A Math Model of a Well Trained Human Operator Performing a ...

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1974

A Math Model of a Well Trained Human Operator Performing a A Math Model of a Well Trained Human Operator Performing a

Tracking Task. Tracking Task.

Thomas Mahoney Perkins Jr Louisiana State University and Agricultural & Mechanical College

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Xptox Unfveretty MfcrofHms300 North Zaab RoadAnn Arbor, MicMaan 43106

74-24,797

PERKINS, Thomas Mahoney, J r . , 1945- A MATH MODEL OF A WELL TRAINED PERFORMING A TRACKING TASK.

The Louisiana S ta te U n iversity and A g r la r t tM l and Mechanical C o lleg e , Ph.D. , 1974 Engineering, chemical

University Microfilms, A XEROX Company, Ann Arbor, n

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RfiCEIVED.

75

A MATH MODEL OF A WELL TRAINED HUMAN OPERATOR

PERFORMING A TRACKING TASK

A D i s s e r t a t i o n

S u b m it te d t o t h e G ra d u a te F a c u l t y o f t h e L o u i s i a n a S t a t e U n i v e r s i t y a n d

A g r i c u l t u r a l a n d M e c h a n ic a l C o l le g e i n p a r t i a l f u l f i l l m e n t o f t h e

r e q u i r e m e n t s f o r t h e d e g re e o f D o c to r o f P h i lo s o p h y

i n

The D e p a r tm e n t o f C h e m ic a l E n g in e e r in g

by

Thomas M ahoney P e r k i n s , J r .B . S . , L o u i s i a n a S t a t e U n i v e r s i t y , 1967 M .S . , L o u i s i a n a S t a t e U n i v e r s i t y , 1971

May, 1974

ACKNOWLEDGMENTS

T h is w o rk w as p e rfo x m e d u n d s r t h e a b l e d i r e c t i o n o f

D r . J . A . P l a n c h a r d . H is a s s i s t a n c e , e n c o u ra g e m e n t an d

c o n f id e n c e i n t h i s p r o j e c t i s g r e a t l y a p p r e c i a t e d . I w o u ld

e s p e c i a l l y l i k e t o th a n k my w i f e , S a r b a r a , who h a s h e lp e d

s u p p o r t a n d e d i t t h i s p a p e r .

O th e r p e o p le who h a v e h e lp e d i n my e f f o r t t o w r i t e t h i s

p a p e r a r e M r. J im P a u l s o n , who k e p t t h e c o m p u te r i n w o rk in g

o r d e r a n d h e lp e d a n s w e r my m any q u e s t i o n s c o n c e r n in g t h e

S ig m a 's s o f t w a r e , a n d M r. P a t P i c k e n s , who h e lp e d w i t h h i s

com m ents p e r t a i n i n g t o c o m p u te r g r a p h i c s .

i i

ABSTRACT

P e r fo rm a n c e o f a w e l l - t r a i n e d human o p e r a t o r e n g a g e d i n

a t r a c k i n g t a s k i s a n a l y z e d . The d e v ic e c o n t r o l l e d b y th e

o p e r a t o r c o n s i s t s o f a m a n u a lly p o w e re d , n o n l i n e a r a im in g

d e v ic e w i t h o p t i c a l f e e d b a c k u s e d t o t r a c k a t h r e e d im e n s io n a l

t a r g e t . M a th e m a t ic a l m o d e ls o f t h e hum an o p e r a t o r a n d t h e

a im in g d e v ic e a r e d e v e lo p e d , t h e n u s e d t o c a l c u l a t e a l ig n m e n t

e r r o r s f o r a v a r i e t y o f t a r g e t t r a j e c t o r i e s . T r a in e d s t u d e n t

o p e r a t o r s t r a c k e d s i m u l a r t a r g e t s w i t h a h y b r i d s i m u l a t i o n o f

t h e a im in g d e v i c e . The r e s u l t s o f b o t h s t u d i e s w e re a n a ly z e d

i n d i c a t i n g t h a t t h e m u l t i - m o d a l o p e r a t o r m o d e l d e v e lo p e d w as

s u p e r i o r t o t h e l i n e a r c o n t in u o u s m o d e l . I n a d d i t i o n , t h e

hum an o p e r a t o r m odel p o s s e s s e s g o o d s t a b i l i t y a n d c a n b e

u s e d t o p r e d i c t t r a c k i n g e r r o r s o v e r a w id e r a n g e o f t a r g e t

t r a j e c t o r i e s .

