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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
MSC INTERNAL NOTE NO. 66-FM-131
November 4, 1966
AS-503A/504A REQUIREMENTS FOR
THE RTCC: GENERALIZED ITERATOR
*
By William E. Mooro
Analytical Mechanics Associates
4
\
MISSION PLANNING AND ANALYSIS DIVISION
MANNED SPACECRAFT CENTER
HOUSTON, TEXAS
*
NA?A - Manned Spacecraft Center
RELEASE APPROVAL
K Type of Document
Internal Note
2. Identification
Page 1 of 1 Pages
Chief , Mission Planning and
Analysis Division
4. Title or Subject
AS-503A/!?01jA. Requirements for the RTCC: Generalized Iterator
3.FROM:
Division Mission Planning and Analysis
Branch Mission Analysis
Section Analytical Mechanics Associates
Date of Paper
ll/li/66
5. Author(s)
William E. Moore
6. Distribution
Number of
Copies
Addressees
Special Handling
Methods
This is a change to distribution on Release Approval dated,
This Is an addition to distribution on Release Approval dated,
si on Chief
Ass
i stan
ropriate
na ger
Signature of
Date
3 NCV 1968
Date
b. Change or Addition made by
Date
JO
See attached memorandum
2 -
S’- (o-L /
7
9. Location of Originals;
MSC Form 199 (Rev Dec 63 )
TO
FROM :
x subject:
OPTIONAL rOftM NO. 10
MAY I HI EDITION
GiA r*4fi (*i cmJ.ioi*ii.*
UNITED STATES GOVERNMENT
Memorandum
See list attached
FM/Mission Planning and
Analysis Division
Generalized Iterator Formulation for the RTCC
The enclosed M3C Internal Note No. 66-FM-131 defines formulation for the
Generalized Iterator subprocessor. The subprocessor has several applica¬
tions in both the 503 find 50b systems. The logic as defined is independent
of the application; the appropriate "setup” logic must be contained in the
supervisor, which calls this subprocessor.
Applications for which this subprocessor will be required include the
following:
date: 3 NOV 1966
66 -FM 51-388
1. Full mission optimization
a. Earth orbit (TLl)
b. Translunar coast (MCC)
2. Return-to-earth aborts
(including TEl)
3. LOI targeting
m*
Frank, Chief
Mission Analysis Branch
The Flight Software Branch concurs with the above recommendations.
Flight Software Branch
APPROVED BY:
Johh\P. Mayer, Chief
Mission Planning and
Analysis Division
Enclosure
*
0010-100
Buy U.S. Savings Bonds Regularly on the Fay roll Savings Flan
Addressees:
IBM/R. Hanrahan (20)
FC/J. D. Hodge
/G. S. Lunney
FM/J. P« Mayer
/H. W. Tindall
/C. R. Huss
/M. V. Jenkins
/j. F. Dalby
/J. P. Bryant (5)
/R. P. Parten
/Branch Chiefs
FM6/R. Regelbrugge
FM5/ r * Ernull (6)
CC l
TRW/J. T. Reid (4)
/C. Evaige (2)
Philco/R. D. Harrington
Belleomm/R. Wagner (5)
EG/Robert Duncan (3)
GV5/M. Rahman
PAl/W. Lee
PD4/A. Cohen
PM3/R. Battey
Technical Information Division, Library (2)
FA/C. C. Kraft, Jr.
/S. A. Sjoberg
/R. G. Rose
FC/T. Weichel (2)
FL/R. L. Thompson
FS/H. E. Clements
FM3/T. Carter (10)
FM5/Mission Analysis Branch (8)
FM5:MPFrank;bc
akllAiAUttitfait ^^-fcfai^MtltitdiJLaUAilAi iAruui-
UNCLASSIFIED
MSC INTERNAL NOTE NO. 66-FM-131
fc
PROJECT APOLLO
AS-503A/504A REQUIREMENTS FOR THE RTCC:
GENERALIZED ITERATOR
By William E. Moore.
