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NPS  ARCHIVE 
1997.09 
WONNACOTT,  W. 


NAVAL  POSTGRADUATE  SCHOOL 
Monterey,  California 


THESIS 


MODELING  IN  THE  DESIGN  AND  ANALYSIS  OF  A 
HIT-TO-KILL  ROCKET  GUIDANCE  KIT 

by 

W.  Mark  Wonnacott 


September,  1997 


Thesis  Advisor: 
Second  Reader: 


Conrad  F.  Newberry 
Louis  V.  Schmidt 


Thesis 
W757 


Approved  for  public  release;  distribution  is  unlimited. 


DUDLEY  KNOX  LIBF 

NAVAL  POSTGRADUATE  SCHOOL 


DUDLEY  KNOX  LIBRARY 

NAVAL  POSTGRADUATE  SCHOOL 

MONTEREY,  CA  93943-5101 


REPORT  DOCUMENTATION  PAGE 


Form  Approved 
OMBNo.  0704-0188 


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1.  AGENCY  USE  ONLY  (Leave  blank) 


2.    REPORT  DATE 

September  1997 


3.  REPORT  TYPE  AND  DATES  COVERED 


Master's  Thesis 


4.  title  and  subtitle  MODELING  IN  THE  DESIGN  AND 

ANALYSIS  OF  A  HIT-TO-KILL  ROCKET  GUIDANCE  KIT 


6.    AUTHOR(S) 

Wonnacott,  W.  Mark 


5.  FUNDING  NUMBERS 


7.  PERFORMING  ORGANIZATION  NAME(S)  AND  ADDRESS(ES) 

Naval  Postgraduate  School 
Monterey,  CA  93943-5000 


8.  PERFORMING  ORGANIZATION 
REPORT  NUMBER 


9.  SPONSORING  /  MONITORING  AGENCY  NAME(S)  AND  ADDRESS(ES) 


10.  SPONSORING  /  MONITORING 
AGENCY  REPORT  NUMBER 


11.  SUPPLEMENTARY  NOTES 

The  views  expressed  in  this  thesis  are  those  of  the  author  and  do  not  reflect  the  official  policy  or 
position  of  the  Department  of  Defense  or  the  U.S.  Government. 


12a.  DISTRIBUTION  /  AVAILABILITY  STATEMENT 

Approved  for  public  release;  distribution  unlimited. 


12b.  DISTRIBUTION  CODE 


13.  ABSTRACT  (maximum  200  words) 

This  thesis  presents  several  computer  models  used  in  the  design  and  analysis  of  a  Hit-to-Kill  Rocket 
Guidance  Kit  (HRGK).  The  HRGK — proposed  as  an  inexpensive  add-on  kit — has  the  potential  of  converting 
unguided  2.75"  diameter  rockets  into  precision  weapons  against  non-tank  targets.  A  Naval  Postgraduate  School 
design  team  recently  participated  in  a  nation-wide  graduate  student  competition  for  the  design  of  such  a  kit.  The 
design  and  analysis  process  led  the  author  to  develop  and  use  various  computer  models  and  simulations.  This 
thesis  documents  three  distinct  types  of  computer  models  found  useful  in  the  design. 

The  first,  operational  effectiveness  modeling,  established  the  cost  effectiveness  of  the  NPS  HRGK.  The 
second  was  related  to  the  preliminary  sizing  of  various  design  aspects — ensuring  the  proper  flow-down  of  system 
requirements  into  design  specifications.  The  third  was  a  six-degree  of  freedom  (6DOF)  simulation,  developed  to 
perform  detailed  analyses  of  the  HRGK's  performance. 

Although  the  models  presented  in  this  thesis  pertain  to  the  HRGK,  the  basic  principles  apply  to  the  design 
or  evaluation  of  other  missile  systems,  and  this  thesis  provides  general  insights  regarding  the  benefits  and 
limitations  of  computer  modeling  in  missile  design. 


14.  SUBJECT  TERMS 

Rocket  Guidance  Kit,  Missile  Design,  Missile  Modeling  and  Simulation 


16.  NUMBER  OF  PAGES 

106 


16.  PRICE  CODE 


17.  SECURITY  CLASSIFICATION 
OF  REPORT 

Unclassified 


18.  SECURITY  CLASSIFICATION 
OF  THIS  PAGE 

Unclassified 


19.  SECURITY  CLASSIFICATION 
OF  ABSTRACT 

Unclassified 


20. 


LIMITATION  OF 
ABSTRACT 


UL 


NSN  7540-01-280-5500 


Standard  Form  298  (Rev.  2-89) 


u 


Approved  for  public  release;  distribution  is  unlimited 


MODELING  IN  THE  DESIGN  AND  ANALYSIS  OF  A 
HIT-TO-KILL  ROCKET  GUIDANCE  KIT 

W.  Mark  Wonnacott 
B.S.M.E.,  Brigham  Young  University,  1989 

Submitted  in  partial  fulfillment  of  the 
requirements  for  the  degree  of 


MASTER  OF  SCIENCE  IN  AERONAUTICAL  ENGINEERING 


from  the 


NAVAL  POSTGRADUATE  SCHOOL 
September   1997 


DUDLEY  KNOX  LIBRARY 

NAVAL  POSTGRADUATE  SCHOOL 

MONTEREY,  CA  93943-5101 


DUDLEY  KMOX  LIBRARY 

^ADUATE  SCHOOL 

,-5101 


ABSTRACT 


This  thesis  presents  several  computer  models  used  in  the  design  and  analysis  of  a 
Hit-to-Kill  Rocket  Guidance  Kit  (HRGK).  The  HRGK — proposed  as  an  inexpensive  add- 
on kit — has  the  potential  of  converting  unguided  2.75"  diameter  rockets  into  precision 
weapons  against  non-tank  targets.  A  Naval  Postgraduate  School  design  team  recently 
participated  in  a  nation-wide  graduate  student  competition  for  the  design  of  such  a  kit. 
The  design  and  analysis  process  led  the  author  to  develop  and  use  various  computer 
models  and  simulations.  This  thesis  documents  three  distinct  types  of  computer  models 
found  useful  in  the  design. 

The  first,  operational  effectiveness  modeling,  established  the  cost  effectiveness  of 
the  NPS  HRGK.  The  second  was  related  to  the  preliminary  sizing  of  various  design 
aspects — ensuring  the  proper  flow-down  of  system  requirements  into  design 
specifications.  The  third  was  a  six-degree  of  freedom  (6DOF)  simulation,  developed  to 
perform  detailed  analyses  of  the  HRGK's  performance. 

Although  the  models  presented  in  this  thesis  pertain  to  the  HRGK,  the  basic 
principles  apply  to  the  design  or  evaluation  of  other  missile  systems,  and  this  thesis 
provides  general  insights  regarding  the  benefits  and  limitations  of  computer  modeling  in 
missile  design. 


VI 


TABLE  OF  CONTENTS 

I.  INTRODUCTION 1 

A.  BACKGROUND 1 

B.  NPS  HRGK  DESIGN 2 

C.  THESIS  SCOPE  AND  OUTLINE 4 

H.  COST  OF  OPERATIONAL  EFFECTIVENESS  ANALYSES 7 

A.  SIMPLIFYING  ASSUMPTIONS 7 

B.  COST  PER  KILL 10 

C.  KILLS  PER  SORTIE 15 

D.  COST  PER  MISSION 15 

E.  OBSERVATIONS 20 

IH.  PRELIMINARY  SIZING  ANALYSES 23 

A.  LENGTH,  WEIGHT,  AND  CENTER  OF  GRAVITY 23 

B.  MANEUVERABILITY 26 

C.  SEEKER  FIELD  OF  VIEW 39 

D.  COMMENTS  AND  OBSERVATIONS 46 

IV.  SIX-DEGREE  OF  FREEDOM  SIMULATION 51 

A.  SIMULATION  OVERVIEW 51 

B.  SIMULATION  COMPONENTS 54 

C.  EQUATIONS  OF  MOTION 55 

D.  NPS  HRGK  RESULTS 58 

E.  OBSERVATIONS 60 

V.  CONCLUSIONS  AND  RECOMMENDATIONS 61 

APPENDIX  A.  MISSILE  AERODYNAMICS  NOMENCLATURE 63 

APPENDIX  B.  NOTES  ON  MATLAB  AND  SIMULINK 65 

APPENDIX  C  MATLAB  CODE  LISTINGS. 67 

LIST  OF  REFERENCES 89 

INITIAL  DISTRIBUTION  LIST 91 


vu 


Vlll 


LIST  OF  FIGURES 

1.  Sample  Normal  Plane  View  for  a  12  mil,  One-Dimensional  Dispersion  at  5  km 
Downrange 3 

2.  Schematic  View  of  the  NPS  HRGK  Design 5 

3 .  Projected  Area  for  Light  Armored  Vehicles 9 

4.  Single-Shot  PK  and  Number  of  Rockets  Required  per  Target  as  a  Function  of 
CEP 13 

5 .  Relative  Cost  per  Kill  against  Various  Targets 14 

6.  AH-64  Apache  Attack  Helicopter  Sorties  Required  to  Achieve  Mission  Objective.  . .  19 

7 .  Number  of  Weapons  Fired  to  Achieve  Mission  Objective 19 

8 .  Number  of  Targets  Killed  by  Category,  Weapon,  and  Loadout  Option 20 

9 .  Physical  Size  and  Center  of  Gravity  Design  Space 25 

10.  Graphical  Representation  of  Maneuverability  Model  Inputs 27 

1 1 .  Closed-Form  Solution  for  Maneuverability  Specifications  with  a  Stationary  Target. .  29 

12.  Moving  Target  Scenario  for  Maneuverability  Analysis 31 

13.  Maximum  Turn  Rate  Required  to  Hit  Target  (95%  Probability) 33 

14.  Minimum  Turn  Radius  Required  to  Hit  Target  (95%  Probability) 33 

1 5 .  Plane- View,  Free-Body  Diagram  for  Steady-Turn  Maneuver 35 

16.  Required  Thrust  and  Angle  of  Attack  for  Maximum  Turn  Rate  or  Minimum  Turn 
Radius 38 

17.  Required  Field  of  View.  Results  for  95%  Probability  of  Target  Staying  in  Field  of 
View  with  Worst  Case  Target  Motion 40 

18.  Target  Distribution  Model 42 

19.  Illustrative  Case  for  Probabilistic  Seeker  Field  of  View  Analysis 43 

20.  Possible  Target  Locations 44 

2 1 .  Alternative  Method  for  Computing  Target  Distribution  Area  within  Seeker  Field  of 
View  Limits 45 

22.  Probability  Density  Functions,  (a)  Rocket  cross- track  error,  (b)  Target  cross-track 
position,  and  (c)  Probability  of  acquisition 45 

23.  Probability  of  Target  in  Field  of  View  for  Maximum  Launch  Range  (6  km) 47 

24.  Sensitivity  Analysis  of  the  Steady-Turn  Maneuver  Model  for  Thruster  Sizing 49 

25.  HRGK  6DOF  Simulation  Overall  Block  Diagram 52 

26.  Detailed  View  of  the  Rocket  Dynamics  Block 53 

27.  Detailed  View  of  the  Parameters  and  Coefficients  Block 53 

28.  Future  6DOF  Architecture 54 

29.  HRGK-Equipped  Rocket  Ballistic  Trajectories 59 

30.  Sustained  and  Pulsed  Thruster  Maneuvers 59 


IX 


LIST  OF  TABLES 

1 .  Measures  of  Merit  and  Their  Associated  Objectives  and  Requirements 3 

2.  COEA  Target  Set  with  Assumed  Dimensions,  Desired  PK,  and  P^ 9 

3.  Single-Shot  Probability  of  Kill  Computer  Model  Inputs 11 

4.  Cost  per  Kill 14 

5.  Kills  per  AH-64  Apache  Sortie  against  Light  Armor  Targets 15 

6.  Weight,  Length,  and  Center  of  Gravity  Model  Inputs 24 

7.  Maneuverability  Model  Inputs 26 

8.  Steady-Turn  Maneuver  Computer  Model  Inputs 36 

9.  Seeker  Field  of  View  Analysis  Inputs 41 


XI 


Xll 


ACKNOWLEDGMENT 

I  am  grateful  for  the  support  of  my  thesis  advisor,  Dr.  Conrad  Newberry,  and  the 
second  reader,  Dr.  Lou  Schmidt,  for  their  help  and  guidance.  Dr.  Robert  Ball,  Aeronautics 
and  Astronautics  Department,  and  Major  Vince  Tobin  and  Captain  Pat  Mason,  U.S.  Army, 
were  also  very  helpful  with  the  cost  of  operational  effectiveness  portion  of  this  thesis.  I  am 
very  appreciative  of  the  sacrifices  of  Major  Boaz  Pomerantz,  Israeli  Air  Force;  Major 
Silvino  L.  Silva,  Brazilian  Air  Force;  and  Lieutenant  Nigel  A.  Nurse,  U.S.  Navy  who 
served  on  the  NPS  Hit-to-Kill  Rocket  Guidance  Kit  Design  Team. 

I  am  indeed  indebted  to  my  management  at  the  Naval  Air  Warfare  Center  Weapons 
Division  and  the  NAWCWPNS  Long-Term  Fellowship  Committee  for  their  continued 
support  of  my  studies.  I  wish  to  specifically  acknowledge  Tomma  Bersie,  Doug  Savage, 
John  Freeman,  Lee  Gilbert,  Paul  Homer,  and  Ron  Derr  for  allowing  me  this  time  at  the 
Naval  Postgraduate  School.  Special  thanks  also  go  to  Ted  Fincher  and  Ken  Morton  for 
inspiring  me  to  pursue  a  graduate  education. 

But,  my  most  tender  and  heart-felt  gratitude  is  reserved  for  my  sweet  wife  Diane 
and  our  dear  children:  Andrea,  Jared,  Nathan,  Melissa,  Marsha,  Jennifer,  and  Kayla. 
Without  their  love  and  sacrifice,  this  thesis  would  not  have  been  possible. 


xiu 


XIV 


I.     INTRODUCTION 

During  the  first  half  of  the  1996-1997  Naval  Postgraduate  School  (NPS)  academic 
year,  a  team  of  aeronautical  engineering  graduate  students  was  assembled  to  perform  the 
conceptual  design  of  a  hit-to-kill  guided  rocket  kit  (HRGK).  (The  kit  attaches  to  existing 
unguided  rockets  to  give  them  precision  strike  capability.)  As  part  of  the  design  team,  the 
author  developed  several  computer  codes  or  models  to  facilitate  the  design  and  analysis  of 
the  HRGK.  Since  the  completion  of  the  design  project,  many  of  these  codes  have  been 
expanded  or  refined. 

This  thesis  documents  the  theory  used  in  the  development  of  the  improved 
computer  models.  It  also  describes  the  process  of  using  the  models  and  presents  the 
modeling  results  for  the  NPS  HRGK  design.  These  descriptions  provide  an  example  of  the 
type  of  modeling  that  can  be  used  in  the  design  and  analysis  of  missiles. 

This  introductory  chapter  provides  background  information  about  the  HRGK  and 
pertinent  details  of  the  NPS  HRGK  design.  It  also  outlines  the  remaining  chapters  of  the 
thesis. 

A.    BACKGROUND 

In  the  mid  1990s,  US  Army  Aviation  identified  the  need  for  a  low  cost,  precision 
kill  weapon  system  for  use  against  soft  or  light-armor  targets  (such  as  trucks,  armored 
personnel  carriers,  artillery,  air  defense  systems,  command  posts,  amphibious  landing 
vehicles,  or  patrol  boats).  The  objective  of  such  a  system  would  be  to  minimize  the  cost  per 
kill  for  these  types  of  targets.  The  2.75-inch  rocket  (Mark-66)  with  a  unitary,  high- 
explosive  (HE)  warhead  (M151)  and  a  point  detonation  fuze  (M423)  was  subsequently 
identified  as  a  promising  candidate  for  this  low-cost  mission. 

These  unguided  rockets  cost  only  about  $  1 ,000  per  unit,  and  large  inventories  are 
already  available  to  the  military  services.  The  2.75-inch  rockets  are  launched  from 
helicopters,  fixed-wing  aircraft,  surface  combat  vehicles,  and  naval  vessels.  U.S.  Army 
Helicopters  carry  the  rockets  in  either  a  7-  or  19-tube  launcher  pod  as  part  of  the  Hydra-70 
weapon  system.  The  Hydra-70  system  includes  the  weapon  (rocket,  fuze,  and  warhead), 
launch  pod,  and  an  armament  management  system.  With  this  system,  the  rockets  can  be 
fired  in  single-shot  mode  or  in  various  sized  ripple  or  salvo  shots.  The  rockets  can  carry 
various  warheads  or  payloads  (including  HE,  kinetic  energy  flechettes,  multi-purpose 


submunitions,  illumination  flare,  marking  smoke,  and  chaff)  with  various  fuzing  options 
(including  remotely  set  timed  burst  and  super-quick  or  delayed  contact  fuzes). 

The  Mark-66  rocket  with  warhead  and  fuze  is  spin  stabilized  to  provide  some 
degree  of  accuracy.  The  rocket  uses  a  fluted  single  throat  nozzle  to  achieve  approximately  a 
10  Hz  spin  by  the  time  it  exits  the  launcher.  By  motor  burnout,  the  spin  rate  has  increased 
to  approximately  35  Hz.  After  motor  burn  out,  the  rocket's  wrap-around  fins  de-spin  the 
projectile  and  cause  a  significant  spin  rate  (15-20  Hz)  in  the  reverse  direction.  The  spinning 
characteristics  of  the  rocket  played  a  significant  role  in  the  NPS  design  as  will  be  discussed 
in  the  following  section. 

The  baseline  rocket  configuration  chosen  for  the  kit  (HE  warhead  and  super-quick 
contact  fuze)  is  an  area  suppression  weapon.  The  unguided  rocket  has  a  total  12 
milliradians  (mill),  one-dimensional  dispersion,  meaning  that  for  every  thousand  meters  of 
down  range  fly  out  a  12  m  error  results  (1  a  or  68%).  For  example,  a  nominal  5  km  range 
launch  would  result  in  a  normal-plane1,  radial  miss  of  up  to  60  m  (5  x  12) — with  39% 
probability  (see  Figure  1).  This  level  of  accuracy  is  far  from  that  required  for  single-shot, 
precision  kills.  To  achieve  point  target  accuracy,  the  rocket  needs  to  be  fitted  with  some 
type  of  guidance  and  control  kit.  The  conceptual  NPS  HRGK,  described  in  the  next 
section,  was  designed  to  serve  this  purpose. 

B.        NPS  HRGK  DESIGN 

The  NPS  HRGK  was  designed  to  meet  certain  requirements.  The  following 
subsections  summarize  these  requirements  and  the  pertinent  aspects  of  the  NPS  HRGK 
design. 

1.         HRGK  Design  Specifications 

The  NPS  HRGK  design  requirements  were  prepared  by  the  American  Institute  of 
Aeronautics  and  Astronautics  (AIAA)  Missile  Systems  Technical  Committee  and  were  used 
for  the  1996/1997  Graduate  Team  Missile  Design  Competition  sponsored  by  the  AIAA  and 
Northrop  Grumman  Corporation  [Ref.  1].  Table  1  summarizes  the  HRGK's  measures  of 
merit  with  their  associated  objectives  and  pertinent  requirements. 


1  For  the  purposes  of  this  thesis,  all  accuracies  are  in  terms  of  the  plane  normal  to  the  rocket  trajectory 
rather  than  in  the  ground  plane.  This  convention  is  typical  for  guided  missiles. 


Elevation  Error 
(68%  of  hits) 


±60  m 


Cross  Track  Error 
(68%  of  hits) 

CE:  circular  error 

CEP:  circular  error  probable 


±60  m- 


90%  CE 
(2.15  a) 


50%  CE- 
(1.18  a) 


60mR(1a) 
(39%  CE) 


-CEP 


Figure    1.      Sample   Normal    Plane    View  for   a    12    mil,    One-Dimensional 
Dispersion  at  5  km  Downrange.  [(12  m/  1000  m)5000  m  =  60  m  (la)]. 


Table  1 .   Measures  of  Merit  and  Their  Associated  Objectives  and  Requirements. 


Measure   of  Merit  and   Objective 

Specific    Requirements 

Cost 

Maximize  Units  Bought 

•  $10,000  Avg.  Unit  Cost  (5,000  Unit  LRIF) 

•  Maximum  $1  Billion  Total  5  Year  Production  Buy 

Current    Systems    Computability 

Maximize  Operability  & 
Reduce  Life  Cycle  Cost 

•  Existing  Rocket,  Fuze,  Warhead,  &  Launch  Pod 

—  72  inch  total  system  maximum  length 

—  loaded  pod  center  of  gravity  between  lugs 

•  Laser  Designator  Compatibility 

Accuracy 

Provide  Hit-to-Kill  Accuracy 

•  0.5  m  CEP 

•  60  mph  target  speed 

Weight 

Maximize  Kits  Deliverable  via  Airlift 

•     30  pound  Total  System  Maximum  Weight 

Maximum   Range 

Maximize  Platform  Standoff  Survivability 

•     6  km  (or  More)  Maximum  Range 
(Maneuverability  at  Maximum  Range) 

Minimum    Range 

Maximize  Launch  Acceptable  Region 

•     1  km  (or  Less)  Minimum  Range 

(Guidance,  Fuzing,  and  Launch  Transients) 

Adverse  Weather  Performance 

Maximize  Operability  and  Combat  Flexibility 

•     99%  Worldwide  Weather  Capable 

Time  to  Target 

Minimize  Time  (Improve  Survivability) 

•     16  seconds  or  Less  to  5  km  Range 

'  Low  Rate  Initial  Production 


2.         NPS   Design   Highlights 

The  following  paragraphs  briefly  describe  the  major  features  of  the  NPS  HRGK 
design.  The  design  team's  proposal  gives  a  more  detailed  description  of  the  design  [Ref. 
2]. 

The  HRGK  attaches  over  the  fuze  at  the  front  end  of  a  2.75  inch  rocket.  It  utilizes  a 
strapped-down,  four-quadrant  silicon  detector  for  semi-active  laser  spot  homing.  The 
HRGK  employs  three  hot  gas  reaction  jets  (thrusters)  to  control  the  rocket' s  attitude  and 
trajectory.  A  solid  propellant  burning  in  an  insulated  pressure  chamber  generates  the 
thruster  gas.  The  kit  uses  a  look-up  table  scheme  to  control  the  thrusters  based  solely  on  the 
seeker  outputs.  Fast-cycle  solenoid  valves  open  and  close  the  "on-off"  thrusters.  The 
thruster  nozzles  are  inter-spaced  between  three  fixed,  wrap-around  canards.  The  canards 
provide  increased  maneuverability  (by  reducing  the  rocket's  static  stability)  and  help 
constrain  any  jet  and  air  stream  interaction  effects.  The  canards  also  contribute  significantly 
to  the  HRGK's  roll  stability. 

Thrust  and  radial  needle  rollers  effectively  isolate  the  NPS  HRGK  from  the 
spinning  motion  of  the  rocket.  Simulations  show  the  kit's  roll  rate  to  be  less  than  9  degrees 
per  second  after  the  first  seconds  of  flight.  The  kit's  low  spin  rate  and  the  simplistic  pursuit 
navigation  scheme  eliminate  the  need  for  inertial  sensors  and  greatly  reduce  the  kit's  signal 
processing  requirements. 

As  shown  in  Figure  2,  the  basic  kit  is  19.0  inches  long.  It  has  a  diameter  of  2.75 
inches  (with  canards  wrapped  in)  and  weighs  7.0  pounds.  The  NPS  HRGK  attaches  to  the 
2.75  inch  rocket  between  the  M151  high-explosive  warhead  and  M423  point  detonation 
fuze.  The  all-up  round  is  68.75  inches  long  and  weighs  30  lb.  The  guided  rocket  is 
compatible  with  the  Hydra-70  rocket  system. 

C.        THESIS  SCOPE  AND  OUTLINE 

The  NPS  HRGK  design  was  achieved  and  verified  with  the  aid  of  several  original 
computer  codes  and  models.  The  codes  and  models  developed  by  the  author  and  described 
in  this  thesis  fall  into  the  following  three  major  categories: 

•  cost  of  operational  effectiveness  analyses, 

•  preliminary  sizing  analyses,  and 

•  six-degree  of  freedom  simulation. 


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The  following  three  chapters  provide  descriptions  of  the  modeling  processes  in  the 
respective  categories.  Each  of  these  chapters  (1)  includes  a  description  of  the  modeling 
methodology,  (2)  gives  example  results  specific  to  the  NPS  HRGK  design,  and  (3) 
discusses  observations  regarding  the  general  use  of  the  models.  The  final  thesis  chapter 
summarizes  these  conclusions  and  provides  recommendations  for  additional  work  with  the 
computer  models.  Appendices  that  contain  the  computer  code  listings  and  additional 
background  have  also  been  included.  The  appendices  include  a  brief  description  of  missile 
aerodynamics  nomenclature  (Appendix  A)  and  a  discussion  concerning  the  use  of  the 
computer  software  Matlab®  and  Simulink®  (Appendix  B). 

In  general,  this  thesis  is  not  intended  to  be  read  from  cover  to  cover.  Instead  this 
thesis  is  offered  as  a  reference  to  possible  analysis  types  and  modeling  methods  for  use  in 
future  conceptual  design  of  missile  systems. 


®  Matlab  and  Simulink  are  registered  trademarks  of  The  Math  Works,  Inc. 


II.     COST  OF  OPERATIONAL  EFFECTIVENESS  ANALYSES 

Cost  of  operational  effectiveness  analysis  (COEA)  must  be  an  important  part  of  the 
design  process.  First,  COEAs  provide  justification  for  the  design  based  on  operational 
effectiveness  cost  savings.  But,  just  as  importantly,  COEA-type  analyses  provide  the 
designer  with  focus  and  insight  into  the  truly  important  aspects  of  the  design.  Basic  COEA 
models  with  many  simplifying  assumptions  can  be  used  early  in  the  process  to  focus 
design  decisions.  As  the  design  matures,  refined  models  can  be  used  to  evaluate  major 
design  iterations. 