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TABLE 0 7 CONTENTS

P a g e

ACKNOWLEDGMENTS ................................................................................................. i i

A B S T R A C T ................................................................................................................ i i i

L IST OP T A B L E S ......................................................................................... VI

LIST OP P IG U R E S ......................................................................................... V II

CHAPTER

I . IN TR O D U C TIO N .......................................................................... 1

I I . BACKGROUND IN FO R M A TIO N ................................................ 3

2 . 1 . I n t r o d u c t i o n .................................................................... 32 . 2 . T e r m i n o l o g y ......................................................................... 32 . 3 . P h y s i o l o g i c a l A s p e c ts o f t h e Human

O p e r a to r ........................................................................ 42 . 4 . M a th e m a t ic a l R e p r e s e n t a t i o n o f t h e Human

O p e r a t o r ........................................................................ 142 .4 1 . L i n e a r C o n s t a n t C o e f f i c i e n t

M o d e l s .............................................................. 142 .4 2 . S a m p le -D a ta M o d e l s .................................... 172 .4 3 . A d a p t iv e M anual C o n t r o l ..................... 192 .4 4 . M u lt i-M o d a l M o d e l s .................................... 222 .4 5 . A p p l i c a t i o n o f O p t i m i z a t io n T e c h

n i q u e s i n M an-M ach ine S y s te m s . 242 . 4 6 . O th e r M o d e l s ................................................... 24

I I I . TRACKING SYSTEM D E SC R IPT IO N ...................................... 26

3 . 1 . I n t r o d u c t i o n .................................................................... 263 . 2 . The Gun M o d e l .................................................................... 263 .3 . C a l c u l a t i o n o f A lig n m e n t E r r o r s ...................... 293 . 4 . Gun P a r a m e te r D e t e r m i n a t i o n ................................ 333 . 5 . I n p u t S i g n a l D e s c r i p t i o n ..................................... 403 . 6 . S e n s i t i v i t y A n a l y s i s o f t h e Gun S y s te m . 48

IV . REAL TIME SIMULATION OF THE TRACKING SYSTEM . . 52

4 . 1 . I n t r o d u c t i o n .................................................................... 52

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4 . 2 . T he H y b r id C o m p u te r . . . . . . . . . . . 524 . 3 . T he H y b r id P r o g r a m s .................................................. 54

V. THE HUMAN OPERATOR M O D E L ........................................... 66

5 . 1 . I n t r o d u c t i o n ................................................................. 665 . 2 . The T r a c k in g M o d e ....................................................... 66

5 .2 1 . T he L in e a r Human O p e r a to r . . . . 665 .2 2 . The N o n l i n e a r O p e r a t o r ................... 75

5 . 3 . V e r i f i c a t i o n o f t h e T r a c k in g Mode M odel . 1195 . 4 . T he A c q u i s i t i o n M o d s ...................................................138

5 .4 1 . D e r iv a t io n o f S w itc h in g T im es • . 1385 .4 2 . A c q u i s i t i o n a t C r o s s o v e r .................141

5 . 5 . V e r i f i c a t i o n o f t h e A c q u i s i t i o n ModeM o d e l ............................................................................. 142

5 . 6 . The D i g i t a l S i m u l a t i o n ..............................................157

V I . EXAMINATION OF RESULTS AND CONCLUSIONS . . . . 161

6 . 1 . E x a m in a t io n o f E x p e r im e n ta l R e s u l t s . . . 1616 . 2 . C o n c l u s i o n s ............................................................................162

B IB L IO G R A PH Y ........................................................................................................... 164

APPENDICES

A . CALCULATION OF ALIGNMENT E R R O R ...............................168

A - l . D e v e lo p m e n t o f C o o r d in a te R o t a t i o n E q u at i o n ........................................................................................168