■Analytical Mechanics Associates
November 4, 1966
MISSION PLANNING AND ANALYSIS DIVISION
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
*
MANNED SPACECRAFT CENTER
HOUSTON, TEXAS
*
Approved:
IhPLJi
M. P. Frank III/Chief
Mission Analysis Branch
Approved
Mission Planning and Analysis Division
\
UNCLASSIFIED
CONTENTS
r*
*
Section Page
1. SUMMARY. 1
\
2 . INTRODUCTION. 1
3. FUNCTIONAL FLOW. 2
* ^
'* %
4. FUNCTIONAL FLOW CHART..'. k
5. DETAILED FLOW. 8
3 • 1 Input. 8
5.2 Initialization. 8
5*3 Residual computation. . 9
5*4 Iteration loop.10
5-5 Barrier checking. 12
5.6 Partial derivative matrix calculation . 12
5*7 Matrix multiplication subroutine.13
5.8 Simultaneous equation solves.13
5»9 X-sizing subroutine... 13
t
6 . DETAILED FLOW CHART. 15
\
iii
i
V
AS-503A/50l<A REQUIREMENTS FOR THE RTCC:
GENERALIZED ITERATOR
By William E. Moore, AMA
1. SUMMARY
This note describes the program logic for the iteration scheme to
he used in the Real Time Computer Complex for AS-503A/504A. It is a
general parameter adjustment scheme which finds the values of the con¬
trol parameters to achieve a given mission or part of a mission and may
optimize the parameter if instructed.
2. INTRODUCTION
The program for an iterative method which designs the optimum mission
that satisfies a given set of constraints is described in this internal
note. It is sometimes no small task to find any set of initial condi¬
tions and control parameter values satisfying a set of constraints.
The iterative method solves both of these problems, first obtaining a
mission that satisfies the constraints (select mode) and then adjusting
the initial conditions and control values until a function of these
variables achieves a minimum or maximum value (optimizing mode).
The method is completely general; it does not depend on the mission,
the initial conditions, control parameters, constraints, or variable to
be optimized. The values of the initial conditions and control para¬
meters appear as a vector of independent variables. The values of the
constraints and of the variable to be optimized (that is, the variable
to be maximized or minimized) appear as a vector of dependent vari-
ables. The actual subroutine that computes the dependent variables
from the independent variables is referred to as the trajectory computer.
The particular trajectory computer to which this iteration scheme ap- n
plies defines the mission to be obtained or optimized.
A constraint may have either of two forms. It may be an equation,
in which a quantity must have a given value, or it may be represented
by an inequality. The distinction between these two forms is very im¬
portant, and is the basis of the success of the method.
2
4
The functional flow of the program is described in section 3 find
diagramed in section 4* Details of the program are described in sec¬
tion 5 and diagramed in section 6 for those who are interested. Since
most of the program is devoted to logic and decisions^ there are few
complex equations» These are incorporated into the flow chart. The
The reference contains the mathematical details of the method.
r
k
3. FUNCTIONAL FLOW
The inputs to the load module include four quantities associated
with each independent variable - a switch to designate if the variable
is achieved^ a first guess (which is to be iteratively updated as the
method proceeds^ a weighting (scaling) factor^ and a step size.
The dependent variables are divided into three classes;
1, Class 1,- Those representing equality constraints
*
2, Class 2.- Those representing inequality constraints
3, Class 3.- The one representing the variable to he optimized
Five numbers are provided to the load module for each dependent
variable. One is a switch to designate if the variable is active, two
define the desired values of the constraint for the variable, one desig¬
nates the class of the variable, and one is a weighting factor which
applies only if the variable is of class 2.
To start the computation, the actual weight for each dependent
variable and several miscellaneous quantities are computed. The tra¬
jectory computer is called, and the first guesses are used as the
values of the independent variables. Residuals and the length of the
weighted residual vector Rj. are calculated. Class 2 variables which
are acceptable are deleted from the weighted residual vector.