The  COEA  modeling  developed  for  the  NPS  HRGK  provided  the  justification  for  a 
HRGK-like  system.  The  overall  objective  of  the  HRGK  was  to  minimize  the  total  cost  per 
kill  against  appropriate  targets.  From  the  analysis,  guided  rockets  showed  a  decreased  cost 
per  target  kill  and  increased  target  kills  per  sortie  when  compared  to  alternative  weapon 
systems.  Additional  modeling  showed  that  guided  rockets  resulted  in  a  reduction  of  sorties 
and  weapon  costs  required  to  achieve  a  specific  mission  objective.  The  final  NPS  HRGK 
was  an  attempt  to  provide  the  lowest  possible  cost  per  kill  design. 

The  COEA  modeling  discussed  in  this  chapter  provides  several  different  levels  of 
results.  The  low  level  results  of  the  modeling  include  single-shot  probabilities  of  kill  and 
the  number  of  weapons  required  per  kill  as  a  function  of  circular  error  probable  (CEP). 
These  values  are  then  used  to  derive  the  cost  per  kill  and  the  sorties  per  mission  results. 
The  number  of  weapons  required  per  kill  results  also  provide  some  additional  insights  that 
was  critical  to  the  NPS  HRGK  design. 

The  following  section  describes  the  simplifying  assumptions  used  in  the  NPS 
HRGK  COEA  models.  Subsequent  sections  provide  more  detailed  descriptions  of  the 
model  methodologies  and  the  results.  The  chapter  ends  with  a  discussion  of  some 
observations  concerning  the  use  of  COEA  modeling  in  design.  (Listings  of  the  computer 
codes  used  in  the  NPS  HRGK  COEA  are  included  in  Appendix  C.) 

A.        SIMPLIFYING   ASSUMPTIONS 

Several  simplifying  assumptions  were  made  to  facilitate  the  early  NPS  HRGK 
COEA  modeling.  These  assumptions  included  defining  guided  and  unguided  weapon 
trajectories  and  costs,  establishing  a  target  set  with  desired  probabilities  of  kill  (PK)  for  each 


target,  and  estimating  each  target's  vulnerability  in  terms  of  the  conditional  probability  of 
kill  given  a  hit  (P^w). 

1.  Weapon  Accuracies 

The  accuracies  for  the  unguided  rocket  were  taken  as  a  function  of  launch  range  and 
were  based  on  a  12  milliradian  accuracy  [Ref.  1].  As  a  baseline  for  the  comparison,  a  mid- 
range,  3-km  shot  was  used  with  a  resulting  42.4  m  normal-plane  CEP  (36  m  1- 
dimensional,  standard  deviation  multiplied  by  1.1774 — see  Figure  1,  for  example).  For  the 
guided  rockets,  three  different  normal-plane  CEP  accuracies  was  assumed  for  comparison 
purposes — 0.5  m,  1.0  m,  1.5  m,  and  3.0  m. 

For  both  the  unguided  and  guided  rockets,  a  20°  terminal  dive  angle  (relative  to  the 
horizontal)  was  assumed  as  a  best  estimate1 .  The  shallow  dive  caused  an  increase  in  the 
rocket's  ground  plane  down-track  error  but  did  not  affect  the  cross-track  error.  These 
effects  were  accounted  for  implicitly  in  determining  the  probability  of  hitting  the  target. 

For  simple  comparison  purposes,  the  AGM-114  laser  Hellfire's  accuracy  was 
assumed  to  be  such  that  if  it  was  shot  at  a  target  it  would  achieve  the  desired  killed. 

2.  Target  Set 

The  effectiveness  analysis  used  four  representative  targets  (mobile  air  defense  unit, 
armored  personnel  carrier,  support  vehicle,  and  patrol  boat).  These  targets  were  each 
assumed  to  be  simple  blocks  of  appropriate  length,  width,  and  height  dimensions  (see 
Figure  3).  Table  2  lists  the  four  targets  with  their  associated  dimensions  [Ref.  3,  4,  5,  and 
6].  The  dimensions  are  representative  for  targets  found  in  the  references.  Table  2  also  lists 
a  desired  PK  for  each  of  the  targets.  These  desired  probabilities  were  assumed,  based  on  the 
typical  importance  of  the  target. 

The  probability  of  each  target  being  hit  (PH)  was  computed  as  the  joint  probability 
of  the  rocket  being  within  both  the  top-to-bottom  and  the  side-to-side  dimensions  of  the 
target.  The  projected  dimensions  of  the  targets  (normal  to  the  flight  trajectory)  were 
computed  for  both  a  head-on  and  a  broad-side  attack.  The  two  attack  approaches  gave  the 
extremes  in  the  target's  presented  area  as  shown  in  Figure  3.  The  Pw's  for  the  two  extreme 
cases  were  computed  and  then  equal-weight  averaged  as  an  estimate  of  the  PH  for  uniformly 
distributed  attack  approaches  (0°-360°). 


'  Later  analysis  showed  that  unguided  rockets  may  have  even  more  shallow  dive  angles,  making  them 
even  more  inaccurate  in  the  ground-plane. 


1 

2m 

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^j\9.4m2\ 

-^    2.5  m     |__ 

'".■ 

15.0  m2 

J""^5: 

5.5  m 

Figure  3.    Projected  Area  for  Light  Armored  Vehicles  (head-on, 
45°,  and  broad-side  attacks  with  20°  dive  angle). 


Table  2.  COEA  Target  Set  with  Assumed  Dimensions,  Desired  PK,  and  P^. 

Target    Description 

Dimensions 
L  x  W  x  H 

Desired  PK 

P  KIH 

Mobile  Air  Defense  Unit  (Light  Armor) 

5  x  2.5  x  2  m 

95% 

90% 

Armored  Personnel  Carrier  (APC) 

5  x  2.5  x  2  m 

60% 

80% 

Support  Vehicle  (Large  Truck) 

8  x  2.5  x  2.5  m 

50% 

100% 

Patrol  Boat 

14  x  4.5  x  1.5  m 

75% 

70% 

3.  Target  Vulnerabilities 

For  each  target  a  "cookie-cutter"  model  [Ref.  7]  was  used  in  determining  the  single- 
shot  PK  (PKss).  With  the  cookie  cutter  model,  the  target  suffers  a  kill  (according  to  an 
assumed  P^)  if  and  only  if  it  is  actually  hit.  "Near  misses"  do  not  provide  any  PK.  Thus 
the  value  for  PK  was  the  simple  product  of  P^  and  PH  .  The  assumed  values  of  P^H  are 
best  estimates  and  are  shown  in  the  last  column  of  Table  2.  These  high  values  of  PKm 
reflect  the  facts  that  (1)  the  targets  were  relatively  soft,  (2)  the  rocket  is  actually  impacting 
the  target  to  score  a  hit,  and  (3)  the  desired  type  of  kill  (firepower,  mobility,  catastrophic, 
etc.),  which  was  not  specified,  may  only  require  minor  damage  to  the  target. 

4.  Weapon  Costs  and  Additional  PK's 

The  per  unit  cost  of  the  guided  rocket  concepts  was  assumed  to  be  $10,000 — the 
upper  threshold  cost  from  the  AJAA  design  contest,  and  the  unguided,  unitary  warhead 
round  (M151)  was  assumed  to  cost  $1,000  [Ref.  1].  A  multipurpose  submunitions 
(MPSM)  round  was  assumed  to  cost  $2,000  and  have  a  10%  PK  against  non-tank  targets. 
The  Hellfire  missile  was  assumed  to  have  a  per  unit  cost  of  $40,000  [Ref.  8]  and  a  single- 
shot  kill  capability  against  all  four  representative  targets  (PK  -  80%  for  tanks  and  PK  = 
100%  for  all  others). 


5.         Additional   Assumptions 

Additionally,  the  assumption  was  made  that  all  weapons  functioned  during  launch, 
flyout,  and  fuzing;  with  100%  reliability.  The  aimpoint  was  assumed  to  be  the  centroid  of 
the  target  block  without  any  bias  errors  in  designation  or  flyout.  It  was  also  assumed  that 
all  rocket  accuracy  errors  had  a  bivariate  normal  distribution  with  independent,  equi- 
variance  (a2)  distributions  along  any  two  orthogonal  axes  normal  to  the  flyout. 
Furthermore,  the  errors  were  assumed  independent  from  rocket  to  rocket.  Thus,  the  CEP 
was  1.1774  a  and  the  PK  for  n  multiple  rockets  was  1  -  (l-PKsingieshot)D- 

The  simplifying  assumptions  discussed  in  this  section  are  reasonably  valid  under 
many  conditions.  The  last  section  of  this  chapter  (II.E.  Observations)  includes  a  brief 
discussion  on  the  affect  of  some  of  the  assumptions. 

B.        COST  PER  KILL 

The  overall  objective  of  the  HRGK  was  to  rninimize  the  total  cost  per  kill  against 
appropriate  targets.  To  determine  the  cost  per  kill,  the  probability  of  kill  per  single  weapon 
had  to  be  established  using  the  assumptions  discussed  above.  This  single-shot  probability 
of  kill,  PKss,  with  the  additional  assumptions  above,  could  then  be  used  to  determine  the 
number  of  weapons  required  to  achieve  the  desired  PK.  From  the  number  of  weapons 
required  per  kill  and  the  assumed  cost  per  weapon,  the  estimated  cost  per  kill  could  then  be 
established.  The  process  for  determining  the  cost  per  kill  and  the  results  for  the  HRGK 
analysis  are  discussed  in  the  following  subsections. 

1.         Methodology 

The  methodology  for  determining  the  cost  per  kill  involves  determining  the  PKss  for 
each  weapon  against  each  target  and  then  determining  the  number  of  weapons  required  to 
achieve  the  desired  PK.  A  computer  program  was  written  in  Matlab  to  compute  the  PKss 
and  number  of  weapons  required  per  kill  as  a  function  of  CEP.  This  program,  which  also 
computes  a  relative  cost  per  kill  tradeoff  plot,  is  described  in  the  following  paragraphs. 

a.  Model  Inputs 

Only  a  small  number  of  inputs  are  required  for  the  single-shot  probability  of 
kill  computer  model.  These  inputs  can  be  divided  into  descriptions  of  the  target  and  of  the 
weapon  trajectory.  Table  3  lists  these  two  types  of  required  inputs. 


10 


Table  3.    Single-Shot  Probability  of  Kill  Computer  Model  Inputs. 


Input   Name 

Input    Description 

Target 

L,  H,  and  W 

Effective  Target  Length,  Height,  and  Width  (unit  of  length) 

Pkh 

Probability  of  Kill  given  a  Hit  within  the  Effective  Target  Dimensions 

PkD 

Desired  Probability  of  Kill  for  the  Target 

Weapon  Trajectory 

dive 

Dive  Angle  Relative  to  Horizontal  (degrees) — 9 

CEP  min 

Minimum  CEP  to  Be  Used  in  Computations  (unit  of  length) 

CEP  max 

Maximum  CEP  to  Be  Used  in  Computations  (unit  of  length) 

b.         Single-Shot  Probability   of  Kill 

The  first  step  in  estimating  the  single-shot  probability  of  kill  is  to  establish 
the  end-on  and  broad-side  projected  edges  of  the  target  normal  to  the  weapon  trajectory. 
The  side-to-side  dimensions  for  the  two  cases  are  simply  X,  =W  and  X2  =  L,  for  the  end- 
on  and  broad-side  cases,  respectively.  The  top-to-bottom  dimensions  are  given  by  the 
following  equations: 


K  =  //cos0  +  Lsin0 


(end-on) 


Y2  =  tfcos0  +  Wsin0 


(broad-side) 


Next,   the  probability   of  hit,   PH,   was   determined  by  estimating   the 

probability  of  the  weapon  striking  within  the  projected  edges  of  the  target — the  joint 

X  Y 

probability  of  being  within  ±—  and±—  of  the  aimpoint,  where  i-  1  or  2  depending  on 

the  approach  case.  Given  the  assumed,  independent  bivariate  nature  of  the  weapon  errors, 
the  joint  probability  could  be  computed  as  the  product  of  the  two — cross-track  (X- 
direction)  and  elevation  (y-direction) — probabilities.  The  erf  function  in  Matlab  is 
defined  as  the  following: 

P  =  -%=\e-,2dt, 

VtJo 

so  the  joint  probability  described  above  is  given  by  the  following  Matlab  expression 
where  the  argument  to  erf  is  normalized  by  V2 : 


11 


X:    .Y\  JXI     1 


PH\  ±-±,±-t   =erf  -±-f=-  -erf 


K 


2  '     2  J  i,  2  V2crJ  1,2  V2<T 

and  where  i  =  1  or  2  and  cr  is  the  CEP  I  \  All  A.  The  total  Pw  is  then  the  equal  weighted 
average  of  />„  and  P„  .  In  other  words,  PH  =  (PH  +  PH  )/2. 

With  the  probability  of  hit,  PH,  and  the  input  P^H,  the  single-shot 
probability  of  kill  is  estimated  as  the  product  of  the  two. 

P     =  P      ■  P 

c.         Number  of  Weapons  per  Kill 

With  the  Pfc5  and  the  assumption  that  flyout  errors  for  each  weapon  shot  are 
unbiased  and  independent,  the  number  of  weapons  required  to  achieve  a  desired  PK  can  be 
calculated. 

The  probability  of  the  target  surviving  a  single  shot  is  given  as  the 
complement  of  PKss,  \-PKss.  The  probability  of  surviving  n,  independent  shots  is  that 
quantity  raised  to  the  nth  power,  (l-PKssT-  Thus,  the  probability  of  kill  for  n,  independent 
shots  is  the  complement  to  the  latter  probability  of  survival,  namely: 

The  number  of  shots,  n,  to  achieve  a  desired  PK  can  be  determined  by 
solving  the  above  equation  for  n  as  is  shown  in  the  following  equation. 

ln(l-PK) 


n  = 


HI-PkssY 


d.         Accuracy  and  Relative  Cost  Trade-off  Chart 

Knowing  the  number  of  weapons,  n,  required  to  achieve  a  desired 
probability  of  kill  (PK)  against  a  specific  target,  the  relative  weapon  costs  to  give  equal  cost 
per  kill  can  be  computed.  For  example,  if  two  weapons  (A)  with  a  0.5  m  CEP  or  six 
weapons  (B)  with  a  2.0  m  CEP  would  be  required  to  achieve  a  desired  PK,  then  if  the  cost 
of  weapon  B  was  one-third  the  cost  or  less  of  weapon  A ,  weapon  B  would  have  a  lower 
cost  per  kill  against  the  specified  target. 

In  the  computer  model,  weapon  costs  are  normalized  relative  to  the  cost  of 
the  minimum  input  CEP,  CEP_min,  weapon.  Also,  the  number  of  weapons  required  to 
achieve  a  desired  PK  are  considered  only  as  integer  values. 


12 


e.  Cost  per  Kill  Calculations 

The  absolute  cost  per  kill  can  easily  be  computed  from  the  results  of  the 
computer  model  described  above.  The  cost  per  kill  for  a  specified  target  and  weapon  is 
simply  the  integer  number  of  weapons  required  to  achieve  the  desired  PK  against  that  target 
multiplied  by  the  assumed  cost  of  that  weapon. 

2.         Results 

The  results  of  the  computer  model  described  above  are  shown  for  the  targets  and 
weapons  described  above  in  section  II.A.  Simplifying  Assumptions.  The  results  include 
both  the  PKss  and  the  cost  per  kill. 

a.         Direct  Model  Outputs 

The  combined  single  shot  probability  of  kill  and  the  number  of  weapons 
required  to  achieve  a  desired  PK  are  shown  as  a  function  of  CEP  in  Figure  4.  Various 
assumed  accuracies  for  the  unguided  rockets  are  also  marked  on  the  plot  with  their 
corresponding  launch  ranges. 


Figure  4.   Single-Shot  PK  and  Number  of  Rockets  Required  per  Target  as  a  Function  of  CEP. 
(PKss  curves  slope  down  from  upper  left;  number  of  rockets  curves  slope  up  to  upper  right.) 


13 


Figure  5  shows  the  relative  weapon  costs  for  equal  cost  per  kill  as  a 
function  of  CEP.  In  this  chart  the  weapon  costs  are  relative  to  the  0.5  m  CEP  weapon  cost. 
From  the  chart  it  appears  that  a  weapon  with  a  2  m  CEP  (B)  would  have  an  improved  cost 
per  kill  against  all  targets  if  the  weapon  cost  were  less  than  33%  of  the  0.5  m  CEP  weapon 
(A).  Weapon  C,  at  half  the  cost  of  A,  has  the  highest  cost  per  kill  at  4  m  CEP. 


2  3  4 

Normal-Plane  Circular  Error  Probable  (m) 


Figure  5.  Relative  Cost  per  Kill  against  Various  Targets.  Lettered  design  points  (A,  B, 
and  C)  are  shown  for  illustration  only. 

b.         Cost  per  Kill  Results 

From  the  number  of  weapons  required  per  target  kill,  the  costs  per  kill  for 
each  of  the  weapons  can  be  computed.  The  results  for  the  HRGK  analysis  based  on  the 
single  assumed  guided  rocket  cost  are  shown  in  Table  4. 

Table  4.  Cost  per  Kill. 


Unguided 
Rocket 

Guided 

Rocket 

Target 

0.5    m 

1.0    m 

1.5    m 

3.0    m 

Hellfire 

Mobile  Air  Defense  Unit 

$2,219k 

$20k 

$20k 

$40k 

$120k 

$40k 

Light  Armor 

$764k 

$10k 

$10k 

$20k 

$50k 

$40k 

Large  Truck 

$294k 

$10k 

$10k 

$10k 

$20k 

$40k 

Patrol  Boat 

$467k 

$20k 

$20k 

$20k 

$40k 

$40k 

14 


C.         KILLS  PER  SORTIE 

A  shortcoming  of  the  cost  per  kill  analysis  previously  presented  is  illustrated  in  the 
following  example.  A  weapon  with  a  2.0  m  CEP  (such  as  B  in  Figure  5)  and  a  quarter  of 
the  cost  of  a  0.5  m  CEP  weapon  has  a  lower  cost  per  kill  than  the  0.5  m  CEP  weapon 
(such  as  A  in  Figure  5).  However,  it  would  take  three  times  as  many  of  the  less  accurate 
(but  inexpensive)  weapons  to  achieve  the  same  number  of  kills  against  mobile  air  defense 
targets.  The  number  of  weapons  available  for  a  given  mission  would  be  limited  by  the 
number  of  launch  platforms  sorties.  The  importance  of  kills  per  sortie  is  completely  ignored 
in  the  cost  per  kill  data.  A  kills  per  sortie  analysis  helps  to  illustrate  the  importance  of 
precision  in  strike  warfare. 

The  kills  per  sortie  analysis  performed  for  the  HRGK  was  based  on  AH-64  Apache 
attack  helicopter  sorties.  The  helicopters  were  assumed  to  have  either  pure  or  mixed 
weapon  payloads  or  loadouts.  The  pure  loadouts  were  either  16  Hellfires  or  76  rockets 
(four  19-tube  rocket  launchers).  The  mixed  loadouts  combined  eight  Hellfires  with  a 
limited  number  of  unitary  warhead  (M151)  rockets  (guided  or  unguided)  and  additional 
MPSM-equipped  rockets.  The  number  of  kills  per  sortie  were  calculated  based  on  the  PKss 
data  in  Figure  4  and  the  assumed  P^'s  from  the  sub-section  II.A.4.  Weapon  Costs  and 
Additional  PK's.  The  results  are  tabulated  in  Table  5  for  the  light  armor  target  case. 

Table  5.   Kills  per  AH-64  Apache  Sortie  against  Light  Armor  Targets. 


Weapon 

Maximum    Load 

Mixed  Load 

Hellfire 

16  Hellfires 

16  Kills 

8  Hellfires  &  38  MPSM 

9.90  Kills 

HRGK  (0.5m) 

76  HRGKs 

60.7  Kills 

8  Hellfires,  24  MPSM,  &  14  HRGKs 

20.4  Kills 

HRGK  (1.0m) 

76  HRGKs 

52.5  Kills 

8  Hellfires,  24  MPSM,  &  14  HRGKs 

18.9  Kills 

HRGK  (1.5m) 

76  HRGKs 

38.7  Kills 

8  Hellfires,  24  MPSM,  &  14  HRGKs 

16.3  Kills 

HRGK  (3.0m) 

76  HRGKs 

15.1  Kills 

8  Hellfires,  24  MPSM,  &  14  HRGKs 

12.0  Kills 

Unguided 

76  Rockets 

0.09  Kills 

8  Hellfires,  24  MPSM,  &  14  Rockets 

9.22  Kills 

D. 


COST  PER  MISSION 


A  final  type  of  modeling  performed  by  the  author  for  the  NPS  HRGK  design 
combines  the  attributes  of  the  cost  per  kill  and  the  kills  per  sortie  analyses.  This  analysis 
assesses  the  cost  (in  terms  of  sorties  and  cost  of  weapons  fired)  to  achieve  a  mission 


15 


objective.  The  following  subsections  outline  the  analysis  and  present  the  results  for  the 
HRGK. 

1.  Mission    Definition 

The  cost  per  mission  analysis  was  based  on  a  specific  assumed  mission  scenario. 
This  scenario  defined  the  target  matrix  with  a  set  priority  on  the  target  types  as  well  as  a 
mission  objective.  The  scenario  also  defined  the  attacking  helicopter  loadouts.  The  assumed 
scenario  in  defined  in  the  following  paragraphs. 

a.  Target  Matrix 

The  assumed  target  was  an  assembly  area  with  the  following  targets  types 
listed  by  priority  and  with  the  target  quantities  in  parentheses: 

•  Mobile  Air  Defense  Units  (8) 

•  Main  Battle  Tanks  (40) 

•  Light  Armored  Vehicles  ( 1 50) 

•  Support  Vehicles  (250) 

The  mission  objective  was  to  destroy  half  of  the  vehicles  in  each  combatant  category. 

b.  Weapon   Loadouts 

The  assumed  initial  strike  force  was  a  flight  of  five  AH-64  Apache  attack 
helicopters.  As  additional  helicopters  were  needed  they  were  added  to  the  attack  force  one 
at  a  time.  Each  Apache  carried  a  nominal  load  of  eight  Hellfires  and  two  19-rocket  launcher 
pods.  Each  pod  (which  could  be  divided  into  zones)  was  loaded  with  twelve  M261  MPSM- 
equipped  rockets  and  seven  guided  or  unguided  M151  unitary  warhead  equipped-rockets. 
[Ref.  9] 

The  limited  number  of  HRGK-equipped  weapons  (14)  per  helicopter  was 
based  on  the  assumption  that  only  six  to  eight  laser-designated  flyouts  could  be  made  from 
a  firing  position  before  the  helicopters  would  be  forced  to  relocate.  With  eight  Hellfires  and 
14  HRGKs,  three  firing  positions  would  be  required  for  the  attack.  This  was  considered  to 
be  the  upper  limit  for  a  deep  strike  mission.  The  MPSM  rockets  would  not  require  laser 
designation,  yet  they  would  still  be  effective  against  clusters  of  light  armor,  material,  or 
personnel.  [Ref.  9] 

2.  Example  Calculation 

The  following  provides  an  example  of  the  process  used  to  model  the  mission 
engagement.  The  1 .0  m  CEP  case  is  illustrated. 


16 


a.  First  Five  Helicopters 

The  weapons  loadout  of  the  first  five  helicopters  consists  of  the  following: 

•  40  Hellfire  Missiles  (each  at  a  tank) 

•  40  x  0.8  =  32  tank  kills 

•  70  HRGK  Rockets  (16  at  air  defense  units— 2  per  unit,  54  at  light  armor) 

•  1 6 — 8  air  defense  unit  kills  with  PK  =  95%) 

•  54  x  0.6909  =  37.31  light  armor  kills 

•  120  MPSM  Rockets  (120  x  0.1  =  12  kills  divided  between  non-tank  targets) 

•  6  light  armor  kills 

•  6  support  vehicle  kills 

In  summary,  after  five  helicopters,  8  air  defense  units  are  killed  (exceeding  objective),  32 
tanks  are  killed  (exceeding  objective),  43.31  light  armor  vehicles  are  killed  (31.69  short  of 
the  objective),  and  6  support  vehicles  are  killed  (exceeding  objective). 

b.  Sixth    Helicopter 

The  following  delineates  the  use  of  the  sixth  helicopter's  weapons: 

•  8  Hellfire  Missiles  (each  at  a  tank) 

•  8  x  0.8  =  6.4  tank  kills 

•  14  HRGK  Rockets  (each  at  light  armor) 

•  14  x  0.6909  =  9.67  light  armor  kills 

•  24  MPSM  Rockets  (24  x  0.1  =  2.4  kills  divided  between  non-tank  targets) 

•  1 .2  light  armor  kills 

•  1 .2  support  vehicle  kills 

In  summary,  after  the  sixth  helicopter,  8  air  defense  units  are  killed  (exceeding  objective), 
38.4  tanks  are  killed  (exceeding  objective),  54.18  light  armor  vehicles  are  killed  (20.82 
short  of  the  objective),  and  7.2  support  vehicles  are  killed  (exceeding  objective). 

c.  Seventh    Helicopter 

The  following  is  for  the  seventh  helicopter: 

•  8  Hellfire  Missiles  (2 — each  at  a  tank,  6  at  light  armor) 

•  2x0.8  =  1.6  tank  kills 

•  6  x  1 .0  =  6  light  armor  kills 

•  14  HRGK  Rockets  (each  at  light  armor) 

•  14  x  0.6909  =  9.67  light  armor  kills 

•      24  MPSM  Rockets  (24  x  0. 1  =  2.4  kills  divided  between  non-tank  targets) 

•  1 .2  light  armor  kills 

•  1 .2  support  vehicle  kills 

After  the  seventh  helicopter,  8  air  defense  units  are  killed  (exceeding  objective),  40  tanks 
are  killed  (exceeding  objective),  71.05  light  armor  vehicles  are  killed  (3.95  short  of  the 
objective),  and  8.4  support  vehicles  are  killed  (exceeding  objective). 


