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WERIFICATION AND FEASIBILITY STUDY 
OF A MICRO-COMPUTER BASED 
BALLISTICS ALGORITHM 


John Thomas Ertischweiger 





DUDLEY, KNOX LIBRARY 
NAVAL POSTGRADUATE SCHOON 


NAVAL POSTGRADUATE SCHOOL 


Monterey, California 





HE S45 


VERIFICATION AND FEASIBILITY STUDY 
OF A MICRO-COMPUTER BASED 
BALLISTICS ALGORITHM 


by 


John Thomas Ertlschweiger II 


December 1976 





Thesis Advisor: U. R. Kodres 


Approved for public release; distribution unlimited. 








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5. TYPE OF REPORT & PERIOD COVERED 
Master's Thesis 
December 1976 


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REPORT DOCUMENTATION PAGE 


laid, vsing 9 aixcoen, bie mantl ae ie ucts 


4. TITLE (and Subtitle) - 

Verification and Feasibility Study of 
A Micro-Computer Based Ballistics 
Algorithm 


7. AUTHOR(e) 
















John Thomas Ertlschweiger ITI 


















10. PROGRAM ELEMENT, seo due TASK 


AREA & WORK UNIT NUMBERS 






9. PERFORMING ORGANIZATION NAME ANO ADDRESS 


Naval Postgraduate School 
Monterey, California 93940 












11. CONTROLLING OFFICE NAME AND ADORESS 












Naval Postgraduate School 


12. REPORT DATE 
December 1976 
. MONITORING AGENCY NAME @ AODORESS(I/ different from Controlling Office) 5 


1S. SECURITY CLASS. (of thie report) 
Naval Postgraduate School Uneladasttica™ + 


Monterey, California 93940 
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SCHEOULE 













16. DISTRIBUTION STATEMENT (of thie Report) 






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186. SUPPLEMENTARY NOTES 







19. KEY WORDS (Continue on reveree aide if neceeeary and identify by block number) 






20. ABSTRACT (Continue on reveree side if neceeeary and identify by block number) 
The radical cost reductions in computer hardware brought 
about by large scale integration (LSI) has motivated this 
feasibility study which explores the use of the INTEL 8080 as 
a ballistics computer in a distributed micro-computer based air- 
borne tactical weapons system. 
The results show that software floating point arithmetic 









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(Page 1) S/N 0102-014- 6601 | 


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20. Abstract (Cont'd) 





using a sixteen bit mantissa is sufficiently accurate for 
solving the ballistics problem. 

Experimental data failed to show that the mathematical 
model accurately predicts the weapon's behavior. Either the 
instrumentation to record the release data was inaccurate, or 
the ballistics tables do not accurately predict the actual 
behavior of falling weapons. 


DD Form 1473 


an 4 
S/N 0102-014-6601 SECURITY CLASSIFICATION OF THIS PAGE(When Date Entered) 


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Verification and Feasibility Study 
of a Micro-Computer Based 
Ballistics Algorithm 


by 


John Thomas Ertlschweiger II 
Lieutenant, United States Navy 
B.A., University of Virginia, 1969 


Submitted in partial fulfillment of the 
requirements for the degree of 


MASTER OF SCIENCE IN COMPUTER SCIENCE 


from the 


NAVAL POSTGRADUATE SCHOOL 
December 1976 


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DUDLEY KNOX LIBRARY 
NAVAL POSTGRADUATE SCHOO! 


ABSTRACT 


The radical cost reductions in computer hardware 
brought about by large scale lateprarton (LSI) has motivated 
this feasibility study which explores the use of the INTEL 
8080 as a ballistics computer in a distributed micro-computer 
based airborne tactical weapons system. 

The results show that software floating point arithmetic 
using a sixteen bit mantissa is sufficiently accurate for 
solving the ballistics problem. 

Experimental data failed to show that the mathematical 
model accurately predicts the weapon's behavior. Either the 
instrumentation to record the release data was inaccurate, 
or the ballistics tables do not aeeNRETY predict the 


actual behavior of falling weapons. 







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TABLE OF CONTENTS 


INTRODUCTION -------------------0------------ 10 
A. FEASIBILITY STUDY = a oor cere---- ti 
B. VERIFICATION --------2---------2-2--"------ 11 
C. PRESENTATION OF THE THESIS -------------- 12 
‘BACKGROUND -e----- <n errr n nnn rere e nn nennne=- 14 
A. THE NAVIGATION COMPUTER ----------------- 14 
B. THE BALLISTICS COMPUTER ----------------- 16 
C. THE EXECUTIVE COMPUTER ------------------ 16 
D. THE COMMUNICATIONS SCHEME --------------- 17 
BALLISTICS PROBLEM -------------------------- 19 
A. THE MATHEMATICAL MODEL ------------------ 19 
B. BASIC ASSUMPTIONS ----------------------- 20 
C. DERIVATION OF DOWN RANGE TRAVEL --------- 73h 
MIRCO-COMPUTERS IN AIRBORNE TACTICAL 
WEAPON SYSTEMS -----------2------- 92ers enen-- 26 
A. THE QUESTION OF ACCURACY ---------------- 26 
B. THE QUESTION OF SPEED ------------------- 29 
VERIFICATION OF DOWN RANGE TRAVEL ----------- 32 
A. THE BALLISTICS ALGORITHM - FORTRAN 

VS. PIM ----- 222-20 nnn nnn nn nnn nnn nnn a2 
B. OBSERVED DATA VS. BALLISTICS TABLES ----- 33 

1. Predicted Down Range Travel --------- 33 


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3. Observed Slant Range ---------------- 40 

a. Method for DRT Calculation ------ 40 

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4. Hit Distances ----------------------- 43 

VI. PRESENTATION OF RESULTS --------------------- 44 
A. FORTRAN VS. PIM ------------------------- 44 


B. BALLISTICS TABLES VS. FORTRAN RESULTS --- 46 


C. EXPERIMENTAL DATA VS. FORTRAN ALGORITHM - 48 


VII. CONCLUSION ---------------------------------- 50 
APPENDIX A ---------------------------------------- 52 
APPENDIX B ---------------------------------------- 111 
APPENDIX C ---------------------------------------- 170 
APPENDIX D ---------------------------------------- 182 
APPENDIX E ---------------------------------------- 197 
FORTRAN VERSION of the BALLISTICS ALGORITHM ------- 210 
PIM VERSION of the BALLISTICS ALGORITHM ----------- 223 
LIST OF REFERENCES -------------------------------- 246 
INITIAL DISTRIBUTION LIST ------------------------- 247 







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LIST OF TABLES 
AMOUNT OF TIME BALLISTICS PROGRAM SPENDS 
IN THE FLOATING POINT PACKAGE -------------- 
SUMMARY OF FORTRAN VS. PIM RESULTS --------- 


SUMMARY OF BALLISTICS TABLES VS. 
FORTRAN RESULTS ---------------------------- 


SUMMARY OF BALLISTICS TABLES VS. 
ACTUAL DELIVERY DATA ----------------------- 








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LIST OF FIGURES 


ORGANIZATION OF MULTIPLE PROCESSOR AIRBORNE 


TACTICAL WEAPONS SYSTEM ---------------------- 18 
EFFECTS OF BALLISTICS TABLES ERROR 

SENSITIVITIES FOR HEIGHT --------------------- 35. 
DESIRED: ERROR SENSITIVITIES FOR 

BALLISTICS PROBLEM --------------------------- 35 
NO WIND SOLUTION TO DOWN RANGE TRAVEL -------- 39 


WIND CORRECTED SOLUTION TO DOWN RANGE 
TRAVEL --------------------------------------- 39 


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ACKNOWLEDGEMENTS 


The author wishes to express his sincere gratitude 
and thanks to Lieutenant Gordon Eubanks, USN, whose 
technical skill and expertise in the field of micro- 
computer programming and hardware trouble shooting proved 
to be an invaluable aid in compiling the data presented 
in the appendices. 

Also special thanks are due to Kathryn B. Strutynski 
for her help in developing a software package to convert 
the three byte floating point data into IBM's four bate 
floating point format. 

Last but certainly not least, thanks go to my wife, 
Beverlea, and children Kimberly and Tom, who persevered 
many lonely weekends and nights, and who provided the 
encouragement and inspiration required throughout the 


preparation of this document. 







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I. INTRODUCTION 


Military airborne tactical weapon systems have been 
designed and implemented primarily to aid the aircrew in 
performing their mission with accuracy and speed. This 
usually means that a shorter time is spent over target 
which increases the survivability of both the weapon 
systems Aer cain and the aircrew. 

The system presently employed on board the AGE neue 
of the Navy's attack aircraft, utilizes an IBM 4 PI series 
mini-computer to perform two major functions. 

1. Navigation. 

2. Solution of the ballistics problem. 

In addition, several other related functions are performed 
by the system. 

3. It provides steering commands. 

4. It provides real-time display of sensor information. 

5. It provides release pulses to the weapon at the 


appropriate time. 