B . PERSPECTIVE P R O JE C T IO N ...................................................171

C . COMPUTER LISTING OF PERSPECTIVE PROJECTIONPROGRAM....................................................................................... 184

D. THE GUN MODEL............................................................................ 198

D - l . D e r i v a t i o n o f S c a le d Gun E q u a t io n s . . . 198D -2 . D e v e lo p m en t o f L o g ic - S i g n a l f o r E l e c t r o n i c

S w i t c h ........................................................................ 201

E . PAPER TAPE CONVERSION PROGRAM....................................208

F . COMPUTER LISTING OF HYBRID PR O G R A M .....................210

G. LINEARIZATION OF THE GUN M O D E L .............................. 223

H . COMPUTER LISTING OF DIGITAL SIMULATION . . . . 225

V I T A ................................................................................................................................255

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L IST OF TABLES

T a b le P ag e

1 . The V a r ia t io n , o f R e a c t io n T im e w i th A g e ............... 10

2 . Gun P a r a m e t e r s ........................................................................... 41

3 . A i r c r a f t M an eu v ers T ype I .................................................. 42

4 . A i r c r a f t M a n eu v e rs Type I I ............................................. 43

5 . Optim um P a r a m e te r s f o r DSP Human O p e r a t o r M odel . 106

D - l . S c a l i n g T a b le o f t h e Gun M odel <A zim u th ) . . . . 200

D -2 . S c a l i n g T a b le o f t h e Gun M odel ( E le v a t i o n ) . . . 203

D -3 . P o t S e t t i n g s f o r t h e A n a lo g M o d e l ................................. 205

D -4 . D e s c r i p t i o n o f L o g ic S i g n a l ................................................ 206

D -5 . T r u th T a b le f o r L o g ic S w i tc h in g S i g n a l ...................... 207

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L IST OF FIGURES

I l l u s t r a t i o n o f C o m p e n sa to ry a n d P u r s u i tT r a c k in g D is p la y s ..................................................................

T he V i s u a l C o m p e n sa to ry T r a c k in g S y s te m . . . .

The V i s u a l P u r s u i t T r a c k in g S y s te m . . . . . .

The E f f e c t s o f T r a i n i n g o n T r a c k in g P e r f o rm a n c e . .......................... .......................... .....

R ange o f I n d i v i d u a l D i f f e r e n c e s U s in g a H andw h e e l t o O vercom e F o r c e s a t D i f f e r e n t R o t a t i o n S p e e d s ....................... . .........................................

D e s c r ib i n g F u n c t io n B lo c k D ia g ra m . . . . . . .

T y p ic a l S e t - u p f o r R em nan t M in im iz a t io n . . . .

S a m p le d -D a ta M odel o f Human O p e r a t o r i n Compens a t o r y T r a c k in g .......................................................................

M a jo r A d a p t iv e F u n c t io n s i n M anual C o n t r o l . .

C o s t e l l o '8 D ual-M ode M odel . . . . . . . . . .

Gun M odel ............................................................................................

Gun P o s i t i o n i n I n e r t i a l C o o r d in a te S y s te m . .

S i t e P a t t e r n Gun C o o r d in a te S y s te m . .....................

A n g le M is a l ig n m e n t Gun C o o r d in a te S y s te m . . .

S i m p l i f i e d T r a c k in g S y s te m ..............................................

R o o t L o cu s D iag ram o f S i m p l i f i e d T r a c k in g - S y s te m I .......................................................................................

R o o t L o cu s D iag ram o f S i m p l i f i e d T r a c k in g - S y s te m I I .......................................................................................