17 


d.  Subsequent    Helicopters 

For  any  subsequent  helicopters  the  weapons  would  be  used  as  follows: 

•  8  Hellfire  Missiles  (each  at  light  armor) 

•  8  x  1 .0  =  8  light  armor  kills 

•  14  HRGK  Rockets  (each  at  light  armor) 

•  14  x  0.6909  =  9.67  light  armor  kills 

•  24  MPSM  Rockets  (24  x  0. 1  =  2.4  kills  divided  between  non-tank  targets) 

•  1.2  light  armor  kills 

•  1.2  support  vehicle  kills 

This  gives  18.87  light  armor  kills  per  sortie;  therefore,  the  number  of  sorties,  after  the 
seventh,  to  reach  the  objective  of  75  light  armor  kills  is  3.95/18.87  =  0.21,  and  the  total 
number  of  sorties  required  for  the  mission  is  7.21. 

e.  Summary 

After  7.21  sorties  the  following  number  of  weapons  have  been  fired  with 
the  indicated  assumed  weapon  cost: 

•  57.67  Hellfires  ($2.3M) 

•  100.9  HRGKs  ($1.0M) 

•  173.0  MPSMs  ($0.35M) 

for  a  total  weapon  cost  of  $3.66M.  The  total  targets  killed  were: 

•  8  air  defense  units 

•  40  tanks 

•  75  light  armor  vehicles 

•  8.65  support  vehicles. 

3.         Results 

Figures  6,  7,  and  8  illustrate  the  results  for  the  HRGK  mission  analysis.  Figure  6 
shows  the  number  of  sorties  required  to  achieve  the  mission  objective.  With  an  assumed 
cost  per  sortie,  these  numbers  could  be  directly  applied  to  the  weapon's  cost  per  kill.  The 
inclusion  of  a  helicopter  attrition  rate  would  also  make  the  cost  per  kill  more  realistic.  Note 
that  the  1 .0  m  CEP  rocket  does  not  require  many  more  sorties  than  does  the  0.5  m  CEP. 

Figure  7  shows  the  number  of  weapons  fired  during  the  mission  for  each  of  the 
different  weapon  cases.  The  total  weapon  cost  in  millions  of  dollars  is  also  indicated  on  the 
graph. 

Figure  8  summarizes  which  weapons  killed  which  targets.  It  is  interesting  to  note 
that  with  accurate  HRGKs,  the  more  expensive  Hellfire  missile  could  be  used  almost 
exclusively  on  tanks.  On  the  other  hand,  when  unguided  rockets  were  used,  nearly  all  the 


18 


target  kills  were  from  the  Hellfire,  and  the  unguided  unitary  (M151)  rocket  only  registered 
0.242  light  armor  and  0.474  support  vehicle  kills. 


Sorties  Required 

Ol                     O                    Ol 

—^^m 

^    CD 

JkJkb 

■ 



0.5m             1.0m              1.5m           3.0m        unguided 
Rocket  Type 

Figure  6.   AH-64  Apache  Attack  Helicopter  Sorties  Required  to  Achieve  Mission  Objective. 


0.5m 


1.0m  1.5m  3.0m 

Rocket  Type 


unguided 


Figure  7.  Number  of  Weapons  Fired  to  Achieve  Mission  Objective.  Total  weapon  costs 
indicated  on  graph  in  millions  of  dollars.. 


19 


0.5m 


1.0m 


1.5m 


3.0m 


unguided 


Rocket  Type 


Figure  8.   Number  of  Targets  Killed  by  Category,  Weapon,  and  Loadout  Option. 


E. 


OBSERVATIONS 


Simple  COEA  type  modeling  is  essential  to  the  design  process.  Besides  assuring 
that  the  design  can  be  cost  effective,  the  results  from  early  analyses  provide  focus  for  the 
design  team.  COEA  results  can  show  general  trends  that  might  otherwise  go  unnoticed. 
COEA  type  modeling  continues  to  be  useful  throughout  the  design  process  as  refined- 
model  analyses  can  help  in  verifying  design  decisions. 

An  example  of  how  unexpected  information  can  come  from  a  COEA  study  occurred 
in  the  NPS  HRGK  analysis.  Figure  4  (page  13)  shows  a  "knee  in  the  curve"  around  1  m 
CEP.  That  is  the  approximate  CEP  where  the  P^/s  begin  to  drop  off  significantly  and  the 
number  of  weapons  needed  per  kill  begins  a  steep  increase.  From  the  figure,  it  appears  that 
weapons  achieving  a  CEP  much  smaller  than  1  m  are  not  needed  against  the  assumed 
targets.  Therefore,  as  a  result  of  the  COEA  modeling,  designs  that  achieve  tighter  than  1  m 
CEPs  at  significantly  higher  costs  are  eliminated  from  consideration. 

On  the  other  hand,  COEA  results  can  be  very  sensitive  to  underlying  assumptions. 
The  possibility  of  invalid  assumptions  can  make  COEA  modeling  too  unreliable  for 
confident  design  decisions.  For  example,  small  changes  in  the  low-level  target  set 
assumptions  can  result  in  important  changes  in  PKss  values.  These  values,  in  turn,  will 


20 


impact  sortie  rates.  Likewise,  high-level  assumptions  like  the  mission  definition  used  in  the 
sorties  per  mission  analysis  will  also  impact  the  final  COEA  results.  The  validity  of  the 
assumptions  involved  in  both  low-  and  high-level  models  must  be  continuously  assessed. 

Specific  examples  of  COEA  sensitivities  to  underlying  assumptions  in  the  NPS 
HRGK  models  include  two  opposing  possibilities.  If  the  assumed  target  set  were  expanded 
to  include  a  smaller  point  target,  the  "knee"  of  the  P^  curve  would  shift  to  a  smaller  CEP. 
Then  a  more  expensive,  0.5  m  CEP  weapon  might  become  more  cost  effective  then  less 
expensive,  1  m  CEP  rockets.  Conversely,  if  the  centered,  unbiased  aimpoint  assumption 
(discussed  on  page  10)  were  not  valid,  the  general  cost  effectiveness  advantage  of  the  high 
precision  rocket  would  become  smaller,  and  guided  rockets  with  slightly  larger  dispersion 
could  become  more  cost  efficient. 

Ideally,  COEA  studies  will  include  sensitivity  analyses  to  determine  the  range  of 
results  depending  on  changes  in  the  assumed  inputs.  With  these  sensitivities  in  mind,  the 
designer  can  use  the  COEA  models  as  powerful  design  tools. 


21 


22 


III.     PRELIMINARY  SIZING  ANALYSES 

Several  aspects  of  any  design  require  preliminary  sizing.  For  the  NPS  HRGK 
design,  the  physical  dimensions  (length,  weight,  and  center  of  gravity),  the  thruster,  and 
the  seeker's  field  of  view  each  needed  to  be  sized  early  in  the  design  process. 

The  author  developed  several  computer  codes  to  allow  the  sizing  of  the  kit,  its 
control  mechanisms,  and  the  seeker's  field  of  view.  These  codes  each  use  top-level 
requirements  to  generate  lower-level  design  specifications.  This  flow  down  of 
specifications  helps  to  assure  that  the  final  design  meets  the  system  requirements. 

The  following  sections  describe  the  methodology  used  in  developing  the  computer 
models  and  describe  some  results  specific  to  the  NPS  HRGK  design.  The  chapter  ends 
with  observations  regarding  the  models  and  the  flow-down  of  design  requirements. 

A.    LENGTH,  WEIGHT,  AND  CENTER  OF  GRAVITY 

The  HRGK  design  weight  and  center  of  gravity  affects  the  overall  center  of  gravity 
of  the  guided  rocket.  The  guided  rocket's  center  of  gravity  is  a  critical  design  consideration 
in  that  it  affects  the  rocket's  stability  in  flight.  Additionally,  the  guided  rocket's  center  of 
gravity  must  be  such  that  a  fully  loaded  launcher  pod's  center  of  gravity  is  kept  between  the 
pod's  two  mounting  lugs.  Furthermore,  the  length  and  weight  of  the  HRGK  are 
constrained  by  limits  specified  for  the  overall  length  and  weight  of  the  guided  rocket. 

A  Matlab  code  was  written  by  the  author  to  help  visualize  the  relationships 
between  the  HRGK  size  and  the  rocket's  center  of  gravity.  The  code  also  maps  the  useable 
design  space  bounded  by  the  maximum  length  and  weight  as  well  as  the  launcher's  center 
of  gravity  constraints.  This  code  is  briefly  described  and  some  results  are  presented  in  the 
following  subsections.  The  code  listing  is  included  in  Appendix  C. 

1.         Methodology 

The  computer  code  used  for  the  length,  weight,  and  center  of  gravity  analysis  is 

based  on  the  simple  equation  defining  a  composite  center  of  gravity,  XCG,  based  on  the 
component  weights,  Wt,  and  component  centers  of  gravity,  xCG  .  Namely, 


XCg  ~ 


23 


a.  Inputs 

The  user-defined  parameters  in  the  code  are  either  design  or  constraint 
parameters  as  listed  in  Table  6.  Additional  model  parameters  needed  for  the  model  include 
the  weight  and  center  of  gravity  locations  for  the  live  and  fired  rocket  and  for  the  empty 
launcher.  The  length  of  the  rocket  and  the  location  of  the  pod's  front  lug  are  also  required. 
These  parameters  for  the  Hydra-70  rocket  system  were  available  from  [Ref.  10  and  11]. 

Table  6.  Weight,  Length,  and  Center  of  Gravity  Model  Inputs. 


Input   Name 

Input    Description 

Design  Space  Parameter 

Wkl 

Range  of  kit  weights  (evenly-spaced  vector)  (lb) 

Lkl 

Specific  kit  lengths  to  be  plotted  (vector)  (in) 

Dkl 

Specific  kit  densities  to  be  plotted  (vector)  (lbm/in3) 

Cgkl 

Assumed  kit  center  of  gravity  (fraction  of  kit  length  back  from  nose) 

dia 

Assumed  kit  diameter  (in) 

Constraint  Parameters 

Lmax 

Maximum  total  length  (in) 

Wmax 

Maximum  total  weight  (lb) 

CgL 

Pod  forward  Cg  limit  (fraction  of  lug  spacing  back  from  front  lug) 

The  kit's  length  is  defined  as  the  length  extending  forward  from  the  tip  of 
the  rocket's  fuze  to  the  forward  tip  of  the  kit.  The  kit's  center  of  gravity  is  based  on  that 
length;  therefore,  if  the  kit's  center  of  gravity  were  behind  the  tip  of  the  fuze,  the  value  for 
Cgkl  would  be  greater  than  unity. 

b.         Outputs 

The  computer  code  creates  a  plot  of  the  rocket's  center  of  gravity1  as  a 
function  of  guidance  kit  weight.  The  design  space  is  marked  with  various  curves  indicating 
user  specified  guidance  kit  densities  and  lengths.  With  any  two  parameters  defined  (kit 
weight,  density,  length,  or  guided  rocket  center  of  gravity),  the  other  two  parameters  are 
fixed  on  the  plot.  The  usable  design  space  is  bounded  by  the  maximum  total  length  and 
weight  as  well  as  the  loaded  launcher's  center  of  gravity  constraints.  The  code  generates  the 
described  plots  for  both  the  live  and  the  fired  rocket  cases. 


Measured  in  calibers  (missile  diameters)  from  the  forward  tip  of  the  guidance  kit. 


24 


2.         Results 

Figure  9  shows  the  output  from  the  computer  model  for  the  live  rocket  case.  The 
solid  lines  represent  the  case  where  the  kit's  center  of  gravity  is  located  at  the  kit's  defined 
mid-length.  The  cross-hatched  markings  around  the  lines  allow  for  the  kit's  center  of 
gravity  to  travel  within  the  middle  third  of  the  kit's  defined  length.  The  upper  edge  of  the 
cross-hatched  regions  represents  the  kit's  center  of  gravity  being  two-thirds  of  the  length 
back  from  the  nose,  and  the  lower  edge  is  for  the  center  of  gravity  at  the  one-third  mark. 
The  bold  lines  represent  the  indicated  design  constraints.  The  NPS  HRGK  design  point  (7 
pounds  and  13.625  inches  longer  than  the  fuze)  is  marked  on  the  plot  as  a  reference. 


CD 


^vKwwouu^1^.' 


0.15  lbm/in3 


4  5 

Guidance  Kit  Weight  (lb) 


Figure  9.  Physical  Size  and  Center  of  Gravity  Design  Space.  Kit  length  is  defined  from  the 
tip  of  the  rocket  fuze  forward.  The  launcher  center  of  gravity  constraint  is  tighter  than 
specified  for  illustrative  purposes. 

From  the  plot  it  appears  that  the  loaded  launcher's  center  of  gravity  requirement 
(keeping  the  pod's  center  of  gravity  behind  the  launcher's  front  mounting  lug)  is  not 
binding  for  most  conditions.  Indeed,  for  Figure  9,  the  constraint  was  tightened  to  keeping 
the  pod's  center  of  gravity  within  the  middle  third  of  the  14  inch  lug  spacing  so  that  the 
constraint  could  even  be  seen  in  the  plot. 


25 


B. 


MANEUVERABILITY 


Another  computer  model  was  developed  in  Matlab  to  defme — based  on  the  system 
requirements — the  HRGK's  maximum  maneuverability  specifications.  An  additional 
computer  program  uses  a  steady  maneuver  to  determine  the  angles  of  attack  and  thruster 
forces  needed  to  achieve  the  specified  maneuverability.  This  latter  code  sizes  the  HRGK's 
thrusters.  The  following  subsections  describe  these  computer  models  and  their  results. 

1.  Turn  Rate  and  Turn  Radius  Methodology 

The  maneuverability  analysis  determines  the  maximum  turn  rate  in  g's  or  minimum 
turn  radius  in  kilometers  needed  to  hit  a  target  over  various  launch  and  seeker  acquisition 
ranges.  The  two-dimensional  model  used  in  this  analysis  is  described  in  this  subsection. 

a.         Model  Inputs 

The  maneuverability  model  requires  several  user-defined  inputs.  These 
inputs  are  listed  and  described  in  Table  7  and  are  illustrated  graphically  in  Figure  10. 


Table  7.    Maneuverability  Model  Inputs. 


Input   Name 

Input    Description 

Launch  and  Seeker  Acquisition  Ranges 

Rmin,  Rmax 

Minimum  and  maximum  range  to  target  at  launch  (km) 

n 

Ivacq 

Specific  seeker's  maximum  target  acquisition  range  (vector)  (km) 

Unguided  Errors 

Pnt 

Launcher  pointing  error  at  time  of  launch — la  (degrees) 

B 

Ballistic  unguided  flyout  error — la  (milliradians) 

Weapon  System  Delays  and  Limitations 

GD 

Guidance  delay  (time  after  launch  before  guidance  is  possible)  (sec) 

k> 

Target  identification  delay  (from  initial  acquisition  until  start  of  turn)  (sec) 

sF 

Straight  final  flight  distance  (terminal  non-maneuvering  zone)  (m) 

Missile  and  Target  Motion 

Time 

Time  values  for  missile  range  and  velocity  time  profiles  (vector)  (sec) 

Dist 

Missile  range  profile  (distances  corresponding  to  Time)  (vector)  (m) 

VM 

Missile  velocity  profile  (velocities  corresponding  to  Time)  (vector)  (m/s) 

VT 

Target  speed  (mph) 

Statistical  Confidence 

sig 

Number  of  sigma's  confidence  (e.g.,  sig=2  gives  ±2a  or  95%  confidence) 

26 


VT 


First  Time  Seeker 
Detects  Target 


$  =  sig  ■  ^JPnt2  +  B2 


No  Guidance  Possible 
(guidance  time  delay,  Go) 

1 


Figure  10.   Graphical  Representation  of  Maneuverability  Model  Inputs. 


27 


The  two-dimensional  model  assumes  that  the  rocket  flies  a  straight  path 
toward  the  initial  target  location  but  with  an  azimuth  error  angle,  6.  This  angle  is  a 
statistical  combination1  of  the  launcher  pointing  error  and  the  ballistic  fly  out  dispersion 
error  of  the  rocket.  After  both  the  initial  guidance  delay2,  GD,  and  after  the  rocket  is  within 
the  assumed  seeker  acquisition  range,  R  ,  of  the  target;  the  rocket  can  begin  a  turn  toward 
the  target — correcting  any  cross-track  errors  due  to  6  or  target  motion. 

A  target  identification  delay  time,  tD,  between  the  first  detection  of  the  target 
and  the  start  of  the  turn  can  also  be  included  in  the  model.  The  distance  associated  with  tD, 
SD,  is  based  on  the  rocket  velocity  at  the  time  of  first  detection  (or  the  end  of  guidance 
delay,  whichever  is  later).  The  model  also  allows  for  a  terminal  non-maneuver  zone — some 
radius,  SF,  from  the  target.  For  modeling  purposes,  this  zone  around  the  target  forces  the 
rocket  to  have  completed  its  turn  to  the  target  and  align  itself  for  a  final  straight  approach. 

b.  Stationary    Target — Closed-Form    Solution 

With  a  stationary  target,  the  inputs  described  above  can  provide  a  closed- 
form  solution  for  the  maneuverability  specifications.  This  process  is  laid  out  in  Figure  1 1 . 

The  equations  illustrated  in  Figure  11  provide  the  maneuverability 
requirements  in  terms  of  both  required  turn  radius,  rT,  and  turn  rate,  nmax.  (They  also 
provide  the  solution  to  the  minimum  required  seeker  field  of  view — FOV.  This  part  of  the 
analysis  will  be  described  more  fully  in  a  later  section.) 

For  the  ±2o  unguided  error  case  (sig  =  2),  the  resulting  maneuverability 
values  (rT  and  nmai)  could  be  interpreted  as  the  biggest  turn  radius  or  slowest  turn  rate  that 
would  still  ensure  a  95%  probability  of  hitting  a  stationary  target.  In  other  words,  any 
larger  or  slower  turns,  than  those  specified,  would  not  give  a  95%  confidence  with  the 
provided  inputs.  On  the  other  hand,  tighter  or  faster  turns  would  give  a  higher  than  95% 
confidence.  Therefore,  if  the  system  is  specified  to  95%  confidence,  the  computed  values 
are  the  maximum  turn  rate  and  minimum  turn  radius  required. 

c.  Moving   Target 

The  basic  equations  shown  in  Figure  1 1  can  be  applied  to  the  moving  target 


1  The  standard  deviation,  a,  of  6  is   the  square  root  of  the  sum   of  the  variances  Pnf  and  B2 — 

I         2  2 

a  =  ~4Pnt    +  B   — where  Pnt  and  B  are  the  standard  deviations  for  the  pointing  error  and  the  ballistic 
dispersion  errors,  respectively.  The  angle  6  is  some  user-defined  multiple  {sig)  of  a. 

2  The  range  associated  with  GD  is  calculated  based  on  the  user-supplied  range  time  profile  for  the  rocket. 


28 


n 


Field  of  View  and  Turn  Rate  Calculations 


x  =  R  sin  0        y  =  R  cos  6 


Rd  =  y~  ^Racq  ~  x*  =>  VMD 


2  2 

acq  ~  x     ~ 

sd^(vMD)(td) 


z  =  Jx2+(y-Rr)' 


FOV  =  sin-1  - 

0  =  --FOV 

y      2 


law  of  cosines: 

d2  =  (sf  +r?\  =  r2  +z2-2rt  zcos</> 


2         2 
z    ~sf 

rt  = — 

2  z  cos  0 


n 


'M7 


max 


^00 


Figure  1 1 .   Closed-Form  Solution  for  Maneuverability  Specifications  with  a  Stationary 

Target. 


29 


scenario  shown  in  Figure  12.  The  moving  target  scenario  assumes  a  "worst-case"  target 
motion.  The  target  moves  with  a  crossing  velocity  in  the  direction  opposite  the  flyout  error. 
Then,  after  it  is  detected,  the  target  turns  in  at  45°  making  the  rocket's  turn  tighter. 

The  moving  target  scenario  precludes  a  closed-form  solution  of  the 
maneuverability  specifications.  However,  the  equations  in  Figure  1 1  can  be  used  with  an 
iterative  step  at  each  of  the  flight  milestones  (target  detection,  start  of  turn,  and  end  of  turn). 
The  Matlab  code  listing  in  Appendix  C  uses  this  approach.  The  following  description 
provides  an  example  of  the  iterative  process  used  in  the  computer  code. 

After  RD  is  calculated  using  the  appropriate  stationary  target  equations  (see 
Figure  11),  the  time  required  to  fly  that  distance  is  available  from  the  range  time  profile. 
Using  the  time  and  the  target  speed,  the  new  target  location  can  be  computed.  The  value  for 
RD  can  then  be  re-calculated  based  on  the  updated  target  location.  These  iterations  continue 
until  the  change  in  RD  between  iterations  is  less  than  some  set  tolerance  (such  as  1  m). 

2.         NPS  HRGK  Turn  Rate  and  Turn  Radius  Analysis 

This  subsection  discusses  the  maneuverability  analysis  performed  specifically  for 
the  design  of  the  NPS  HRGK.  The  following  paragraphs  describe  the  inputs  used  for  the 
computer  model  and  present  the  analysis  results. 

a.  NPS  HRGK  Model  Inputs 

The  NPS  HRGK  analysis  used  the  full  gamut  of  specified  launch  ranges  ( 1 
to  6  kilometers)  and  various  seeker  acquisition  ranges  (0.75,  1.0,  1.15,  2.0,  3.0,  and  6.0 
km).  The  assumed  la  values  for  launcher  pointing  error  and  ballistic  dispersion  were  3? 
and  12  mils,  respectively.  (Using  a  95%  confidence  level — sig  =  2,  the  angular  flyout  error 
was  internally  computed  to  be  6.16°.)  The  target  speed  was  60  mph  per  the  system 
requirements.  Initial  guidance  was  delayed  as  a  linear  function  of  launch  range  (0.5 
seconds  for  1  km,  up  to  1.0  second  for  6  km).  The  target  identification  delay  was  0.625 
seconds  (arbitrarily  chosen  to  correspond  to  5  laser  pulses  at  8  Hertz),  and  the  terminal 
non-maneuvering  distance  was  arbitrarily  chosen  to  be  200  meters. 

The  rocket's  velocity  and  distance  time  profiles  used  in  the  analysis  were 
based  on  an  early  simulation  of  a  HRGK-equipped  rocket  using  a  ballistic  trajectory. 
Specifically,  the  simulation  used  the  150°F  propellant  thrust  profile  [Ref.  1 1]  and  a  generic, 
hemispherical-nosed  guidance  kit  with  three  canards.  The  aerodynamic  coefficients  and 
derivatives  used  in  the  simulation  were  determined  using  the  software  code,  Advanced 
Design  of  Aerodynamic  Missiles — ADAM  [Ref.  12] . 


30 


Target  at  Detection 
(target  begins  45°  turn) 


Target  at 
Rocket 
Launch 


Target  Position 
(when  rocket 
starts  turn) 


Start  of 

Rocket 

Turn 


Assumptions: 

•Target  has  a  crossing  velocity 

•After  target  detection,  target  turns  45c 


Unguided  Flyout  Azimuth  Error 
(Initial  Pointing  and  Dispersion) 


Launch  Helicopter 


Figure  12.    Moving  Target  Scenario  for  Maneuverability  Analysis. 

b.         NPS  HRGK  Analysis   Results 

Based  on  the  inputs  described  above,  the  maximum  turn  rate  and  minimum 
turn  radius  specifications  for  the  NPS  HRGK  are  shown  in  Figure  13  and  Figure  14, 
respectively.  The  results  are  shown  as  a  function  of  launch  range  for  several  seeker 
acquisition  ranges.  The  analysis  results  show  that  as  long  as  the  seeker  can  initially  detect 


31 


the  target  at  1.15  kilometers  or  more,  the  hardest  g-loading  maneuver  for  the  rocket  would 
be  less  than  15  g's  and  tightest  turn  radius  would  be  greater  than  1.83  kilometers. 

The  maximum  turn  rate  is  a  strong  function  of  launch  range  with  short  range 
launches  requiring  higher  g-loading  maneuvers.  This  is  primarily  due  to  higher  rocket 
speeds  at  shorter  ranges.  For  example,  after  flying  0.5  km,  the  rocket's  velocity  is  1.8 
times  higher  than  it  is  after  coasting  out  to  5  km.  For  equal  turn  radii,  a  1.8  factor  increase 
in  velocity  yields  a  3.2  (or  1.82)  factor  increase  in  turn  rate.  The  guidance  delay  also 
increases  the  turn  rate  at  shorter  launch  ranges.  Target  identification  delays  primarily  impact 
the  shorter  seeker  acquisition  range  cases — longer  delays  causing  higher  turn  rates. 

Unlike  the  maximum  required  turn  rate,  the  minimum  required  rum  radius  is 
almost  independent  of  rocket  speed1.  Once  the  launch  range  places  the  target  initially 
outside  the  seeker's  acquisition  range,  the  turn  radius  curves  remain  fairly  flat  with 
increasing  launch  range.  For  example,  with  a  1.15-km-acquisition-range  seeker,  the 
minimum  required  turn  radius  changes  less  than  59%  across  the  spectrum  of  launch  ranges 
(1.83  km  at  1  km  launch  to  2.90  km  at  6  km  launch).  Conversely,  if  the  seeker  has 
sufficient  detection  range  to  take  advantage  of  it,  the  longer  range  launches  allow 
significantly  wider  turns.  For  example,  only  a  gentle  12.9  km  turn  radius  is  required  with  a 
6  km  seeker  acquisition  range  and  a  6  km  launch  range. 

By  either  measure  (turn  rate  or  turn  radius),  the  most  stressing  case  for  the 
maneuverability  requirement  is  the  short  range  launch,  because  of  rocket  speed  and  the  lack 
of  reaction  time.  The  seeker's  acquisition  range  can  also  significantly  impact  the  HRGK 
maneuverability  requirements — shorter  range  seekers  forcing  higher  maneuverability 
requirements.  Conversely,  long  range  seekers  can  alleviate  some  maneuverability 
requirements  by  detecting  the  target  earlier  and  allowing  gentler  turns. 