A. FEASIBILITY STUDY 
This thesis will attempt to prove the feasibility of 


implementing an airborne tactical weapons system using 


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micro-computers. Two important questions must be answered 
in order to establish the feasibility of using a micro- 
computer in a tactical weapons system: 

1. Is the micro-computer accurate enough? 

2. Is the micro-computer fast enough? 
The accuracy problem was approached by executing the ballis- 
tics algorithm on an IBM system 360 using a 21 to 24 bit 
mantissa in the standard floating point format and comparing 
the results of the same algorithm executed on an INTEL 8080 
micro-processor with a 16 bit floating point mantissa. The 
question of speed was answered by ee the ballistics 
algorithm for numerous weapon types and initial conditions 


and observing the elapsed clock time. 


B. VERIFICATION 

The second aspect of this thesis was to verify that the 
ballistics Piceciclineorcasnended to published tables as 
well as experimental data. The NAVAIR 01-1C-1T-1 ballistics 
tables were used to compare time of fall and down range 
travel against the results of the FORTRAN version of the 
ballistics algorithm. A total of 1813 different initial 
release conditions were examined spanning 18 different 
weapon types, various dive angles from +10 degrees to -60 


degrees, altitudes from 500 feet to 15,000 feet, and air- 


speeds from 300 knots to 650 knots. 


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The source of experimental data was a set of data cards 
recorded by the bombardiers of an A-6E Squadron over a 
period of one year. The data for each bomb drop consists 
of 24 various computer readouts at the instant the weapon 
is released from the aircraft as well as the hit coordinates 
of the weapon. This information is used to determine the 
initial conditions for the ballistics equations. Unfortun- 
ately a critical parameter, the dive angle, was recorded 
only to the nearest degree. Consequently an error analysis 
was conducted to determine the maximum error which could be 


expected from the rounding of the output data. 


C. PRESENTATION OF THE THESIS 

Chapter II explains the organization of a distributed 
micro-computer airborne tactical weapons system and dis- 
cusses how the output of each subsystem is integrated with 
the entire system. The ballistics problem and the deriva- 
tion of the differential equations used to-describe the 
mathematical model are discussed in detail in Chapter III. 
Since no analytical solution exists for these equations, a 
simplified version of the model is solved analytically to 
gain insight. 

Chapter IV explores the feasibility of using micro- 


processors in an airborne tactical weapons system. 


12 





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Aspects of accuracy and speed are examined. Factors in 
attempting verification of the ballistics algorithm and 
experimental anes mick the NAVAIR 01-1C-1T-1 ballistics 
tables are contained in Chapter V. 

Chapter VI presents the results of this thesis. The 
conclusions and recommendations of the author concerning this 
thesis are written in Chapter VII. 

Appendix A contains the output from the FORTRAN program 
comparing the FORTRAN and PLM versions of the ballistics 
algorithm. Appendix B presents the same comparison between | 
the FORTRAN version of the ballistics algorithm and the 
ballistics tables. Appendix C is a listing of the experi- 
mental data as they were recorded from the cockpit readouts 
of various A-6E aircraft. Appendices D and E are the results 
of the experimental delivery data compared against the 
FORTRAN version of the ballistics algorithm (approximating 
the ballistics tables), using two sets of drag coefficients. 

The ballistics algorithm used in this thesis is the 
SIGMA version of the BOEING ALGORITHM modified by the Naval 
Weapons Laboratory at China Lake, California. The algorithm 
was further modified at the Naval Postgraduate School in 
Monterey, California for eventual implementation on the 
ballistics processor of the multiple micro-processor tactical 


weapons system. 


13 





II. BACKGROUND 


To provide a better understanding of the role the 
ballistics processor plays in the multiple processor 
System, a brief overview of the proposed system will be 
discussed. The computer system is composed of three micro- 
processors: a navigation computer, a ballistics computer, 
and an executive computer. Each machine is dedicated to 
the process to which it is assigned instead of sharing 
resources of a single processor as in the present opera- 


tional systems. 


A. THE NAVIGATION COMPUTER 

The navigation computer is a basic element to all 
tactical systems. In present operational systems, the 
navigation program is executed periodically to update the 
present position by the change in position since the last 
time increment. The navigation computer utilizes input 
from four major sensor instruments as its primary source 
of information. 

1. The Inertial Navigation System. 

2. The Doppler Radar. 

3. The Air Data Computer. 


4. The Search Radar, 


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The inertial navigation system provides heading 
(azimuth), attitude (roll and elevation), and velocity 
increments in the X, Y, and Z directions. The doppler 
radar is a velocity sensor that utilizes the doppler shift 
principle bh measure ground track speed and drift angle. 
The ground speed and drift angle derived from the doppler 
radar along with the true heading from the inertial naviga- 
tion system are used to calculate the direction and magni- 
tude of the wind. The air data computer uses the ambient 
static pressure and ram air pressure from the pitot tube 
to calculate arude 9f 4 static pressure, pressure altitude, 
and mach number. Outputs from the air data computer are 
used to damp the raw velocities from the inertial system. 
The search radar provides target aximuth, range, and eleva- 
tion signals to the navigation computer. The search radar 
elevation along with the aircraft elevation from the inertial 
system are used to measure the radar depression angle (look 
down angle to target from flight path vector). The depres- 
sion angle and search radar slant range are used to compute 
ground distance to the target independent of altitude. 

As the navigation computer calculates new incremental 
distances for each time increment, the present position 
is continuously updated. After each update, the executive 


computer is interrupted and the current value is passed to 


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it. If the navigation computer is functional, the aircrew 
will have current present position information independent 


of the status of the executive computer. 


B.. THE BALLISTICS COMPUTER 

The ballistics computer is provided with the most current 
estimates of position and velocity of the aircraft. It is 
also provided with the weapon type selected by the aircrew. 
The ballistics algorithm computes the down range travel 
and time of fall for the weapon based on the airspeed, dive 
angle and altitude of the aircraft. The ballistics algorithm 


will be discussed in depth in Chapter III. 


C. THE EXECUTIVE COMPUTER 

The executive computer displays the information generated 
by both the ballistics and navigation computers. The execu- 
tive computer is also responsible for issuing steering 
commands to the autopilot and firing pulses to the weapons 
release mechanism. The most important task the executive 
Seater performs is to extrapolate a predicted weapon release 
point based on a time history of the position and velocity 
of the aircraft. Thus, even though the other two computers 
“can operate independently, the aircrew cannot make a computer 


delivery without the executive computer. 


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D. THE COMMUNICATIONS SCHEME 

Because Be the inherent hierarchy between the computers 
involved, the master-slave type of multiprocessing is the 
most suitable and simple form to implement. The navigation 
and the ballistics computers act as peripheral devices of 
the executive computer, resulting in a one way interrupting 
scheme. The only computer which has to be interrupted is 
the executive computer. The navigation and ballistics 
computers are the dedicated slaves which asynchronously 


interrupt the executive computer. 


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BALLISTICS COMPUTER 


FIGURE 1. ORGANIZATION OF MULTIPLE PROCESSOR AIRBORNE 
TACTICAL WEAPONS SYSTEM 


18 


III. BALLISTICS PROBLEM 


Since the earliest days of aerial warfare, cha hesee of 
the ballistics problem has been to drop a bomb from an air- 
borne weapon platform and to consistently hit a target. The 
problem still exists today; however, with the aid of a 
computer the problem can be quickly and accurately solved. 
This necessitates the development of a mathematical model 
which approximates the actual path a weapon travels through 
space. Statistically, projectiles have been shown to follow 
predictable paths which behave very much like freely falling 


bodies described by Isaac Newton. 


A. THE MATHEMATICAL MODEL 

The ballistics problem can be described as a body 
falling through space according to Newton's second law of 
motion. The mathematical model or equation governing the 
trajectory of a ballistics projectile is a second order 


differential equation: 


ma =mg -c |vi[v C2) 


where: m = mass of the body 


acceleration vector 


1) 
Hl 


acceleration due to gravity 


oq 
Hl 


19 





c drag coefficient 


v = velocity vector 
The term e( Ivf v) is the drag due to the air resistance of 
the body and is proportional to the square of the velocity. 
At the time the projectile is released from the aircraft 
it will have an initial position and velocity. The differ- 
ential equation together with the initial conditions 
uniquely determines the trajectory of the projectile. 
This particular mathematical model was chosen for two 
reasons: 
1. It approximates reality accurately. 
2. The Navy publishes range and time of fall 
information in tables (NAVAIR 01-1C-1T-1) 
which uses this model. 
The latter is the primary reason for using the ballistics 
tables as a standard or guideline in the verification of 


any new mathematical models or program implementations. 


B. BASIC ASSUMPTIONS 

Unfortunately there is no known analytical solution for 
this set of differential equations. However, a simpler 
problem does have an analytical solution. Thus, for the 
sake of simplicity and to aid in the discussion of this 


solution, let us first assume: 


20 





1. Level, non-accelerated flight. 
2. Bombs are not ejected from the bomb rack. 
3. No forward firing ordnance. 
4. Drag/wind resistance is negligible. 
5. Time of release is time = 0. 
In addition, this discussion also assumed: 
6. The Earth is flat and non-rotating. 


