some typical trajectories are given in Figures 11-4 and 11-5.
A modification to this system was made which gives promise of better
performance. In place of feeding the position signals Oy and Oz to the
saturation blocks, te sum and difference, (Cy + Oz) and (Oz - 0y) are
the inputs to the saturation blocks. The block diagram of this system is
given in Figure II-6. Figure II-7 is a phase-plane plot showing the bounda-
ries of the no-fire region. Figure II-8 is a typical trajectory for this system.
11
Thrusters 12,15 Thrusters 11,16/
Thrusters 9,14
Figure II-3. Y-and-Z-axis RCJ AStitude Control System Block Diagram
12
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Comparing the data presented in Figures II-4 and II-8, it can be
seen that the modified system is slower but the fuel consumption (based
on both the ~y and z errors being less than the deadband value) is re-
duced. Further work is needed to determine if the rate-ledge and feed-
back gains can be changed so that the modified system has comparable
speed of response with less fuel consumption. It should be noted that
with this modification the (~z + *y) controller and the (4y - 4z) con-
troller are decoupled even when the saturation limits are active; thrusters
11, 12, 15, 16 are active for (~z + *y) errors and thrusters 9, 10, 13, 14
are active for (z - *y) errors.
III. THE CONTROL MOMEX!T GYRO ATTITUDE CONTROL SYSTEM
The control moment gyro (CMi) attitude control system also
controls the angles Ox, y , Oz. The control system simulated in this
study uses the optimal steering law proposed in [4]. The CMG cluster
arrangement is shown in Figure III-1. Part A of this section gives the
block diagram and system equations in matrix form as taken from [4].
Part B is a general state-variable analysis to determine the general
structure of the input and feedback matrices that will decouple the
X, Y, and Z axes.
Part A. System Simulation Using the Optimal Steering Law [4]
A block diagram of the system is shown in Figure III-2. With the
outer gimbals clamped, the outer gimbal rates 63(1), 63(2), 63(3) are
zero. The equations for the reaction moments, in matrix form, are:
.MRXV
MRYV_
RVi
;1(1)-
[A] 1(2) , where
L 61(3)(III-1)
sin(f1(1))cos(63(1))
[A] = -sin(61(1))sin(63(1))
L
cos(6 1(2))
sin(6 1(2)) cos(63(2))
-sin(61(2))sin(63(2))
-sin(61(3))sin(63(3))
cos(61(3)) (III-2)
sin(6l(3)) cos(63(3))
The steering law [TS] gives the commanded gimbal rates
commanded moments.
in terms of the
18
19
.o0*60
4m02H
0HOD
.riFr4
?J .6
0/-Jw
_r
<
:t
O-
W°
20
E1
0r.,) co
C,
0i-4u0 -ewn
:ESl~
Fu
21
c ~~M]
* ~~~~~~~~~~~~~~~IIcl (2)c = [Ts] 4CY(III-3)
* II~~S 61(3)c Mc
Assuming that the actual gimbal rates are equal to the commanded gimbal
rates, the relationship between the commanded moment and the reaction
moment is:
-MRXV NCXV'
MRY
v
= [A] [Ts ] MCYV (III-4)
_RV MCZV
It is desirable that the reaction moment equal the commanded moment,
which requires that:
[A] [TS] - [I], where[I] is the identity matrix. (III-5)
The steering law is thus given by
[Ts] = LA] 1 , wherelA] 1 is the inverse of the matrix A. It
should be noted that [A] is singular at some operating conditions.
There is one vehicle control loop for each of the three axes. A
rate-plus-position feedback is used for each of the vehicle control
22
loops. The vehicle control law output is processed to obtain the moment
command. This moment command is compared to the actual moment and
the error is used together with the optimal steering law, to provide the
proper reaction moment on the vehicle.
A digital computer simulation of the three axis CMG attitude con-
trol system has been written and is included in the Appendix.
Part B. A State-Space Analysis of the CMG System (Clamped Mode) for
Axis Decoupling
The object of this analysis is to write the system equations in
state-space form and determine the form of the input matrices and the
feedback matrices that will minimize or eliminate the coupling between
the X, Y, and Z axes. In the first phase of the study the torque
motor dynamics are neglected but are included in the second phase.
Phase 1. Torque Motor Dynamics Neglected
The open-loop system is shown in Figure III-3. Let the states and
the inputs be defined as
FXl xt Ul [11.
x = ; the input, u= u2[]2
12 (111-6)
x3 =y L 13.
X I
4 y
5 z
X6
23
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ZC:
P.4 00c.
a)o Cd
crbe o
I-4 00 0qa~ aJ
Cu
0o 0Cu
r--0 00CJ-0 0'.40ooCu
,-4
-c -o
-I n
N
'n
0 n~
~
0~~
r4 -4~~..
I
r- 4 ,
c i
0
'54 ..
&-I-
0
fi) '0
-
e~~~~~~~~~~~~--
I X
>e *
! *k+
a
*vk
t X
>
1
I x
>
N
I430- -&
-G~i
II V� �p E
24
Then the state-space equation is
o
x = [Ao] x + [B] u
where
0
0
0
C
0
0
1 O. 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
[B] =
6x3
0
1
0
0
0
0
To decouple
u= [Ts] Ic +3x3
0 0
0 0 [J] [A]3x3 3x3
0 0
1 0
0 0
0 16x3
the axes let
[F1] x where ~c =3x6
(III-9)
* cxcx]= desired output angles and (III-10)
cCYOcy.
*i
[Ts] and [F1] are decoupling matrices.
[A ]=
(III-7)
(III-8)
25
Then
x = {[Ao] + lB]I[F1]} x + [B] [Ts] ~c
For the axes to be decoupled,
0 0
Y21 0
0 0
0 Y42
0 0
0 0
[B] [Ts] must have the form
0
0
0
0
0
Y63
where the y's are nonzero.
Thus [B][Ts] 0
Y21
0
0
0
0
1
0
0
0
0
0 0
0 0
0, 0
Y42
0
0
0
0
1
0
0
0
[J] [A] [Ts]
(III-11)
0
0
Y6 3
0
0
0
0
0
1
(III-12)
1
26
[J] is a diagonal matrix of full rank, so this condition can be
satisfied only if [A][Ts] is a diagonal matrix of full rank.