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F ig u r e P a fP

3 -8 Gun M odel (A z im u th C h a n n e l) w i t h L i m i t e r s . . . 38

3 -9 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th N um ber1 0 2 44

3 -1 0 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th N um ber 1 . 49

3 -1 1 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th N um ber 5 . 44

3 -1 2 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th N um ber1 0 4 7

3 -1 3 R e sp o n se C u rv e s f o r Two U n s ta b le S y s te m s . . . 30

3 -1 4 R e s u l t s o f S e n s i t i v i t y A n a l y s i s ................................... 31

4 -1 H y b r id S i m u l a t io n : O p e r a t o r i n L oop . . . . . 99

4 -2 H y b r id S i m u l a t io n o f C o m p e n sa to ry T r a c k in g . . 94

4 - 3 D i g i t a l F low C h a r t .................................................................. 37

4 -4 A n a lo g S i m u l a t io n o f G u n ................................................... 34

4 - 5 P e r fo rm a n c e V e rs u s E x p e r i e n c e ........................................ 41

4 -6 T r a c k in g P e r fo rm a n c e o f A c tu a l O p e r a t o r I . . . 4 9

4 - 7 T r a c k in g P e r fo rm a n c e o f A c tu a l O p e r a t o r I I . . 44

4 -8 T r a c k in g P e r fo rm a n c e o f A c t u a l O p e r a t o r I I I . . 44

5 -1 E r r o r V e rs u s E r r o r R a t e ........................................................ 67

5 -2 T r a c k in g o f S t a t i o n a r y T a r g e t K » 5 ........................... 49

5 - 3 T r a c k in g o f S t a t i o n a r y T a r g e t K * 2 0 ...................... 44

5 -4 T r a c k in g w i t h L i n e a r O p e r a to r I ..................................... 79

5 -5 T r a c k in g w i th L i n e a r O p e r a t o r I I ................................ 73

5 -6 T r a c k in g w i t h L i n e a r O p e r a t o r I I I ................................ 74

5 - 7 T r a c k in g w i t h N o n -L in e a r O p e r a t o r I . . . . . . 74

5 - 8 T r a c k in g w i t h N o n -L in e a r O p e r a t o r I I .................... 77

5 -9 T r a c k in g w i t h N o n -L in e a r O p e r a t o r I I I .................... 78

5 -1 0 C o n t r o l S y s te m w i t h Human O p e r a t o r S e n s in gA i r c r a f t V e l o c i t y ................................................................... 80

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5 -1 1 T r a c k in g w i t h N o n - L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r I ................................................................... 82

5 -1 2 T r a c k in g w i t h N o n - L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r I I .............................................................. 83

5 -1 3 T r a c k in g w i t h N o n -L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r I I I .............................................................. 84

5 -1 4 T r a c k in g w i t h N o n - L in e a r M odel P l u s F e e dF o rw a rd O p e r a to r I V ............................................................. 85

5 -1 5 T r a c k in g w i t h N o n - L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r V .................................................................. 86

5 -1 6 T r a c k in g w i t h N o n -L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r V I .............................................................. 87

5 -1 7 T r a c k in g w i t h N o n -L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r V I I ............................................................. 89

5 -1 8 T r a c k in g w i t h N o n -L in e a r M odel P l u s F e e dF o rw a rd O p e r a t o r - C o u rs e 1 0 2 .................................... 91

5 -1 9 T r a c k in g M odel o f H unan O p e r a t o r ..................................... 92

5 -2 0 a A v e ra g e Human O p e r a t o r T r a c k in g P e r fo rm a n c e - I(A z im u th E r r o r ) ........................................................................ 95

5 -2 0 b A v e ra g e Human O p e r a to r T r a c k in g P e r fo rm a n c e - I I( E l e v a t i o n E r r o r ) . , . .............................................. 96

5 - 2 1 a A v e ra g e Human O p e r a t o r T r a c k in g P e r fo rm a n c e - I I(A z im u th E r r o r ) ......................... 97

5 -2 1 b A v e ra g e Human O p e r a t o r T r a c k in g P e r fo rm a n c e - I I( E l e v a t i o n E r r o r ) ................................................................... 98

5 -2 2 a A v e ra g e Human O p e r a t o r T r a c k in g P e r fo rm a n c e -I I I (A z im u th E r r o r ) .............................................................. 99

5 -2 2 b A v e ra g e Human O p e r a t o r T r a c k in g P e r f o rm a n c e -I I I ( E l e v a t i o n E r r o r ) . .................................................... 100

5 -2 3 P e r fo rm a n c e o f Human O p e r a t o r M odel - I(A z im u th O p e r a t o r ) ........................................................................ 101

5 -2 4 P e r fo rm a n c e o f Human O p e r a t o r M odel - I I( E le v a t i o n O p e r a t o r ) ................................................................... 102

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P e r fo rm a n c e o f Human O p e r a t o r M odel - I I I (A z im u th O p e r a t o r ) .............................................................