An  additional  benefit  of  the  maneuverability  analysis  is  that  the  minimum 
required  seeker  acquisition  range  is  shown  to  be  1.15  km.  With  shorter  acquisition  ranges 
and  long  launch  ranges,  the  target's  cross-track  position  relative  to  the  rocket  (due  to  6  and 
target  motion)  exceeds  the  seeker's  range.  For  example,  with  a  4.1  km  launch  range,  the 
rocket's  closest  unguided  approach  to  the  target  is  0.75  km  (2a).  For  longer  launch  ranges, 
the  closest  approach  distance  would  be  greater  than  0.75  km;  therefore,  a  0.75-km- 
detection-range  seeker  would  not  have  a  95%  probability  of  detecting  the  target. 


1  Rocket  speed  indirectly  affects  the  turn  radius  since  it  partially  determines  the  distance  the  target  can 
travel  before  it  is  detected  or  impacted. 


32 


16 


w 
3I2 

a> 

"co 
QC 

c 
§     8 

h- 

E 
£ 
cc     4 


t     ,,,,,, 

I.    J 

"                                    ♦ 
i      '    ",                            ♦    • 

V/       *.                              ♦  : 

V         '.                       ♦' 

X             V         *»                 • 

X                   *k                                    • 

,..X ;...>«  .'..   *    ..• ; 

Seeker 
Acquisition 
Range  (km) 

•""0.75 

////////  1 _Q 

■■■■■•■■a  1.15 

Xv  !                      '>.       '.,              *. 

i                                                 1 

1 1 

3  4 

Launch  Range  (km) 


Figure  13.  Maximum  Turn  Rate  Required  to  Hit  Target  (95%  Probability).  Only  with  a 
seeker  able  to  detect  a  target  from  1.15  km  or  more  (Racq  >  1.15)  can  the  maximum 
range  requirement  (6  km)  be  meet. 


0.75  km 


3  4 

Launch  Range  (km) 


Figure  14.    Minimum  Turn  Radius  Required  to  Hit  Target  (95%  Probability).  Seeker 
acquisition  ranges  are  indicated  with  the  corresponding  curves. 


33 


3.         Steady-Turn  Maneuver  Analysis 

Once  maneuverability  specifications  are  set,  a  steady-turn  maneuver  model  can  be 
used  to  size  the  control  mechanism  needed  to  achieve  the  needed  turn  rate  or  turn  radius. 
The  following  paragraphs  describe  a  computer  code  developed  by  the  author  to  perform 
this  task.  Later  a  specific  example  for  a  guided  rocket  is  presented.  The  Matlab  code 
listing  with  the  NPS  HRGK  parameters  is  included  in  Appendix  C. 

a.  Mathematical  Steady-Turn  Model 

The  purpose  of  this  mathematical  steady-turn  model  is  to  determine  the 
thrust  force  and  angle  of  attack  (or  sideslip  angle)  necessary  to  generate  a  constant 
acceleration  turn  in  the  horizontal.  Throughout  the  description,  the  more  familiar 
longitudinal  nomenclature  (loosely  associated  with  the  vertical  plane)  is  used  even  though 
the  turn  is  described  in  the  directional  case  (loosely  associated  with  the  horizontal  plane). 
The  interchange  of  nomenclature  is  justified  since,  for  the  guided  rocket  or  missile  case,  the 
longitudinal  and  directional  dynamics  are  equivalent.  A  useful  listing  of  missile 
aerodynamic  nomenclature  is  included  in  Appendix  A. 

Figure  15  shows  a  free-body  diagram  for  a  horizontal  maneuver.  For  a 
constant  acceleration  turn,  the  sum  of  the  forces  and  moments  can  be  expressed  by  the 
following  equations: 

£  Forces  =  « W  =  N  +  T  =  qACNa a  +  kT 


( 


1 


A 


2& 


pV2AC„ 


a  +  kT 


£  Moments  =  -NXCP  +  TXT  -  Mq  =  0 

=  -{qACNa)Xcp  +  kTXT  -  qAd{cMq  ^j 

=  -{qACNaa)xcp  +  kTXT  -  qAd 

=  -(qACNa  a)Xcp  +  kTXT  -  qAd[cMq  ^j 


,    Mq  2V  , 


-  a 


1  ^  1 

—pV2ACNa   XCP  +  TkXT-—pACMd2ng 


34 


where  k  is  a  thrust  multiplication  factor  (accounting  for  misalignment  of  the  thruster  and  the 
effect  of  jet  and  free  air  steam  interaction)  and  the  other  terms  are  either  defined  by  Figure 
15  or  are  standard  terms  defined  in  Appendix  A.  The  moments  equation  uses  the  fact  that 


rT 


V 


rV2\ 


\azJ 


V  "  V 


Figure  15.    Plane- View,  Free-Body  Diagram  for  a  Steady-Turn  Maneuver  (horizontal,  left 
turn — gravity  force  vector  pointed  into  the  page). 

The  sum  of  forces  and  sum  of  moments  equations  can  be  rewritten  as  the 
following  matrix  equation: 


1 


2£0 


pV2AC„ 


KA-t 


\ 


2*o 


pV2AC„ 


LC/> 


a 


4*o 


nW 


pACMd2ng 


which  gives  the  thrust  magnitude  and  angle  of  attack  in  terms  of  guided  rocket  parameters 
and  the  turn  rate  n.  Using  the  relationship  rT  =  V2/ng,  the  matrix  equation  can  be  expressed 
in  terms  of  the  turn  radius  as  the  following: 

V2W 


1 
2g0 


pV'AC„ 


kX7 


2£0 


pV2AC. 


Lc/> 


a 


4rTg0 


grT 


pV2ACMd2 


35 


b.         Steady-Turn   Computer  Code 

The  author  has  developed  a  computer  code  to  implement  the  mathematical 
model  described  above.  This  code  solves  for  the  thrust  and  angle  of  attack  required  to 
achieve  the  required  turn  rate  or  turn  radius  over  a  range  of  Mach  numbers.  The  code 
requires  the  user-supplied  inputs  listed  and  described  in  Table  8. 

Table  8.    Steady-Turn  Maneuver  Computer  Model  Inputs. 


Input   Name 

Input    Description 

Scenario  Definition 

n  or  rT 

Desired  turn  rate  (g's)  or  turn  radius  (ft) 

type 

Indicator  for  computations  based  either  on  turn  rate  (1)  or  turn  radius  (2) 

alt 

Pressure  altitude  (ft) 

Mmin,  Mmax 

Minimum  and  maximum  Mach  number  for  the  calculations 

Rocket  Configuration 

Wght 

Rocket  total  weight  at  maneuvering  time  (lb) 

Dia 

Rocket  diameter  (for  reference  area  and  length)  (in) 

Xcg 

Rocket  center  of  gravity  (calibers  from  nose) 

Xt 

Thruster  location  (calibers  from  nose) 

Aerodynamic  Parameters  (functions  of  Mach) 

Mach_dat 

Mach  values  for  corresponding  aerodynamic  parameters  (vector) 

CNa_dat 

Rocket  normal  force  curve  slope  (corresponding  to  Mach)  (vector)  (rad1) 

Xcp_dat 

Center  of  pressure  (corresponding  to  Mach)  (vector)  (calibers  from  nose) 

CMq_dat 

"Pitch"  damping  (corresponding  to  Mach)  (vector)  (rad1) 

c.         NPS  HRGK  Example 

The  following  provides  example  solutions  to  the  equations  previously 
developed  for  the  steady-turn  model.  The  examples  are  specific  to  the  NPS  HRGK  with  the 
guided  rocket's  maneuverability  specifications  set  at  15  g's  turn  rate  or  1.83  km  (6,000  ft) 
turn  radius.  The  manual  calculations  are  for  the  NPS  HRGK  design  at  Mach  1.0.  These 
calculations  are  done  first  for  the  turn  rate  specification  and  then  for  the  turn  radius 
specification.  The  aerodynamic  coefficients  are  from  ADAM  [Ref.  12].  All  computations 
are  for  500  feet  pressure  altitude  and  use  the  fired  rocket  center  of  gravity  (most 
maneuvering  takes  place  after  the  1.0  sec  rocket  motor  burn).  Because  no  additional  data  is 
available,  the  value  of  k  (an  adjustment  factor  for  the  thruster  force)  is  assumed  to  be  unity. 


36 


The  manual  calculations  are  followed  by  sample  output  from  the  steady-turn  computer 
code. 

( 1 )        Sample  Manual  Calculation  Based  on  Maximum  Turn  Rate. 


(0.0754^)(lll4f)2(0.0412  ft2)(20.05  rad1) 

zr  L-L  ibf  s2  / 

(0.0754 ^)(ll  14 f)2 (0.0412  ft2)(20.05  rad1) 


1.79ft 


ZrZ,Z  Ibf  s2  / 


15  (22.8  Ibf) 


0.0824  ft 


a 


(0.0754  ^)(0.04 12  ft2)(-5520  rad_1)(0.229  ft)2 15  (32.2  f) 


4(32.2^) 


The  solution  to  the  set  of  equations  is: 


1 

1200^" 

~T 

'    342  Ibf 

~T~ 

"  13.2  Ibf  " 

"13.2  Ibf 

1.79  ft 

_QQ  1  'W  ft 
771    rad  _ 

a 

-3.38  Ibf  ft 

— > 

a 

0.273  rad 

15.6° 

1  1200^ 


1.79  ft    -99.1^ 


(2)        Sample  Manual  Calculation  Based  on  Minimum  Turn  Radius. 

(1114^)2  (22.8  Ibf) 
(32.2^-)(6000  ft) 

(0.0754 ^)(H14f)2 (0.0412  ft2)(-5520  rad_1)(0.229  ft)2 


4(32.2  «-)(6000  ft) 


The  solution  to  the  set  of  equations  is: 


1 

1200^  " 

~T~ 

"     147  Ibf 

~T 

"  5.68  Ibf  " 

"5.68  Ibf 

1.79  ft 

QQ    1    Ibf  ft 

77,1    rad  . 

a 

-1.45  Ibf  ft 

— > 

a 

0.117  rad 

6.71° 

37 


(3)  Sample  Output  from  Steady-Turn  Computer  Code.  Figure 
16  shows  the  output  from  a  computer  code  developed  by  the  author.  The  code  computes 
the  solution  to  the  steady-turn  equations  that  were  manually  solved  in  the  previous 
paragraphs. 


CD 

cc 

Q. 
CO 

-a 
o 

O" 

CD 
DC 


1 

3-4                0.6                    0.8                   1.0 

1.2 

1.4 

1.6 

Af\ 

4U 
20 

3^  J 

0 

0-4  0.6  0.8  .1.0  1.2 

Mach  Number 


1.4 


1.6 


Figure  16.    Required  Thrust  and  Angle  of  Attack  for  Maximum  Turn  Rate  (15  g's — 
dashed  line)  or  Minimum  Turn  Radius  (1.83  km — solid  line)  (NPS  HRGK  design). 

From  the  data  plotted  in  Figure  16,  the  difference  in  the  two 
methods  of  specifying  maneuverability  requirements  is  apparent.  For  example,  at  subsonic 
speeds,  the  model  predicts  that  the  rocket  could  only  achieve  a  15  g  maneuver  with  very 
high  thrust  and  extremely  high  angle  of  attack.  On  the  other  hand,  the  requisite  1.83  km 
turn  radius  can  be  achieved  at  subsonic  speeds  with  a  moderate  angle  of  attack  and  low 
thruster  force. 

Fortunately,  15  g  maneuvers  are  not  required  when  the  missile  has 
coasted  down  to  subsonic  speeds  (see  long  range  side  of  Figure  13,  page  33).  Clearly,  the 
reasonable  curve  for  sizing  the  thruster  is  the  rninimum  turn  radius  curve  (solid  line).  This 
seems  reasonable  since  the  turn  radius  requirements  remained  fairly  flat  over  the  entire 
gamut  of  launch  ranges  (see  Figure  14). 


38 


Designing  to  the  minimum  turn  radius  and  given  the  fact  that  the 
rocket  has  slowed  down  to  nearly  Mach  1.3  by  the  first  kilometer  of  flight,  the  thruster  can 
be  sized  to  provide  10  to  15  pounds  of  thrust.  A  design  with  higher  thrust  early  in  the  flight 
would  be  ideal,  since  the  short-range,  high-speed  scenario  is  the  most  demanding. 

4.         Probabilistic  Approach 

The  maneuverability  analysis  performed  in  the  previous  subsections  was  based  on 
the  statistical  (probabilistic)  fly  out  of  the  rocket  but  a  deterministic,  worst-case  target 
motion.  Another  approach  to  determining  the  needed  level  of  maneuverability  would  be  to 
apply  a  statistical  model  to  the  target  motion  as  well  as  the  fly  out.  This  latter  approach 
would  give  valuable  insight  into  the  reasonableness  of  the  worst-case  approach  taken 
above.  This  type  of  purely  probabilistic  approach  was  needed  for  the  seeker  field  of  view 
analysis,  which  is  described  next. 

C.        SEEKER  FIELD  OF  VIEW 

The  seeker's  field  of  view1  was  another  early  consideration  for  the  NPS  HRGK 
design.  Again,  this  design  specification  could  be  derived  from  top-level  system 
requirements  and  simple  mathematical  models.  A  possible  approach  to  determining  the 
needed  seeker  field  of  view  was  alluded  to  in  the  maneuverability  analysis  section.  This 
section  will  briefly  discuss  that  approach  and  its  results.  Then  a  probabilistic  approach  with 
its  results  will  be  discussed  in  greater  detail. 

1.         Analysis  with  "Worst-Case"  Target  Motion  Model 

The  two-dimensional  scenario  described  in  subsection  EQ.B.l.  Turn  Rate  and  Turn 
Radius  Methodology  provides  a  means  for  calculating  the  needed  HRGK  seeker  field  of 
view.  The  equations  shown  in  Figure  1 1  (page  29),  can  be  used  to  determine  the  field  of 
view  that  keeps  a  stationary  target  within  the  seeker's  field  of  view  with  some  specified 
level  of  confidence  (for  example,  95%)  up  until  the  target  can  be  detected  and  a  turn  toward 
the  target  can  be  initiated.  As  discussed  in  the  earlier  subsection,  the  equations  shown  in 
Figure  1 1  can  be  solved  iteratively  to  determine  the  required  field  of  view  for  a  moving 
target  (see  Figure  12,  page  31). 


1  The  seeker's  field  of  regard — meaning  the  total  area  covered  by  a  fixed  or  scanning  seeker — is  the  actual 
consideration,  but  for  a  strapped-down  seeker,  the  field  of  view  and  field  of  regard  can  be  used 
interchangeably.  This  thesis  uses  the  term  field  of  view. 


39 


Unfortunately,  the  target  motion  model  used  for  the  maneuverability  analysis  is  a 
worst-case  example.  (The  target  crosses  at  maximum  velocity  in  the  opposite  direction  of 
the  statistical  flyout  and  pointing  errors.  Furthermore,  after  detection,  the  target  turns  in  at 
45°  which  creates  the  need  for  a  still  larger  field  of  view.  See  Figure  12.) 

The  required  field  of  view  results  shown  in  Figure  17  for  the  NPS  HRGK  were 
obtained  using  the  worst-case  target  motion  model  and  the  computer  code  inputs  discussed 
in  subsection  III.B.2.  NPS  HRGK  Turn  Rate  and  Turn  Radius  Analysis  (page  30).  The 
values  are  for  half  field  of  view  and  are  shown  as  a  function  of  launch  range  and  for 
various  seeker  acquisition  ranges.  For  example,  with  a  maximum  launch  range  and  a  seeker 
acquisition  range  of  1.5  km,  a  ±48°  seeker  field  of  view  would  be  needed  to  assure  (95%) 
that  the  target  stays  in  the  field  of  view  up  until  a  turn  can  be  initiated.  (With  a  stationary 
target,  only  a  ±29°  seeker  field  of  view  would  be  needed.) 


Racq  =  6  km 


3  4 

Launch  Range  (km) 


Figure  17.  Required  Field  of  View  Results  for  95%  Probability  of  Target  Staying  in  Field 
of  View  with  Worst  Case  Target  Motion.  (Corresponding  field  of  view  requirements  for  a 
stationary  target  are  indicated  with  dashed  lines.) 

From  the  curves  in  Figure  17,  it  is  clear  that  longer  launch  ranges  require  larger 
fields  of  view  (after  longer  fly  outs,  the  rocket's  cross-track  error  and  the  target's  motion 


40 


away  from  the  rocket  will  be  larger).  Shorter  acquisition  range  seekers  also  require  larger 
fields  of  view. 

Unfortunately,  the  indicated  field  of  view  sizes  for  long  range  launches  and  short 
range  seekers  are  unattainably  large,  especially  for  a  strapped-down,  semi-active  laser  spot 
homing  seeker  like  that  on  the  NPS  HRGK.  The  worst-case  target  motion  model  is 
unacceptable,  and  so,  a  probabilistic  approach  must  be  taken. 

2.         Analysis  with  Probabilistic  Target  Motion  Model 

The  probabilistic  target  motion  model  used  for  the  seeker  field  of  view  analysis  is 
based  on  a  distribution  of  target  positions.  In  this  analysis,  the  probability  of  the  seeker 
acquiring  a  target  is  derived  from  an  assumed  seeker  field  of  view  and  the  probabilistic 
distributions  for  the  rocket  and  the  target.  The  methodology  for  defining  a  seeker  field  of 
view  specification  with  this  approach  is  developed  in  this  subsection.  The  resulting 
specifications  for  the  NPS  HRGK  are  also  presented.  The  Matlab  code  developed  by  the 
author  to  perform  this  analysis  is  included  in  Appendix  C. 

a.  Computer  Model  Inputs 

The  inputs  required  for  the  probabilistic  seeker  field  of  view  analysis  are 
listed  and  described  in  Table  9. 


Table  9.   Seeker  Field  of  View  Analysis  Inputs. 


Input   Name 

Input    Description 

Seeker  Descriptions 

FOVmin,  max 

Minimum  and  maximum  seeker  half  field  of  views  (deg) 

Specific  seeker's  maximum  target  acquisition  range  (vector)  (km) 

Unguided  Errors  and  Weapon  System  Delays 

Pnt 

Launcher  pointing  error  at  time  of  launch — la  (degrees) 

B 

Ballistic  unguided  flyout  error — lo  (milliradians) 

GD 

Guidance  delay  (time  after  launch  before  guidance  is  possible)  (sec) 

tD 

Target  identification  delay  (from  initial  acquisition  until  start  of  turn)  (sec) 

Missile  and  Target  Motion 

RL 

Range  to  target  at  launch  (single  value)  (km) 

Time 

Time  values  for  missile  range  and  velocity  time  profiles  (vector)  (sec) 

Missile  range  profile  (distances  corresponding  to  Time)  (vector)  (m) 

Vx           V-r 
"  Tmin'    *  Tmax 

Minimum  and  maximum  target  speed  (mph) 

41 


b.         Target  Motion  Model 

The  probabilistic  seeker  field  of  view  analysis  uses  a  simple  distribution 
model  for  target  motion.  The  target's  speed  is  assumed  to  be  constant  and  uniformly 
distributed  between  the  user-supplied  inputs,  VTmin  and  VTmax.  Its  bearing,  <f),  is  also 
assumed  to  be  constant  and  to  be  uniformly  distributed  between  0°  and  360°.  Figure  18 
shows  a  plane  view  of  possible  target  locations  after  time,  tT  (the  flyout  time  before  the 
rocket  starts  its  turn  to  the  target).  The  figure  also  includes  the  corresponding  probability 
density  function  for  the  target's  cross-track  position,  x.  This  function  provides  the 
probability  of  the  target  being  at  some  specific  cross-track  position,  xT,  or  between  two 
cross-track  limits,  x,  and  x2.  (The  v-axis  indicates  the  direction  of  the  rocket  flyout.) 


original 

target 

location. 


trVm 


illustrative  target  location 


Pt(xt) 
(probability:  target 
at  cross-track  Xi) 


iPr{x) 


tr  Vjmax 


area  of  possible 
target  locations 


(b) 


Pt  (xuxs) 
(probability:  target 
between  X2  and  X2) 


Figure  18.    Target  Distribution  Model,  (a)  Plane  view  of  possible  target  positions  and  (b) 
Cross-track  probability  density  function,  Pj(x) 

c.         Analysis    Methodology 

The  methodology  used  to  compute  the  probability  of  the  target  being  within 
the  HRGK  field  of  view,  has  previously  been  documented  by  the  author.  The  following 
paragraphs  briefly  summarize  the  methodology.  In  these  paragraphs,  the  assumption  is 
made  that  if  the  target  is  within  the  HRGK's  field  of  view,  it  will  be  acquired;  therefore,  the 
desired  quantity  is  the  probability  of  acquisition,  Pacq.  The  referenced  technical 
memorandum  should  be  consulted  for  a  more  complete  description  of  the  theory  and  the 
limitations  of  the  simplifying  assumptions.  [Ref.  13] 

Figure  19  shows  a  sample  case  for  the  probabilistic  model  and  shows 
several  of  the  variables  used  in  the  model  computations.  To  obtain  the  probability  of  the 
target  being  within  the  seeker  field  of  view  (x,  <  xT<  x2),  the  independent  probabilities  of 


42 


the  rocket  and  target  cross-tracks  must  be  computed.  The  rocket  cross-track  computation 
assumes  a  normal  distribution  of  9 — which  depends  only  on  the  independent,  normally 
distributed  pointing  and  ballistic  flyout  errors,  Put  and  B,  and  the  target's  position  depends 
on  the  target  motion  model  described  above.  The  cross-track  distribution  for  the  rocket  is 
evaluated  for  the  assumed  range  to  detection  (RD),  and  the  target  distribution  is  evaluated 
for  the  assumed  time  at  start  of  turn  (f^  time  required  for  the  rocket  to  fly  to  RT). 


Figure  19.  Illustrative  Case  for  Probabilistic  Seeker  Field  of  View  Analysis.  The  rocket  is 
shown  at  the  assumed  position  RD. 


The  probability  of  the  rocket  being  at  cross-track  position  xR  is 

1 


p*w = 


-x2 
2a2 


<7V2tt 


where  a  =  RDtan0  =  RD^]tan2(Pnt)  +  B2 .  Because  of  the  assumed  target  distribution,  the 
target  has  equal  probability  of  being  anywhere  in  the  possible  region  (ATOT)  shown  in 
Figure  20.  Therefore,  the  probability  of  the  target  being  within  the  field  of  view  limits  (jc; 
and  x2)] — the  shaded  area,  A,  in  the  figure — is  the  ratio  of  the  seeker  acquired  area,  A,  to 
the  total  possible  area,  AT0T.  The  seeker  acquired  area,  A,  can  be  expressed  as  the  area 
between  the  limits  (*;  and  x2)  and  inside  the  outer  circle  (radius  =  r2)  minus  the  area 
between  the  limits  and  inside  the  inner  circle  (radius  =  r,).  The  total  possible  area,  AT0T,  is 
K{r2  ~  r\\> tnus' tne  desired  probability  is  given  as: 


The  field  of  view  limits  are  x,  =  xR  -  F  and  x2  =  xR  +  F  where  the  offset,  F  =  Racq  sin(FOV). 


43 


rTyXl,X2)  - 


"n 


OT 


sign(x2)min(r2\x2\)  sign(x2)min(rl,\x2\) 

2    j-sjr2-x2dx    -      2    j^r2-x2dx 

sign(x,  )  min(r2  ,|  xx  \ )  sign(x^ )  min(  r,  ,|  x,  | ) 


where  the  limits  of  the  two  integrations  are  set  to  never  exceed  the  radius  of  the  outer  or 
inner  circle,  respectively. 


possible  target  locations 
within  field  of  view 


n  =  tr  VT> 


min 


Atot 


r2=  trVjmax 
all  possible  target  locations 


Figure  20.  Possible  Target  Locations.  The  ratio  of  the  area  (A)  bounded  by 
the  field  of  view  limits — x,  and  x2 — and  the  total  possible  area  (ATOT)  is  the 
probability  of  the  target  being  within  the  two  limits. 

An  alternative  calculation  of  the  seeker  acquired  area,  A,  is  illustrated  in 
Figure  21.  The  calculation  method  relies  on  the  fact  that  since  the  probabilities  are 
sy metrical  about  the  y-axis,  the  calculations  only  need  to  be  made  for  positive  values  of  xR  ; 
thus  x2  will  always  be  positive.  The  method  involves  calculating  the  area,  A , ,  bounded  by 
the  circles,  the  lower  field  of  view  limit  (x2),  and  the  y-axis  as  given  by  the  equation: 

/]  =min(r2,£2) L2=nuB(,r1,x2  ) 

A,  =2       j^jr22-x2dx-2       j^jr2-x2dx 


( 


=  L,Vr22-Z12+r22sin-ip   -Z^^-^+l 


(i   \ 


2sin-! 


Vr2 


Li 


Kr\J 


Next,  the  area  A2  is  calculated.  This  area  is  bounded  by  the  circles,  the  lower  field  of  view 
limit  (jc,),  and  the  y-axis,  and  is  given  by  the  equation 


44 


l}  =min(r2,|j||) Lt  =min(r,  ,|  X|  | ) 

A2  =    2       \^r2-x2dx-    2       \ -^r2  -  x2  dx 


=  L^^jr2  -L]  +  r2  sin  '   —   -  L4 s\r2  - L24  +  r2  sin  ' 


' L  ^ 


Finally,  the  area  A2  is  subtracted  from  A,  if  jc,  is  greater  than  zero  (Figure  21  (a))  or  added 
to  A,  if*,  is  less  than  zero  (Figure  21  (b)) — A  =  A,  -  sign(x,)  A2. 


£z 

*<&& 

^/r 

® 

;/    * 

(b)^ 

1 

r  /  / 

r"Xi 

Figure  21.     Alternative  Method  for  Computing  Target  Distribution  Area  within  Seeker 
Field  of  View  Limits.  (A  =  A,  -  sign(x,)A2.) 