In addition, [Ao] + [B][F1] must have the form
ai
a2 1
'0
0
0
0
a1 2
a2 2
0
0
0
0
0
0
a 3 3
a'4 3
0
0
0
0
a3 4
a4 4
0
0
0
0
0
0
a5 5
a6 5
0
0
0
0
a5 6
a66
where same a's in each sub-block are nonzero. Since [Ao] has the above
form, then [B][F1 ] must also have the above form.
Thus [Bj][Fl1] = a11
a 2 1
0
0
0.
0
a 1 2
ac2 2
0
0
'0
0
0
0
a 3 3
a 4 3
0
0
0
0
a3 4
a 4 4
0:
0
0
0
0
0
a5 5
a6 5
0
0
0
0
a5 6
a6 6
(III-13)
o 0 o
1 0 0
0 0 0
0 1 0
0 0 0
0 0 1.
[J] [A] [F1] (III-14)
27
This implies that [A][F1,] must have the form
[B11
10LO
B1 2
0
0
0
B2 3
0
0
B2 4
0
0
0
B3 5
o
B36
where a B
Thus
(1)
(2)
in each row must be nonzero to have a feedback signal.
the conditions for axes decoupling are that
[A][Ts] be a diagonal matrix
IA][F1 ] have the form
Bll B1 2 0 0 0 0
0 0 B2 3 B2 4 0 0
0 0 0 0 B3 5 B36
Note that if
B1 2 = a constant
B24 =
B36 = 'I II I
times B1 1
" B2 3
" B3 5
then the feedback is proportional-plus-rate and the decoupling can be
achieved as shown.
x
28
where [F 1 ] =
LO
Alx 0
0o 1
0 0
0o 0
Aly 0
0 1
01
0j
Alij
If [A] [Ts] = [I], the identity matrix, then this implements the optimal
control law of [4].
Phase 2. Torque Motor Dynamics Included
Assuming that the torque motors can be represented as first-order
systems with time-constants, Tm, the open-loop diagram is as shown in
Figure 111-4 where u1 , u2 , u3 are the input signals to ,the torque motors.
The states and the input vectors are
x
;X~~~uX.x . ''
[M = |; u2 = input to the torque motors
z
61'12
613
Then the state-space equation is
[__ [A--0] -[--- ---xl [ 1 -.
-I to r{|tfo ---_I+IITm Jm
(III-15)
(III-16)
29
I m
r-4 M
LM
I
I X
Px
XC _I
*^
(oo,.,0
0
$4J
I P4J.
0O4 .
no H
ax
I:
.~ o
1-4bo H
0as.
30
To decouple the axes let
u = ITs] Ac + [[F1] : [F2 ]] ]
= [Ts] Oc + [F1]x + [F2 ]z
x = [Ao]x + [B]z
z = m[ F2 - I]z + -[F 1] + 1 m[Ts]jc
Assume that [A], [Ts] , [F1 ] , and [F2] are constant matrices for small
changes about a steady-state operating point. Then
x = [Ao] x + [B] z (III-21)
= {[Ao] 2 +L[B][F1 ]}x+ {[Ao][B] + 1 [B][F2 - I]}z + 1 [B][Ts]Ic (III-22)Tm Tm Tm
To decouple the axes, [B][Ts] must have the same form as in Part A and
as before [A][Ts] must be a diagonal matrix.
Continuing,
... 1 3 + L B [ F A 1 1x {[Ao] 3 + T[B][F1][A.] + m[A ][B][F1 ] + Tm[B][F2 I][Jl}x +
{[A][B] Tm. Tm T-2
{[AJ2 (B] + L [B][F1 ][B] + L [A ][B][F2 - I] +iT 2 [B][F2 - I]2}z +Tm Tm 2 T Ts]+B(II23
{1L[A.] [B] [Ts] +L 2 [B] [F2 I I][TsI}.tc +L[B ][Ts ];c' ~ (III23)Tm Tm2 Tm
Thus
(III-17)
(III-18)
(III-19)
(III-20)
31
To decouple the axes,
form
1 lB] TS ]T[AQ I [B] [Ts]I
1 T--m [ I F
2- I] [Ts] must have the
Y1 1
Y21
0
0
0
0
0
0
Y32
Y4 2
0
0
0
0
0
0
Y5 3
Y63
; a Yij in each sub-blockis nonzero
The previous condition stipulates that [B]I[Ts]have this form. It is easy
to show that [Ao][B][Ts]has this form. Therefore [B][F2][Ts]must also
have this form. This condition is satisfied if [F2] is equal to a con-
stant times the identity matrix.
Let [F2 - I] = k2[I] (III-24)
Tk2 Z + T [F1 ] X + [Ts] c- Tm - Tm Tm (111-25)
[B]Z = k2. [B]Z + [B][Fl] X + [B][Ts] cTm Tm Tm
and solving (111-19) for [B] Z gives
[B] Z = k2 X'+ [B[F k2 [A X + [B]Ts Tm Tm Tm - Tm
then
and
(III-26)
(111-27)
32
This substituted:into (111-21) yields
x = {[Ao] + k2[I]} X + {[BI[F1] _ k2 [Ao]} X + [B][Ts] ~c (III-28)Tm - Tm M Tm
This equation yields an additional requirement for decoupling; [B][F1]
must have the form
0111i
a 2 1
0
O
0
0
a 1 2
a 2 2
0
0
0
0
0
a3 3
a4 3
0
0
0
0
a 3 4
a 4 4
0'
0
0
0
0
0
a5 5
a6 5
0
0
0
0
a5 6
a6~
where some a's in each block are nonzero.
This result is the same as that derived in
[A][F1] must have the form
B1 2
0
0
0
B2 3
0
0
B2 4
0
Part A and implies that
0
0
B3 5
o]
B3B
as was the case in Part A
To summarize,
the axes is that
the necessary structure of the matrices to decouple
.I
LO
33
(1) [A][Ts] must be a diagonal matrix
(2) [F2 ] must be a constant times the identity matrix
(3) [A][F1 ] must have the form
Bll B1 2 0 0 0 0
0 0 B23 B24 0
0 0 0 B35 B3
The system with feedback is shown in Figure III-5.