The iteration loop begins by establishing a new set of nominal
values of the independent variables. If, at any stage, a class 2 vari¬
able is close to a boundary, this boundary will act as a barrier. Con-
no Y.f? sses t0 a "barrier-checking procedure, which
?! c ? ndltlon -» and f orces the value of the variable away,
the variable to class 1, if in the
optimize mode. Next the partial derivatives of all the dependent vari-
ables with respect to each independent variable are computed by the
SGCdn u me wioa •
3
Using these, the weights, the„ residuals, and the current values of
a parameter called 1, increments fpr the independent variables are com¬
puted. These are compared to the step sizes used in the partial deriva¬
tive calculation. The value of X is increased and increments are re¬
computed until the increments are of appropriate size. Next the latest
increments are added to the independent variables, and the new values
are used in the trajectory computer. The residual vector computer is
again called to obtain a new value of R_. This is compared with the
previous value, and, if it is not smaller, the value of X is increased
and the increments are recomputed until it is.
f
The increases of X have been of relatively gross size (multipli¬
cation by 8). An attempt is made to find a value of X, one-half or
one-quarter as big, which might also be appropriate. Finally, the value
of X is decided, and the values of the independent variables are the
ones used in the next nominal trajectory. The iteration counter is
advanced, X is reduced in value, and control passes back to the beginning
of the loop.
#
*
The select mode ends when all of the class 1 and 2 variables satisfy
their constraints. Then the variable to be optimized is given a weight
and introduced into the problems. The optimizing mode ends when no fur¬
ther change may be made in the values of the independent variables, or
when the maximum number of interations for that mode is exceeded.
The results are the final values of the independent variables, and
their associated dependent variable.
4. FUNCTIONAL FLOW CHART
RESIDUAL
COMPILATION
"~T
immuzAim
l
ADVANCE
ITERATION
COUNTER
EXIT
5
»
&
V
PaqeZof4
TRAJECTORY
COMPUTER
REINITIALIZE
h(\t4of 4
8
5. DETAILED FLOW
*-
5.1 Input
, The load module is capable of handling up to twenty candidates
for independent and up to twenty candidates for dependent variables.
For each of the independent variables, four input quantities are re¬
quired. These are a switch to show if the variable is active or not,
a first guess of the value of the variable, a weight, and a step size
to be used in partial-derivative computations. The latter two are
needed only when the variable in question is active.
All of the dependent variables can be thought of as having desired
values. For consistency, the value for each variable is represented by
two numbers, a lower and an upper limit of acceptability. For variables
in class 1, these limits are the desired value in the equation decreased
and increased, respectively, by a suitably small tolerance. For vari¬
ables in class 2, several situations can arise. If the inequality is
two-sided, the limits are the obvious values from it. If the inequality
is one-sided, the dependent variable in question may have a natural
limit on the other side which can be used in combination with the given
limit. If even this fails, the interval of acceptability must be chosen
so that it contains exactly one- solution to the problem described by
all the other constraints and the variable to be optimized. For a vari¬
able in class 3, the notion of acceptability reduces to a convenient
fiction, and the limits (usually having the same value) designate the
direction in which the variable to be optimized should move. If its
value is smaller than the limits, the iterative adjustments will in¬
crease it,* if its value is larger, the adjustments decrease it.
Thus, each dependent variable will have five input quantities -
a switch to show whether the variable is active or not, lower and upper
limits, the class designator, and the weighting factors for class 2
variables.
Other inputs are used for housekeeping. The procedure should know
how many iterations are to be tried in the select mode and in both modes
before terminating. Furthermore, there should be a switch to allow
starting in the optimize mode if desired.
5.2 Initialization
t
The switch must be examined to determine how many of each set of
variables are active and to establish an array of indices for each set.
One of these arrays permits the values of the active independent variables
t
9
V
A
to "be assigned the proper locations for the trajectory computer to use;
the other array allows the values ^of the active dependent variables
candidates to be arranged next to 'each other after the trajectory com¬
puter is finished. The two processes involved are called the open-ranks
and close-ranks operations, respectively. Each call to the trajectory
computer must be preceded by a call to an open-ranks subroutine and fol¬
lowed by a call to a close-ranks subroutine. Hereafter, the term "tra¬
jectory computer" will always refer to the set of all three subroutines.
After'resetting the iteration counter, the trajectory computer is
referenced, using the input first guesses as the values of the independent
variables. If this trajectory cannot be computed, the whole procedure
is abandoned.