P e r fo rm a n c e o f Human O p e r a t o r M odel - IV( E l e v a t i o n O p e r a to r ) ........................................................

P e r fo rm a n c e o f D6P Human O p e r a t o r M odel - I (A z im u th O p e r a t o r ) ..............................................................

P e r fo rm a n c e o f DSP Human O p e r a t o r M odel - I I ( E l e v a t i o n O p e r a to r ) ........................................................

P e r fo rm a n c e o f DSP Human O p e r a t o r M odel - I I I (A z im u th O p e r a t o r ) ..............................................................

P e r fo rm a n c e o f DSP Human O p e r a t o r M odel - IV ( E l e v a t i o n O p e r a to r ) .........................................................

Mean T r a c k in g E r r o r fro m M odel f o r S y s te m 3 -1 (A z im u th O p e r a t o r ) .............................................................

Mean T r a c k in g E r r o r fro m M odel f o r S y s te m 3 -1 ( E l e v a t i o n O p e r a to r ) ........................................................

Mean T r a c k in g E r r o r fro m M odel f o r S y s te m 3- I I (A z im u th O p e r a t o r ) ..............................................................

Mean T r a c k in g E r r o r fro m M odel f o r S y s te m 3 - I I ( E l e v a t i o n O p e r a to r ) ........................................................

Mean T r a c k in g E r r o r f ro m M odel f o r S y s te m 2 -1 (A z im u th O p e r a to r ) .....................

Mean T r a c k in g E r r o r fro m M odel f o r S y s te m 2 -1 ( E l e v a t i o n O p e r a to r ) ........................................................

Mean T r a c k in g E r r o r fro m M odel f o r S y s te m 2 - I I (A z im u th O p e r a t o r ) .............................................................

Mean T r a c k in g E r r o r f ro m M odel f o r S y s te m 2 - I I ( E l e v a t i o n O p e r a to r ) .........................................................

A z im u th E r r o r v s T im e ( F l i g h t P a th #1 -S tu d y #2) ..................................................................................

E l e v a t i o n E r r o r v s Tim e ( F l i g h t P a t h #1 -S tu d y #2) . . . . . . . . . . . . .....................

A z im u th E r r o r v s T im a ( F l i g h t P a t h #5 -S tu d y #2) ..................................................................................

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5 -3 8 E l e v a t i o n E r r o r va T im e ( F l i g h t P a th #5 -S tu d y # 2 ) .............................................................................................123

5 -3 9 A s im u th E r r o r v s T im s ( F l i g h t P a th #6 -S tu d y # 2 ) ............................................................................................. 124

5 -4 0 E l e v a t i o n E r r o r v s T im e ( F l i g h t P a th #6 -S tu d y # 2 ) ............................................................................................. 125

5 -4 1 A z im u th E r r o r v s T im e ( F l i g h t P a th #11 -S tu d y # 2 ) ............................................................................................. 126

5 -4 2 E l e v a t i o n E r r o r v s T im e ( F l i g h t P a th #11 -S tu d y # 2 ) ......................................... 127

5 -4 3 A z im u th E r r o r v s Tim e ( F l i g h t P a th #16 -S tu d y # 2 ) .............................................................................................128

5 -4 4 E l e v a t i o n E r r o r v s T im e ( F l i g h t P a th #16 -S tu d y # 2 ) ....................................... 129

5 -4 5 A z im u th E r r o r v s T im e ( F l i g h t P a t h #6 -S tu d y # 3 ) .................................. 130

5 -4 6 E l e v a t i o n E r r o r v s T im e ( F l i g h t P a t h #6 -S tu d y # 3 ) ........................................................................................... 131