With proportional-plus-rate feedback the system takes the form
of Figure 111-6. In this form the [F 2 ] matrix, a k2[I] matrix, does not
aid in the axes decoupling. However, if this diagram is redrawn in
terms of the original set of variables in a modified form, Figure III-7,
where it is assumed [Ts] is non-singular, then since it is desirable
that [Ts][A] = [I] [Ts] = [A], the system can be implemented as
shown in Figure III-8. This is the system implemented in [4] with the
objective that [Ts][A] = [I]. However, this cannot be accomplished under
all conditions since [A] is singular (and thus [Ts]) at some operating
conditions. This problem is minimized in 14] by proper selection of the
angles at which the outer gimbals are clamped and by placing a limit on
the minimum value of Idet [A]| used in the computation of ITs].
34
lae~U)
-e U)0 U)
0 a -I.0I0'-4 UII,-Ii-I-Hr.k
u-e-
-9-
I.-4Cdtoon 0)0)0oTo
-to
00
I4-4 CI 0a
*1-4.J00 a
I -a
-U
0
oCd
'0)
0)
35
Figure III-7. Closed-loop System; proportional-plus-rate Feedback; Clamped Mode
Figure III-8. Closed-loop System; Proportional-plus -rate Feedback; Clamped Mode
IV. COMBINED REACTION CONTROL JET AND CONTROL1M MFT GYRO ATTITUDE CONTROL SYSTEM
The reaction control jet (RCJ) attitude control system and the
control moment gyro (CMG) attitude control system are combined to
operate as a single attitude control system. The objective was to
mate the systems so that the torques generated by each system are
never in opposition, speed of response is improved, and fuel consump-
tion by the RCJ system is reduced.
The RCJ system uses the sum and difference feedback control described
in Section II of this report. The CMG system uses the same sum and dif-
ference feedback signals as are generated in the RCJ system, ex, Ey, Cz.
A block diagram of the CMG system is shown in Figure IV-1. The satu-
ration blocks in the position feedback paths are used to place the CMG
torque reversal lines in the center of the RCJ no-fire regions. The phase-
plane plot of the no-fire boundaries of the X-axis and a typical response
are shown in Figure IV-2. A phase-plane plot of the no-fire region of
the sum (~z + y) and the difference (oz - 4y) systems and a typical
response are shown in Figure IV-3. The same data plotted for each axis
is shown in Figure IV-4.
The CMG system torques aid the RCJ system torques whenever the RCJ's
are firing and the CMG torques tend to hold the RCJ system inside the no-
fire regions. A digital computer program to simulate the combined systems
is described in the Appendix.
36
37
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V. COMPUTATION OF DISTURBING TORQUES DUE TOMDTION OF THE TELEVISION CAMERA
The proposed CARD spacecraft has an on-board television camera which
is used during orbiting or landing to view the terrain and proposed landing
sites. The TV camera and its mount are constructed to permit rotation
about two axes. The purpose of this section is to develop a procedure for
computing the torques about the spacecraft center-of-mass produced by motion
of the television camera.
The general shape, dimensions, and mass of the TV camera and its mount
are scaled from Figure 4.5 and Figure 4.83 of Reference 1. Based on these
figures the TV camera and mount are assumed to have the general form and
dimensions of Figure V-1. It will be treated as a rigid, homogeneous mass.
The XY plane is a plane of symmetry and the XYZ coordinates of Figure V-l
are parallel to the spacecraft XYZ coordinates.
The additional assumptions made are:
1. The spacecraft (main portion) angular velocities are small compared
to the relative angular velocities of the television camera.
2. The motion of the system (main portion plus the TV camera) center-
of-mass due to TV camera rotation may be neglected in computing
inertia characteristics.
Part A. Derivation of Torque Equations*
The inertia matrix of the TV camera and mount is
*This approach was outlined by Dr. John E. Cochran, Assistant Professorof Aerospace Engineering, Auburn University.
41
42
.... L N-I
!
I1~c
, 2 1 Y ..-I 00
I4 l4S- 1.ft-*
2M1 = 2.49 lb-sec /ft
M2 = 1.72 lb-sec2 /ft
-'. I* Xi No C M
l, ,-- - , -
X1 = .575 ft
Y1 = 1.56 ft
Z 1 =0
x2 = 0=0
2 = .575ft
Z2 =0
Figure V-1. Television Camera and Mount
I" · I
S44i
X -0
Y14- - -
II I
A
_no s_~2. 33 ft.
I
-O�P, -j
43
[I] = XX Ixy 0
xY Iyy (V-l)
O 0 Izz
where certain cross-product terms are zero since the XY plane is a plane
of symmetry. The direction the TV camera points is described in terms
of two Euler angles; first a rotation Tc about the Y axis and second a
rotation Ocabout the Zc axis as is shown in Figure V-2.
Let wl, w2, w3 be the angular velocities of the TV camera about the
Xc Yc Zc axes respectively. Then in matrix form the angular momentum
vector is
[H] = IXX IXY 0 1 wl[H]XcYcZc IXY Iyy w
2(V-2)
0 0 I z w3
The torque vector is
[T~~~~~~I]xcYcc z=cTIT] xcy=cZc- [T dIH]xcyczc (V-3)dt
[TJ
Performing the indicated operation 15], 16] yields
'y~~~~~~~~~~~~-
[Ti] Ixxwl- I I w2 + Izzw2w3 + IXY w1 w3 - Iyy w2w3.