7. The gravitational attraction, g, is constant. 


C. DERIVATION OF DOWN RANGE TRAVEL 
Rewriting equation (1) with respect to the time deriva- 


tives of the position vector, say u, the equation becomes: 


ei ae iar od Ch ae (2) 


By letting u = (x,y), equation (2) is transformed into a 
system of two equations describing motion in a two dimen- 


sional coordinate system. 

mx |< \ x? Piet | x (3) 
o -2 2 2 

my = -mg + Vx ty iE (4) 


Initial conditions for the system are derived from the re- 


lease position and velocity vectors of the aircraft: 


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Position Velocity 


(tg) = me(t,)i x(to) = r,(to) 


y(to) = ry(to) —-¥(tq) = Fy(tp) 


As a result of the assumption that air resistance is 
negligible and the aircraft's velocity and position deter- 
mine the initial conditions, equations (3) and (4) are 
simplified to 


initial conditions 


mx = 0 x(0) = r,(0) x(0) = r,(0) 
my = -mg y(0) = £(0) y(0) = r,(0) 


The solution of these two equations is obtained by dividing 
by the constant m, integrating twice with respect to time, 


and solving for the constants of integration. 


face dt fo dt => x(t) 
[aces dt fe, dt => x(t) ; : 


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ke 


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ct 
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re 
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therefore, 
x(t) = r(0)t ate (0) (5) 


where: 


Ze 





x(t) = total down range travel after release. 
r (0) = initial displacement x(0) of weapon 
at the time of release, usually 0. 


EC O)t = down range travel due to initial velocity. 
[yo dt [rs dt > y(t) 
[ue dt frectey dt > y(t) 


“ct + Cy 


sheen +c t+ ¢ 


q 2 
when t = 0: a y(0) = (0) 
e50= yO) = r, (0) 
therefore, 
y(t) = -bet’ + Ey(o)e + x, (0) (6) 
where: 


y(t) = the height above the ground at any time t. 

r(0) = initial altitude y(0) of weapon at time of release. 

ry(0)t = altitude loss/gain due to aircraft's initial 
vertical velocity. 


Beet = altitude lost due to gravity. 


If the time from weapon release to weapon impact, or 
time of fall, is known, the ballistics problem is reduced 
to determining the down range travel, x(t*), given its 


initial-velocity, =2 (00) and the time of fall, t*. Time of 


Z3 


— «Ofte z rreok 





egeao las 





fall can be found by setting equation (6) to O and solving 


for the positive root of t. 
ket? +r (0)t +x (0) =0 
¥ yi 
‘Using the quadratic formula: 


t* -[s - b2 - ia | 2a 


where: 
a = -g/2 
b = ry (0) 
c = = 8°) 
55 ONG Ore 
g 


By substituting time of fall, t*, into equation (5) 
x(t*) = r (0)t* + r (0) 
x x 


down range travel, DRT, can be calculated. 


£(0) + VCE (0))? + 2er_ (0) 
DRT = x(t*) = r,(0) pe SEES ae NS) Seen eae ee r,,(0) 
& 
Since the coordinate system is arbitrarily placed, 


assume initial displacement, r (0) ; to be zero at the time 


of release. Also, since level non-accelerated delivery is 


24 





assumed, initial vertical velocity, r(0), is zero. The 
expression for down range travel now becomes: 


DRT = (0) 2ry (0) 


e 
In reality the problem must take into account the con- 
straints placed on the problem earlier. Wind, drag, 
vertical velocity, and non-level delivery prarmeters make 
the solution more difficult. Solution in this case is 
accomplished numerically using a second order Runge-Kutta 


scheme on a digital computer. 


23 






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IV. MICRO-COMPUTERS IN AIRBORNE TACTICAL WEAPON SYSTEMS 


The question of whether or not micro-computers are 
feasible in an airborne tactical weapon systems environment 
was approached by first translating the ballistics algorithm 
into two high level languages, FORTRAN -and PIM. (See Ref. 
3). The FORTRAN version was executed on the IBM system 360 
and the PIM version was executed on the INTEL 8080 micro- 
processor. Thus, if the same algorithm is executed on two 
different machines, the INTEL 8080, whose floating point 
mantissa has 16 bits and the IBM 360, whose floating point 
mantissa has 21-24 bits, then the differences in results 
can only be attributed to the difference in the precision 


of the two machines. 


A. THE QUESTION OF ACCURACY 

The solution of the ballistics problem requires solving 
a set of four differential equations numerically on a digi- 
tal machine. This necessitates numerous arithmetic opera- 
tions, including multiplication and division. 

Since micro-computers presently lack hardware multiply 
and divide functions, a software package capable of perform- 
ing floating point arithmetic operations must be used. 


However, this requires a significant amount of additional 


26 






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computing time in the solution of the ballistics problem. 

As a result an alternate design in computer architecture 

was explored which utilizes three INTEL 8080 micro-processors 
instead of one, as in the mini-computer systems. In this 
multiple micro-processor system, each processor is dedicated 
to each of the primary functions of the system: executive, 
navigation, and ballistics computations. 

The floating point mathematical package used in conjunc- 
tion with the ballistics micro-processor uses a three byte 
binary representation with a 16 bit mantissa and an 8 bit 
exponent. The mantissa is left justified so that the most 
significant bit is always on and need not be stored, giving 
a full 16 bits of precision. The exponent is expressed as 
a power of 2 where the most significant bit serves as the 
sign bit. This three byte number is used instead of a con- 
ventional four byte scheme in order to reduce the time needed 
to perform the calculations necessary to solve the ballistics 
problem. 

IBM's system 360 computer utilizes a four byte floating 
point hexadecimal number with a 21 to 24 bit mantissa and 
an 8 bit exponent. Since this method requires the first 
byte to have the value 1 to F (hexadecimal), the precision 
of the mantissa can vary between 21 bits (when the leading 


three bits are zero) and 24 bits. The exponent is expressed 


27 





nae tot 
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ae 9 





as a power of 16 and also contains the sign bit. An 
obvious advantage the IBM 360 has is that it does floating 
point arithmetic in hardware which makes it approximately 
two orders of magnitude faster than the software version. 

The INTEL 8080 micro-computer uses 8 bit operations and 
has the option of using double precision (16 bit) operations. 
The double precision operation permits multiplications and 
division to be performed as sequences of 16 bit additions 
and subtractions. Then, if the mantissa is kept left 
justified (16 bit precision), the double precision feature 
can be used to maintain 16 bits of precision throughout the 
calculation. If more precision is desired, such as 21 or 
24 bits of precision, a quantum jump in execution time can 
be expected because of the additional computer cycles 
required. 

It is the intention of this thesis to show that the 
loss of one byte of precision will not significantly affect 
the results of the ballistics solution. The accuracy with 
which a weapon is delivered depends greatly upon the 
accuracy and precision of the sensor supplied information. 
The loss of one byte of precision only affects the sixth 
most significant digit which is an order of magnitude more 
precise than most of the input sensors on board attack 


aircraft. This then is the motivation to compare the 


28 





results of a FORTRAN ballistics algorithm executed on the 
IBM system 360 with the same algorithm translated into PIM 


and executed on the INTEL 8080 micro-computer. 


B. THE QUESTION OF SPEED 

The second area of interest is the question of speed. 
At the time this thesis was written, an LSI (large scale 
integration) "chip" existed which could perform the floating 
point multiply and divide operations at the cost of $270. 
However, due to budgetary constraints this equipment was not 
readily available for experimentation. The hardware float- 
ing point "chip'"’ can execute approximately 100 times faster 
than the software floating point package. For example, a 
multiply operation in the software package takes approxi- 
mately 600 microseconds to execute whereas the hardware 
package executes a multiplication in 6 microseconds. A 
hardware multiply and divide operator was also developed 
and constructed as a micro-computer course project at the 
Naval Postgraduate School and was demonstrated to function 
at 60 microseconds. | 

By interrupting the program during execution and record- 
ing the location of the program address register, it was 
determined that the ballistics program spends about 92% of 


its execution time in the floating point package. According 


29 





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to Jupin (Ref. 3) the execution time of each solution was 
proportional to the calculated time of fall. These results 
were confirmed _ executing nearly 1800 separate calculations. 
-The time to calculate the predicted release point proved to 
be about 10% of thélealculated timalof fall, *Laahares 

(Ref. 2) was able to show that the ballistics algorithm was 
fast enough for certain initial conditions, however for high 
airspeeds and low altitude release conditions his extrapola- 
tion technique was not usable. 

A substantial amount of the time the program spends in 
the floating point package (about 92%) is spent in the 
multiply and divide procedures. Using either the commercially 
produced "chip" or the locally constructed hardware multiply 
and divide operator, a significant reduction in execution 
time would result. Although this thesis will not answer the 
question of speed with an unqualified yes, it supports the 


finding that the ballistics processor is fast enough. 


30 





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NUMBER OF PER CENT OF 
TOTAL INTERRUPTS TIME PROGRAM 
NUMBER OF PROGRAM WAS IN|SPENT IN 
INTERRUPTS FLOATING POINT| FLOATING POINT 
ATTEMPTED 


PACKAGE 





TABLE 1. AMOUNT OF TIME BALLISTICS PROGRAM SPENDS IN 
THE FLOATING POINT PACKAGE 


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V. VERIFICATION OF DOWN RANGE TRAVEL 


Two separate verifications of down range travel were 
made, using the ballistics tables as a "standard." First, 
the FORTRAN and PIM versions of the mathematical model, 
previously discussed, were tested against the ballistics 
tables. Second, observed data was compared against the 
FORTRAN version for accuracy in time of fall and down 
range travel. The FORTRAN version was executed on the 
IBM 360 (32 bit machine), while the PIM version was executed 
on the INTEL 8080 micro-computer utilizing a 24 bit floating 


point mathematical package on an 8 bit machine. 