Next certain computations have to be performed once for all itera¬
tions. First, average values for each dependent value are calculated
half-way between the limits. The weight for each class 1 variable is
obtained from the deviations of the limits from the average value as
follows. Let y - and y + &., be the lower and upper limits, respec¬
tively. Then 2
-40
j
are the weights w.. These weights are averaged
J
using a crude geometric mean to get a number w to be used for the
class 2 variables. Each class 2 variable has a weight equal to its
weight factor times w . The class 3 variable is assigned a weight
” * Q *
based on the value resulting from the trajectory computer. If there
are k, dependent variables of class 1 and if y - y is the residual of
the variable to be optimized, the weight is (lO*j (2~^) k, / (y - y)^
-32 o
if starting in the select mode, or (2 J ) f k^ / (y - y) if starting
in the optimize mode. The factor f is the weight factor associated
with the class 3 variable.
#
Another calculation pertaining to class 2 variables obtains 0.2 %
of the widths of their intervals. These numbers are used during the
iterations to test how near a variable is to one of its limits. Finally,
2ft
the parameter , is set to a small value, 2 • ,
*
5»3 Residual Computation
The residual vector, weighted residual vector, and its length are
now computed in the residual package. The residual vector is the set
of differences between each dependent variable's value y and its
✓
10
average value y. as derived above. Now the class designators c. have
J r j
been assigned the value 1 for class 1 and 3 variables and 0 for class
2 variables. Thus, the weights w. are replaced by c.w. = w. for those
J J J 0
variables within the given limits; for variables outside' the limits.
the weights w. = w, are retained. The weighted residual vector is
J J
* /„ - \ _. ., ,_ c g 1 / - \2 where n is the
w. Cy. “ yJ; and its length PL = £ w (y. - y.)
j _i J Jj . i .1 .1
j~ = 1
total number of dependent variables.
If, while checking the values of the dependent variables, it should
be found that all except the variable to be optimized are within their
intervals, and if the iterations are still in the select mode, that
mode is terminated. If there is to be an optimize mode, the weight on
the variable to be optimized is reinitialized, and the iterations pro-
cede in the optimize mode. If there is no variable to be optimized,
the procedure is finished.
The initial trajectory is the first in a sequence of trajectories
designated a "lost good trajectory". As the iterations are performed,
a new good trajectory is one that is better than the last good trajec¬
tory, in the sense that its value for IL is smaller than that of the
old one.
5•4 Iteration Loop
Each trajectory is defined by the values of the independent vari-
Qbles used to get it. Thus the values of the independent variables
used to obtain the last good trajectory of a previous iteration must be
saved separately from the current values of the independent variables,
and are the nominal values of the independent variables. This, then,
is the first step of each iteration, the transferral of the last good
values into the nominal position.
*
There follows a lengthy procedure of barrier checking, in which
each class 2 variable is examined to see where its value is in relation
to its acceptable interval, and appropriate action is taken to treat the
intervals. A discussion of this portion of the load module is post¬
poned until a later section.
At this point, the iteration counter is advanced, and a test is
made to see if all the allowable iterations have been used in either
mode.
11
*
4
*
Next, the partial derivative matrix subroutine is called, followed
hy a call to a specialized matrix multiplication subroutine, which pro¬
cesses the partial derivative matrix P, the dependent variable weights
m m
w , and the weighted residual vector to get P WP and P WAy, where w
is the matrix having the weights w along its diagonal and Ay is the
/ ^
residual vector (so that wAy is the previously computed weighted
residual vector).
Now, using the current value, of X and the independent variable
weights, and a vector of adjustments Ax.^ to the independent variables is
obtained by referring to a simultaneous equation solver. If no solu¬
tion can be obtained by the equation solver, X is multiplied by 8, and
the vector of adjustments is recomputed.
If there is a solution, the absolute value of each component is
tested against 65536 times the corresponding step-size. To guard a-
gainst wild excursions of the independent variables, if any component
is too big, X is multiplied by 8 and the vector of adjustments is
recomputed. In this way, the value of \ builds up until all components
are small enough.
Now the independent variables are incremented by their adjustments
and a new trajectory computation is done. The resulting dependent
variables are used in the residual computation. If the new value of
is not smaller than the old value, then X is multiplied by 8, and
new increments in the independent variables are computed as above.
This procedure continues until the new value of II finally is smaller
than the old value.