T2 = Ixyw2- Ixy wl - Izzwlw3 + Ixx w1w3 - Ixy w2 w3 (V-4)
T3 Izz w3- + IyyWlW2 - Ixx W lW2 + Ixy 2
44
y
Yc
0C
zc
'Pc
z
(1) rotate c about Y axis(2) rotate Oc about zc axis
xcyczc
are TV camera axes
c
xC
Figure V-2. Television Camera Coordinates
X
45
where
w { sin e
w2~~~ =w2 = T'c cos Oc (V-5)
w3 Oc
and
W1 . c sin Oc + c c cos 0c
c cos c c Oc sin E) (V-6)_lyt3_ ~ ec _ W O~~c
Let Tx, Ty, Tz be the torques about the XYZ axes. The transformation
is
_Tx cos 'c Cos Sc - cos 'c sin Oc sin 'c Tl
[Ty = sin Oc cos Oc 0 T2 (V-7)
Tz -sin Yc cos Oc sin Tc sin Oc cos YC T3
Performing the indicated operations results in the following expressions
for the disturbing torques about the spacecraft center-of-mass, written in
terms of the TV camera and mount inertias and the Euler angles Oc and Tc
Tx = .IXX- Iyy sin 2 c - Ixy cos 2c (Cos c) c +
{Izz sin Tc} Oc + .
{(IXX - Iyy) cos 2 0c + Izz + 2IXY sin 20c}(Cos Yc) ;c Oc +
{ixy cos 20c
- (Ix- Iyy) sin 2Sc } (sin Tc) c2 (V8)2
T {Ixx sin2 Oc + I cos Oc I sin 20 c } c +
y~~~~~~~~~ .XT Siyy Cy
{(Ixx- Iyy) sin 20 c - 2Ixy cos 20 c } Tc Oc(V-9)
z {Ixy cos 20(c - + (- Ixx + I) sin 2ec}(sin Tc) c +2°'
{-2IXy sin 2C + Iyy cos 20c IXX cos 20c-Izz}(sin c)COC
+ Izz (cos Tc) c +
-I + Iyy sin 2 c + Ixy cos 2 c } (cos c c (V-10)
2
Part B. Computation of the Inertias for the TV Camera and Mount
The TV camera and mount are considered as two rectangular blocks with
centers-of-mass at C1 and C2 respectively, as shown in Figure V-1. The
equations to compute the inertias of rectangular blocks are well-known
[6] and yield for M 1, about C1
2 22Ixly1 = Ml(b + c ) = .28 lb-ft-sec2
12
Iylyl= M(a2 + c 2 ) = 1.27 lb-ft-sec2
12
I = Ml(a2 + b2) = 1.27 lb-ft-sec2zlzl
12
where M1 = 2.49 lb-sec2 /ft
a = 2.33 ft
b = 0.82 ft
c = 0.82 ft
47
For M2 about C2,2 t 2 ,
= M2(b2 + c2) = 0.285 lb-ft-sec2x2x2 2
12
yy2
=M2(c2 2 a2 )- 2 + a= a2 3.285 lb-ft-sec2
12
I = M2(a2 + b 2 ) = 0.378 lb-ft-sec2Z2z2 2
12
where
M2 = 1.72 lb - sec2/ft
a =1.15 ft
b = 1.15 ft
c = 0.82 ft
The parallel-axis theorem [6] is used to compute the moments-of-in-
ertias of the two bodies about C. This yields
Ixx Ixlx + Ix2x+ MY' + M Y2 7.193 lb-ft-sec2xx Xl1 ~ 2~ M1Y1 2 2
I = + I + MX1 = 2.34 lb-ft-sec2yy yYlYl Y2Y2
I I 1 1 + I + M1 (Y1 + X1 ) + M2 (Y2 ) = 9.05 lb-ft-sec2
z zi Zl1 z2 z2 2Y
The non-zero cross-product term is
Ixy = M1 X1 Y1 = 2.22 lb-ft-sec
where the values of X1, Yl, Y2 are listed on Figure V-1.
48
Part C. Computation of Disturbing Torques
The inertias computed in Part B are substituted in (V-8), (V-9),
(V-10). The resulting equations are
..
Tx = {2.425 sin 20c - 2.22 cos 20C}(Cos Tc)TC +
9.05 (sin c)0 c +
{4.85 cos 20c + 9.05 + 4.44 sin 20c}(cos 'c)icOc +
{2.22 cos 20c - 2.425 sin 22C}(sin dc) c2 (V-11)
T = {7.19 sin2 Oc + 2.34 cos2 ac - 2.22 sin 2c}c +y
{4.85 sin 20c 4.44 cos 20c}'cOc (V-12)
Tz= {2.22 cos 20c - 2.425 sin 20c}(Sin "c)"c +
9.05 (cos Tc ) O c +
{-4.44 sin 20c - 4.85 cos 2 0c - 9.05}(sin Tc)cOc +
{-2.425 sin 20c + 2.22 cos 2c}(cos c)c2 (V13)
To obtain numerical values for the disturbing torques it is necessary
to assume scanning rates and accelerations for the TV camera. It is
assumed that these values are bounded by
-10°/sec< -c' c < + 10°/sec (.174 rad/sec)
-10 /sec2 < 0 cS 'c < + 10 /sec2 (.174 rad/sec2)
Interest is in those conditions which result in large disturbing torques.
49
The entries in Table 1, except for the one designated "upper bound" (but
not least upper bound) were computed for selected values of Oc and Tc
by assigning scanning rates and accelerations so all terms are of the same
sign and thus additive.
The entry in Table 1 designated the "upper bound" was computed by
maximizing (or minimizing) each term in (V-11) with respect to Oc and
assigning the polarity of the scanning rates and accelerations so all
terms are positive. Next Tc was determined by solving DTx = 0.
D c
The following illustrates this procedure:
Referring to (V-11) let
fl(Oc) = 2.425 sin 2 0 c - 2.22 cos 2 0c
afl(Oc) = 0 yields Oc = -23.75° and f1 (-23.75° ) = 3.287a0c
f2(0c) = 4.85 cos 2 0c + 9.05 + 4.44 sin 20c
af2 (ec) = 0 yields 0 = +21.250 and f2 (21.25°) = 15.62
cc
f3(0 c) 2.22 cos 20 c - 2.425 sin 2ec
af3 (ec) = 0 yields Oc = -23.75° and f3 (-23.75°) = 3.287
aDoc
Next the scanning rates and accelerations are chosen so all terms are
positive. This yields
Tx= (3.287)(.174) cos Tc + (9.05)(.174) sin Tc +
(15.62)(.174)2 sin s Tc + (3.287)(.174)2 sin c (V-14)
50
Evaluating aTx = 0 yields Tc = 58°. This value gives
Alc
Tx = 1.97 lb-ft-sec2
upper bound
The disturbing torques computed are comparable in magnitude to
the torques available from the RCJ thrusters which for comparison purposes
are listed at the bottom of Table 1. The conclusion to be drawn is that
for the dimensions, mass, and scanning rates and accelerations assumed,
the disturbing torques are significant. Equations (V-8), (V-9), (V-10)
and the procedure outlined for computing an upper bound on the disturbing
torque can be used to study the effect of changes in the parameters of
the TV camera and mount.