A. THE BALLISTICS ALGORITHM. - FORTRAN VS. PIM 

A straight forward comparison between the FORTRAN version 
and the PIM version of the ballistics algorithm was made 
contrasting the down range travel and time of fall. An 
input/output interface was written to the PIM program so 
that data could be read from a floppy disk and the results 
written onto the same device. The floating point package 
was also modified in order to execute on the INTEL 8080 
and a logic error in the multiply procedure was corrected. 
The FORTRAN program (Ref. 3) was virtually unchanged, 


however a statement was added to the TRAJ subroutine to 


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patch a logic error affecting the second stage trajectory 


calculation. 


B. OBSERVED DATA VS. BALLISTICS TABLES \ 
Since the ballistics tables are considered a "standard" 
against which various types of ballistics results are com- 
pared for validity, a comparison between observed results, 
obtained from the A6-E experimental data, and the ballistics 
Pabliéa (NAVAIR 01-1C-1T-1) was desired to establish a 
correlation between the two. However, several problems 
were encountered in making the comparison. 
1. Predicted Down Range Travel 

The initial conditions of the observed data are 
not compatible with those of the ballistics tables. The 
observed data has initial conditions composed of various 
dive angles, altitudes, and airspeeds, whereas the ballis- 
tics tables' initial conditions are multiples of 50 and 
100 for altitude and airspeed, and zero for dive angle 
(only considering level delivery). Two apparent solutions 
to this difficulty are (1) the error sensitivity tables 
and (2) interpolation of the ballistics tables. However, 
both have disadvantages. 

The error sensitivity tables failed to help because 


the corrections are based on maintaining a constant sight 


53 





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line (mil setting) rather than keeping down range travel 
constant. For example, in a level delivery situation the 
error sensitivities table assumes a constant sight line 
and varies the down range travel by affecting corrections 
to the altitude. Theverores 4= che areveiae is higher 
than planned and the sight line is maintained, the hit 
will be short. However, the problem using observed data 
requires a constant down range travel to target, applying 
correction for altitude and airspeed. Therefore, the 
error sensitivity tables could not effectively be used in 


this case. 


34 










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HIGH RELEASE HEIGHT 


Ley 
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RON TS Bs 
<p SS 
\ PLANNED RELEASE HEIGHT 
os 
N Be ; 
x \ LOW RELEASE HEIGHT 
Nias 3 ~ 
TiS 
Ny iar 
\ > 8 
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SHORT TARGET LONG 
FIGURE 2. 


EFFECTS OF BALLISTICS TABLES ERROR SENSITIVITIES FOR HEIGHT 


HIGH RELEASE HEIGHT 


~ PLANNED RELEASE HEIGHT 


< 
‘ LOW RELEASE HEIGHT 





ALT Sa x x 


SHORT TARGET LONG 


FIGURE 3. DESIRED ERROR SENSITIVITIES FOR BALLISTICS PROBLEM 


35 





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The ballistics tables are not linear. Second and 
even third order interpolation would be required to deter- 
mine down range travel on anything other than a cardinal 
altitude or airspeed. This process is extremely tedious 
and time consuming and was abandoned as impractical. 

As a result, an accurate substitute for the ballis- 
‘tics table value for down range travel was sought. Using 
the ballistics tables for the MK-76 mod-5 practice bomb, 

96 data points were selected from all dive angles, air- 
Beeceda. altitudes, and tested against the FORTRAN ballis- 
tics program. Time of fall and down range travel compared 
favorably with a mean error of 0.23% for down range travel 
and 0.11% for time of fall. This was considered accurate 
enough ~ be used as an approximation for the ballistics 
tables' value for down range travel. 

2.,.Wind Effect 

The wind plays an important and rather subtle role 
in determining the total down range travel. First, assuming 
a no wind condition, the down range travel, DRT, is computed 
in the direction of the true heading. Since ho ead is 
present to deflect the projectile, the true heading and 
ground track will coincide and the down range travel along 


the true heading and ground track will be equal. 


36 


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Now consider the wind. If a coordinate system 
(x,y,z) is introduced such that the air does not move with 
respect to the coordinate system, then an aircraft in this 
air mass has velocity with respect to the air mass and its 
heading will be the true heading. Because the air mass 
moves with respect to the ground with the wind velocity, 
w = (w,, W., W,), the Earth fixed coordinate system 


(x',y',z') is related to(x,y,z) by the following equations: 


x’ =xt wt 
L a fy aw 
yi y yt 

' = 
Zz & 7 We 


Considering only the horizontal wind, the vertical 
wind, wz, becomes zero. Down range travel can now be deter- 
mined given an initial altitude, z, and air velocity, V, 
and will be in the direction of the true heading. To 
determine the point (x',y') in the Earth fixed coordinate 


System at which the weapon lands requires: 


Ma 
Hl 


=(€to£) + wi * (tof) 


< 
i 


y(tof) + ae * (tof) 


For example, given the same initial conditions, down range 
travel will be computed, as before, in the direction of the 


true heading. However the aircraft will drift with the 


37 


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wind and will actually move across the ground on a differ- 


ent heading, called ground track. Since DRT is computed. 


along true heading, the projection of DRT onto the ground 


track is that distance the projectile will travel due to 


the initial conditions alone. 


PROJECTED DRT = cos(TH-GT) * DRT 


The down range component (along the ground track) of DRT 


due to wind is computed as follows: 


where: 


X-COMP 


-cos(WDIR-TH) * WKTS * TOF 1.6867 


X-COMP = down range component of DRT due to wind 
WDIR = true wind direction 

TH = true heading 

WKTS = wind speed in knots 

TOF = time of fall in seconds 


1.6867 = conversion from knots to ft/sec 


This distance added to the PROJECTED DRT gives the WIND 


CORRECTED DRT which is the total DRT the projectile will 


travel in a moving air mass. 


WIND CORRECTED DRT = X-COMP + PROJECTED DRT 


38 





x 





FIGURE 4. 
NO WIND SOLUTION TO DOWN RANGE TRAVEL 





FIGURE 5. WIND CORRECTED SOLUTION TO DOWN RANGE TRAVEL 


39 





3. Observed Slant Range 


The experimental data taken from the A-6E aircraft 
during actual anes conditions provides many useful delivery 
parameters, sensor readings, and intermediate calculations. 
However, the one piece of information needed to make the 
desired comparison, down range travel, was missing. But 
various other parameters were available, and DRT could be 
reconstructed by several different methods. The problem 
was to determine which method is the most accurate. 

a. Method for DRT Calculation 

Slant range to the target and search radar 
depression angle (look down angle from flight path vector) 
is made available by the search radar. Down range travel 


simply becomes 


DRT = cos(DEPANG) * SR 
where: 
DEPANG = search radar depression angle 


SR = search radar slant range to target 


A second method takes advantage of the aircraft's 
present position altitude and target altitude to compute 


vertical separation and down range travel. 


VERT SEP = PPA - TGT ALT 


DRT = cos arsin(VERT SEP / SR) * SR 


40 





where: 
VERT SEP = vertical separation 
PPA = inertial derived present position altitude 


TGT ALT = target altitude 


A third and final method of computing down 
range travel uses the vertical separation generated by the 


ballistics program on board the A-6K aircraft. 
DRT = cos arcsin(VERT SEP / SR) * SR 


b. Error Analysis 
An error analysis was conducted to determine 

which method would produce the most accurate value for 
DRT. This would yield a maximum error bound which can be 
expected in DRT due to this method of calculation. The 
third relationship proved to have the smallest error bound 
and was eventually used to reconstruct the down range 
travel from the experimental data. 


For example, the maximum error bound on the 


relationship 
DRT = cos arsin(VERT SEP / SR) * SR 


is the sum of the partial derivatives of DRT multiplied by 


their tolerances. 


41 





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The third method of computing down range travel will serve 


as an example for this procedure. 
DRT = cos arsin (VERT SEP / SR) * SR 


The partial derivative of DRT with respect to VERT SEP 


becomes: 
OQ DRT f SR 


© VERT SEP (eee ae 


The partial derivative of DRT with respect to ALT becomes: 


OpDRT _ “ALT 


pe (sR)* - (ALT) “ 


the error bounds are: 


VERT SEP = +0.5 feet (rounded to the nearest foot) 
SR = +5.0 feet (rounded to the nearest 10 feet) 
Therefore, the maximum error bound that can be expected from 


rounding error of actual delivery data becomes: 


Aprr = —0:5(SR)_+ 5.0(ALT) 


\/ (SR) ~ (ALT) 


This analysis was performed using all three 
methods of constructing down range travel. The third 


method was found to be the most accurate with a maximum 


42 





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a 
Ts 
+? 
. 





error bound of 5.2 feet and was the method used to construct 
the down range travel from the freeze data. 
4. Hit Distances 

The hit coordinates of the experimental data were 
not utilized because the eoreuree bases its calculations 
on the location of the search radar cursors. If the cursors 
are not properly placed, the weapon, most likely, will not 
hit the target. However, for the slant range measured by 
the search radar set, the down range travel and time of 
fall will be calculated accurately and the weapon will hit 


the ground in the proximity of the cursor placement. 