Now,  the  probability  of  the  rocket  being  at  a  specific  cross-track  and  the 
target  being  within  the  HRGK's  field  of  view  can  be  expressed  as  the  joint  probability 
Pacq(xR)  =  P(xR,  x,<xT<  x2)  =  PR{xR)P1{x1,x2).  The  total  probability  of  the  rocket  having 
the  target  in  its  field  of  view  for  any  flyout  is  the  integral  of  this  joint  probability.  These 
probabilities  are  illustrated  in  Figure  22.  The  actual  method  for  computing  the  probability  of 
acquisition  is  described  in  the  following  paragraphs. 


min.  target 
position,  -rz 


Extreme  values 
at  x  =  ±(i2+F) 


Pacq(Xfi)  = 
Pfl(Xfl)-Pr(Xi,X2) 


•  acq\X) 

•acq  =  J 'acq  \X) 


(c) 


Figure  22.    Probability  Density  Functions,  (a)  Rocket  cross-track  error,  (b)  Target  cross- 
track  position,  and  (c)  Probability  of  acquisition. 


45 


d.  Computer   Code  Description 

The  Matlab  computer  code  developed  to  determine  the  probability  of  target 
acquisition1  includes  a  Matlab  script  file  and  a  Matlab  function.  (Appendix  B  briefly 
describes  the  difference  between  script  files  and  functions.)  The  script  file  sets  the  user 
inputs,  determines  the  rocket's  cross-track  error  standard  deviation,  sets  the  seeker  offset 
(F),  calculates  the  integral  of  the  joint  probability  \P  R{x)  P  j{x±F)dx  from  zero  to  an  upper 
bound,  and  plots  the  results.  The  Matlab  function  is  called  by  the  integration  scheme  in  the 
script  file  and  returns  values  for  the  joint  probability  (Pacg(x)  =  P/?(x)P7(;t±F)).  These 
functions  are  all  performed  over  a  range  of  fields  of  view  and  for  several  seeker  acquisition 
ranges.  Both  the  script  file  and  the  function  are  included  in  Appendix  C. 

e.  NPS   HRGK  Results 

The  computer  code  results  for  the  NPS  HRGK  design  case  are  shown  in 
Figure  23.  The  results  are  for  the  maximum  launch  range  case  (6  km)  as  this  is  the  most 
stressing  case.  All  other  input  parameters  are  set  as  in  previous  analyses.  The  target  speed 
is  uniformly  distributed  between  20  to  60  miles  per  hour.  The  results  show  that  a  95% 
probability  of  having  the  target  in  the  field  of  view  at  time,  ?p  can  be  obtained  with  fields  of 
view  slightly  smaller  than  those  that  would  be  required  for  the  same  results  against 
stationary  targets.  For  example,  with  a  seeker  acquisition  range  of  1.5  km,  a  ±25°  seeker 
would  provide  a  95%  probability  of  acquisition  against  the  probabilistic  target  model.  But, 
against  a  stationary  target,  a  ±29°  seeker  would  be  required  for  the  same  confidence  (Figure 
17,  page  40).  The  probabilistic  moving  target  model  results  in  a  smaller  field  of  view 
because  of  the  possibility  of  the  target  moving  in  the  same  direction  as  the  rocket  cross- 
track  error  is  allowed.  This  possibility  increases  the  probability  of  having  the  target  within 
the  field  of  view,  even  with  a  slightly  smaller  field  view. 

D.         COMMENTS  AND  OBSERVATIONS 

In  concluding  this  chapter  on  preliminary  sizing  analyses,  several  observations  are 
appropriate.  These  include  (1)  a  discussion  of  the  benefits  and  data  that  came  from  the 
computer  models,  (2)  a  word  of  caution  about  the  limitations  of  the  models,  and  (3)  a 
listing  of  additional  work  that  could  be  done  to  expand  the  usefulness  of  the  preliminary 
sizing  models. 


Actually  the  probability  of  the  target  being  within  the  cross-track  limits  of  the  seeker's  field  of  view. 


46 


Figure  23.   Probability  of  Target  in  Field  of  View  for  Maximum  Launch  Range  (6  km). 

1.         Additional  Data  from  Modeling 

The  primary  purpose  of  the  computer  models  discussed  in  this  chapter  was  to  give 
the  designer  a  better  understanding  of  how  the  system  requirements  affect  different  design 
specifications.  However,  the  codes  not  only  facilitate  the  "flow-down"  of  requirements,  but 
they  also  provide  additional  insights  into  the  design  of  the  system.  In  the  case  of  the  NPS 
HRGK  design,  several  observations  were  made  from  the  use  of  the  preliminary  sizing 
codes.  The  following  paragraphs  site  some  examples. 

From  the  length,  weight,  and  center  of  gravity  analysis  it  was  determined  early  in 
the  design  that  the  launcher  center  of  gravity  constraint  is  not  binding  for  any  design  within 
the  maximum  weight  and  length  constraints.  Thus,  the  kit's  center  of  gravity  can  be  placed 
to  provide  the  best  combination  of  performance  and  packaging  without  concern  for  the 
launcher's  center  of  gravity.  In  a  general  case,  a  designer  can  use  this  type  of  knowledge  to 
move  through  design  iterations  without  making  calculations  regarding  the  non-binding 
constraints.  Then,  as  the  final  design  is  approached,  calculations  can  be  made  to  verify  the 
preliminary  assessments. 

Another  good  example  of  additional  information  gained  from  the  models  is  the 
minimum  required  seeker  acquisition  range.  The  required  seeker  range  was  inadvertently 


47 


discovered  by  trying  several  seeker  acquisition  ranges  in  the  maneuverability  analyses.  The 
final  seeker  range  specification  could  be  derived  based  on  any  of  the  three  target  motion 
types  (stationary,  worst-case,  or  uniformly  distributed). 

The  maneuverability  and  field  of  view  analyses  provided  other  general  insights  into 
the  effect  of  the  system  requirements.  For  example,  from  the  analyses  it  became  apparent 
that  the  minimum  launch  range  requirement  defines  the  maneuverability  specification  and 
that  the  maximum  launch  range  requirement  drives  the  seeker  field  of  view  specification. 
This  insight  is  useful  in  determining  design  tradeoffs  with  relaxed  system  requirements. 
For  example,  if  the  minimum  range  requirement  for  the  NPS  HRGK  were  increased  to  2 
km,  the  effect  on  the  seeker  would  be  minimal,  but  the  affect  on  the  thruster  design  would 
be  significant.  Knowing  this  the  designer  can  determine  the  value  of  relaxing  the  system 
requirement. 

2.         Limitations  of  Models 

Despite  the  usefulness  of  the  codes  discussed  in  this  chapter,  some  caution  must  be 
applied  when  using  these  and  other  mathematical  or  computer  models.  Model  outputs  are 
sometimes  skewed  or  erroneous  because  of  (1)  the  nature  of  the  model  with  its  simplifying 
assumptions  or  (2)  because  of  the  model's  sensitivity  to  incorrect  inputs.  Examples  of  both 
these  error  sources  are  abundant  in  the  NPS  HRGK  analyses. 

For  example,  both  the  maneuverability  and  the  field  of  view  analyses  used  a  two- 
dimensional  model  for  a  three  dimensional  problem.  Implied  in  these  models  is  the 
assumption  that  the  azimuth-related  maneuverability  and  field  of  view  requirements 
outweigh  the  elevation-related  requirements.  In  reality,  the  HRGK  must  execute  significant 
maneuvers  in  both  yaw  and  pitch  (at  the  same  time)  to  prosecute  a  target  as  it  appears  in  the 
seeker's  field  of  view.  Furthermore,  the  field  of  view  sizing  may  well  be  defined  by  the 
requirement  to  look  down  at  a  target  rather  than  the  requirement  to  look  to  the  left  or  right 
for  a  target.  These  possibilities  are  not  accounted  for  in  the  two-dimensional  model. 

Another  example  of  potentially  incorrect  modeling  concerns  the  appropriateness  of 
the  target  motion  model.  The  constant  heading  and  speed  with  uniform  distributions  may  be 
an  appropriate  target  motion  model  for  short  time  of  flight  attacks  against  sluggish  ground 
targets.  However,  this  type  of  target  motion  model  would  not  be  appropriate  in  a  surface- 
to-air  scenario  where  the  target  may  have  evasive  maneuver  capability.  In  general,  any 
target  motion  model  must  allow  for  a  reasonable,  worst-case  evasive  maneuver. 


48 


I  I  I 


'  ''-,!.. ' 


0.4       0.6       0.8      1.0       1.2       1.4    1.6 

Mach  Number 


0.4      0.6      0.8       1.0       1.2       1.4     1.6 

Mach  Number 


Figure  24.   Sensitivity  Analysis  of  the  Steady-Turn  Maneuver  Model  for  Thruster   Sizing, 
(a)  CM  ±  100%  and  (b)  XCP  ±  3%. 

Finally,  Figure  24  illustrates  the  importance  of  sensitivity  analysis  and  the  potential 

for  erroneous  results  due  to  incorrect  inputs.  The  figure  shows  data  from  the  control 

thruster  sizing  analysis  with  variations  in  the  steady-turn  model  parameters.  The  plot  on  the 
left  (a)  shows  the  effect  of  ±100%  changes  in  CM  .  Doubling  or  zeroing  the  parameter 

produces  only  minor  changes  in  required  control  thrust.  Clearly,  the  thrust  required  is  very 
insensitive  to  changes  in  CM  .  On  the  other  hand,  the  plot  on  the  right  shows  the  impact  of 

±3%  changes  in  xCP.  The  small  changes  in  this  input  parameter  affect  the  required  control 
thrust  significantly. 

Another  point  of  interest  illustrated  in  the  above  example  is  the  importance  of 
parameter  selection.  In  the  NPS  HRGK  case,  the  parameter  xCP  is  measured  from  the  nose 
of  the  guided  rocket  and  varies  between  28.5  and  31.8  inches  as  a  function  of  Mach 
number.  Small  percentage  changes  in  xcp  result  in  large  percentage  changes  in  the  rocket's 
static  margin  (the  small  distance  between  the  xcp  and  the  xCG).  In  fact,  the  ±3%  change  in 
xcp  produces  between  a  ±22.7%  and  ±86.5%  change  in  static  margin.  Although  either  xCP 
or  static  margin  could  be  used  as  model  inputs,  the  model  sensitivity  to  percent  changes  in 
the  two  parameters  would  vary  significantly. 

3.         Additional  Work 

The  computer  models  developed  by  the  author  could  be  modified  to  produce  new  or 
refined  models.  The  addition  of  probabilistic  target  motion  to  the  turn  rate  and  turn  radius 
code  would  provide  useful  data.  The  analysis  of  elevation  plane  considerations  would  also 
be  helpful  in  the  guided  rocket  design.  A  model  that  could  estimate  both  the  required 
vertical  field  of  view  and  pitch  axis  maneuverability  would  help  in  verifying  the  thruster 


49 


50 


IV.     SIX-DEGREE  OF  FREEDOM  SIMULATION 

This  chapter  addresses  the  use  of  detailed  computer  modeling  in  the  missile  design 
process.  The  chapter  describes  a  six-degree  of  freedom  (6DOF)  simulation  developed  by 
the  author  for  the  analysis  of  the  NPS  HRGK.  The  HRGK  6DOF  is  a  major  modification 
to  a  simulation  developed  by  Professor  Robert  (Gary)  Hutchins,  Electrical  and  Computer 
Engineering  Department,  Naval  Postgraduate  School,  and  his  1996  "Navigation,  Missile, 
and  Avionics  Systems"  class.  The  simulation  was  developed  in  Simulink  using  Matlab 
functions  to  perform  the  simulation's  computations.  The  listings  of  the  Matlab  functions 
are  included  in  Appendix  C. 

Currently,  the  6DOF  code  requires  more  than  20  minutes  to  simulate  a  15  second 
rocket  flight1.  To  speed  up  the  simulation,  some  of  the  simulation  features  are  not  yet  fully 
implemented,  but  the  author  is  in  the  process  of  converting  the  simulation  to  a  more 
efficient  computation  structure.  This  structure  uses  the  native  Simulink  "S-function"  and 
was  introduced  to  the  author  by  Mr.  Allen  Robins;  Dynamics  and  Controls  Section,  Naval 
Air  Warfare  Center  Weapons  Division,  China  Lake,  California.  (Appendix  B  briefly 
describes  the  different  computation  structures  the  can  be  used  in  Simulink.)  Preliminary 
tests  by  the  author  have  shown  that  the  use  of  the  S-function  structure  can  cut  simulation 
run  times  by  a  factor  of  nearly  30. 

This  chapter  begins  with  an  overview  of  the  simulation  architecture  and  then 
addresses  the  functions  of  the  major  simulation  components.  A  section  of  the  chapter  is 
devoted  to  the  development  of  the  guided  rocket  equations  of  motion.  And  another, 
presents  simulation  results  used  in  the  design  of  the  NPS  HRGK.  The  chapter  ends  with  a 
few  observations  concerning  the  use  of  detailed  simulations  in  the  missile  design  process. 

A.         SIMULATION  OVERVIEW 

The  HRGK  6DOF  simulation  is  designed  to  be  used  with  an  initialization  script  file. 
This  Matlab  file  is  executed  prior  to  running  the  simulation.  It  sets  up  simulation 
parameters  such  as  vectors  with  the  rocket's  aerodynamic  coefficients  (functions  of  Mach) 
and  time  profiles  for  mass  properties  and  the  main  rocket  thrust.  The  file  also  sets  the  initial 
conditions  for  all  the  simulation  states.  When  run,  the  simulation  uses  a  user-selectable 


1  Silicon  Graphics,  Inc.  Indigo  II  workstation 


51 


integration  scheme  to  simulate  the  rocket  flight.  The  simulation  writes  several  output 
vectors  to  the  Matlab  workspace.  These  outputs  include  the  flight  time,  target  miss 
distance,  the  guided  rocket's  states  (U,  V,  W,  P,  Q,  R,  North,  East,  Altitude,  (p,  6,  and 
y/),  and  their  time  derivatives  with  respect  to  the  appropriate  coordinate  frames. 

Figure  25  shows  a  top-level  block  diagram  of  the  HRGK  6DOF.  Figure  26  and 
Figure  27  respectively  show  detailed  views  of  the  "Rocket  Dynamics"  and  the  "Parameters 
and  Coefficients"  blocks  as  they  are  currently  implemented. 


0- 


Clock 


55 


time 


flight  time 


2.75"  GUIDED  ROCKET  SIMULATION 

TgtN 


on 


0*Rlce 


timer  noise     timer 


foM-' 


Guidance 

Delay 

Switch 


off 


WsT"*^.      :        phi 

SkVNofses    ;  v\ 


r... ,, 


angle 
Off  Bore 


TgtE 


Tgth 


Target 
Dynamics 


' Sj 


Angle 


'•j 


Thruster 
Cmds 


Controller 


Seeker  / 
Filter 


's> 


MslN» 
MslE 


Mslh, 


phi  skii 


theta , 


psi 


V 


LOS's 


True  Horiz  Line  of  Sight 


Wind  Gust  Models 


Rocket         True  Geometry 
Dynamics 


True  Vert  Line  of  Sight 


|  Dmiss    | 
Miss  Dist 


Demux 


Figure  25.  HRGK  6DOF  Simulation  Overall  Block  Diagram.  (Dashed  blocks  are  not  yet  fully 
implemented.) 

The  blocks  in  Figure  25  with  dashed  lines  are  not  yet  fully  implemented.  For 
example,  the  thruster  control  inputs  are  not  currently  generated  based  on  seeker  inputs,  but 
rather,  the  thruster  commands  must  be  "hard  coded"  into  the  conttoller  block. 

Inside  the  Rocket  Dynamics  block,  the  time  derivatives  of  the  rocket's  states  are 
computed  by  a  Matlab  function.  The  state  derivatives  are  then  integrated  and  feedback  into 


52 


the  function  and  other  parts  of  the  simulation.  The  rocket's  aerodynamic  and  mass  property 
parameters  are  determined  using  lookup  tables  in  the  Parameters  and  Coefficients  block. 


Thruster  1 


1 


Thruster  2 


T1 


Thruster  3 


T2 


T3 


Thrust  dat 


Thrust  History 


Rkt  X 


Rocket  State  Vector 


Rocket  Rate  Vector 


Rkt  Xdot 


Matlab 
Function 


1/s 


ROCKET  DYNAMICS 


FlatEarth 
MotionEqns 


Integrator 


Mach 


■£: 


*>-T- 


Matlab 
Function 


Mach  No. 


Params  &  Coefs 


u.v.w, 

P.Q.R 

phi 

c 

theta 

X 

E 

CD 

Q 

*   6 

-^.P_rocket 

1 

Pn 

2 

Pe 

3 

Rocket 
States 


Figure  26.  Detailed  View  of  the  Rocket  Dynamics  Block. 


X 

Z3 
5 

* — I  Mass_dat 

I IME  &  MACH 

1 

Mass  History 
*-|    CG.dat    | 
CG  History 

DEPENDENT 

—  DADAft/ICTCDC 

Mach 

V 

rArlAlVIt  I  tno 

*— |    JxR_dat 

Rocket  Jx 
« — |    JxK_dat 

Xcp 

h 

/ 

Guidance  Kit  Jx 

*^         C 

IdO-Bun 

ling 

1 

4 — I     Jy_dat 

^A 

Param& 
Coefs 

Jy  History 
«a — |     R_Pdot 

*^ 

Thrust 

X 

rocket  roll  accel 

A 

c 

dO-Coas 

sting 

CNa 

/ 

/ 

CMq 

Clp 

Figure  27.  Detailed  View  of  the  Parameters  and  Coefficients  Block. 


53 


Figure  28  shows  the  overall  architecture  for  the  simulation  currently  under 
development.  Most  of  the  components  of  this  architecture  are  described  in  the  following 
section. 


*— » 

target  motion 

Geomeiry 

Environment                                           | 

•  ••                                             i 

— 

relative 
LOS 

translation 
matrix 

absolute 
positions 

clock 

atmosphere 

gravity 

wind 

Missue                   r- 

parameters  & 
coefficients 

non-linear 
equations 
of  motion 

^ 

Mach 

i 

i 

I 

i  i 

I 

- 

seeker 

forces  & 
moments 

i 

control 
logic 

IMU 

ii 

i 

Figure  28.  Future  6DOF  Architecture. 


B. 


SIMULATION  COMPONENTS 


The  simulation  shown  schematically  in  Figure  28  is  made  up  several  components. 
These  components  are  very  briefly  described  in  the  following  subsections. 

1.  Environment 

The  environment  block  simulates  the  environment  in  which  the  rocket  operates. 
This  block  produces  the  simulation  time  and  provides  constant  gravity.  It  also  computes  the 
atmospheric  properties  needed  to  calculate  the  Mach  number  and  dynamic  pressure.  The 
block  uses  the  Dryden  wind  gust  model  to  produce  wind  gusts  in  pitch  and  yaw. 

2.  Missile 

The  missile  block  simulates  the  dynamics  of  the  guided  rocket.  The  block  has 
several  sub-components.  These  blocks  are  divided  to  align  as  closely  as  possible  with 
actual  subsystems.  The  missile  block  uses  external  signals  and  data  to  produce  realistic 
seeker  and  IMU  signals.  From  these  signals  control  inputs  are  determined  and  applied  to 


54 


the  forces  and  moments  sub-component.  The  forces  and  moments  component  computes  all 
the  body  frame  forces  on  the  rocket.  From  these  forces  and  moments  the  dynamics  of  the 
body  can  be  determined  by  the  flat-earth  equations  of  motion. 

3.  Target  Dynamics 

The  target  block  provides  the  target  location  and  motion  to  the  relative  geometry 
block.  This  data  allows  the  line  of  sight  (between  the  rocket  and  the  target)  and  the  line  of 
sight  rate  data  to  be  calculated. 

4.  Relative  and  Absolute  Geometry 

The  relative  geometry  block  uses  the  target  and  rocket  dynamics  to  solve  the 
navigation  equations  (described  in  the  following  section).  From  the  navigation  equations, 
the  absolute  and  relative  position  of  the  rocket  and  target  can  be  can  be  computed.  The  line 
of  sight  and  line  of  sight  rates  are  also  provided  by  the  geometry  block. 

C.        EQUATIONS  OF  MOTION 

This  section  contains  a  brief  derivation  of  the  flat-earth  equations  of  motion  as  they 
apply  specifically  to  the  HRGK  6DOF.  The  derivation  is  adapted  from  Stevens  and  Lewis 
[Ref.  15]. 

1.         General  Matrix-Form  Equations 

Starting  with  generic  flat  earth  equations  of  motion: 

F 

vB  =  -£2Bvg  +  BBg'Q  +  —  (Force  Equation) 

m 

cog  =  —J~  QgJ(dg  +  J~  TB  +  HR  (Moment  Equation) 

6  =  E(Q>)(£>B  (Attitude  Equation) 

T 

Pned  =  ^ByB  (Navigation  Equation) 

where  the  vB  is  the  velocity  vector,  [U  V  W]T;  gg  is  the  gravity  vector  at  the  surface  of  the 

earth  (adjusted  for  the  earth  rotation),  [0  0  9.81m/s]T;  FB  is  the  force  vector,  [Fx  Fy  FJT; 
coB  is  the  angular  rate  vector,  [P  Q  R]T;  TB  is  the  torque  vector,  [L  M  N]T;  HRi  to  be 
defined  later,  is  a  vector  to  account  for  the  gyroscopic  effects  of  the  spinning  rocket  motor; 
O  is  the  Euler  angle  vector,  [<J)  6  V|/]T;  pNED  is  the  north-east-down  frame  position  vector, 
[pN  pE  pD]T;  Qq  is  the  body  frame  angular  rates  cross-product  matrix, 


55 


nB  = 


BB  is  the  rotation  matrix  from  the  North-East-Down  frame  to  the  body  frame, 


0 

-R 

Q~ 

R 

0 

-p 

-Q 

p 

o  _ 

BB  = 


10  0 

0     cos0      sin</> 


cos0    0    -sin0 
0        1         0 


0    -sin0    cos0_||_sin0     0     cos0  J       0 
J  is  the  inertia  matrix  (with  symmetry  assumed  about  the  x-axis), 


cosi/a      sini/f     0 
-sini/f    cosy/    0 
0        1 


J  = 


0 


0 
0 


0 
0 


0      /, 


and  the  Euler  angle  function  is 


£(0)  = 


1     tan0sin0    tan#cos0 

0        cos0  -sin<p 

sin0  cos(f> 


cosO  cos  6 

The  state  vector  is  [vBT  coBT  FT  pNEDT]T.  Now  the  forces  and  moments  need  to  be  defined 
for  the  HRGK  case. 

2.         Forces  and  Moments 

The  aerodynamic  forces  and  moments  are  functions  of  Mach  number  and  can  be 
expressed  by  the  coefficients  and  their  derivatives.  The  following  summarizes  the  forces 
and  moments  computed  in  HRGK  6DOF: 


/ 


Fx=T-qA 


r2 


CD  + 

V  "a 


Fy  =  Ty+qA(-CNaP) 


(Axial  Force) 

(Side  Force) 
(Down  Force) 


(C,  d    \ 


L  =  qA  d 


\  113 


+  TR       (Rolling  Moment  on  HRGK) 


56 


M  =  TZXT  +  qAd 


N  =  TyXT  +  qAd 


Cm  OCXrp  + 


'N, 


CP 


CMad 

— —Q 

w 


\ 


Cm  d 

Cm    uXcp  H R 


\ 


(Pitching  Moment) 


(Yawing  Moment) 


where  T  is  the  rocket  motor  thrust;  Ty  and  Tz  are  the  y  and  z  components  of  the  reaction 
thruster  forces,  Tz  =  r,cos0  +  T2cos(<(h-2/3k)  +  r3cos(</H-4/37c)  and  T  =  T^siiKp  + 
r2sin( 0+2/371)  +  r3sin(0+4/37t);  and  TR  is  the  rolling  torque  exerted  on  the  HRGK  by  the 
spinning  rocket  motor, 

TR  =  idXRPR 

where  fi  is  the  coefficient  of  friction  for  the  bearings,  Jx  is  the  axial  moment  of  inertia  for 
the  rocket  alone,  and  PR  is  the  rocket  motor's  spin  axis  angular  acceleration  (adapted  from 
Ref.  [11]).  The  state  PR  has  to  be  added  to  the  state  vector,  and  the  gyroscopic  affect  of  the 
spinning  rocket  is  added  to  the  moment  equation.  The  effect  can  be  accounted  for  by  adding 
the  vector  HR  to  the  right-hand  side  of  the  moment  equation,  where  HR  is 

T 


H.= 


0 


,RJxBPR 


Q'x.Pr 


3.         HRGK  Scalar  Flat-Earth  Equations 

From  the  general  matrix  equations,  the  HRGK-specific  scalar  form  of  the  equations 
can  be  written  in  terms  of  the  defined  forces  and  moments. 

a.  Force  Equations 

U  =  RV-QW-g0sine  +  ^L 

m 


V  =  -RU  +  PW  +  g0sm<l)cosd  +  ^- 

m 


W  =  QU-PV  +  g0  cos0cos0  +  -^ 

m 


Moment  Equations 


57 


Q  = 

(    j) 

R  = 

'LA 

A    ) 

PR 


J..      7. 


RPD 


N     Jx 

Jy  Jy 


Kinematics   Equations 

0=  P  + tan  0(0  sin  0  +  /?  cos  0) 


0  =  QcosQ-  R  sirup 

,COS0 


i//  =  Qsirut)  +  R 


cos  6 


d.         Navigation   Equations 

PN  =  £/cos#cosyf  +  V (sin 0 sin 6 cost//  -cos 0 sin y/)  +  W(cos0sin0cosy/,  +  sin0siny/') 

/^  =  £/cos0siny/'  +  V(sin0sin0sini//  +  cos0cosy/)  +  W(cos0sin0siny/-sin0cosi//') 
PD  =  -Usin6+  Vsin0cos#  +  Wcos0cos0 


D. 


NPS   HRGK   RESULTS 


The  force  and  moment  equations,  as  well  as  the  equations  of  motion,  are 
implemented  in  a  Matlab  function  which  is  used  by  the  HRGK  6DOF  simulation.  The 
following  subsections  show  two  examples  of  how  the  simulation  was  used  in  the  design. 