51
Tx Calculations
Oc Tc ;c ic cTorque Torque
de2 deg deedeg/saec de&/sec de/gsec2desec2 lb-ft
-23.75 20 10 10 0 0 .2900
21.25 0 10 10 0 0 .469
45.0 0 10 10 0 0 .3021
21.25 18.6 10 10 10 10 .9477
-23.75 81 10 10 10 0 1.6953
0 63.9 10 10 10 -10 1.8294
-23.75 63.9 10 10 10 -10 1.8742
--- 58 10 10 10 -10 1.97*
T Calculationsy
11.65 --- 10 10 --- 10 .223
21.25 ..--- 0 --- 10 .2595
-30.85 --- 10 10 --- 10 1.16125
Tz Calculations
21.25 90 10 10 0 10 .4729
-23.75 26.8 -10 10 10 10 1.87688
* Upper-bound
note: RCJ torques are:
Tx = 1.8 lb-ft
Ty = 1.28 lb-ft
Tz = 1.28 lb-ft
Table 1. Disturbing Torques due to Motion of Television Camera
VI. CONCLUSIONS AND RECOMMENDATIONS
One objective of this study was to design a highly reliable attitude
control system for experiment pointing, station keeping, docking and
separation from the target for the CARD space vehicle. The design
presented herein is based on using a RCJ attitude control system and/or
a CMG attitude control system. The combined system was designed so
that the torques from each system are never in opposition and to minimize
cross-coupling of the axes.
The RCJ attitude control system uses a no-fire region similar to
that employed on the Skylab program [2]. A modification was introduced
which results in more predictable phase-plane trajectories for the
Y-axis and Z-axis control systems and shows promise of improved
performance. In essence, in place of controlling *y and oz, the Y-axis
attitude and Z-axis attitude respectively, the feedbacks are arranged
so that (0z + *y) and (-z " 0y) are being controlled. If the phase-
plane trajectories are plotted as (Alz z + Aly y) vs (Oz + *y) and
(Alzfz - A y ) vs (z - * ) rather than z vs z and y vs y then(Alzz -ly'y z y y y
the trajectories can be superimposed on the no-fire boundaries. More
test runs need to be made to evaluate this system.
The CMG system simulated for this study uses the optimum steering
law described in [4]. This steering law was chosen for convenience;
the approach to designing the system was general and could have employed
52
53
any steering law. The state-space analysis of the CMG system led to
the required structure of the input matrix (steering law) and of the
feedback matrix if the system axes are to be decoupled. The optimum
steering law and proportional-plus-rate feedback are shown to be
special cases. Additional work is needed to determine if this approach
will lead to better steering laws and feedback schemes.
In the attitude control system using both the CMG's and the RCJ's
the feedback for the Y-axis and the Z-axis are the sum and difference
signals. The torque reversal line of the CMG system was designed to
be in the middle of the RCJ system no-fire region. With this design
the two systems never produce opposing torques. Additional phase-plane
trajectories are needed to fully evaluate this system and other CMG
torque reversal lines need to be investigated.
Another objective of this study was to derive a procedure for
computing the disturbing torques due to.the motion of the on-board
TV camera. The equations for the torques in terms of the TV camera
and mount inertias and scanning rates and accelerations were derived.
A simplified procedure for computing the upper bound (but not the least
upper bound) of the torques was outlined. More precise computations
can be made once the camera requirements are better defined.
REFERENCES
[1]. "A Feasibility Study of Unmanned Comet and Asteroid Rendezvousand Docking Concepts Using Solar Electric Propulsion (CARD)"Vol. II, Concept Analysis, Planning and Cost, Final Report underContract NAS8-27206, Northrup Services, Inc., October, 1971.
[2]. A. C. McCullough, C. C. Rupp, "Skylab Nested Attitude ControlSystem Concept", Proceedings of the ION National Space Meeting onthe Space Shuttle, Space Station, Nuclear Shuttle Navigation,February 23-25, 1971.
[3]. D. W. Russell et.al., "Study of Limit Cycle Behavior For An On-OffAttitude Control System Using Linear Signal Mixing", TwelfthTechnical Report, Contract NAS8-2484, Auburn Research Foundation,October, 1964.
[4]. J. S. Boland, III., e t.al., "Study of an Improved Attitude ControlSystem", Twenty-fifth Technical Report, Contract NAS8-20104, AuburnUniversity, December, 1970.
[5]. D. A. Wells, "Theory and Problems of Lagrangian Dynamics", SchaumOutline Series, McGraw-Hill Book Company, 1967.
[6]. H. Goldstein, "Classical Mechanics", Addison-Wesley, 1950.
54
ATTITUDE CONTROL SYSTEM SIMULATION PROGRAM
RCJ Attitude Control System - Linear Signal Mixing - Sum and DifferenceFeedback
and/or
CMG Attitude Control System- Optimum Steering Law - Sum and DifferenceFeedback
Input Data
DELT Integration step size, seconds
PRDEL Print increment, seconds
FINTIM Total simulation time
2INERTX x-axis moment-of-inertia, lb-ft-sec
INERTY y-axis moment-of-inertia, lb-ft-sec2
CUTOFF Minimum absolute value of det [A]
GSTOP Gimbal stop, radians
FPX x-axis position feedback gain
FPY y-axis powition feedback gain
FPZ z-axis position feedback gain
AMX x-axis acceleration due to RCJ thrustersdeg/sec
AMY y-axis acceleration due to RCJ thrustersdeg/sec
AMZ z-axis acceleration due to RCJ thrustersdeg/sec.