43 


' 
4 





Pe 


r i] 





VI. PRESENTATION OF RESULTS 


A. FORTRAN VS. PIM 

Appendix A contains the results of comparing identical 
ballistics algorithms: a FORTRAN program which is a Naval 
Postgraduate School modification of the Naval Weapons Center 
BOEING algorithm, and a PIM version of the same algorithm 
(Ref. 3). The difference between the procedures is that 
the FORTRAN program uses the standard IBM 21 to 24 bit 
mantissa for its floating point number, whereas the PIM 
version uses a 16 bit mantissa. 

A summary of the results is presented in Table 2. 
With the exception of weapon number five (MK-76 MOD-5) 
the feted average difference in down range travel (DRT) 
was 1.6 feet and the maximum absolute difference in DRT 
was 17 feet, which occurs when DRT is 8,332 feet. Weapon 
number five is suspected to have a coefficient error, 
although none have been discovered. 

Overall these results indicate that the sixteen bit 
mantissa is sufficiently accurate to perform the ballistics 


algorithm. 


44 


sol 3aJ 













































AVERAGE 
PER CENT 


AVERAGE 
PER CENT 
IN DRT IN TOF IN DRT IN TOF 

(feet) (seconds) (feet) (seconds) 


Te [esi [eons [om | oan 
ee 
ssl etpaans onan om la 
Po [sow eos fos [oss 
ane soos 
Ps [owe ones [oar | oa 
Pia [ots [om [ose | or 
ee Oe 
[a [owe fos Pour Fa 
Pa femur foo [eae |e 


TABLE 2. SUMMARY OF FORTRAN VS. PIM RESULTS 


MAXIMUM 
PER CENT 


MAXIMUM 
PER CENT 








45 











0.0 


B. BALLISTICS TABLES VS. FORTRAN RESULTS 

| The ballistics tables were compared against the results 
of the FORTRAN version of the ballistics algorithm. Appen- 
dix B contains the results for a variety of initial condi- 
tions and weapon types. The results indicate substantial 
discrepancies between the ballistics tables and the FORTRAN 
program. The latest version of the Naval Weapons Center's 
ballistics algorithm (including revised drag coefficients) 
still does not resolve these differences. Table 3 gives a 
summary of the data in Appendix B. Weapon number five 
(MK-76 MOD-5), which was used in the SONS data 
shows reasonable accuracy for the range of parameter 


values used in the experiment. 


46 






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- AVERAGE 







AVERAGE MAXIMUM MAXIMUM 



























PER CENT PER CENT PER CENT PER CENT 
IN DRT tN TOP IN DRT IN TOF 
(feet) (seconds) (feet) (seconds) 


: 
pee [oo [oan [ose [oe 
Pas [esas [owas [ase | aa 
Pare [ein [oon [oa | 
Pin [ear [aeons [oa 
Pa fase fase [ rae | ae 


* used invalid drag and mach coefficients 


TABLE 3. SUMMARY OF BALLISTICS TABLES VS. FORTRAN RESULTS 


47 





C. EXPERIMENTAL DATA VS. FORTRAN ALGORITHM 

Experimental data gathered by bombardiers from Naval 
Air Station Whidbey Island at Boardman bombing range is 
given in Appendix C. Because the ballistics table and 
the FORTRAN algorithm agree reasonably well, the FORTRAN 
algorithm was used in place of the ballistics tables for 
convenience. The down range travel and time of fall were 
calculated from: 

1. The experimental data. 

2. The FORTRAN algorithm with the old drag coefficients. 

3. The FORTRAN algorithm with the new drag coefficients. 
The calculations are described in section (5.B.2, Wind 
Effect), and the results are summarized in Table 4. 

The substantial discrepancy between the experimental 
data and the results of the ballistics algorithm cannot be 
dismissed. Because the ballistics algorithm's down range 
travel agrees with the official Navy ballistics tables to 
within 0.2%, the 12% discrepancy in DRT leads to the con- 
clusion that either the instrumentation on many different 
aircraft indicated erroneous readings, or the behavior of 
the weapons is substantially different from the behavior 
described by the ballistics tables. Additional data under 
more precise initial conditions would have to be gathered 


before any definite conclusions can be drawn. 


48 


> 





OLD NEW 
COEFFICIENT COEFFICIENTS 


AVERAGE ABSOLUTE 
DIFFERENCE IN 247.00 247.00 
DOWN RANGE TRAVEL 


AVERAGE ABSOLUTE 
DIFFERENCE IN OU. E299 0.2573 
TIME OF FALL 


AVERAGE PER CENT 
DIFFERENCE IN £L973 12.435 
DOWN RANGE TRAVEL 


AVERAGE PER CENT 
DIFFERENCE IN 
TIME OF FALL 





TABLE 4. SUMMARY OF BALLISTICS TABLES VS. ACTUAL 
DELIVERY DATA 


49 


VII. CONCLUSION 


The radical cost reductions in computer hardware 
brought about by large scale integration (LSI) have intro- 
duced an opportunity to construct micro-computer based 
airborne tactical systems which reduce the hardware costs 
by at least an order of magnitude. To establish the 
feasibility of constructing such a system requires that © 
two questions be answered in the affirmative. 

1. Is the computation sufficiently accurate? 

2. Is the computation fast enough to satisfy real 

time requirements? 

This study concentrated on the first question. As the 
results indicate, the 16 bit floating point mantissa is 
sufficiently accurate for the ballistic calculations. 

As a byproduct, the BOEING-Naval Weapons Center algo- 
rithm was compared with the published ballistics tables. 
Although some of the weapons displayed close agreement, 
others revealed substantial discrepancies which remain 
unresolved. 

The most significant and unexpected finding is related 
to the experimental data generated by the bombardiers based 


at Naval Air Station Whidbey Island. If the initial 


50 


-~ 
1 





+ : 
Le 





conditions recorded by the aircraft's instruments are used 
to predict where the weapon would impact the ground, then 
ike ballistics tables predict that the weapons land con- 
sistently more than 10% short of where they actually landed. 
Either the recorded initial conditions are incorrect or the 
ballistics tables do not predict reality for this weapon. 
Although the micro-computer is substantially slower in 
executing arithmetically complex tasks when compared to a 
mini-computer, several micro-computers can be used as 
dedicated machines for specific tasks. Such a distributed 
System can operate sufficiently fast to solve the real time 


problen. 


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APPENDIX A 


This appendix compares the results of the FORTRAN and 
PIM versions of the ballistics algorithm. The absolute 
difference in down range travel and time of fall is 


presented. 


Sy 


cmv, 






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WEAPON COEFFICIENTS FOR IDNO 10 


ALT 


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CAHOON OPM OCOFORFNOV OF SOIR HORA HONMN 
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NSEANONMNENPANMNO ROOD RANA OOOODeS 

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WEAPON COEFFICIENTS FOR 
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FHOORFROnHORR NANG 
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COEFFICIENTS FOR IDNO 17 


WEAPON 


ALT 


TAS 


DEG 


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DOM DMO FOMNAHANONPMODMODNOM ADO SPO Fao 

NOM OHPHAOIDASAM OMS SIA ADP ASTNOMEDONM 4 

PSY OO GS OE SEO AD See MN ne 
Ss aS aes St st AI NN ON 


ND OMMSNOMONOND OMODHANAMVNDDOMME 
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Ss SAaSteeeaedsaest OH OM MH OM OM TO = Ss eS 


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110 


APPENDIX B 


This appendix compares the NAVAIR 01-1C-1T-1 Ballistics 
Tables with the results of the FORTRAN version of the 
ballistics algorithm. The difference in down range travel 


and time of fall is presented. 


Tat 


@ LTUagtA 








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@eeoesve@etsrteeensvee#esvtsete eeeee#eseeeevwvrestseetesee#enst ee? tees e ee @ @ 
See et pe aie Se aa can erence bn noe neve ol rs 

= sa t= NN A A Nn ni nnn nN SN 


®Corxpneereeeoee ere ere eee er eevee eee ee @ 
ooocoococsocoooqoc“esoooeoecooe00o0o0o oo 
ao0co0q0oo0qoq0o0000000090dM0OcOCOOCOON oo ooo Oo So 
WLAN S OMOWMO DWNNOWMMNNONOMNMNO OO OCOMNOOMOMMNOQNOOW 
AT FOI PN MOA EN NILA ALN AO NLD NID 019 00 9 BOB 09) CO 0) WO OD 099 0 0) OV OY 


ecoeoevreoereoet ov eet eeovreeteeeeevee eevee eevee eee 
elolelelololelelolololelololclolololelclololclelololelolelolelolololololelojlole) 
WOAWOOWMNMNODOMNOOMMNOOMNN COMNOOWN OSMINOOMMOOMNEO 


ALT 


TAS 


DEG 


TOMOM NORIO SENOS EONS OE NORIO SEO 

MO AA FUVO DO DM ON OO MM OLA WV CO MM $9 ON OC 

eereoereeoe ote eee eee eee eee eee @ 
! | f tit | | | | I 


ORM DOOCWNODAIMNONE ME NONNAMM MFO 
SOL ONS OAS OOS OS NO Eee cae 
eeoeotreoereeoer ee ee eee eee eee ee @ 


oo SOHONOYONOaS SdAdAddomdts 
a 


eeoeeoeoeeeer oe eee eer oe eee eee o 
MODOOOLPFONNAHSHOnOLT ANOS ORONO 
D ANSMNMMNRAO | ON | aeANANOssS WM 
1 I ! ! ! ! 1 1 ! fi Fotis it 