Meanwhile, each time the independent variables are incremented,
and the resulting values are compared with the original values. If
no change is detected, the increments have been choked to negligible
values. When this happens in the select mode, it indicates that no
solution to the problem as presented exists anywhere in the neighbor¬
hood of the current values of the independent variables, which consti¬
tute the best choice of these values in the least squares sense. When
this happens in the optimize mode, the optimum value of the variable
to be optimized has been obtained, and the current values of the inde¬
pendent variables are the answer to the original problem.
The X-sizing subroutine is now called to further refine the value
of X in an attempt to take larger corrections to the independent vari¬
ables. The last good trajectory from this subroutine becomes the new *
nominal trajectory and control returns to the beginning of the itera¬
tion loop.
fc
5*5 Barrier Checking
* *
»
This procedure is carried out only for the nominal trajectory
available at the beginning of each iteration. It pertains only to the
class 2 dependent variables, which will, for the rest of this section,
be called "variables".
In the select mode, search is made for variables which are inside
the acceptable interval, but only just inside, that is, they are within
0.2$ of full range of one of the limits. When a variable has such a
value, a move procedure begins. The limit near the value of the vari¬
able is temporarily replaced by its opposite limit, thus shrinking the
interval of acceptability to zero length. The residual vector and its
length are then recomputed. The move counter, which has been continu-
illy reset at every iteration, now begins to count, and control passes
—-■to the basic iterations loop. As the iterations proceed, additional
•ariables may have values which would start a move procedure. If this
ippens, the limits are treated in the same way, and the move counter
* jins to count from 1 again. Finally when the move counter reaches 6,
—- the limits are restored to their values, as originally input, and
e move counter reverts to being reset at every iteration.
In the optimize mode, there are two procedures, a lock pi’ocedure,
for variables already locked, an unlock procedure. The lock pro-
:re is as follows: If any variable has a value outside the accept-
* limits, it may be greater than the upper limit or less than the
•r limit. In the former case, the value of the lower limit is re-
'-3 by the value of the upper limit; in the latter, the value of the
•r limit takes the place of the upper limit. In either case, the
?ble is treated like a class 1 variable in the residual computations.
:50th the number of iterations during which the value of a previously
'• variable is inside the input limits and the number of iterations
•.which the value is outside the limits are known. If any time after
' the inside exceeds the outside by three or more, the unlock pro¬
restores the original input limits. When the limits are restored,
iteration the required value is taken to be 1.6% of full range
- original interval of acceptability. Thereafter, the interval
: to its original form, and the variable is again treated‘like '
•’ variable in the residual computation.
5*6 Partial Derivative Matrix Calculation
'•dependent variable value x. is changed successively, and
4 *
• { by th< ; corresponding step-size. The trajectory com-
erencea, using the new set of independent variables thus
yu^ujyu^
Mid
i
formed. The resulting values of the dependent variables will be de¬
noted by y, (x J + Ax.). Let y. (x.) be the nominal values of the depen-
J 1 i J **■
dent variables. Then, the elements of the partial derivatives matrix P
5y, (x i + Ax^ -yjO^)
are P,, *=
ji
5x,
«L
Ax.
l
Thus, there are as many calls to the trajectory computer as there
are independent variables, each one generating one column of P. If any
of the references to the'trajectory computer fails, a step in the reverse
direction is attempted, and if its trajectory succeeds, the required
partial derivative may be obtained from it. If steps in both directions
fail, the whole procedure must halt.
5.7 Matrix Multiplication Subroutine
The inputs to this routine are a matrix P, which has n rows, a
vector of n weights, wj, and a vector of n weighted residuals WAY.
Each row of P is multiplied by its corresponding weight, wl, giving a
matrix WP a^d vector (WAY), both of which are multiplied ^ by the
transpose P . This is done, in the usual way, by multiplying each of
the columns of WP and WAy by each of the columns of P.
5*8 Simultaneous Equation Solves
The best method of solving simultaneous equations is to use a
straight-forward elimination technique, followed by a back solution.
If at any stage, division by a number small enough to cause underflow
or division by zero would be required, the remaining equations may be
interchanged. If all attempts at this fail, the subroutine returns
with an error indication.