RLDGX x-axis rate ledge, deg/sec
55
APPENDIX
56
RLDGY y-axis rate ledge, deg/sec
RLDGZ z-axis rate ledge, deg/sec
PDB Deadband, deg
TQMTRL Torque motor rate limit, rad/sec
PHIX Initial value of x-axis angle, deg
PHIY Initial value of y-axis angle, deg
PHIZ Initial value of z-axis angle, deg
DPX Initial value of x-axis rate, deg/sec
DPY Initial value of y-axis rate, deg/sec
DPZ Initial value of y-axis rate, deg/sec
DEL 11 Initial value of 611, deg
DEL 12 Initial value of 612, deg
DEL 13 Initial value of 613, deg
DEL 31 Clamped value of 631, deg
DEL 32 Clamped value of 632, deg
DEL 33 Clamped value of 633, deg
57
RtAL MXCMND0tPYCMNC,MZCMND,MXCMDE,MYCMDE,MZCMDEtMRX, RY,MKLt TRXC,INERTX, INERTYINERTZd,I!M2,P3,Mi1 INT,V2INT,M31Nr
CUpMC01 RAOIAN,INERTXINERTY,INERTZtI-FPX,FPY,FPZ,FDPX,FDPYFCPZCO)MMOhN/B I/LDCT r1!DLDT12tLD T13,MXCMND ,MXCMDE ,MR X, MRY,MRZ, PTRXD,
$ CDIlIC,COT12C,CDT13C,DEL3lRDEL32R,DEL33R,CUI)fF,GSTCP,$ EX,EY,EZtDODPX1tCDPYTtDPZTtTRXV,TRYVTRZVCUtMCN/B4/AIXAlY,AlZPRX,PRYPRZOELT =0.02PRDEL =0.2FINT!=60.INERTX=1127.INERTY=2947.INtERTZ=2918.CUTOFF= .05GSTOP=3.FPX=7CC.FPY=1000.FPL=lCOO.AMX=.2C8AMY= . C 391AMZ=.0378RLCGX=.172RLCGY=.172RLCGZ=.172PDB=.3X 1=4.*PCB*AMX+RLDGX**2YI=4.*PCB*AMY+RLDGY**2Z}=4.*PCB*AMZ+RLOGZ**2X2=8.*APMX*(2.*AMX*PDB**2-PDB*RLDGX**2)Y2=8.*APY*(2.*AMY*PDB**2-PCB*RLDGY**2)Z2=8.*AMZ*(2.*AMZ*PDB**2-PDB*RLDGZ**2)PRX=(XIt+SQRT(Xl**2-X2))/(4.*AMX)PRY=(Y1+SCRT(Yl**2-Y2))/(4.*AMY)PRZ=( ZI+SQRT(Zl**2-Z2))/(4.*AMZ)AIX=SCRT((PRX+PDB)/(2.*AMX))AIY=SCRT( (PRY+PDB)/(2.*AY))AIZ=SCRT((PRZ+PDB)/(2.*APMZ))WRITE (6,20)AlX,AIYAIZWkITE (6,30)PRX,PRYPRZ
20 FORMAT(Il',AIX =',E13.5,4X,'lAIY =',E13.5,4X,'A1Z =',E13.5/)30 FORMAT(' ','PRX =',tl3.5,4X,'PRY =',E13.5,4X,'PRZ =',E13.5////)
PHIX=5.PH IY=C.PhIZ 1=O.DPX =0.DPY =0.DPZ =0.DEL 1=0.
58
DEL 12=0.CEL13=C.MI INT=C.M2 INT=C.M3INT=O.CtL31=45.DEL32=45.DEL33=45.RADIAN=57.29578DELlRl=CEL11/RAOIANOEL12R=CEL12/RADIANDEL13R=CEL13/RADIANDEL31R=DEL31/RADIANDEL32R=DEL32/RADIANDEL33R=CEL33/RADIANTELST=-DELT/2.TIME=C.OFULL=C.DO 9CC I=l,ICCOCOOIF(TIME.GT.FINTIM)GO TO lC00CALL FCN(TIMEPHIXDPX,PHIY,DPY,PHIZ,DPZ,$ CELllR,CEL12R,DEL13R,MlINT,M2INtP3INT,$ FCNlFCN2,FCN3,FCN4tFCN5,FCN6,$ FCN7,FCN8,FCN9,FCNIO,FCNlI,FCNl2)
SUM=PH IY+PHILZOIFF=PFIY-PHIZDSUM=AIY*OPY+AIZ*DPZDDIFF=AIY*DPY-AIL*DPZDDPXT=O.DDPYT=O.DDPZT=O.FUtLA=O.IF{A8S(EX).LT.PD8) GO TO 100CDPXT=CDPXT+SIGN(AMXEX)FUELA=FUELA+4.*DELT
l0O IF(ABS(EY).LT.PDB) GO TO 110DDPYT=CDPYT+.5*SIGN(AMY,EY)CDPZT=CDPZT+.5*SIGN(AMZEY)FUELA=FUELA+2.*DELT
110 IF(A8S(EZ).LT.PDB) GO TO 120DDPYT=CCPYT-.5*SIGN(AMY,EZ)CDPZT=CCPZT+.5*SIGNIAMZEZ)FUELA=FUELA+2.*DELT
120 CONTINUEABl1=AES(DEL1lR)AB12=AeS(DOLL2R)AbI3=ABS(DEL13R)TF(ABil.GE.GSTOP.CR.AB12.Gt.GSTOP.OR.AB13.GE.GSTOP)GO TO .10
59
IF(A8S(CDPXT).GT.AfX/4.0) GO TO 300IF(A2S(CDPYT).GT.AMY/4.0) GO ro 3COIFIAES(CDPZT).GT.AMZ/4.0) GO TO 3COIF(TIOE.LT.TEST) GO TO 400TLST=TEST+PRDELGO TO 310