Pt ONOONNDOENOMOENMOFON DONO 
OWONOOHFONOMNSINS NOS SsasSsasaurNnnes 
eoeerevereoee eo eee eeee eee ee eee eo 


ocooo°;eqcoococoo0oo eS Sees 


eeoeceeoeeeeoeee ee eee eee eee 8 ee 
ONWINE OD LPSSAODM APeh=TOMre FMOO 
DY MOU COM OD AD OM AFM APIE OAH 
DANUNPODMS Ma VNSEOMNEE DONMMMNOY AO 
PVA AIO NO AIS OF MID MAD SONI EI AMO AIO AO On 


DO FOFPASF AHODMMAMSAOMOLRKSANSEHE MN 
NF DOWN AM OHODANDATFPOMIOMMANNODA 
eeee<esvse302seegense20282828202080220270892808208089 0 @ 
DOM MON DODOMMOMAODFPOMANMAM a 

MANN N OM OH OM HS Teter Saran 


eoeveevwo ere eoew ee ee eee ee eee eee @ 
MFANTOMNSTODODADONEONANNSTOAEAOS 
DNFSAODAPMODNAA-SMIMMANE OM FNDOOO 
DDAUFOCDOMNAE OSM DON ANAND HY aO 
MANMAYMONS OSMAN AINaEMANMAMAMM > 


ONE OPFRMOUNOSOAREMONMNAINUOOM ONO 
eeoeeveeev oor ee eo ere eer ee ee oo 8 @ 
Fe ee ee ee Ne OS 

OTN NN OM PF tata te rem 


WW OOWMNOOWMMOOMMNOOMMOOMNNOOCWWU 





te ore A ?y niin a 
»eteees eee 
Joc oS owd 

‘ | 











wile) Tid Wyse ay! vue 


epaey's ee & ye + 
» h ee Ls 
dcde moooeS 7 alitian hire 


ee 
*2e es : 


oc 4 aboe 


- sae oie) see e 8 eee 
= = es eee erage 


ea am rie Sy OO 
rc eye* ~\eytenaranon Deed 
vm en wr 


erect 
>» ¥ wrrcere rT 
>% Torys anos 

ont ._ naneaywy ww 


2 
~,. 









7 sien & 8 cane eer 8 0 


d Wong ae i 

arent: me} Laeeaeky vi 
A 3 AAA Se ecto 
yh 4 OP ee ee Nee means 


= AD OREN A? tee Ge eee . 
‘ Wig tbe tl & ie are v 
. * ‘ais 54 5 24 8 *** 
5 ip enka tee 


WEAPON COEFFICIENTS FOR IDNO 21 


4.0000000 


ALT 


TAS 


DEG 


MONO DDHOOND DIO OM ANDINA MIA MALIA M OP At 
DUDOMNONDOMNNNDDOM MONE MAMOOSSIEMOM™ MON 
eeoeeeseteesteeeeeteeereet ese eet eeees @ 
Sete SN ON ON tt OO SILI ett mt et LO SP tt ett OY 
Jim Jem Joes Jas) To Je Fee Veet aC FRG TMM Vt a et YOR Via J TUT OCR RR eo Paul 0 


ODNOONNODUFMOM ANE DENOSIFNDANUMNNOO 
MOA DM FOND FOS DODAHONMMHDOMONMAO TiN 
eoeerevreereeoeeereoerere er eee eevee ee eee 
SAO ONONANANASNNNNNNN MON OMNN MANLY 


eoceoeoeoereeoe ee eee eee 
NEIMOFADOMIMN SM NOOIM AN 
Mt CO COM AMO SHO OOM tM 
a ee i a i U Brace 


-148. 

-9l.e 
-136. 
-167. 


OHFOREEEEDODNODASTOPOONOMADNRAO 
See eR eRe ee PO ee ee 

eoees2eeee®@ eeosveeeestespte~eesterte? © 8 @ 
ocococoNoNoNoNnocoocoScacccecco 


0.04 
ca 


eeovceoeoeeoereeeoeo reer eee eee ee eee eee 
OMOMFPNOOLETIPAROMNDOSTOONILNTARDMODO 
NWNOEASLOREANOO SOOT NIFNONNSIPRORS SS 


DDOPMMNNSEMONDONWNMESODOAIM$SFODRSE OMA 
NDOOMH MYHMOM ALA SIS OMNOM THONDONNTOO 
@eeeeevsse98seepeeepeeeeseee08802202080802080280808080 © 
NWIMOMMS SH MAMNOWN SC OS P AO esas ANONNAHN 
Set eat et Qt et ete Sete StU SIN SN aN 


0. 


eeee 
MM ADOO DM NSANO OSA OOM VAM LAD ON IND in 
SOE LAM MALO OSL EEO DNDLTOAMMOOO- ODD 
DOMADOMODOPNS CHOOOMFNEONDOT HO 
SPUD 011 OO SF OLA LA PLN FLA DED UNO 0 SF PLT 


DSF OMASFODADHAPFPORDASPAUMASP?ODVODa ON 
SOND SHOE NN SAMMMOSE ME ODENPOMNOCOMAIN 
eebe2e@eeveeeveeeveeeeveeveeveeeerweeeeeee tee 
NOANOMOFTE MMM MOMOMAOAHDOKBOKNO RSet 


SIN SN SIN tA ert ert ai Set ISIN SIN AN aN SN 


®eet@eoeeeeeeevpteereeeteetrteseeseteeeeee @ 

oocooccooocooq0oco0oo0q0q000o0oe0scoSeo00sco 

oCoooocoocooooco9oooe ooo oo0ooo0oo 

NOWONOMONOMOSSDOOVCOONDSOOCONnNcTco0 

mM OM FH BN age ie ORS NS ES ENCE) 
a x 


eevee ererere eee eee ee eer eo ee ee 8 oe 
lelolelelolelelololelelelololololeloleloe)lelelelololsjolelolea) 
DOWMNN CONN OOCMNDOMNOOMNOOMNN COMNMNDO 
FFP FLAN IVD FF FPO AY FFF PIN NISS SPL 


®eeteeetrtesveeseeereetrteeeresee?eeeevrkeesee?te®@ 
oQgocoocoqco0cq0o90cocooo0ooooco°o0e so o0cooetod 
tt a et tt SA St Seat SSI OOOO 

Fim Te Vine ie a Tate ji RR VN ial cat J 


165 






ie 
% 


J Axor iO rue’ . 4: - ; ji 
Day ae 5 all a a 
fereuy ee * * 


LAP ab bee SiGe oe - 
iy tet} : i) i 


HT? 


mf 
i 
~O 
© 4 
o a i ke ifelp 
. dee 
. ~ a. > 
1) 
. ae] 
=: 
» a 
° Ag eae - 
: r 5 2 Ooo tose 
c won FF 


LV 2 





ALT 


TAS 


DEG 


COOPO Pee OEE AHOON 
DONSOAHIMNSFOMANINMO 
eeetstevrerteen3sproeesee02e28?ekeeree#seteteee8se eee @ 
OW AQ S40 OSS SsaMMNDOasaeaeannqoa 
TOUS UT UESUR SU UsetinaUis Ui Ce eet 


CM) COMLAS MOPFIAMN SH ODDDANMOSASR ONS 
NEMVONONGIN AAA aI G oN Lora 
@eeesecrieee1eseteeet ee @ ®eeeeee21e%#*e%e¢?9e?e¢0 
AAMOFMAFALSMAAMST AGF MI MAF a 
= ' aN 4 nN 


eeeseeseeeeeeeeeeesteee ee ee @ @ 
DADO POTIADDSNS Seat FIOM OWE OT 
OCW FOMMPODFSUIEE NWN POO arin 
PET gee eT maT TTT 


ASONNOESOMNLFAMOONSRADADNMOAD 

NNUMONDDO-DFOODORTNESRONNONO 

ecervoeeseeoereoeree eo ee ese eeo eee ee 

ASO BOC O SSH FOOCO Ss SANNOOO SAAN 
! 


eoeoeveveeoeoeer eer eee ee eee ee ee eo 
WOM FRHODD OS OMINSR MAHOU DONA 
WALA OD NNAODRDONOMINPFONING FONINFOM AY 
ABAARSTNOSNE ST PODTFOPOONE SRNR ST 
LVN PSEA PLA FLA FM PPS PFEMMMSEMS OSL 


NANMMODODOEEODOONNOMDAMMINOM ON FAI 
DFPONANFPRAAMNNDNODOOFVNOAMODOS 
@eeeeoeesveeeseenveeststeresenrteeee7en1t 8 teee?t@¢ @ 
AFADONOTNE SAE NDAONNN SA FOMmnNe 
SNS SSNS AN SN SO SIM SHHAMAaAMan 


FONOWMOMNFOD PFADMNADMANNDO$SANO 


ALA LALO ODO SPaANVSARDMOSONDODNM OM 
SAtPOtPNONNDODO ONO EOFtHMHMNMN 
eoeoeoeevee eee eoee ese eee eee ee ee @ 
OM AAO FOMNANOM ADODANGHOOMASTOW 
ANENAN NANGFINANAN MAMAN MeO 


eeceoevreeoreeve eer eeere sve ee Fe eee ® 
coooocoocecocooc|ecsoooecesoooa 
Soooqcoooooocoeocoooocoooecoeoso 
ODWMOLAWIINVUIN OD QUIN OLALA A LALALALALALA © OL 
COLD OYUN HO OO PM SP FOF OSPRTOFOROSTONO 


eoeoeoer eee ere eee eee eee eee eee @ 
elololelolelolololelololololelelolololelololejlojolo) 
UMNO OWMNOOWMMNODOLANOOMNNOOMNOOWMW 
LEV EDN SP SPP PL VV SPS SPP LVS EVN SPS SPP LOLLY 





















eh yt ae aes 
Biueves Aen tot 
pir tie 4, we S 
mm Ge try Go) 

Yi 


* oe ee ea eee 


4A Aye 


aerate OTe Pir 


a* ° > «6 &8 *** 
es as —<-nWAG ruth bp 
5 ACR f DG jiweeas 
t fin & PI ates 
Sir ent I Gee Sire wr 


cf ACS fu aoe ¢ ae 

+ SeoGme Ie 9 Se esis ve > 
=a ee sowee-e0e.0e® e+ ee) 
COFn (St ben Ano ~ (Ae wONIOK ad 


Pry eee OP) bee OD bee PI tee law. (ey ee £7 hme et whe? 