1.  Ballistic  Flight  and  Maximum  Range  (3DOF) 

Early  in  during  the  NPS  HRGK  design  process,  the  HRGK  simulation  was 
simplified  to  only  include  the  three  dimensions  of  the  rocket's  longitudinal  dynamics.  (The 
gyroscopic  effects  of  the  spinning  rocket  were  also  ignored.)  Using  the  simplified  3DOF, 
the  results  shown  in  Figure  29  were  obtained.  These  results  confirmed  that  the  HRGK- 
equipped  rocket  could  meet  the  time  to  target  and  the  maximum  range  requirements. 

2.  Pulse  Width  Modulated  Turn  Rate  (6DOF) 

Later  in  the  design,  the  affect  of  thruster  pulse  width  modulation  was  investigated 
using  the  HRGK  6DOF.  Two  simulations  were  run.  The  first  simulated  the  missile  flight 


58 


Figure  29.  HRGK-Equipped  Rocket  Ballistic  Trajectories.  Ground  launch  at  sea 
level  with  various  launch  elevation  angles.  Solid  lines  are  for  cold  rocket  motor, 
dashed  lines  are  for  hot  rocket  motor,  and  flight  times  are  indicated. 

with  a  constant  thruster  force  from  the  appropriate  side  only.  The  second  used  a  pulsed 
sequence  of  all  three  thrusters,  but  with  a  longer  pulse  on  the  appropriate  side.  The  results 
for  the  two  runs  are  shown  in  Figure  30.  From  the  analysis,  it  was  determined  that  constant 
thrust  does  not  generate  satisfactory  maneuvers.  (In  fact,  in  the  simulation,  the  rocket 
began  "cartwheels"  without  making  substantial  changes  in  cross-track).  On  the  other  hand, 
the  pulsed  control  generated  an  adequate  maneuver  while  maintaining  a  somewhat 
reasonable  side  slip  angle,  /?. 


X-Trk=12.1  m 

Constant  Thruster  Maneuver 

$             "$ 

//          11 

to 

c 

n                     r\                              r\    j. 

it      -^i— ^— ^ 

a 

'a 

0 

"t, C?'"'                       W                           "» 

CO 

CL 

p  =  9.7°                       p  =  29.2° 

p  =  52.1° 

P  =  79.7° 

o 

CO 

h- 

co 

o 

Pulsed  Thruster  Maneuver 

X-Trk  =  13.6  m 

O 

(with  short  opposing  pulses) 

P  = 

=  32.3° 

p  =  3.6°                    P  =  12.1°                  P  =  22.3° 

0                               50                              100 

1 50 

200 

Down -Track  Range  (m) 

Figure  30.  Sustained  and  Pulsed  Thruster  Maneuvers.  Flight  paths  and  headings  indicated  for 
a  0.8  second  maneuver. 


59 


E.         OBSERVATIONS 

As  designs  become  more  refined  and  leave  the  conceptual  phases,  high  fidelity 
simulations  becomes  more  important.  In  the  case  of  the  NPS  HRGK,  the  6DOF  simulation 
was  helpful  in  performing  some  detailed  analyses.  Nevertheless,  the  time  required  per 
simulation  run  precluded  the  use  of  the  simulation  in  many  cases. 

Fast,  modular  simulations  that  can  be  easily  modified  for  various  designs  are 
desirable  design  tools.  In  many  cases,  lower-order  simulations  (3DOF  or  5DOF)  meet  all 
the  needs  of  the  designer.  For  example,  the  3DOF  version  of  the  HRGK  simulation  had 
sufficient  fidelity  to  be  used  in  the  maximum  range  and  time  to  target  analyses. 
Unfortunately,  other  analyses  required  a  6DOF  so  that  the  roll  of  the  seeker  and  the 
gyroscopic  effects  of  the  spinning  rocket  could  be  observed. 

If  the  restructuring  of  the  HRGK  simulation  provides  the  expected  improvement  in 
run  time,  this  simulation  would  become  a  much  more  useful  design  tool.  With  run  times  of 
only  a  minute  or  so,  several  design  iterations  could  be  evaluated  quickly. 


60 


V.     CONCLUSIONS  AND  RECOMMENDATIONS 

The  purpose  of  this  thesis  is  to  provide  documentation  for  the  several  computer 
codes  developed  by  the  author  during  the  design  and  analysis  of  the  NPS  HRGK.  The 
thesis  is  also  intended  to  provide  insights  concerning  the  use  of  modeling  in  the  design  and 
analysis  processes.  Several  germane  points  can  be  made  in  conclusion. 

First,  models  with  differing  levels  of  complexity  are  appropriate  during  different 
phases  of  the  design.  Models  used  in  the  analysis  of  conceptual  and  preliminary  designs  are 
usually  the  simplest  while  those  models  used  in  design  refinement  are  more  complicated. 
COEA-related  modeling  is  essential  throughout  the  design  process  to  ensure  that  the  design 
meets  the  system's  cost  effectiveness  goals. 

Second,  simplistic  models  can  be  used  to  ensure  the  flow  down  of  system 
requirements  into  design  specifications.  In  this  way,  the  models  help  ensure  a  design 
driven  by  the  requirements.  Often  these  simple  models  provide  great  insight  into  additional 
aspects  of  the  design  problem  allowing  several  critical  specifications  to  be  defined  in  the 
earliest  phases  of  the  design. 

Third,  modeling  in  the  design  process  should  include  the  appropriate  mix  of 
probabilistic  and  worst-case  conditions.  These  two  types  of  models  can  provide  very 
different  results  which  must  be  judiciously  evaluated. 

Finally,  knowing  the  sensitivity  of  a  model  to  variations  in  parameters  is  critical  to 
the  proper  use  of  the  model  in  decision  making.  Sensitivity  analyses  or  "what-if '  studies 
must  be  performed  to  ensure  the  modeling  is  providing  reliable  data. 

In  addition  to  these  concluding  points  several  recommendations  for  follow-on  work 
are  also  appropriate.  First,  the  design  and  analysis  of  the  NPS  HRGK  would  have  been 
facilitated  by  simple  Matlab  codes  that  allow  the  early  aerodynamic  sizing  of  the  kit.  Such 
a  program  would  be  helpful  in  evaluating  the  effect  of  nose  shape  and  canard  size  and 
shape  on  the  range  of  the  HRGK-equipped  rocket. 

Additional  work  is  also  needed  to  include  a  probabilistic  target  motion  model  in  the 
maneuverability  analysis  code.  The  development  of  models  that  evaluate  the  required  pitch 
maneuverability  and  the  required  elevation  field  of  view  would  also  be  useful. 

Finally,  the  modification  of  the  HRGK  simulation  to  make  it  a  generic  6DOF 
simulation  would  be  very  useful  for  future  missile  design  work.  The  Simulink  architecture 
can  offer  fast  modular  simulations  that  can  be  easily  modified  to  meet  future  users'  needs. 


61 


62 


APPENDIX  A.    MISSILE  AERODYNAMICS  NOMENCLATURE 

Many  of  the  aerodynamic  coefficients,  stability  derivatives,  and  various  other  terms  and 
symbols  used  in  this  thesis  are  defined  below.  The  nomenclature  closely  follows  that  used 
by  Chin  [Ref.  16]  with  some  adaptations  like  that  used  by  Blakelock  [Ref.  17]  and  by 
Howard  [Ref.  18].  It  is  worth  noting  two  significant  differences  between  terms  associated 
with  missiles  and  those  associated  with  traditional  aircraft.  First,  most  missile  coefficients 
are  referenced  to  the  missile's  diameter  or  its  cross-sectional  area  (normal  to  the 
longitudinal  axis)  rather  than  a  wing  chord  or  a  planform  area.  Second,  normal  and  axial 
force  coefficients,  C^  and  C^,  (aligned  to  the  body-axis  system)  are  used  rather  than  the 
lift  and  drag  force  coefficients,  Cl  and  Cr>  (in  the  stability  axis  system).  When  the 
missile's  angle  of  attack  (a)  is  small  there  is  very  little  difference  between  the  two  systems, 
but  for  significant  angles  of  attack,  the  relationship  given  in  the  following  equations  should 
be  used. 

Ca  =  Cd  cos  a  -  Cl  sin  a 
CN  =  CL  cos  a  -  CD  sin  a 
Reference  Dimensions 

A  or  Sn    Cross-sectional  area  of  missile  body  (5.94  in2) 

d  Diameter  of  missile  body  (2.75  in) 

L  Missile  length  (68.75  in) 

Xcg  Distance  from  nose  tip  to  center  of  gravity  on  missile  (27  to  3 1 .7  in) 

Flight  Conditions 

a         angle  of  attack 

P  side  slip  angle 

M         Freestream  Mach  number 

Vt         Freestream  air  speed 

p  Freestream  air  density 

q  Dynamic  pressure  ( 4  pVt  ) 


63 


Aerodynamic  Coefficients* 

CT         Coefficient  of  lift, 

qsn 

N 
CN       Normal  force  (N)  coefficient, 


qsn 

M 

Cm       Pitching  moment  (M)  coefficient, 


qSnd 
C  a        Axial  force  (A)  coefficient, 

qsn 

CDn      Drag  force  coefficient  at  zero  lift 

Cjxj     Coefficient  of  drag  due  to  normal  force  (DN),  — — 

Static  and  Dynamic  Derivatives* 

op 

Cw       Normal  force  coefficient  per  radian  of  angle  of  attack,  — — 
a  da 

op 

CM       Pitching  moment  coefficient  per  radian  of  angle  of  attack,  — — 
a  da 

—   — -M 
d  J  da 

Missile  "States" 

£/,  V,  W      Body  axis  velocities  (jc,  v,  and  z-axes) 

P,  (>,/?        Body-axis  angular  rates  (about  the  x,  y,  and  z-axes) 

0, 9,  i//         Euler  Angles  (roll,  pitch,  yaw) 

pn>pbpd    Flat  Earth  Positions  (North,  East,  Down) 

h  height  {-PD) 


Directional  coefficients  and  derivatives  {e.g.;  CY,  C^,  CY  ,  C  ^  ,  C  ^  )  are  assumed  equal  to  their 
longitudinal  counterparts  {e.g.;  CN,  CM,  CNot,  CMa,  CMq;  respectively). 


64 


APPENDIX  B.     NOTES  ON  MATLAB  AND  SIMULINK 


A.         MATLAB 

Matlab  is  a  high-performance  numeric  computation  and  visualization  computer 
software  package.  It  is  essentially  a  high-level  programming  language  that  is  especially 
tailored  for  the  manipulation  of  matrices,  (see  Matlab  Reference  Manual  and  Matlab 
User 's  Guide) 

Matlab  commands  can  either  be  invoked  in  an  interactive  mode  at  the  keyboard  or 
through  Matlab  script  files.  Another  type  of  file  is  the  Matlab  function.  Both  the  script 
and  function  files  are  labeled  with  an  "m"  extension  at  the  end  of  the  file  name  and  are 
commonly  called  M-files. 

A  script  M-file  can  run  as  a  stand  alone  "batch"  file  invoking  Matlab  commands 
when  the  file  is  executed.  On  the  other  hand,  a  function  M-file  is  called  from  within  script 
files  or  from  the  keyboard.  A  function  can  be  called  with  or  without  arguments  and  can 
return  computed  values  when  completed. 

Variables  declared  or  set  in  a  script  file  or  at  the  keyboard  are  available  throughout 
the  Matlab  "workspace."  These  variables  are  common  and  can  be  accessed  by  other  script 
files.  On  the  other  hand,  variables  declared  in  a  function  M-file  are  local  variables  and  are 
not  available  in  the  workspace.  Furthermore,  functions  do  not  have  access  to  workspace 
variables  unless  the  variables  are  passed  to  the  function  or  are  declared  as  "global"  in  both 
the  workspace  and  in  the  function. 

The  standard  syntax  used  in  the  first  line  of  a  function  M-file  is 

function  y  =    funciu) 
where  "function"  is  the  Matlab  function  indicator,  y  is  the  variable  that  will  be  passed 
back,  u  is  the  input,  and  func  is  the  name  of  the  function  (func.m). 

Files  that  are  written  in  C  or  FORTRAN  can  be  converted  into  executable  Matlab 
files  with  a  ".mex"  extension  by  using  the  "cmex"  or  "fmex"  commands,  (see  Matlab 
External  Interface  Guide)  Mex-files  have  the  advantage  of  running  faster  than  normal  M- 
files. 


65 


B.         SIMULINK 

Simulink  is  a  dynamic  system  simulation  software  package  and  an  extension  to 
Matlab.  Simulink  offers  many  ready-made  blocks  that  can  be  assembled  graphically  to 
model  a  dynamic  system.  Two  special  Simulink  blocks  allow  tremendous  flexibility  in 
modeling  systems,  (see  Simulink  User's  Guide) 

The  first  is  the  Matlab  function  block.  This  block  when  placed  in  a  block  diagram 
will  execute  the  commands  in  the  indicated  M-file  function.  So  for  example,  the  equations 
of  motion  for  a  system  could  be  placed  in  a  M-file  function.  This  function  would  have 
inputs  from  simulink  that  include  the  states  and  the  control  input.  The  function  would  then 
output  the  states'  time  derivatives  which  could  then  be  integrated  and  then  passed  back  the 
Matlab  function. 

This  method  runs  extremely  slow! 

The  second  more  useful  block  is  the  S-function  block.  This  block  calls  either  a  M- 
file  or  a  Mex-file  that  is  specifically  structured  to  run  efficiently  in  Simulink  (see  Simulink 
1.3  Release  Notes).  Chapter  4  of  Simulink  1.3  Release  Notes  outline  the  procedure  for 
creating  Mex-files  from  C  codes  for  use  in  fast  running  Simulink  models. 

Additionally,  simulink  models  that  do  not  contain  M-file  references  (that  use  Mex- 
files  in  the  S-function  blocks)  can  be  turned  into  stand-alone,  accelerated,  executable  code 
using  Simulink's  Real-Time  Workshop  (see  Real-Time  Workshop  User's  Guide). 


66 


APPENDIX  C.     MATLAB  CODE  LISTINGS 


This  appendix  provides  listings  of  the  following  Matlab  computer  codes  which 
were  discussed  in  this  thesis: 

•  Single-Shot  Probability  of  Kill 

•  Length,  Weight,  and  Center  of  Gravity 

•  Turn  Rate  and  Radius 

•  Steady-Turn  Maneuver 

•  Seeker  Field  of  View 

•  6DOF  Codes 
-Initialization  Code 
-Aerodynamics  Data 
-Equations  of  Motion 
-Line  of  Sight 
-Mach 

The  user-defined  inputs  in  the  codes  are  denoted  with  bold-face  type. 
C.         SINGLE-SHOT  PROBABILITY  OF  KILL 

%  Single  Shot  Pk  using  "Cookie  Cutter"  methodology 

%  W.  Mark  Wonnacott,  Naval  Postgraduate  School,  May  1997,  corrected/revised   Aug  1997 

n%%%%nnHn%%iNPOTs 

%  Terminal  Dive  Angle  (deg  relative  to  horizontal) 

dive   «   20; 

%   CEPs    to   be   used   in   computations    (for   a   single   CEP   set   CEP_min   egual    to  CEP_max) 

CEP_min      =       .5;  %   minimum   CEP    (m)    to   be   used 

CEP.max      =      10  0;  %  maximum   CEP    (m)    to   be   used 

%   Target    Size    (m)  ,    Prob.    of   Kill   given  Hit,    &  Desired   Prob.    Kill 


trgt='Light   Arm.     (AirDef)' 
trgt=' Light   Armor    (APC) ' ; 
trgt= ' Large   Truck ' ; 


L    =    5.5;    H    =    2.0;    W    =    2.5;    Pkh   =.90;    PkD   =.95 

L    =    5.5;    H    =    2.0;    W    =    2.5;    Pkh    =.80;     PkD    =.60 

L    =    8.0;    H    =    2.5;    W    =    2.5;    Pkh    =    1;       PkD   =.50 

L     =     14.;      H     =     1.5;      W     =     4.5;      Pkh     =.70;     PkD     =.75;  trgt='Patrol    Boat' 

%%%%%%%%%%%%%%%!.  NP  UTS 

%    Elevation   and  Crossrange   Projected   Dimensions    (l:Head  on   &    2:Broadside) 

Yl    =    H*cos(dive*pi/180) +L*sin(dive*pi/180)  ; 

XI    =   W; 

Y2    =    H*cos(dive*pi/180)+W*sin(dive*pi/180)  ; 

X2    =    L; 

%CEPs  in  desired  range 

if  CEP_min<=0,  CEP_min=eps;  end  %CEP_min  set  to  allowable  small  number 
if  CEP_max<=0,  CEP_max=eps ;  end  %CEP_max  set  to  allowable  small  number 
if  CEP_min>CEP_max,  temp=CEP_min;  CEP_max=CEP_min;  CEP_min=temp;  end 


67 


CEPs  =  logspacedoglO  (CEP_min)  ,  loglO  (CEP_max) )  ; 

sigmas  =  CEPs/1 .1774;  %corresponding  sigmas, 1-D  miss  distances  (m) 

S  =  sigmas*sqrt  (2)  *2;  %normalizd  half  sigmas  for  Matlab 

%  Probability  of  Hit  (equal  (50%)  prob.  of  broadside  or  head-on  encounter) 

Ph  =  0.5*(erf (Yl./S) .*erf (Xl./S) )  +  0.5* (erf (Y2. /S) . *erf (X2 . /S) ) ; 

%  Probability  of  Single  Shot  Kill 

PkSS  =  Pkh*Ph; 

%  Number  of  Weapons  Needed  to  Achieve  Desired  Prob.  of  Kill 
N  =  log(l-PkD) ./log(l-PkSS) ; 

figure  (1)  %  Single-Shot  Probability  of  Kill 

semilogx  (CEPs, PkSS, '-') ,  grid  on 

axis ( [CEP_min, CEP_max ,0,1]) 

xlabeK'CEP  (m)  '  )  ,  y  label  ('  Single  Shot  Probability  of  Kill') 

figure  (2)  %  Number  of  Weapons  per  Kill 

loglog  (CEPs,N, ' -' ) ,  grid  on 

axis( [CEP_min,CEP_max, 1,1000] ) 

xlabeK'CEP  (m)  ' )  ,  ylabel  ([' Number  of  Rockets  for  Pk  =  '  ,num2str  (PkD)  ] ) 

figure  (3)  %  Relative  Cost  for  Equal  Cost/Kill 

Nc  =  max (1, ceil (N) ) ; 
[X,Y]=stairs(CEPs,Nc(D  ./Nc) ; 
plot  (X,Y, '-' ) ,  grid  on 
axis ( [CEP_min, CEP_max, 0,1]) 
xlabel ( 'CEP  (m)  '  )  ,  ylabel ( 'Relative  Weapon  Cost' ) 

D.        LENGTH,  WEIGHT,  AND  CENTER  OF  GRAVITY 

%  GUIDED  Rocket  Center  of  Gravity  as  a  Function  of  Kit  Weight 
%============================================================ 

%  W.  Mark  Wonnacott,  Naval  Postgraduate  School,  May  1997 

%%%%   Design  Space  Paramters 

Wkl     =     linspace  ( 1,  8  )  ;  %  assummed  kit  mass  range    (lbm) 

Lkl     =  [5,10, 13.625,Lmax-Lr]  ;  %   specific  kit   lengths    (in) 

Dkl     =      [ .  05,  .  07,  .  0865,  .  1,  .  15]  ;  %  specific  kit  densities    (lbm/in~3) 

Cgkl=     1/3;  %  kit  Cg  from  nose    (fraction  of  kit   length) 

dia     =     2.75;  %  kit  diameter    (in) 

area>     pi*  (dia/2 )  A2;  %  kit  cross  section  area    (in~2) 


68 


%%%%    Constraint  Parameters 

Lmax=     72;  %  max  total   length 

Wmax=      30;  %  max  total  weight 

CgL     =     1/3;  %pod  forward  Cg  limit (frac  of   lug  spacing  back   from  front   lug) 


%%%%  Rocket  Parameters 
Lr     =     55.125; 
CgR_live     =     29.96; 
CgR_burn     =     33.55; 
WR_livo     =     22.95; 
WR_burn     =     15.73; 


%  rocket  length 

%  live  rocket  Cg  (inches  from  base) 
%  fired  rocket  Cg  (inches  from  base) 
%  live  rocket  mass  (lbm) 
%  fired  rocket  mass  (lbm) 


%%%%  Launcher  Pod  Parameters 

Llug7   =  43.2; 

Llugl9=   43.4; 

Cg7   =  Llug7-(CgL)*14; 

Cgl9=Llugl9- (CgL) *14; 

Cgpod   =  32.6; 

Wpod7  =  196.2, 

Wpodl9=  516.0; 


%  front  lug  from  back  of  launcher— 7  (in) 

%  front  lug  from  back  of  launcher— 19  (in) 

%  constraint  cg  (e.g.  within  middle  half  of  lug  space) 

%  constraint  cg  (e.g.  within  middle  half  of  lug  space) 

%  loaded  pod  Cg  from  back  w/o  kits  (in) 

%  loaded  pod (7)  mass  w/o  kits  (lbm) 

%  loaded  pod (19)  mass  w/o  kits  (lbm) 


%  (VARIOUS  DENSITY  CURVES  PLOTTED) 
%  ================================ 

Wk  =  Wkl'  *ones(l,  length (Dkl) ) ;    %  matrix  of  kit  masses 

Lk  =  Wk(:  ,l)*(l./(area*Dkl) )  ;     %  matrix  of  kit  lengths  (in) 

Cgk=  Lr+(1-Cgkl) *Lk;  %  kit  Cg  locations  (inches  from  base) 


Mk  =  Wk.*Cgk; 


%  Moment  about  base  due  to  kit  (lbm  in) 


%  centers  of  gravity  are  calculated  from  base  in  inches 
Cg_live  =  (WR_live*CgR_live+Mk) ./ (WR_live+Wk) ; 
Cg_burn  =  (WR_burn*CgR_burn+Mk) . / (WR_burn+Wk) ; 

%  centers  of  gravity  are  changed  to  calibers  from  nose 
Cg_live  =  ( (Lr+Lk) -Cg_live) /dia; 
Cg_burn  =  ( (Lr+Lk) -Cg_burn) /dia; 

%figure 
figure  (1) , 

plot  (Wk(:,l),  Cg_live) ,  grid  on 

title  ('Live  Round  with  Kit') 

ylabel  ('Cg  (Calibers  from  nose) ' ) 

xlabel  ('kit  weight  (lbm)') 

hold  on 


69 


figure  (2)  , 

plot  (Wk( : , 1) ,  Cg_burn) ,  grid  on 
title  ("Fired  Round  with  Kit') 
ylabel  ('Cg  (Calibers  from  nose) ' ) 
xlabel  ('kit  weight  (lbm)") 
hold  on 


%  (VARIOUS  LENGTH  CURVES  PLOTTED) 
%  =============================== 

Lk  =  Lkl; 

Dmin  =  0.  8*min(Dkl)  ; 

Dmax  =  1.2*max(Dkl) ; 

Dk  =  linspace (Dmin, Dmax) ; 

Dk  =  Dk ' *ones ( 1 , length ( Lk) ) ; 

Wk  =  area*Dk( : , l)*Lk; 

Cgk=  Lr+(l-Cgkl)*Lk; 

Cgk=  ones ( length (Wk) , 1 ) *Cgk; 


%  specific  kit  lengths  (in) 

%  lower  limit  on  assumed  density  range 

%  upper  limit  on  assumed  density  range 

%  assummed  kit  density  range  (lbm/in"3) 

%  matrix  of  kit  densities 

%  matrix  of  kit  masses  (lbm) 

%  kit  Cg  locations  (inches  from  base) 

%  matrix  of  Cg  locations 


Mk  =  Wk.*Cgk; 


%  Moment  about  base  due  to  kit  (lbm  in) 


%  centers  of  gravity  are  calculated  from  base  in  inches 
Cg_live  =  (WR_live*CgR_live+Mk)  .  /  (WR_live+Wk)  ; 
Cg_burn  =  (WR_burn*CgR_burn+Mk)  .  /  (WR_burn+Wk)  ; 

%  centers  of  gravity  are  changed  to  calibers  from  nose 
Cg_live  =  (  (Lr+ones  (length (Wk)  ,  1)  *Lk)  -Cg_live)  /dia; 
Cgjourn  =  ( (Lr+ones ( length (Wk) , 1) *Lk) -Cg_burn) /dia; 

figure  (1) , 

plot (Wk, Cg_live) 

plot ([ (Wmax-WR_live) *ones(l,2) ],  [1,20] ) %max  weight  constraint 
figure  (2), 

plot (Wk , Cg_burn) 

plot ( [  (Wmax-WR_live) *ones(l,2)  ]  ,  [1,20] ) %max  weight  constraint 

%  (POD  CENTER  OF  GRAVITY  CONSTRAINT) 
%  ================================== 

Wk  =  Wkl ' *ones (1, 2) ;  %  matrix  of  kit  masses 

%length  from  loaded  pod  (w/o  kits)  cg  to  constraint  Cg  (in) 

LI  =  ones ( length (Wk) , 1) * ( [Cg7,Cgl9] -Cgpod) ; 

%  loaded  pod  weights  (lbm) 

Wp  =  ones ( length (Wk) , 1) * [Wpod7,Wpodl9] ; 

X  =  ones  (length (Wk)  ,1)  *  [7, 19] ;    %  number  of  kits  per  pod 

%  length  from  rocket  nose  back  to  front  lug 

%  (assume  rocket  base  4"  from  back  of  pod) 

Lt  =  ones ( length (Wk) , 1) * [Lr-Llug7+4,Lr-Llugl9+4]  ; 


70 


%  Solve  for  allowable  kit  length 

Lk  =  (1/ (1-Cgkl) ) * ( (LI. *Wp) ./ (X.*Wk)-Lt) ; 

Cgk=  Lr+ (1-Cgkl) *Lk;  %  kit  Cg  locations  (inches  from  base) 

%  center  of  gravity  calculated  from  base  in  inches 
Cg_live  =  ( (WR_live*CgR_live) + (Cgk. *Wk) ) . / (WR_live+Wk) ; 
Cg_burn  =  ( (WR_burn*CgR_burn)  +  (Cgk . *Wk) )  . / (WR_burn+Wk)  ; 

%  center  of  gravity  changed  to  calibers  from  nose 
Cg_live  =  ( (Lr+Lk) -Cg_live) /dia; 
Cg_burn  =  ( (Lr+Lk) -Cg_burn) /dia; 

figure  ( 1)  , 

plot(Wk,Cg_live) 

axis([1.8,7.2,7.8,14]) 

%hold  off 
figure  (2) , 

plot (Wk,Cg_burn) 

axis( [1.8,7.2,6.8,13] ) 

%hold  off 

E.         STEADY-TURN  MANEUVER 

%  This  m-file  computes  thrust  required,  T,  for  certain  load  factor,  n 

%  (g's)  ,  in  a  steady  turn  for  various  Mach  numbers.  The  load  factor  is 

%  achieved  using  a  thruster  forward  of  the  CG. 