3C0O CONTINUEPROOX=CDPXT*PRXPRODY=CCPYT*PRYPRCCZ=CDPZT*PRZIF(PRCOX.LT.C.) WRITE(6,197)IF(PRCCY.LT.O.) WRITE(6,198)IF(PRCCZ.LT.C.) WRITE(6,199)
197 FORMPAT(' ','***X-AXIS GAS THRUSTERS CPPOSE CMGS***')198 FURMAI' ','***Y-AXIS GAS THRUSTERS CPPCSE CMGS***')199 FORMAI(' ','***Z-AXIS GAS THRUSTERS CPPOSE CMGS***')310 WRITE(6,200)TIIEtPHIX,TRXV,CDPXTDPX
WRITE(6,201)PHIY,TRYVtCCPYTDOPYWRITE(6,202)PHIZ,TRZVDODPZTDPZWRITE(6,203)CEL11,DEL12,DEL13,MTRXDWRITE(6,204)SUM,DSUM,DIFFCODIFFWRITE(6,205)FUEL
4CO FUEL=FLEL+FUELACALL INTGRL(TIME,CELT,FCNIFCN2,FCN3,FCN4,FCN5,FCN6,$ FCN7,FCN8,FCN9,FCNlO,FCN11,FCN12,$ PHI[X,CPXtPHIY,DOPY,PHIZDPZ,$ CELIlR,DEL12R,OELI3R,1INT,M2INT,M3INT)OtL11=CELLL1R*RADIANDLL12=CLL12R*RADIANDtEL13=CEL13K*RADIANTIME=TIME+CELT
900 CONTINUEGO TO ICOO
910 WRITE(6,920)920 FORMAT(' ','****GIMBAL ANGLE EXCEEDS GIMBAL STOP****')
WRITE(6,200)TIME,PHIX,TRXV,CDPXT,DPXWRITE(6,201)PHIY,TRYV,CDPYT,CPYWRITE (6,202)PHIZtTRZV,CODPZT,DPZWRITE(6,204)SUMDSUMDIFF,0DIFFWRITE(6,203)CELIl,CEL12,CELI3,PTRXDWRITE(6,205)FUEL
200 FORMAT(' ','TIME =',E12.4,5X,'PHIX =',E13.5,5X,'TRXV =',E13.5,S 5X,'CCPXT =',E13.5,5X,'DPX =',E1I3.5)
201 FORMAT(25X,'PHIY =',E13.5,5X,'TRYV =',E13.5,5X,'DDPYT =',E13.5,$ 5X,'CPY =',E13.5)
202 FORMATI25X,'PHIZ, =',E13.5,5X,'TRZV =',E13.5,5XtOOPZT =',E13.5,$ 5X,'CPZ =',E13.5)
203 FCRMAT(25X,'CELll =',E13.5,5X,'DEL12 =',E13.5,5X,'DEL13 =',El3.5,
60
$ 5X,'MTRXD =:,E13.5)204 FuRMA((25X,'SUM =',EI3.5,5X,'DSUM =',E13.5,5X,'DIFF =',El3.5,
$ 5X,'CDIFF =t',E13.5)205 FORMAT(25X,'FUEL =*,E13.5/)Ic00 STCP
EN C
61
SL)RCLTINt INTGRL(T,H,FCNIFCN2,FCN3,FCN4,FCN5,FCN6,$ FCN7,FCN8,FCN9,FCNlOFCNi1,FCN12,YlY2,Y3,Y4,Y5,Y6,$ Y7,Y8,Y9,Y lO,Yll,Y12)H2=H/2.CALL FCN(T,Yl,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9gYOY,Y0,Yl12,FCNI,FCN2,FCN3,
S FCN4,FCN5,FCN6,FCN7,FCN8,FCN9, FCNIOFCNl1,FCN12)PI=F*FCN1P2=H*FCN2P3=H*FCN3P4=H*FCN4P5=H*FCN5Pb=H*FCN6P7=H*FCN7P8=H*FCN8P9=H*FCN9PIO=H*FCNLOPII=H*FCNllP12=F*FCN12CALL FCN(r+H2,Yl+PI/2.,Y2+P2/2.,Y3+P3/2.,Y4+P4/2.,Y5+P5/2.,
$ Y6+P6/2.,Y7+P7/2.,Y8+P8/2., Y9+P9/ 2.,YIO+PlO/2.,YIL+PII/2.,$ Y12+P12/2.,FCNl,FCN2,FCN3,FCN4,FCN5,FCN6,FCN7,FCN8,FCN9,
$ FCNIO,FCNII,FCN12)Ql=H*FCNl02=H*FCN2Q3=H*FCN3Q4=H*FCN4Q5=H*FCN5Q6=H*FCN6Q7=H*FCN7Q8=H*FCN8Q9=g*FCN9
QIO=H*FCN10QI1=H*FCNIIQ12=H*FCN12CALL FCN( T+H2,YI+QI/2., tY2+Q2/2.,Y3+Q3/2.,Y4+Q4/2.tY5+Q5/2.,$ Y6+Q6/2.,Y7+Q7/2.,Y8+Q8/2.,Y9+Q9/2.,YlO+QI0/2.,Yll+QIll/2.,$ Y12+Q12/2.,FCN[,FCN2,FCN3,FCN4,FCN5,FCN6,FCN7,FCN8,FCN9,$ FCN1O,FCNll,FCN12)RI=H*FCNIR2=H*FCN2R3=H*FCN3R4=H*FCN4R5=H*FCN5R6=H*FCN6R7=H*FCN7R8=H*FCN8R9=H*FCN9RIO=H*FCNIG
62
RII=H*FCNllR12=H*FCN12CALL FCN( +H,YIl+RlY2+R2,Y3+R3,Y4+R4,Y5+R5,Y6+R6,Y7+R7,Y8+R8,