5 -4 7 A z im u th E r r o r v s T im e ( F l i g h t P a th #10 -S tu d y # 3 ) ............................................................................................132

5 -4 8 E l e v a t i o n E r r o r v s T im e ( F l i g h t P a t h #10 -S tu d y # 3 ) ....................................... 133

5 -4 9 A z im u th E r r o r v s T im e ( F l i g h t P a th #15 -S tu d y # 3 ) .................................. 134

5 -5 0 E l e v a t i o n E r r o r v s T im e ( F l i c p i t P a t h #15 -S tu d y # 3 ) ....................................... 135

5 -5 1 A z im u th E r r o r v s T im e ( F l i g h t P a th #20 -S tu d y # 3 ) .................................. 136

5 -5 2 E l e v a t i o n E r r o r v s T im e ( F l i g h t P a th #20 -S tu d y # 3 ) ....................................... 137

5 -5 3 F low D ia g ra m o f t h e M u lt i-M o d a l T r a c k in gM o d e l ............................................................................ 144

5 -5 4 a C o m p a riso n o f A c q u i s i t i o n P e r f o rm a n c e . . . . . 145

x i

F i g u r e P ag e

5 -5 4 b C o m p a riso n o f A c q u i s i t i o n P e - fo rm a n c e . . . . 146

5. 55a C o n p a r is o n o f A c q u i s i t i o n P e r fo rm a n c e . . . . 147

5 -5 5 b C o m p a riso n o f A c q u i s i t i o n P e r fo rm a n c e . . . . 148

5 - 5 6 a C o m p a riso n o f A c q u i s i t i o n P e r fo rm a n c e . . . . 149

5 -5 6 b C o m p a riso n o f A c q u i s i t i o n P e r fo rm a n c e . . . . 150

5 -5 7 P e r fo rm a n c e a t C r o s s o v e r ..................................................151

5 -5 8 P e r fo rm a n c e a t C r o s s o v e r .................................................. 152



5 -6 1 D u a l-M o d e l T r a c k in g D is c o n t in u o u s T a r g e t . . 155

5 -6 2 D u a l-M o d e l T r a c k in g D is c o n t in u o u s T a r g e t . . 156

5 -6 3 F low C h a r t o f D i g i t a l S i m u l a t i o n ........................ 159

A—1 n o t a t i o n o f I n e r t i a l C o o r d in a te S y s te mT h ro u g h A n g le s a n d .........................................................169

B - l P r o j e c t i o n P ro g ra m F low C h a r t ........................................173

B -2 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a t h 101 . . 175

B -3 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th 103 . . 176

B -4 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th N o . 2 . 177

B -5 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a t h N o . 3 . 178

B -6 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a th N o. 4 . 179

B -7 P e r s p e c t i v e P r o j e c t i o n o f F l i g h t P a t h N o . 6 . 180



B -1 0 P e r s p e c t i v e P r o j e c t i o n o f - F l i g h t p a t h N o . 9 . 183

D—1 S c a l e d Gun M o d e l ......................................................................199

X ii

F i g u r e P ag e

D -2 C i r c u i t D iag ram - G e n e r a t io n o f t h e L o g icC o n t r o l S i g n a l (L S V f) ................................... 202

G - l F low D iag ram o f L i n e a r i s a t i o n P r o c e d u r e . . . 224

x i i i

CHAPTER X

INTRODUCTION

B e g in n in g a b o u t 19>t5, a l a r g e num ber o f p a p e r s w e re

p u b l i s h e d i n t h i s c o u n t r y an d i n E n g la n d w h ic h u t i l i z e d c o n

t r o l t h e o r y t o d e s c r i b e t h e r o l e o f t h e human i n t r a c k i n g .