% 

%  The  inputs  include: 

%  Scenario  Definition: 

%  n:    desired  load  factor  (g's) 

%  rt :   desired  turn  radius  (ft) 

%  alt:  altitude  (ft)  (std  day  assumed) 

%  Mmin:  min  Mach  number 

%  Mmax :  max  Mach  number 

%  Rocket  Configuration: 

%  Wght :  rocket  mass  (lbm) 

%  Dia:  rocket  cross-section  diameter  (in)  (for  ref  area  and  length) 

%  Xcg:  rocket  center  of  gravity  in  calibers  from  nose 

%  Xt :    thruster  location  in  calibers  from  nose 

%  Aerodynamic  Parameters: 

%  CNa :   normal  force  coefficient  per  radian  of  alpha  [f(Mach)] 

%  Xcp :   center  of  pressure  location  in  calibers  from  nose  [f(Mach)] 

%  CMq:  moment  coefficient  per  rad/sec  of  pitch  rate  [f(Mach)] 

%  (Aero  Paramaters  are  linearly  interpolated  between  points  in  table) 


71 


%  Code  uses  normal  force  and  pitch  moment  equations: 

%     Fz=nW=N+T=  (Qbar) *S*CNa*alpha  +  T,  and 

%     m  =  0  =  -N*(Xcp-Xcg)  +  T*(Xcg-Xt)  -  CMq*q* (Dia/2V) *Qbar*S*Dia 

%  {since,  q  =  theta-dot  =  V/R  =  V/(V/v2/n)  =  n/V 

%  and  N  =  (Qbar)*S*CNa*alpha} 

%     M  =(Qbar*S*CNa*alpha) (Xcp-Xcg)  +  T* (Xcg-Xt)  -  CMq*n* (Dia~2/V~2) *Qbar*S 

%  The  two  equations  are  solved  for  the  unknowns,  alpha  and  T,  using  several 

%  values  of  Mach  number. 

%  W.  Mark  Wonnacott,  Naval  Postgraduate  School,  May  1997 
clear,  close  all 


%   INPUTS 


% 


%  Scenario  Definition 

n  =  15; 

Rt  =  6000; 

req_type  = 

alt  =  500; 

Mmin=   0.4; 

Mmax=  1.6; 


2; 


%  desired  load  factor  (g's) 
%  desired  min  turn  radius  (ft) 
%  requirement  (l:turn  rate,  2: turn  radius) 
%  std  day  pressure  altitude  (ft  MSL) 
%  minimum  Mach  number  for  calculations 
%  maximum  Mach  number  for  calculations 
%  Rocket  Configuration  (based  on  burned-out,  30  lbm,  68.75"  rocket) 
Wght  =  22.78;  %  rocket  mass  (lbm) 

Dia  =  2.75;  %  reference  length  (in) 

Xcff  =10.0;  %  CG  in  calibers  from  nose 

Xt  =  2.2;  %  thruster  location  in  calibers  from  nose 

%  Aerodynamic  Parameters  (based  on  final  config  68.75"-w/  3  case#8  strakes) 


% 

aero  table  = 


Mach# 
[0 

0 .6614 
866 
9682 


0308 
118 
1.25 
1.4142 
1.6008 
1.8028 
2.0156 

Mach_dat=  aero_table ( : , 1 )  ; 

CNa_dat   =  aero_table ( : , 2 ) ; 

Xcp_dat  =  1 . 03  *aero_table ( 

CMq_dat  =  aero_table ( : , 4 ) ; 


CNa  Xcp  CMq 

15.0616  11.52547  -4166.814 
16.41589  11.32152  -4556.4946 
18.10928  11.01214  -5041.0303 
20.04303  10.36514  -5518.2495 
20.05156  10.35951  -5519.5352 
20.05156  10.35951  -5519.5352 
18.91632  10.45431  -4859.3901 
17.51701  10.5577  -4376.4146 
16.10659  10.68451  -3938.8662 
14.68889  10.88046  -3541.0081 
13.47025  10.99184  -3202.1553 
12.38573     11.05396     -2916.2678]; 

%  Mach  numbers 

%  Body  Normal   Force  per  Radian  Alpha 
,3);  %  Center  of  Pressure  in  calibers   from  nose 

%   Pitching  Moment  per   rad/sec   of   Pitch  Rate 


72 


%   CONSTANTS 
gO  =  32.17; 
TempO  =  518.67; 
PresO  =  14.696*144; 
B_atm  =  0.003566; 
R_air  =  53.35; 
k_air  =  1.4; 


%  Gravity  Constant. 
%surface  temperature  (°R) 
%surface  pressure  (lbf/ft'>2) 
%temp  lapse  rate  (°R/ft) 
%gas  constant  (ft  lb/(lbm  °R)  ) 
%specific  heat  ratio 


%   INTERMEDIATE  CALCULATIONS 

%  Speed  of  Sound 

temp  =  TempO -B_atm*alt; 

pres  =  PresO* ( 1-B_atm*alt /TempO )" (1/R_air/B_ 

rho  =  pres/ (R_air*temp) ; 

Vs  =  sqrt(k_air*R_air*temp*gO) ; 

%  Rocket  Config 

d  =  Dia/12;  % 

S  =  pi*d~2/4;  % 

XT  =  (Xcg  -Xt)*d;  % 

XP_dat  =  (Xcp_dat-Xcg)*d;  % 

Fz  =  n  *  Wght;  % 

%  Mach  numbers  and  speeds 

%  adjust  min  and  max  Mach  to  table  values 
Mmin  =  max (min (Mach_dat ), Mmin) ;  % 

Mmax  =  min(max(Mach_dat) ,Mmax) ;  % 

kmax  =50;  % 

Mach  =  linspace (Mmin, Mmax, kmax) ;  % 


%  temperature  at  altitude  (°R) 
atm)  ;  %pressure  @  alt  (lb/ft"2) 

%  density  at  altitude  (lbm/ft~3) 
%  speed  of  sound  (ft/sec) 

reference  length  (ft) 

reference  area  (ft~2) 

thruster  location  from  CG  (ft) 

center  of  pressure  location  from  CG  (ft) 

normal  force  (lb) 


min  Mach  number  to  be  used 
max  Mach  number  to  be  used 
number  of  Mach  numbers  to  be  used 
vector  of  Mach  numbers  to  be  used 


%   THRUST  AND  ALPHA  CALCULATIONS  FOR  EACH  MACH  NUMBER 
for  k  =  1 : kmax 
V  =  Mach(k)*Vs; 
OS  =  S  *  l/(2*gO)*rho*V"2; 
CNa  =  interpl(Mach_dat,CNa_dat,Mach(k) ) 
Xcp  =  interpl(Mach_dat,XP_dat,Mach(k) ) ; 
CMq  =  interpl(Mach_dat,CMq_dat,Mach(k) ) 
A  =  [QS*CNa,       1; 

-QS*CNa*Xcp,  XT] ; 
if  req_type  ==  2, 

B=  [(V~2*Wght)/(gO*Rt) ; 
(1/2) *QS*CMq*d~2/Rt] ; 
else 

B=  [Fz;  (l/2)*QS*CMq*n*gO*(d/V)"2]; 
end 

X  =  inv(A)*B; 
a(k)  =  X(l)*180/pi; 
T(k)  =  X(2); 
end 


%  rocket  speed  (ft/s) 

%  (dyn  pres-lb/ft"2) (ref  area-ft"2)  (lb) 

%CNa  for  Mach# 

%Xcp  for  Mach# 

%CMq  for  Mach# 


%  RHS  vector  for  turn  radius 


%  RHS  vector  for  turn  rate 

%  solution  vector 

%  alpha  in  degrees 

%  required  thrust  in  pounds 


73 


subplot  (2,1,1) 

plot  (Mach,T, 'r ' ) ,  grid  on 

xlabel ( ' Mach  Number ' ) 

ylabeK 'Required  Thrust  (lbf) ' ) 

hold  on 
subplot  (2,1,2) 

plot  (Mach, a, 'r ' ) ,  grid  on 

xlabel ( 'Mach  Number' ) 

ylabeK 'Required  alpha  (deg) ' ) 

hold  on 

F.         SEEKER  FIELD  OF  VIEW 

%  This  m-file  calculates  the  probability  of  a  target  being  in  the 

%  seeker  field  of  view,  FOV  (deg) ,  for  for  several  seeker  acquistion 

%  ranges  (Racq)  and  fields  of  view.  Only  single  launch  range  is  used. 

% 

%  inputs  include:   Launcher  Pointing  Error,  Pnt  (deg)  -  1  sigma 

%  Missile  unguided  ballistic  error,  B  (mils)  -  1  sigma 

%  Missile  distance  time  profile,  (km)  (function  of  fly  out  time) 

%  Guidance  time  delay,  Gd  (sec) 

%  Time  delay  for  target  ID,  td  (sec) 

%  Min  and  Max  Target  Speed  (mph) 

%  also  required  are: 

%  Seeker  acquistion  ranges  (km)  -  (vector) 

%  Min  and  Max  FOV  (deg) 

%  Launch  Range  (km) 

%  Mark  Wonnacott,  April  1997 
%  Naval  Postgraduate  School 
clear,  close  all 

%%  Target  Description 

VTmin  =20;  %  (mph)  min  target  speed 

VTmax  =60;  %  (mph)  max  target  speed 

%%  Missile  Configuration  Inputs 
%  Launcher  Pointing  Error 

Pnt  =3;  %  (deg-  1  sig)  initial  pointing  error 

%  Unguided  Missile  Ballistic  Error 

B  =  12;  %  (mills-  1  sig)  meters  error  per  km  flight 

%  Guidance  Delay 

Gd  =  1.0;  %  (sec)  max  range  delay  (no  guidance  for  1  sec) 

%  Delay  time  to  ID  target 

td  =  5*1/8;  %  (sec)  5  laser  pulses  received  at  8  Hz 


74 


%  Seeker  Field  of  View 

FOVmin  =  .  5; 

FOVmax  =  45; 
%  Seeker  Acquisition  Range 

Racq  =  [1.15,1.5,2,3,6] ; 
%  Launch  Range 

RL  =  6; 
%  Missile  Speed  Table 
Time  =  [0,0.22,0.89,1.02,1.67,...  %flight  time  (s) 

2.52,4.96,7.91,11.24,14.88,18.8,22.99,27.3] ; 
Dist  =  [0,0.01,0.2,0.27,0.6,1,2,3,4,5,6,7,8];        % flight  dist  (m) 


%  (deg)  min  half  FoV 

%  (deg)  max  half  FoV 

%  (km)  parametric  seeker  acquistion  ranges 

%  (km)  maximum  launch  range 


%%  Inputs  converted  to  usable  variables 

VTmin2  =  VTmin/2237; 

VTmax2  =  VTmax/2237; 

PntR  =  Pnt*pi/180; 

Bkm  =  B/1000; 

kmax  =  length  (Racq) ; 

mmax  =  30; 

FOV  =  linspace (FOVmin, FOVmax.mmax) *pi/180; 

Rg  =  interpl (Time, Dist, Gd) ; 


%  min  target  speed  (mph->km/s) 

%  max  target  speed  (mph->km/s) 

%  initial  pointing  error  (deg->rad) 

%  X-trk  error  (km  per  km  flight) 

%  number  of  seeker  acq.  ranges 

%  number  of  FOV's 

%  FOV's  (deg->rad) 

%  unguided  flyout  range  (km) 


Pacq  =  zeros (kmax, mmax) ; 


%  pre-set  Pacq  matrix  size 


for  k  =  1 : kmax 
for  m  =  l:mmax 

D  =  Racq(k)*cos(FOV(m) ) ; 
F  =  Racq (k)*sin (FOV (m) ) ; 
if  RL-Rg  >  D 

Rd  =  RL  -  D; 
else 

Rd  =  RL  -  Rg; 
end 

tD  =  interpl (Dist, Time, Rd) ; 
tT  =  tD  +  td; 

Rt  =  interpl (Time, Dist, tT) ; 
s  =  Rt*sqrt  (tan  (PntR)  A2+(BkmP2) 
rl  =  tT  *  VTmin2; 
r2  =  tT  *  VTmax2; 


%  calculations  for  each  Racq 
%  calculations  for  each  FOV 

%  detection  range  (trgt  @  FOV  limit) 

%  max  x-trk  offset  for  FOV 

%  range  flown  before  detection 

%  range  flown  when  guided  flight  begins 


%  time  at  detection 

%  time  at  start  of  turn 

%  distance  flown  at  start  of  turn 

%1  sig  X-trk  error  at  start  of  turn 

%  min  target  motion  at  start  of  turn 

%  max  target  motion  at  start  of  turn 


L5  =  min(5*s,  r2+F)  ;  %  integration  upper  limit 

%%%  Integration  of  Pr (Xr) *Pt (Xr-F,Xr+F) 

%%%  from  0  to  L5  done  using  'quad8'  (Pacq  function  in  'ProbAcq') 
Pacq(k,m)  =  2*quad8 ( 'ProbAcq' , 0,L5, 0.01, [] ,F,s,rl,r2) ; 


end 


end 


75 


FOV  =  FOV*180/pi;  %  field  of  view  changed  back  to  degrees 

figure 

plot  (FOV, Pacq),  grid  on 

title  (['Vt  =  [  '  ,num2str (VTmin)  ,  '  ->  '  , num2str (VTmax)  ,  '  ]  mph']) 

legend (num2str  (Racq(l)  )  ,num2str  (Racq(2)  )  ,num2str  (Racq(3)  )  , num2str  (Racq(4)  )  , num2str (Rac 

q(5))) 

ylabel    ('Probability  of  Target   in  FOV), 

xlabeK'half  FOV    (±deg)  '  ) 

ProbAcq  Function 

function  Pacq  =  ProbAcq(Xr,F,s,rl,r2) 

%  Returns  the  product  of  the  assumed  probability  of  the  rocket  being 

%  approximately  at  a  certain  cross-track  position,  Xr,  and  the 

%  probability  of  the  target  being  between  ±F  to  either  side  of 

%  the  rocket  position  (seeker's  FOV  limits). 

%  The  probabilities  are  based  on  the  1-sigma  rocket  cross-track  error— s 

%  and  the  min  and  max  possible  target  cross-track  motion— rl  and  r2 . 

% 

%  This  function  has  to  take  Xr  as  a  vector  and  output  Pacq  as  a  vector! ! ! 

%  W.  Mark  Wonnacott,  Naval  Postgraduate  School,  1997 

if  rl  ==  0;  rl  =  0.0001;  end  %  avoid  division  by  zero 

Prkt  =  l/(s*sqrt(2*pi) ) *exp(-Xr.~2. / (2*s~2) ) ;   %normal  distribution  of  rocket 

%  Find  probability  of  target  in  FOV  extremes  (Pt(xl,x2))  for  each  Xr 
for  k  =  1: length (Xr) 

x  =  Xr(k);  %  step  through  each  rocket  x-trk  position 

XI  =  x  -  F;  %  extreme  bound  of  FOV 

X2  =  x  +  F;  %  extreme  bound  of  FOV 

if  XI  >  r2  %  weapon  farther  out  than  possible  for  target 

Ptgt(k)  =  0; 

elseif  XI  <  -r2  %  all  possible  target  locations  inside  FOV 

Ptgt(k)  =  1; 

else 

%  compute  area  bounded  rl  &  r2  circles,  fly-out  axis,  and  X2 

LI  =  min(X2,r2);  %  limit  of  intgrt'n  for  r2  circle 

L2  =  min(X2,rl);  %  limit  of  intgrt'n  for  rl  circle 

Ala  =  Ll*sqrt(r2~2-Ll~2)  +  r2~2*asin(Ll/r2) ;  %bounded  area  in  r2  circle 

Alb  =  L2*sqrt(rl~2-L2~2)  +  rl~2*asin(L2/rl) ;  %bounded  area  in  rl  circle 

Al  =  Ala-Alb; 


76 


%  compute  area  bounded  rl  &  r2  circles,  fly-out  axis,  and  XI 

L3  =  min(abs(Xl) ,r2) ; 

L4  =  min(abs(Xl) ,rl) ; 

A2a  =  L3*sqrt(r2/S2-L3~2)  +  r2A2*asin(L3/r2) ; 

A2b  =  L4*sqrt(rl"2-L4"2)  +  rl"2*asin(L4/rl) ; 

A2  =  A2a-A2b; 

A  =  Al-sign(Xl)*A2; 
Ptgt(k)  =A/  (pi*(r2"2-rl"2) ) ; 
end 
end 

Pacq  =  Prkt.*Ptgt*100; 
return,  end 

G.        6DOF  CODES 

1.         Simulation  Initilization 

%  File  Initializes  Simulation 

clear 

data_out  =  [2001,5] ; 

%%%  Missile  INITIAL  CONDITIONS  (6  DOF  Flight) 

%%%   (u,v,w,p,q,r,phi,  theta,psi,p_rocket,pn,pe,ph) 

%  ground  launch  initial  conditions 

Msl_Xo  =  [0;0;0;0;0;0;0;10*pi/180;0;0;0;0;0] ; 

%  target  pointing  after  12  sec  flight  (10°  elev  launch)  trgt  §  5000m 

%Msl_Xo  =  [285;0;0;0.03;  .  01; 0;0; -.383 ; 0; -125 . 6;4230;0;310] ;%theta  =0.0246 

%  straight  and  level  cruise  after  12  sec  flight 

%Msl_Xo  =  [300;0;0;0;0;0;0;0;0;-125.6;0;0;300]; 
%%%  Target  INITIAL  CONDITIONS  (6  DOF  Flight) 
%%%   (Pn,  Pe) 

Tgt_Xo  =  [5000, -00]; 

T_Vn  =  0; 

T_Ve  =  0 

Talt  =  0 


Aerodat    %  weight  and  aero  data  file  for  2.75"  HRGK-equipped  rocket 


77 


2.    Aerodat 

%  This  File  Contains  Look-up  Table  Data 

%  2.75"  Rocket  with  Final  Design  Guidance  Kit 

global  gO  TempO  PresO  B_atm  R_air  S  d  Sd  Xt  mu  rho  c; 

%CONSTANTS  &  CONVERSIONS 

%%%  Conversions 

in2m  =  39.37;  % inches  per  meter 

lb2kg  =  2.2046;  %pounds  per  kilogram 

%%%  Gravity 

gO  =  9.81;  %  Gravity  Constant,  g'0,  in  m/s^2. 

%%%  Atmospheric  Data: 

TempO  =  288.15;  %surface  temperature  (K) 

PresO  =  101330;  %surface  pressure  (Pa) 

B_atm  =  0.00650;  %temp  lapse  rate  (°C/m) 

R_air  =  287.0;  %gas  constant  (N  m/ (kg  K) ) 

%  ===================================================== 

%  Guidance  Package  Parameters 

%  ===================================================== 

radius  =  2.75/2/in2m;     %Guidance  Package  Radius  (m) 

GP_Mass0  =  7/lb2kg;       %Initial  Guidance  Package  Mass  (kg) 
GP_Massf  =  7/lb2kg;       %Final  Guidance  Package  Mass  (kg) 
GPJLngth  =  13.625/in2m;   %Guidance  Package  Length  (m) 

Xt  =  6.6/in2m;  %Thruster  Location  from  Nose  (m) 

mu  =  0.006;  %Rocket  to  Kit  Coefficient  of  Friction 

GP_CG0  =  8.5/in2m;       %Initial  Guidance  Package  CG  from  nose  (m) 
GP_CGf  =  8.5/in2m;       %Final  Guidance  Package  CG  from  nose  (m) 

GP_Jxx0  =  ...  %Aprox. Initial  Axial  Moment  of  Inertia  (kg  m^2) 

GP_MassO*radiusA2/2; 
GP_Jxxf  =  ...  %Aprox. Final  Axial  Moment  of  Inertia  (kg  m~2) 

GP_Massf *radius"2/2  ; 

GP_Jyy0  =  ...  %Aprox.  Initial  Transverse  M.  of  I .  (kg  m~2) 

GP_Mass0*(GP_CG0-GP_Lngth/2)~2  +  GP_MassO*GP_Lngth~2/12; 

GP_Jyyf  =  ...  %Aprox. Final  Transverse  M.  of  I.  (kg  m"2) 

GP_Massf*(GP_CGf-GP_Lngth/2)  A2  +  GP_Massf*GP_Lngth'N2/12; 

FlyTime  =  30;  %Approximate  Flyout  Time  Bound  (sec) 


78 


%  ===================================================== 

%  Rocket  Parameters  from  IHSP  89-289,  Mk-66  Data  Book 
%  ===================================================== 

d  =  2.75/in2m;  %Rocket  Diameter  (m) 


R_MassO  =  22.95/lb2kg; 
R_Massf  =  15.73/lb2kg; 
R_Lngth  =  55. 125/in2m; 


%Initial  Rocket  Mass  (kg) 
%Final  Rocket  Mass  (kg) 
%Rocket  Length  (m) 


R  CGO 


R_Lngth-29.96/in2m  +  GP_Lngth; 


R  CGf 


%Initial  Rocket  CG  from  nose  (m) 


%Final  Rocket  CG  from  nose  (m) 


R_Lngth-33.55/in2m  +  GP_Lngth; 


R_JxxO  =  26.2/(lb2kg*in2nT2) ; 
R_Jxxf  =  1 9 . 7 / ( lb2  kg*  in2m~  2 ) ; 


%Initial  Axial  Moment  of  Inertia  (kg  m^2) 
%Final  Axial  Moment  of  Inertia  (kg  m/v2) 


R_JyyO  =  6248/(lb2kg*in2nT2) ; 
R_Jyyf  =  5008/(lb2kg*in2nr2) ; 


%Initial  Transverse  M.  of  I.  (kg  m^2) 
%Final  Transverse  M.  of  I.  (kg  m"2) 


BurnTime  =  1.05; 


%Average  Rocket  Motor  Burn  Time  (sec) 


Thrus 

tHot_dat= . 

[0.00 

0 

0.05 

1600 

0.07 

1700 

0.10 

1690 

0.15 

1600 

0.20 

1530 

0.25 

1500 

0.30 

1490 

0.35 

1510 

ThrustCold_dat= 

[0.00  0 

0.05 

1150 

0.07 

1300 

0.1 

1380 

0.15 

1400 

0.2 

1400 

0.25 

1400 

0.3 

1390 

0.35 

1400 

0.40 
0.45 
0.50 
0.55 
0.60 
0.65 
0.70 
0.75 
0.80 


%Thrust  Time  History  (sec  &  lbf  for  150°F) 
1520  0.85  1800 

1560 
1610 
1650 
1700 
1730 
1750 


0.90 

1700 

0.95 

600 

1.00 

200 

1.05 

40 

1.10 

10 

1.15 

0 

FlyTime  0]  ; 

1780 
1800 

%Thrust  Time  History  (sec  &  lbf  for  -50°F) 
0.4   1380  0.85  1680 


0. 

.45 

1350 

0. 

.5 

1340 

0 

,55 

1350 

0 

.6 

1350 

0 

.65 

1400 

0 

.7 

1440 

0 

.75 

1500 

0 

.8 

1590 

0.9 

1760 

0.95 

1750 

1.00 

1600 

1.05 

700 

1.1 

100 

1.15 

0 

FlyTime  0]  ; 

ThrustHot_dat ( : , 2 )  =  ThrustHot_dat ( : , 2 ) *g0 /lb2kg; 
ThrustCold_dat ( : , 2 )  =  ThrustCold_dat ( : , 2 ) *g0/lb2kg; 


79 


%  Reference  Area/Length  and  Mass  &  Moment  Time  Histories 
%  ===================================================== 

S  =  pi* (d~2) /4;  %  Cross-Sectional  Area  (Ref  Area-nT2) 

Sd  =  S*d;  %  Ref  Area  x  Ref  Length  (m~3) 


t  =  [0;  BurnTime;  FlyTime] ; 


%time  vector  for  time  histories 


Mass_dat  =  [t,    [R_MassO+GP_MassO 
R_Mas  s  f  +GP_Mas  s  0 
R_Massf+GP_Massf ] ] ; 


%  Mass  Time  History  (sec  &  kg) 


cg0= (R_CG0*R_Mass0+GP_CG0*GP_Mass0) / (R_MassO+GP_MassO) 
cgl= (R_CGf *R_Massf+GP_CG0*GP_Mass0) / (R_Massf+GP_MassO) 
cg2= (R_CGf *R_Massf +GP_CGf *GP_Massf ) / (R_Massf +GP_Massf ) 
CG_dat  =  [t,   [cgO  %  CG  History  (sec  &  m  from  nose) 

cgl 

cg2]]; 


JxO  =  R_JxxO+GP_JxxO 
Jxl  =  R_Jxxf+GP_JxxO 
Jx2  =  R_Jxxf+GP_Jxxf 
Jx_dat  =  [t,   [JxO 
Jxl 
Jx2]]; 


%  Axial  M.  of  I.  History  (kg  m~2) 


JxR_dat  =  [t,  [R_JxxO;  R_Jxxf;  R_Jxxf ] ] ; 
JxK_da  t  =  [ t ,  [ GP_ JxxO ; GP_ JxxO ; GP_Jxxf ] ] ; 


JyO  =  R_Jyy0+R_Mass0*(R_CG0-cg0)~2+GP_Jyy0+GP_Mass0*(GP_CG0-cg0)'N2 
Jyl  =  R_Jyyf+R_Massf*(R_CGf-cgl)~2+GP_Jyy0+GP_Mass0*(GP_CG0-cgl)"2 
Jy2  =  R_Jyyf+R_Massf*(R_CGf-cg2)"2+GP_Jyyf+GP_Massf*(GP_CGf-cg2)A2 
Jy_dat  =  [t,   [JyO  %  M.  of  I.  History  (kg  m~2) 

Jyl 

Jy2]]; 


%  Rocket  Roll  Acceleration  History 
R  Pdot  =  . . . 