5.9 X-sizing Subroutine
*
When a value of X has been arrived at by the above process, it is '
tested to see if a slightly smaller value with bigger increments might
be even better. Thus, the latest values of the independent variables
become the values for the last good trajectory, but we continue to work
from the values we previously had, still using them as nominal values.
X is replaced by x/2, and the increments are again computed. These are
tested by the same procedures as before, comparison with a multiple of
/
14
the step-size and comparison with the value from the last good tra¬
jectory. If they pass the tests, thtese latest values of the independent
variables now define the last good trajectory, X is replaced by X/2,
and the whole process is repeated once more. If either of the two
values of X fails to produce a better trajectory, the last good tra¬
jectory is the result of the subroutine.
*
t
* i--i.Jl-flUJiiiidUL-.iUai..
6. DETAILED FLOW CHART
15
ENTER
INPUTS
SWITCHES , FIRST GUESSES (Xj), WEIGHTS (dj ) , SIZES
(XSTEP]) FOR INDEPENDENT VARIABLES
SWITCHES, LOWER LIMITS C^uj), UPPER LIMITS
(Ylj), CLASS DESIGNATORS (Cj) } WEIGHT FACTORS
(bj) FOR DEPENDENT VARIABLES
NUMBER OF ITERATIONS ALLOWED IN SELECT MODE
TOTAL NUMBER OF ITERATIONS ALLOWED
OPTIMIZE SWITCH
SET UP INDICES FOR
INDEPENDENT and
DEPENDENT VARIABLES
l
OPEN \
RANKS /
TRATECT0tN\_^
CLOSE
RANKS
fy. t of 2-0
(
)
P/kGE i
Page 2
17
TURN M0\!£ INDICATOR OFF
N- 3 of 10
*
Page 3
18
#
PaOB 4
TURN OFF SKIP AND BOTH
SIZING INDICATORS
RESET SIZING COUNTER
~ I j
INCREASE ITERATION
COUNTER
/
Page S
20
COMPUTE COEFFICIENTS
OF EQUATIONS
(p r wP + A a) At ~-p'wa y
%
\
/ SOLVE \
{simultaneous)
\ equations /
r yes
TURN MOVE
/NOiCA TOR OFF
SET A-2~ Z?
TUNN S/OP ON
1
1
L
mi
(? Of £0
0
21
HAS
x- SlZINGr
FINISHED WITH
BA 0 /?k
?
YES
HO
OPTIMISING-
?
EX IT
MO
MO
P<j. 7 of 10
22
*
is 1
TURN OFF CONVERGENCE JN&JCAToR
^Ajeu/i c *1 **' ^ X|
CG
\J7
TURN
CONVERGENCE
mbiCAlOR
ON
*
fy. 2 A of Zb
23
✓
Pq. SB of 10
BARRIER computation
*
I
INCREASE
MOVE
COUNTER
TURN
SKIP
OH
l<
V
I
25
/
2 6
*
#
Pq. tlOfZO
/
h J; *■ -*■ .-hm
-iL:l:
*■
TEST ON
K PULL
K PULL,s K PULLfM
/Is
1 YB8J
i YES
y
f?
siy
K PtiLLj :
Bitpuu-^-a.
*
\
fq.IZOfZt
\
t
VC PlAU.;<» Z
PUT . iavto C-Usstt. t
MN6£j
Pq.t3 0f2»
4
29
*
*
4
\
/
fy /$ 20
T .
TK 5 (Vi ’ } <h) 1
RiT j: Xr»rt> CLASS 2 ,
£ VULL- - -5
I
Pq.lfofZO
/
#*
31
/
RESIDUAL COMPUTATION AND CHECKING
?q.lboflO
i
32
i
* XT Yavgj
TURN
CONVERGENCE
INDICATOR
OFF
YES
fy. I7ofio
33
i*
'I
*
YES
f
OUTPUT:
CONVERGENCE INDICATOR
WAY
?
^ EXIT
fy iff of 10
34
V
«
PMTML DERIVATIVE CALCULATION
<r
EXIT
OUTPUT '•
p..
r ii
CftROR
INDICATOR
OPEN
RANKS
trajectory
COMPUTER
CLOSE
ranks
YES
Pq.Z0of2O
REFERENCE
r*
%
\
*
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