I n t e r e s t i n - t h i s a r e a o f c o n t r o l d e c l i n e d d u r in g t h e m id

6 0 * s b e c a u s e im p ro v e m e n ts i n c o m p u te r t e c h n o lo g y l e d many

p e o p le t o b e l i e v e t h a t i n s t r u m e n t a t i o n w o u ld r e p l a c e t h e

hum an o p e r a t o r i n m o st c o n t r o l t a s k s . H o w ev er, t h e v a lu e o f

t h e hum an o p e r a t o r i s b e in g r e d i s c o v e r e d b e c a u s e o f h i s a b i l i t y

t o p e r fo r m a m u l t i t u d e o f t a s k s , h i s a d a p t a b i l i t y u n d e r

v a r y i n g c o n d i t i o n s , h i s r e l i a b i l i t y , an d h i s r e l a t i v e l y low

c o s t .

To d a t e , a d e f i n i t i v e m odel o f t h e human o p e r a t o r h a s

n o t b e e n d e v e lo p e d , i n s t e a d , t h e r e a r e many m o d e ls . M ost o f

t h e d i f f e r e n c e s i n t h e s e m odelB a r i s e fro m c o n s i d e r i n g t h e

o p e r a t o r fro m d i f f e r i n g p o i n t s o f v ie w . A ls o t h e r e may b e a

t r a n s f e r f u n c t i o n f o r e a c h o f t h e v a r i o u s c l a s s e s o f i n p u t -

o u t p u t c o m b in a t io n s . T h u s , f o r a p a r t i c u l a r m an -m ach in e

s y s te m a n d c l a s s o f i n p u t s i g n a l s i t i s p o s s i b l e t o d e s c r i b e

m a th e m a t i c a l ly t h e t r a n s f e r f u n c t i o n o f t h e human o p e r a t o r .

2

Much o f t h e c u r r e n t r e s e a r c h i s c e n t e r e d a ro u n d o b t a i n i n g

a c c u r a t e hum an o p e r a t o r m o d e ls i n o r d e r t o a i d i n t h e d e s ig n

o f m an -m ac h in e s y s te m s a n d t o d e v e lo p b e t t e r t e c h n iq u e s f o r

e v a l u a t i n g m a n u a lly c o n t r o l l e d s y s te m s .

I n t h e s e c o n d c h a p t e r som e o f t h e te rm s u s e d i n t h e

p a p e r a r e d e f i n e d , a n d a few p h y s i o l o g i c a l a s p e c t s o f t h e

hum an o p e r a t o r a r e 'p r e s e n t e d . F u r th e r m o r e , som e o f t h e v a r io u s

m e th o d s u s e d t o m o d e l t h e hum an o p e r a t o r a r e p r e s e n t e d .

I n C h a p te r I I I a m a th e m a t i c a l m o d e l o f t h e c o n t r o l l e d

s y s te m , a n a im in g m ec h an ism , i s d e v e lo p e d . T h is i n c l u d e d

m o d e l in g t h e c o n t r o l l e d e l e m e n t , d e v e lo p in g a way t o c a l c u

l a t e a im e r r o r s , a n d s e l e c t i n g v a l u e s o f t h e m o d e l p a r a m e t e r s .

F i n a l l y , i n c o n j u n c t i o n w i t h d e s c r i b i n g t h e c o n t r o l l e d s y s te m

t h e t r a c k i n g t a s k s a r e d i s c u s s e d .

I n C h a p te r IV t h e h y b r i d c o m p u te r p ro g ra m u s e d i n

o b t a i n i n g t r a c k i n g d a t a i s d o c u m e n te d . T h is d a t a w i l l be

u s e d i n C h a p te r s V a n d VI t o d e v e lo p hum an o p e r a t o r p a r a m e te r s

a n d t o v e r i f y t h e o p e r a t o r m o d e l .

The hum an o p e r a t o r m odel c o n s i s t i n g o f a m u lt i -m o d e

m o d e l i s d e v e lo p e d i n C h a p te r V , T he m odel c o n s i s t s o f a

t r a c k i n g mode a n d a c q u i s i t i o n m o d e s . I n a d d i t i o n t o d e v e lo p in g

t h e hum an o p e r a t o r m o d e l, t h e d i g i t a l s i m u l a t i o n o f t h e

t r a c k i n g s y s te m i s a l s o d i s c u s s e d .