[0 

0 

0.5 

45.0 

0.6 

100.0 

0.9 

15.0 

1.0 

-50.0 

1.2   ■ 

-120.0 

1.4 

-92.0 

1.6 

-34.0 

R_Pdot ( 

:,2)    = 

%  Rocket  Roll  Accel  (rev/s/s) 


1.8 

-8.0 

2.0 

0.0 

2.4 

2.0 

3.0 

7.5 

3.6 

8.5 

4.0 

9.5 

4.4 

7.5 

5.6 

0.0 

5.8 

-4.5 

6.2 

-12.5 

6.6 

-10.0 

7.0 

-7.0 

8.0 

-4.0 

9.5 

0.0 

12. 

1.0 

21. 

0.1] 

R_Pdot(: ,2)*2*pi; 


%  converted  to  rad/s/s 


80 


%  Aerodynamics  Data  from  ADAM ' s  MADR 

Mach_dat  =  ... 

[0  1.0000  1.4142 

0.6614  1.0308  1.6008 

0.8660  1.1180  1.8028 

0.9682  1.2500  2.0156]; 
%Static  Coefficient /Derivative  Components  Based  on  Mach  Number  Vector 

% 

CdOCoast_dat  =    ...   %  Skin,  Wave,  and  Base  Drag  (rocket  coasting) 

[0.52292  0.83218  1.09629 

0.48521  1.0067  1.13922 

0.51417  1.00424  1.16393 

0.75708  1.02549  1.17799]; 

Cd0Burn_dat  =  .  .  .  %  Skin  and  Wave  Drag  (rocket  burning) 

[0.42291  0.73219  0.90267 

0.38521  0.81307  0.95226 

0.41417  0.81061      "  0.99252 

0.65708  0.83187  1.02295]; 

CNa_dat  =  .  .  .  %  Body  Normal  Force  per  Radian  Alpha  or  Beta 

[15.0616  20.05156  16.10659 

16.41589  20.05156  14.68889 

18.10928  18.91632  13.47025 

20.04303  17.51701  12.38573]; 

CP_dat  =  . . .  %  Center  of  Pressure  in  Calibers 

[11.52547  10.35951  10.68451 

11.32152  10.35951  10.88046 

11.01214  10.45431  10.99184 

10.36514  10.5577  11.05396]; 

CP_dat  =  CP_dat*d; 

%Dyanamic  Coefficients /Derivatives  Based  on  Mach  Number  Vector 

% 

CMq_dat  =  . . .  %  Pitching  damping  per  Radian/sec  of  Pitch  Rate 

[-4166.814  -5519.5352  -3938.8662 

-4556.4946  -5519.5352  -3541.0081 

-5041.0303  -4859.3901  -3202.1553 

-5518.2495  -4376.4146  -2916.2678]; 

Clp_dat  =  .  .  .  %  Roll  damping  (HRGK  only)  /Radian/sec  of  Roll  Rate 

[-7.513  -7.513  -6.7529 

-7.513  -7.513  -6.453 

-7.513  -7.2767  -5.5802 

-7.513  -7.0348  -4.7831]; 


81 


3.         Equations  of  Motion 

function  xo  =  MotionEqns (uo) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  This  Function  computes  the  Flat-Earth,  Body  Axes, 

%%%  6-DOF  Dynamics  Equations  for  a 

%%%  Thruster -Controlled  Missile 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  Adapted  for  rolling  airframe,  thrust  control,  and 

%%%  variable  aero  coefficient  and  missile  parameters  by 

%%%  W.  Mark  Wonnacott,  March  1997 

%%%  Naval  Postgraduate  School 

%%%  Adapted  from  the  G.  Hutchins'  Code  for  EC  4330. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  Original  Code  by  R.  G.  Hutchins,   10  December  1996 

%%%  Thanks  to 

%%%         Rob  King  and  Mark  Wonnacott  for  aero  assistance 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  The  State  Vector  is  defined  as: 

%%%     Body-Axes  Velocities 

%%%       x(l)  =  U,   speed  along  missile  axis,  x  direction 

%%%       x(2)  =  V,   speed  of  sideslip,  y  direction 

%%%       x(3)  =  W,   speed  along  z  direction  ("Down") 

%%%     Body-Axes  Angular  Velocities 

%%%       x(4)  =  P,   roll  rate 

%%%       x(5)  =  Q,   pitch  rate 

%%%       x(6)  =  R,   yaw  rate 

%%%     Euler  Angles 

%%%       x(7)  =  phi,    roll  relative  to  vertical  down 

%%%       x(8)  =  theta,   pitch 

%%%       x(9)  =  psi,     yaw 

%%%      Rocket  Rolling 

%%%       x(10)  =  Pr,  rocket  roll  rate 

%%%     Location  Variables 

%%%       x(ll)  =  Pn,   Position  North  of  (0,0,0) 

%%%       x(12)  =  Pe,   Position  East  of  (0,0,0) 

%%%       x(13)  =  h,   Height 

%%%  The  input  vector  is  defined  as : 

%%%       u(l)  =  Tl,   thrust  from  thruster  #1 

%%%       u(2)  =  T2,   thrust  from  thruster  #2 

%%%       u(3)  =  T3,   thrust  from  thruster  #3 

%%%       u(4)  =  T,    thrust  through  the  missile  x  axis 


82 


%%%Other  Missile  Parameter  Inputs (functions  of  Mach  or  time): 

%%%      P(l)  =  mass,   missile  mass  (kg)  [f(t)] 

%%%      P(2)  =  Xcg,  missile  Cg  (m  from  nose)  [f(t)] 

%%%      P(3)  =  Xcp,  center  of  pressure  (m  from  nose)  [f(M)] 

%%%     P(4)  =  JxR  rocket  MofI  about  long  axis  (kg  m~2) 

%%%     P(5)  =  JxK    kit  MofI  about  long  axis  (kg  m~2) 

%%%      P(6)  =  Jy=Jz,  missile  Mofl-y  or  z  axes  [f(t)] 

%%%     P(7)  =  PdotR,   rocket  roll  accel  (rad/s/s) [f (t) ] 

%%%  Aero  Coefficients /Derivatives (functions  of  Mach  or  time): 

%%%   C(l)  =  CdO,zero  alpha  drag  coefficient 

%%%   C(2)  =  CNa, normal  force  /rad  alpha  (lift  curve  slope) 

%%%   C(3)  =  CMq,  pitch  damping  per  rad/sec  pitch  rate 

%%%   C(4)  =  Clp,  roll  damp  per  rad/sec  roll  rate  (HRGK) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  Related  Quantities 

Angle  of  Attack 
Sideslip  Angle 

Resultant  Vert.  Thrust  (up,  -z  dir.) 
Reultant  Side  Thrust  (y  direction) 
Missile  Speed 

Air  Temperature  (K)  [f (altitude) ] 
Air  Pressure  (Pa)  [f (altitude) ] 
Air  Density  (Kg/m~3)  [f (altitude) ] 
Dynamic  Pressure  =  1/2  rho(h)  Vt*2 
missile  length  (m) 
missile  diameter  (m)  (ref  length) 
missile  cross-sectional  area(m^2) (ref  area) 
rocket  to  kit  coefficient  of  friction 
%%%  Aerodynamic  Forces 

%%%   CN  =  CNa*alpha  =>  N=CN*qbar*S+Tz    Normal  Force  (=Lift) 
%%%   CY  =  -CNa*beta  =>  Y=CY*qbar*S+Ty    Sideslip  Force 
%%%   CD  =  CdO  +  or 2 /CNa  =>  D=CD*qbar*S  Drag  Force 
%%%  Aerodynamic  Moments 
%%%   CLbar  =  Clp*P*d/ (2*U) , 

%%%     =>  Lbar  =  CLbar*qbar*S*d  Kit  Roll  Moment 

%%%   CM  =  CNa*alpha(Xcg-Xcp)  +  CMq*Q*d/ (2*U) 
%%%      =>  M  =  CM*qbar*S*d  +  Tz(Xcg-Xt)        Pitch  Moment 
%%%   CNbar  =  CNa*beta(Xcg-Xcp)  +  CMq*Q*d/ (2*U) 
%%%     =>  Nbar  =  CNbar *qbar*S*d  +  Ty(Xcg-Xt)   Yaw  Moment 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%  Declare  Global  Variables 
global  gO  TempO  PresO  B_atm  R_air  S  d  Sd  Xt  mu; 


%%% 

alpha, 

%%% 

beta, 

%%% 

Tz, 

%%% 

Ty, 

%%% 

vt, 

%%% 

Temp, 

%%% 

Pres, 

%%% 

Rho, 

%%% 

qbar, 

%%% 

L, 

%%% 

d, 

%%% 

S=pi*d*2/4, 

%%% 

mu, 

83 


%%%  Define  State  Variables  from  Inputs 

x  =  uo(l:14) ; 

%%%  Body-Axes  Velocities 


U  =  x(l) 
V  =  x(2) 
W  =  x(3) 


%   speed  along  missile  axis,    x 
%  speed  of  sideslip,    y  direction 
%   speed  along  z  direction    ("Down") 


%%%  Body-Axes  Angular  Velocities 


P  =  x(4) 

Q  =  x(5) 

R  =  x(6) 
%%%  Euler  Angles 

phi  =  x(7) ; 

theta  =  x ( 8 ) ; 

psi  =  x(9)  ; 
%%%  Rocket  Rolling 

Pr  =  x(10) ; 


%  roll  rate 
%  pitch  rate 
%  yaw  rate 

%  roll 
%  pitch 
%  yaw 


%  rocket  roll  rate 
%%%  Location  Variables 

Pn  =  x(ll);      %  Position  North  of  (0,0,0) 
Pe  =  x(12);      %  Position  East  of  (0,0,0) 
h  =  x ( 13 ) ;       %  Height 


%%%  Define  Control  Variables  from  Inputs 

u  =  uo(14:17); 

Tl  =  u(l);  %  thrust  from  thruster  #1 
T2  =  u(2);  %  thrust  from  thruster  #2 
T3  =  u(3);  %  thrust  from  thruster  #3 
T   =  u(4) ;      %  thrust  through  the  missile  long,  axis 


%%%  Other  Missile  Parameter  Inputs  (time  or  Mach  dependent) : 

Param  =  uo(18:24) ; 

%missile  mass  (kg) 

%missile  Cg  (m  from  nose) 

%missile  CP  (m  from  nose) 

%rocket  M  of  I  about  long  axis (kg  m^2) 

%kit  M  of  I  about  long  axis  (kg  m/N2) 

%missile  MofI  about  y  or  z  axes (kg  m~2] 

%rocket  roll  acceleration  (rad/s/s) 

%%%  Define  Mach  Number  Dependent  Coefficients/Derivatives 

C  =  uo(25:28) 

%zero  alpha  drag  coefficient 
%normal  force  /rad  alpha  (lift  curve  slope) 
%pitch  damping  per  rad/sec  pitch  rate 
%roll  damping  (kit  only)  per  rad/sec  roll 


mass  =  Param(l) ; 
Xcg  =  Param(2) 
Xcp  =  Param (3) 
JxR  =  Param(4) 
JxK  =  Param(5) 
Jy  =  Param  ( 6 )  ; 
PdR  =  Param(7) 


CdO  =  C(l) 
CNa  =  C(2) 
CMq  =  C(3) 
Clp  =  C(4) 


84 


%%%%%%%  Define  the  Forces  and  Moments 

%%%  Related  Quantities 
%  speed  quantities 

Vxz2  =  UA2  +  VT2; 

Vt2  =  Vxz2  +  VA2; 

Vt  =  sqrt ( Vt2 ) ; 
%  atmospherics 

temp  =  TempO-B_atm*h; 

pres  =  PresO*(l-B_atm*h/TempO) A (gO/R_air/B_atm) ; 

rho  =  pres/ (R_air* temp) ; 

qbar  =  . 5*rho*Vt2; 
%  control  thrusts 

Ty  =  Tl*cos(phi)  +  ... 

T2*cos(phi+2/3*pi)  +  T3*cos (phi+4/3*pi) ; 

Tz  =  Tl*sin(phi)  +  ... 

T2*sin(phi+2/3*pi)  +  T3*sin(phi+4/3*pi) ; 
%  angles  of  attack 

alpha  =  atan2(W,U); 

beta  =  atan2 (V, sqrt (Vxz2 ) ) ; 

%%%  Forces 

CN  =  CNa*alpha;  %  normal  force  coefficient 

Fz  =  -qbar*S*CN  -  Tz;  %  normal  force 

CY  =  -CNa*beta;  %  side  force  coefficient 

Fy  =  qbar*S*CY  +  Ty;        %  side  force 

CD  =  CdO  +  (CNA2+CYA2) /CNa;%  drag  force  coefficient 

Fx  =  T  -  qbar*S*CD;  %  drag  force 

%%%  Moments 

CI  =  Clp*P*d/(2*U+.001) ;       %  roll  moment  coefficient 
Lbar  =  qbar*Sd*Cl  +  mu*JxR*PdR;   %  roll  moment  on  kit 
CM  =  CNa*alpha* (Xcg-Xcp)  +  ...  %pitch  moment  coefficient 

CMq*Q*d/(2*U+.00D  ; 
M  =  qbar*Sd*CM  +  Tz*(Xcg-Xt);   %  pitch  moment 
Cn  =  -CNa*beta*  (Xcg-Xcp)  +  .  .  .  %  yaw  moment  coefficient 

CMq*R*d/(2*U+.001) ; 
Nbar  =  qbar*Sd*Cn  +  Ty*(Xcg-Xt);  %  yaw  moment 


85 


%%%  Compute  the  Time  Derivatives  from  Flat-Earth  Equations 

%%%  Force  Equations 

Ud  =  R*V  -  Q*W  -  gO*sin (theta)  +  Fx/mass; 

Vd  =-R*U  +  P*W  +  gO*sin(phi) *cos(theta)  +  Fy/mass; 

Wd  =  Q*U  -  P*V  +  gO*cos(phi)*cos(theta)  +  Fz/mass; 

%%%  Moment  Equations 

Pd  =  Lbar/JxK; 

Prd  =  PdR; 

Qd  =  (l-JxK/Jy)*P*R  +  M/Jy  -  ( JxR/Jy) *R*Pr; 

Rd  =  (JxK/Jy-l)*P*Q  +  Nbar/Jy  +  (JxR/Jy) *Q*Pr; 

%%%  Kinematic  Equations 

phid  =  P  +  tan(theta) * (Q*sin(phi)  +  R*cos (phi) ) ; 

thetad  =  Q*cos(phi)  -  R*sin(phi); 

psid  =  (Q*sin(phi) +R*cos (phi) ) /cos (theta) ; 

%%%  Navigation  Equations 

Pnd  =  U*cos (theta) *cos (psi)  +  ... 

V* (sin(phi) *sin(theta) *cos (psi)  -  cos (phi) *sin(psi) )  +.. 

W* (cos (phi) *sin(theta) *cos(psi)  +  sin(phi) *sin(psi) ) ; 
Ped  =  U*cos (theta) *sin (psi)  +  ... 

V* (sin(phi) *sin(theta) *sin(psi)  +  cos(phi) *cos (psi) )  +.. 

W* (cos (phi) *sin( theta) *sin (psi)  -  sin(phi) *cos (psi) ) ; 
hd  =  U*sin(theta)  -  V*sin (phi )* cos (theta)  -  ... 

W*cos (phi) *cos (theta) ; 

%%%%%%%%%%  Define  the  output  vector 
%%%   Body-Axes  Accelerations 


xo(l)  =  Ud 
xo(2)  =  Vd 
xo(3)  =  Wd 


%  ace  along  missile  axis,x  direction 
%  ace  of  sideslip,  y  direction 
%  ace  along  z  direction  ("Down") 


%%%   Body-Axes  Angular  Accelerations 


%  roll  ace 
%  pitch  ace 
%  yaw  ace 


xo(4)  =  Pd; 

xo(5)  =  Qd; 

xo(6)  =  Rd; 
%%%   Euler  Angle  Rates 

xo(7)  =  phid;  %  d  phi/dt 

xo(8)  =  thetad;  %  d  theta /dt 

xo(9)  =  psid;  %  d  psi/dt 

%%%   Rocket  Rolling  Equations 

xo(10)  =  Prd;  %  rocket  spin  acceleration 

%%%   Location  Variable  Rates 

xo(ll)  =  Pnd;  %  Position  North  of  (0,0,0)  rate 

xo(12)  =  Ped;  %  Position  East  of  (0,0,0)  rate 

xo(13)  =  hd;  %  Height  rate 


86 


4.    Line  of  Sight 

function  yo  =  Lineof Sight (uo) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  This  Function  computes  the  seeker  frame  LOS, 

%%%  from  the  tangent  plane  coordinates 

%%%  W.  Mark  Wonnacott,  May  1997 

%%%  Naval  Postgraduate  School 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%   INPUTS 

%%%     Position  Differences  Xt-Xm 

%%%     u(l)  =  N,        delta  north 

%%%     u(2)  =  E,        delta  east 

%%%     u(3)  =  H,        delta  height 

%%%     Euler  Angles 

%%%       u(4)  =  phi,     seeker  roll  relative  to  vertical  down 

%%%       u(5)  =  theta,   rocket  pitch  relative  to  horizontal 

%%%       u(6)  =  psi,     rocket  yaw  relative  to  north 

%%%   OUTPUTS  Seeker  LOS 

%%%     yd)  =  alpha,    off-boresight  angle 

%%%     y(2)  =  beta,     angle  from  seeker's  ref.  roll  position  (down) 

%%%     y(3)  =  R,     slant  range  to  target 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  Define  Tangent  Plane  LOS  Offsets 

N  =  uo(l);  %  north  distance  to  target 

E  =  uo(2);  %  east  distance  to  target 

D  =  uo(3);  %  down  distance  to  target 

%%%  Define  Rocket  Euler  Angles 


phi  =  uo(4) 
theta  =  uo(5) 
psi   =  uo(6) 


%  seeker  roll  relative  to  vertical  down 
%  rocket  pitch  relative  to  horizontal 
%  rocket  yaw  relative  to  north 
%%%  Transformation  Matrices  (NED  to  XYZ-body) 
CI  =  [cos (psi) 
-sin (psi) 
0 
C2  =  [cos (theta) 
0 
sin (theta) 
C3  =  [   1 
0 
0 
X=C3*C2*C1* [N;E;D] ; 

Offbore  =  atan2 (sqrt (X(2) A2+X(3) "2) ,X(1) ) ; 
pointing=  atan2 (X(3) ,X(2) ) ; 
R  =  sgrt(X(l)'v2+X(2)A2+X(3)"2)  ; 

yo(l)  =  Offbore;    yo(2)  =  pointing;      yo(3)  =  R; 


87 


sin (psi) 

0; 

cos (psi) 

0; 

0 

1]; 

0 

-sin (theta) ; 

1 

0; 

0 

cos (theta) ] ; 

0 

0; 

cos (phi) 

sin (phi)  ; 

-sin (phi) 

cos (phi) ] ; 

5.    Mach  Number 

function  M  =  Mach(uo) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  This  function  computes  the  Mach  Number  as  a  function  of 

%%%  component  velocities  and  altitude 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  W.  Mark  Wonnacott,  March  1997 

%%%  Naval  Postgraduate  School 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  INPUTS: 

%%%   Body-Axes  Velocities 

%%%     x(l)  =  U,   speed  along  missile  axis,  x  direction 

%%%     x(2)  =  V,   speed  of  sideslip,  y  direction 

%%%      x(3)  =  W,   speed  along  z  direction  ("Down") 

%%%     x(13)=  h,   altitude  (m) 

%%%  Related  Quantities 

%%%     Vt,        Missile  Speed  (m/s) 

%%%      Temp,       Air  Temperature  (K)  as  function  of  altitude 

%%%      Pres,       Air  Pressure  (Pa)  as  function  of  altitude 

%%%      Rho,       Air  Density  (Kg/mA3)  as  function  of  altitude 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%  Declare  Global  Variables 

global  gO  TempO  PresO  B_atm  R_air  S  d  Sd  Xt; 

%%%  Define  Variables  from  Inputs 

%%%     Body-Axes  Velocities 

U  =  uo(l);      %  speed  along  missile  axis,  x  direction 
V  =  uo(2);     %  speed  of  sideslip,  y  direction 
W  =  uo(3);      %  speed  along  z  direction  ("Down") 
h  =  uo ( 13 ) ;     %  height 

Vt  =  sqrt(lT2  +  V"2  +  VT2); 

%  atmospherics 

k  =  1.4; 

temp  =  TempO -B_atm*h; 

pres  =  PresO*  ( 1-B_atm*h/Temp0 )  A  (gO /R_air/B_atm)  ; 

rho  =  pres/ (R_air*temp) ; 

c  =  sqrt (k*R_air*temp) ; 

M  =  Vt/c; 


88 


LIST  OF  REFERENCES 


1.  AIAA  Missile  Systems  Technical  Committee.  "Graduate  Team  Missile  Design 
Competition,  RFP:  Hit-To-Kill  Guided  Rocket  (HGR)."  American  Institute  of 
Aeronautics  and  Astronautics;  Reston,  VA:  June  1996. 

2.  Wonnacott,  W.M;  Pomerantz,  B;  Silva,  S.L;  and  Nurse,  N.A.  "The  Three  and  a  Half 
Rocketeers  Hit-to-Kill  Rocket  Guidance  Kit"  (design  proposal  for  AIAA  graduate 
student  competition).  Naval  Postgraduate  School;  Monterey,  CA:  June  1997. 

3.  Foss,  Christopher  F.  and  Gander,  Terry  J.,  eds.  Jane's  Military  Vehicles  and 
Logistics,  1996-97.  Jane's:   1996. 

4.  Foss,  Christopher  F.,  ed.  Jane's  Armour  and  Artillery,  1996-97  (7th  ed).  Jane's: 
1996. 

5.  Cullen,  Tony  and  Foss,  Christopher  R.,  eds.  Jane's  Land-Based  Air  Defense,  1996- 
97  (9th  ed).  Jane's:   1996. 

6.  Sharpe,  Richard,  Captain  RN,  ed.  Jane's  Fighting  Ships,  1995-96.  Jane's:  1995. 

7.  Washburn,  Alan.  "Notes  on  Firing  Theory."  Naval  Postgraduate  School;  Monterey, 
CA:   1983. 

8.  Nicholas,  Ted  and  Rossi,  Rita,  eds.  US  Weapon  Systems  Costs,  1994  (14th  ed). 
Data  Search  Associates;  Fountain  Valley,  CA:  1994. 

9.  Tobin,  Vince,  Maj.  U.S.  Army  and  Mason,  Pat,  Capt.  U.S.  Army — AH-64  pilots, 
personal  communications.  Naval  Postgraduate  School;  Monterey,  CA:  January 
1997. 

10.  HYDRA-70/2.75  Inch  Rocket  Management  Office.  "Hydra-70  2.75  Inch  Rocket 
System  Information  Handbook."  US  Army  Armament,  Munitions  and  Chemical 
Command;  Rock  Island,  IL:  April  1994. 

11.  Ferguson,  John  H.,  Jr.  and  Garvey,  Paul  B.  "2.75-Inch  Rocket  Motor  Mark  66  Data 
Book:  Characteristics  and  Performance  Data,"  IHSP  89-289.  Naval  Ordnance 
Station;  Indian  Head,  MD:  June  1989. 

12.  Hindes,  John  W.,  Jr.   Using  ADAM  (3rd  ed).  Madison,  AL:  October,  1993. 

13.  Wonnacott,  W.M.  "Initial  Version  of  the  Medium  Range,  Antiship  Weapon  Analysis 
Program— SeaHit,"  NWC  TM  6823.  Naval  Air  Warfare  Center  Weapons  Division; 
China  Lake,  CA:  September  1990. 

14.  Advanced  Systems  Concepts  Office.  "Missile  Sizing  Program  User's  Manual."  U.S. 
Army  Missile  Command;  Huntsville,  AL:  March  1989. 


89 


15.  Stevens,  Brian  L.  and  Lewis,  Frank  L.  Aircraft  Control  and  Simulation.  Wiley;  New 
York:   1992. 

16.  Chin,  S.S.;  Missile  Configuration  Design;  1961:  McGraw-Hill  Book  Company;  New 
York. 

17.  Blakelock,  John  H.;  Automatic  Control  of  Aircraft  and  Missiles;  2nd  Ed;  1991:  John 
Wiley  &  Sons,  New  York. 

18.  Howard,   Richard;  AA3701:  Missile  Aerodynamics   Class  Notes;    1995:      Naval 
Postgraduate  School;  Monterey,  CA. 


90 


INITIAL  DISTRIBUTION  LIST 


1 .  Defense  Technical  Information  Center 

8725  John  J.  Kingman  Rd.,  STE  0944 

Ft.  Belvoir,  VA  22060-6218 

2.  Dudley  Knox  Library  

Naval  Postgraduate  School 

411  DyerRd. 

Monterey,  CA  93943-5101 

3.  Dr.  Conrad  F.  Newberry  

(Code  AA/Ne) 

Naval  Postgraduate  School 
Monterey,  CA  93943-5106 

4.  Dr.  Louis  V.  Schmidt  

(Code  AA/Sc) 

Naval  Postgraduate  School 
Monterey,  CA  93943-5106 

5.  Commander  

Code  471300D  (Attn:  Mr.  W.  Mark  Wonnacott) 
Naval  Air  Warfare  Center  Weapons  Division 

1  Administration  Circle 
China  Lake,  CA  93555-6001 

6.  Commander  

Code  473000D  (Attn:  Mr.  Stephen  F.  Lyda) 
Naval  Air  Warfare  Center  Weapons  Division 

1  Administration  Circle 
China  Lake,  CA  93555-6001 


Mr.  Eugene  L.  Fleeman 

Boeing  North  American,  Inc. 

MS031-FB23 

3370  East  Miraloma  Avenue 

P.O.  Box  3105 

Anaheim,  CA  92803-3105 


91 


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