. 4 ‘ **eéease 
~ SOM “A+ Pe el 
sPor aro wo Pe PY oTe 
c “\ Cay at eo a 


ak 


+ cOoeoV aoe Vie “Ao 
4 . SOAs 
e« ai eee 


= @* *® * ° , « 
[29050 2Coeo] 
sccea clonoocsS 
“COVED ivr Pepe are 
yo, * >t 
— 


WEAPON COEFFICIENTS FOR IDNO 22° 


ALT 


TAS 


DEG 


DO FONVNGOPHMMOHAONDNVDSA™NOPFONDSM SO 
MOM FUE ANANS NM SHO OF DOMAOOMNHOOONR4 
eeoeveevetoe eee eee e ee eer eeoeree eee @ 
NOMOOKNNAHNDOMOTFODODANMODNOMOTO 
eee tle Wn ee a eh a 


DEADONONSOQOOD FHODHAHDNSRDOOONSMNDS 

ONE IO BOO a i st NEO O sol eee 

eeeesoeee#eeosoeecs<«#*#ees28tt?ee%s#e*#et © 6 @ 

ASAONGNGMOMGOOAGMSFOVAGASSNGMS 
' I ett ! | 1 ! 


MOPNTOIANIMOAAM 

ONOMATNOOsORn0 

eoeoeoeveeeoee eee 

lelolelalelojolelelelelelo) 
i ! 


-0. 02 
06 

Ol 
0.14 


eeoee0eeteerteeeee@#enpmteeseeoeeeee#e#t © @e8 @ @ 
FPFEAINEMHOLPOSGLTPVOOADE FVNOMODFPANOAAMH™ Dao 
TEP OASTH EM ONANDOMANAOMAM-MSODOMANNOAaON 
OO) tL OLN OLN OD FPO SPOIL FOO DO ODA OM Mf 
AN AN AN SN AN AN AN SN SN SIN SAMO SAN SN SN AN 


DHOOM OMMSFFHNOONDOSFONDYOOFONANOMO 
OM OWN SNVNAD SIN SALVO FOMNORBONDNEO NSO 
ovoeeeoeeoeoereoeo ero eee eee eee eee ee 8 
FREMTORS FAT SEOTS SOOO ny eating 

s+ = SF A ft A SF Ft St SS OK 


eevveeveerereeoeeeor eevee eereer eee ever 
ADDOMMNDDAMHDODDMONSNNOADONDY OY Lor 
NDDOMONODDODDHOCONVOFNONONE DOG 
THOASFHMEDDHOOSHSMOMNMOROOSINNANODMAO™ 
SINAN ANANSI NO AN AN AN SMONM NM SINAN RAIN 


66 


MOOS OM SPAM MAMODFNONMNANOCANOS NSE 
SAAN OUAMS HS ODI MAO SOP OtMa Age 
eeetee#e?e eeeeee#eeee#2eeg?2e8teee#eeete @ @ @ 
FOF AFOFSPAFAFOFS FSF MMA MAA 
Ce a ee ee ee | 


eeo@esvoeetseeoeeeseee98e8kes2set#ee#eeqe#8gse#e#eeee#eee @ 
ODDT0d0coOoOCoccCeceSeSeseococ0ocoooeo0000oo 
(eleololelelololelolelolololololelololololololelololeleolololec) 
NON DNONANNNNIMAMIINDOODORPODRODOCOM 

= = et st Bt ete AA NNN NN NAN 


Dia te PN ot — “OY ce j 7 r 

Cewnoce “se at) dealt 

eae e@eweeee . . 5 . 

mori. nA Do. a7, Sele —— 
| 


{. tim imi | t way | i—tet » ' 
i "|! ‘ : 
marvin wee oe yiite o> OSGI ~=> oO 
Inmet -ywos MS be Mile Se Gea y 
o«eese ‘ , 4 sees © Oe 9¢°*% 2 9 85 
woo te o. WT - ~ , A Crore. 
j ieryevriree 
* oF & ‘ * + © ee © @2* @& B&O we 
~ od iin ys 9% Vi Os 00% 
i | =“ 2 {Ne . 3h Oe ee 
may 7a |» fie beet | 
' j ’ } ' ' 
— j . > « 1-9 om 
+> -~=s>=668 
7 . * se @ 6.4 
_ Ccocogd 
: yr) F #9 
* * ‘ . ® 6 
—" »~*+ 
A - A] 
= > es 
— + °® ) = 
i= ~ j » de 
: oO — vc - 
e : ® . ’ U * . « 
af Wi ' ot ee t > y e@' 
= ™ 
a *» 8 » *@ 
‘ - yoo 
a ’ oe & ~ 2oe 
- o*/ r » 
- a lll o- 
his . 
. ™ 
4 
-_— « * ~ 
< : : 
+ 





ALT 


TAS 


DEG 


PAI COLLIN AID COM AD SIO FON SAOSIN DEE DAE DOMOM SOM OLSON 
DODMINO Sa OIANANMMNOSLAMONM™ SNOW FOND OO +FaHDNMO 
ecoerereoer oreo eee eee rere ere oe eee rere eeeeeoseesn 
MNAOMAOAHOnBORFOMOSFNONOOOCOOCORONRNTOOCORnOROONO 
PN NE a a ae ane ae Piety HCG Apa 


WOM OSASODMOAMSSFAMMMAMS NOW MAHOM OW AE UP OOOO 

FO WC DIN FAN MSNA ODO LACDO LAD BM CUM oO ee 

ecoveeeere eo eee ere eee eee eee eee eee erro eee ses 
U | I 


s 
eoeveeeeoeoeoe eer eo oe ee eee eee ere eereeeere ee eee 
MASA FANOPSAMUNODCOODDLS SNUNOPAMIE NE DOCOTHOHIDOONMO 
AM-ODON MAM MPILOORMN lH la tAOmM OFtasttia NINNAMN 
ane ' rar. ! tt 


ODAOFPAVUOMMAMNAMNODDOOHNANNNO ODDO 

NAMNNOTOSCOCOHAOVNAEMNNENOOOCOCOONOMNDO 

eoeeveeeoeoeeo eee eee eee eee eee ee ee 
| 


0.55 
00 


eoevoereeereoere ee eevee ee ereoeen ere eee eee eveseerees ee 
MADBOOOSTF OD MMM DWN OO Sa PH NUAAMONOONDO PPS DANO 
DO DFID FR FM AD OANA DIN OO FD DSM SHAADOADAMM OAD 
NANNNNNANSNNNNNNNNN SSRN NNNNNNN SSRI SION 


SN SANNOPS ENE ESO STONNNOODNDROTORORAHNE OTN OO 
MMOMAMNSM™ NODDOOMDAONDMHORAFNONDNOSMOCOREMMAMO+0O 
eee0eeseees2s8e@eseseesgee#se#e?eeteeetee#et e@eeeneee#?8tg 8 @t ©8888 @ 
Baa eee re) DONS ON OA OY OOOO OnOnGws 

onl = dd ed ad ed ced ed et a et ee St et et et tO ested et St HHO St IIA 


eeoeoeeoveevere ee er eee eevee ee ere eee eee ereereeve ee oe 
ODA SFOOFIENE MADSFOODMNSFANFOMIANATHAO FOONOKEMANMNOO 
MPOFODOREOSFAMAMNAOOMNAMNMHADDODNNONSAMODODOM +S 
MOPFDINADQOM MAPA NDOOEDDSEHONASMOMNNAEADEODNOMAS 
NANNNN OM SANSAINNNN NNN SSS NNNNNNNN Nett NN aINAINAAN 


SLA ONE DINO FRDHODNMOFINSFIADNONSOAHOINSFODY OM 
ANDODMNSEOMODMOMONDODNAERNONVSFOABHOGOORE SMAHH000 
eeeeeeeesreesvsee#errpeeensevee*#ese3«ee*ev#ve<e8svsesevsvse888?8f 8 @8@ee8e8ee8see8e 8@ 8 @ @ 
Pa aa gee oe (SOO ONS SEN OOS oS onerous 