$ Y9+R9,YlO+RIC,Yll+RI1,Y12+R12,FCNI,FCN2,FCN3,FCN4,FCN5,FCN6,$ FCN7,FCN8,FCN9,FCNlO,FCNll,FCNl2)SI=H*FCNlS2=H*FCN2S3=H*FCN3S4=H*FCN4S5=H*FCN5S6=H*FCN6S7=H*FCN7S8=H*FCN8S9=H*FCN9S10=H*FCNIOSlI=H*FCNI1S12=H*FCNI2YI=Yl+(Pl+2.*(QI+RlI)+Sl)/6.Y2=Y2+(P2+2.*(Q2+R2)+S2)/6.Y3=Y3+(P3+2.*(Q3+R3)+S3)/6.Y4=Y4+(P4+2.*(Q4+R4)+S4)/6.Y5=Y5+(P5+2.*(C5+R5)+S5)/6.Yb=Y6+(P6+2.*(Q6+R6)+S6)/6.Y7=Y7+(P7+2.*(Q7+R7)+S7)/6.YB=Y8+(P8+2.*(C8+R8)+S8)/6.Y9=Y9+(P9+2.*(Q9+R9)+S9)/6.YIO=YIC+(PIO+2.*(QIO+RlO)+S0IO)/6.YII=YII+(P1+2.*(ClIl+Rll)+SIli)/6.Y12=Y12+(PI2+2.*(Q12+R12)+S12)/6.RETURNEN D
63
SUBKOUTINE FCN(TIMEPHIXDPX,PHIY,CPY,PHIZ,DPL,$ DELIlR,DEL12R,OEL13RtIlINT,M2[NT,M3INT,$ FCN1,FCN2tFCN3,FCN4,FCN5,FCN6,FCN7,FCNH,$ FCN9,FCNIO,FCNllFCNl2)RLAL IXCMNO,HYCMND,MZCMNDMXCMDEPYCMDEMZCHDE,MRX,FRY,MRZ,ITRXC,$ -INERTX,INERTYINERTZtI,M2,P3,MIINT,P2[NT,M3INTCUPPON RACIAN,INERTX,INERTY,INERTZFPX,FPY,FPZFDPX,FDPY,FCPZCOPMCN/Bl/DLCTIl, CLDT12,DLDT13 ,XCHNCMXCMDE,PRX,MRY,tPRL,PTRXC,
DDoI[C,CDT12C,CDTI3C,CEL31R,DEL32RDEL33R,CUTOFFtGSTGP,$ EX,EY,EZ ,DPXT,CDPYTtCCPZT,TRXV,TRYV, TRZVCUPMCN/84/AlX,AIY,AlZ,PRX,PRYPRZAll= SIN(DELIIR)*CCS(DEL31R)A12= CCS(DELI2R)A13=-S IN(DEL 13R)*SIN( DEL33R)A21=-SIN(DELl R)*SIN(DEL31R)A22= SIN(DELI2R)*COS(DEL32R)A23= CCS(DEL13R)A31= CCS(DELIIR)A32=-SIN(DEL12R)*SIN(DEL32R)A33= SIN{CEL13R)*COS(DEL33R)PTRXD= All*A22*A33+A12*A23*A31+A21*A32*A13
$ -A13*A22*A31-A12*A21*A33-A23*A32*AlIIF(ABS(rTRXO).LT.CUTOFF)MTRXD=SIGN(CLTOFF,W[RXO)Tll=( A22*A33-A23*A32)/MTRXDT12=(-A 12*A33+A13*A32)/HTRXDT13=( Al12*A23-A13*A22)/MTRXDT21=(-A21*A33+A23*A31)/M[RXDT22=( AII*A33-A13*A31)/MTRXDT23=(-All*A23+A13*A21)/MTRXDT31=( A21*A32-A22*A31)/MTRXDT32=(-All*A32+A12*A31)/MTRXDT33=( All*A22-A12*A21)/MTRXDDDTllR=5.*MlINTDDTI2R=5.*P2INToCIl3R=5.*M3INICLCTI:=CDTllR*RAODIANCLCT12=ECDTl2R*RAOIANDLCTl3=CDTl3R*RACIANPXF=PHIXPYf=PHIY+PHIZPZF=PFIZ-PHIYIF(ABS(PXF).GT.PRX) PXF=SIGN(PRX,PXF)IF(ABS({PYF).GT.PRY) PYF=SIGN{PRY,PYF)IF(APS(PZF).GT.PRZ) PZF=SIGN(PRZPZF)EX=- ( X*ODPX,+PXF)EY=-(AIZ*CPZ+AlY*CPY+PYF)EZ=-(AIZ*DPZ-AIY*DPY+PZF)WXCMNC=FPX*EX/RADIAN
64
MYCMNC=FPY*IEY-EZ)/RADIANPZCMNC=FPZ*(EY+EZ)/RADIANMRKX=(AII*DCTrllR+Al2*ODDL2R+AI3*DDfI3R)*60.MkY=(A21*CCTllR+A22*CDT12R+A23*DCT13R)*60.MRZ=(A31*CCTI1R+A32*CDTI2R+A33*DDT13R}*60.TRXV=RAOIAN*PRX/INERTX[RYV=RADIAN*MRY/INERTYTRZV=RACIAN*MRZ/INERTZPXCMCE=PXCMND-MRXPYCMCE=PYCMNC-MRYMZCMDE=MZCMNC-MRZODTI1C=(Tll*FXCMDE+TI2*MYCMDE+T13*MZCMDE)/60.DDTI2C=(T21*PXCMDE+T22*PYCMDE+T23*MZCMDE)/60.CDT13C=(T31*PXCMDE+T32*PYCMDE+T33*PZCMDE)/60.D[r11G=2.*CODTilCDDT12G=2.*DDT12CODT13G=2.*CODT13CTCMTRL=4.5/RADOIANIF(ASS(CD[lIG).GT.TQMTRL)CDTI1G=SIGN(TQMTRL,DDTlIG)IF(ABS(EDT12G).GT.TQMTRL)CDT12G=SIGN{TQMTRLDDT12G)IF(ABS(CDTI3G).GT.TQMTRL)DOCTI3G=SIGN(TQHTRL,DDT13G)FCNI=CPXFCN2=IRXV+DDPXTFCN3=CPYFCN4=TRYV+CCPYTFCN5=CPZFCN6=TRZV+DDPLTFCN7=CCTIIRFCN8=CCT12RFCN9=CDTI3RFCN10=ODTlIG-DDTllRFCNlI=CDT12G-CDT12RFCN12=CDT13G-CDT13RRETURNEND