S00 tO St Ot RS St tt St St St St IN St tt et NS at tN 


eeeoereeeree eevee ee eee een eeree @ 
ocooococoocooooseocooococooeoeeoSsds 
elolelolololelololelolololelelololelolololejeojolojololololelololoia) 

LALLA VLA OUD NS O COWMWMMN GIN OD OOWMNNOONOOWMS 
FNANAMN SNMIMNUMNONMN SAMS OF OSOS SOE MSM 


0. 
0. 
0. 
0. 
0. 
OQ. 
0. 
0. 
00. 


elolelolelslolelelolololejelololel=jelololelololojlelojojejeleoloelelelolelojololo) 
AWS OWN COMM OOMN COUN COMM COMNOOMINOOMMN COMM OO 
SV VV LN 0) 09 00.) SSS SN VEN 09.099 09 0 SS SPL LUN 099 09 O99 SPS SL 


4 Hye hy oe 
a 7 —_—C é -— 
s LJ e 6 J ¢ 
- v -eueoyT9 : 
: i Je ee 





am iL ain oat fer 
1 Prime Gey Se A ee 
® «vw e-e'e ©) 6 8:8 


* s ¢* @& « 
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. =.» ‘v 
- ap ae ae 


- ts hd Lad ber CS Ge “OIE 
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cS 
~ a, t 
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ne) 
. ® 
/ or 
re Vite G 
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ALT 


TAS 


DEG 


AODOGCOFNMODMONONW 
DO FHM AM ND Fin FO 
eeoeoeao eee et? eeee e@ 
FMOOCOCHFOnRORORANO 
tt ! ! 


DAYMAMO MO Da SO VO tes 
DAMA NO SBONNAMOO 
eeoeeoereereeoer eevee 
SAG Ee lk 

treed | t t 


eeeoeveve eevee ee ee 
AMMO OTNONOOF-IN YS 
MOWD SLANT NO MOM 
NINES MONS OO-EArO 
NN Sst Sa sistas aed an 


OWODTINDODODNSFNSHON 
ODAAHFANDOSFOOOM 
@eeeoe<este?#028?8?8 @ ee @ 
PMOAPTEODOCOTTOWM 
SIN SI SM SAIS OU OE 


eeeeeese#28 8ee 8 8 68 @ 
ONAMAANOMAMAN OD 
DOMDMNMOtaHEMNAAS 
MANE MOAN DORADO 
ONG wt tt at et Stet St SION 


DODMNNSEODONUMLIAN 
DPrAMOMOHADODOAINS 
eeoee3«e2xtet ee @ee @ 
ODMOAFPERODADROMNSINNY 
SN SAM SIM SO SPAN 


@eeeeoe¢e*ee#8e@ee ee 
ooooocoocoooosoo 
lelelelelelelele\elelelolelo) 
NOOCNMOMOOMNNNOCS 
TOTPOSFODHONOC CHEN 

a ft et x 


169 


seae@¢aeoen ee ee 
a I Se a o/h 
mr oy Sime 
DoE A Om Us Fre 
ye ee bee 


| ee ae ¥ 


7 





APPENDIX C 


This appendix is a listing of the output from the cockpit 
of Ores: aircraft recorded at the aac Hee ezanen 
was released. 

The following is a brief description of the parameter 


headings as they appear on the listing. 


TAS = true airspeed in knots 


TH = true heading in degrees relative to true north 


5 
Ww 


wind direction in degrees true 


A 


wind speed in knots 

GT = ground track in degrees true 

GS = ground speed in knots 

RA = release angle in degrees 

VZ = vertical velocity in knots x 10 

VSEP = vertical separation in feet 

DA = depression angle (search radar) in degrees 
SR = slant range to target in feet 


TOF 


time of fall in seconds 
HIT = his distance from target in feet 


AZ = hit azimuth in clock code 


170 


5 KICwessA 



















wloos efi moti Juqsuo SH to snldald & al 


7 . a l 
froqasw ott sapient ent 358 bgybtopst Jistatia® 


* - 
, . 7 * oO 34159%8— Satud & 8 Bee bd 


j - 
7 
42] ang 6a  tTanqys vena 2 


= so 


ajout nk beequais oun = 


> 


oY gokbaod setae 


ai 7 ’ 4 » ' 44 “ 
b . " ies —~ Soa 


ag513 wit notsset!bh balw = 


co ot Soeqe Balw see | 
sso71geb a) doses Semone 
sol ol bese bauer Se 


a 
osrgsh ol Signs seaaiem 


a: ! 4io0is¥ Ieolstsyv = 


nolijeyeges, lesiazey 


sboeose ot [ist ae 
tes , mort somedeke etd 


(> okt doventam See 





VSEP DA SR TOF HIT AZ 


VZ 


TH WOIR WKTS GT GS RA 


TAS 


eceeoevereeeoeeo oe core ereveer ee ee ee ener seoee ee eee eee 
elolelojlelojeleloelclelolelololclololelelelejlolelololelo)lcleololelelelelelola\elelelolo) 
ANMMOMOM MOMO MMNODOCOMOOMOMMMOOOCOODOOMOMO0oO 
THINNMON NANO DDNNNNNOANONONNNON ST SARANN AN RANAO 


foe) tn) = SS Sees 1 cael eee Peele tet ro 
eeeeeeeeeeeteeteeeee#eeeteteseeettet eget 8eeeeesee 
SODCGCODOCODOOOSCSC OOOO OCOCOCOCCOOCOCOCOOCOCOCOOCOCOOOCOCOOO 
DROSTPFOO NOD NFTONnFtAOPrOVNIoreDONnNDOFornodynno 
a a x a Stn “NS Oa NN Ss NSO 
ta) 


DADO-ODOLTOMNMS DOE NNDOOFDONDOLFONEMARDONSROARHONO 
eoeereoeeoeo eee eevee eee eee eo ree eee ee ee eee eee ereee 


ADHRODA DO WOM OUND MO OOM ODDDODS HOO OF anne 
= we 


eoeeevereeo ec eee ereoeeeereeeeeoee ee ee eee eoere ee eee eee 
lelololelolelololololojolelololololololojelelolelelejelolelelelolelejlolololelelelolo)\=) 
OANMNODOFRFPAMNOOE DO HaMOGDNSIM VOVOEMMNDSAOOWM ORO 
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APPENDIX D 


tne Sep pentiae contains the output which en: ae the 
experimental data with the FORTRAN version of the ballistics 
algorithm using the old set of mach, drag, and weapon coef- 
ficients. The weapon used for this experiment was the MK-76 
MOD-5 25 pound practice bomb. 

The coefficients are all assigned in the DECODE sub- 
routine (see Ref. 3 for further explanation of these 
coefficients). Both sets of coefficients, old and new, 


are summarized here for the reader's convenience. 


Weapon Coefficients 


Old Coefficients New Coefficients 
IREF = 2 IREF = 2 

IBOTH = 1 IBOTH = 1 

ITYPE = -l ITYPE = -1 

DMAX = 3.0 DMAX = 6.0 
CFORM1 = 0.0039077 CFORM1 = .1064453 
CFORM2 = 0.0 CFORM2 = 0.0 
DKG1 = 0.0063648 DKG1 = -.0043918 
DKG2 = 0.0 DKG2 = 0.0 

DM1 = 0.0 DM1 = -.270 

DM2 = 0.0 DM2 = -.270 
VMUZ = 0.0 VMUZ = 0.0 

FN = 0.0 FN = 0.0 

VE = 0.0 VE = 0.0 

SL = 0.0 SE = O20 

DS = 0.0 DS: = 0.0 


182 
















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APPENDIX E 


This appendix compares the experimental data with the 
ballistics algorithm using the new coefficients for drag, 


mach, and weapon type. 


ng7 


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LIST OF REFERENCES 


Grumman Aerospace Corporation, Navy Model A-6E Aircraft, 


Integrated Weapon System Theory, NAVAIR 01-85ADF-2- 
* 10.1, September, 1973. 


de Linhares, T. P. B., Distributed Microcomputer Air- 
borne Tactical System, M.S. Thesis, Naval Post- 
graduate School, Monterey, California, December 
£O75. 


Jupin, J. A., The Ballistics Processor of a Multiple 
Processor Airborne Tactical System, M.S. Thesis, 


Naval Postgraduate School, Monterey, California, 
June 1975. 


McCracken, W. L., Design Study of an Avionics Navigation 
Microcomputer, A.E. Thesis, Naval Postgraduate 


School, Monterey, California, June 1974, 


Tactical Manual Ballistics Tables, NAVAIR 01-1C-1T-1, 
October 1973. 


246 





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INITIAL DISTRIBUTION LIST 


Defense Documentation Center 
Cameron Station 
Alexandria, Virginia 22314 


Library, Code 0212 
Naval Postgraduate School 
Monterey, California 93940 


Department Chairman, Code 52 
Department of Computer Science 
Naval Postgraduate School 
Monterey, California 93940 


Associate Professor Uno R. Kodres, Code 52Kr 
Department of Computer Science 

Naval Postgraduate School 

Monterey, California 93940 


LT John T. Ertlschweiger, USN 


10424 Greenford Drive 
San Diego, California 92126 


247 


No. Copies 


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Thesis | 
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Verification and 
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Verification and feasibility study of 


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