T he f i n a l c h a p t e r c o n t a i n s a n e x a m in a t io n o f t h e e x p e

r i m e n t a l r e s u l t s a n d t h e c o n c l u s i o n s a r r i v e d a t a s a r e s u l t

o f t h e r e s e a r c h .

CHAPTER I I

BACKGROUND AND INFORMATION

2 . 1 . I n t r o d u c t i o n

The r e s e a r c h d e s c r i b e d i n t h i s d i s s e r t a t i o n d e a l s w i th

t h e m a th e m a t ic a l d e s c r i p t i o n o f m an -m ach in e s y s te m s . T h e r e

f o r e , i t i s c o n v e n ie n t a t t h i s p o i n t t o d e f i n e some o f th e

te rm s u s e d i n t h i s p a p e r a n d t o d i s c u s s a few p h y s i o l o g i c a l

a s p e c t s o f t h e human o p e r a t o r . A ls o , some o f t h e v a r i o u s

m e th o d s u s e d t o m o d el th e hum an o p e r a t o r a r e su m m arize d i n

t h i s c h a p t e r .

2 . 2 . T e rm in o lo g y

T h e re a r e b a s i c a l l y tw o t y p e s o f t r a c k i n g t o o l s - com

p e n s a t o r y a n d p u r s u i t t r a c k i n g . 3. The tw o k in d s o f t r a c k i n g

d i f f e r o n ly i n t y p e o f s i g n a l s e n s e d by t h e o p e r a t o r . The

o p e r a t o r r e c e i v e s s e p a r a t e l y t a r g e t a n d c o n t r o l l e d e le m e n t

i n f o r m a t i o n i n p u r s u i t t r a c k i n g , w h e re a s he o n l y r e c e i v e s t h e

d i f f e r e n c e ( a n e r r o r s i g n a l ) i n c o m p e n s a to ry t r a c k i n g .

C o m p e n sa to ry t r a c k i n g . I n c o m p e n s a to ry t r a c k i n g t h e r e

i s a d i s p l a y ( u s u a l l y v i s u a l ) w h ic h h a s tw o i n d i c a t o r s , o n e

f i x e d a n d t h e o t h e r m o v e a b le . The t a r g e t i s r e p r e s e n t e d by

3

t h e m o v e a b le i n d i c a t o r , w h i le t h e c o n t r o l l e d e le m e n t i s r e p r e

s e n t e d by a f i x e d p o i n t a n d i s s u b j e c t t o c o n t r o l b y m a n ip u

l a t i o n o f t h e c o n t r o l d e v i c e s . When t h e tw o a r e s u p e r im p o s e d ,

t h e c o n t r o l l e d e le m e n t i s "o n t a r g e t . " Any d i f f e r e n c e

r e p r e s e n t s an e r r o r , a n d t h e f u n c t i o n o f t h e o p e r a t o r i s t o

e l i m i n a t e o r m in im iz e t h a t e r r o r . An i l l u s t r a t i o n o f th e d i s

p l a y iB shown i n F ig u r e 2 -1 A , w h i l e a b lo c k d ia g ra m o f a t y p i c a l

v i s u a l c o m p e n s a to ry t r a c k i n g s y s te m i s g iv e n i n F ig u r e 2 - 2 .

P u r s u i t T r a c k in g . I n p u r s u i t t r a c k i n g th e l o c a t i o n o f

b o th t h e t a r g e t a n d t h e c o n t r o l l e d e le m e n t a r e know n , s o t h e i r

r e l a t i o n s h i p ( a s i n s p a c e ) c a n be v i s u a l i z e d b y t h e o p e r a t o r .

A p p r o p r i a t e c o n t r o l w i l l a l i g n t h e c o n t r o l l e d e le m e n t w i th

t h e t a r g e t a n d k e e p i t on t a r g e t , e v e n th o u g h t h e t a r g e t

c o n t i n u e s t o m ove. A p u r s u i t d i s p l a y i s i l l u s t r a t e d i n

F ig u r e 2-1B an d a b lo c k d ia g ra m o f a v i s u a l p u r s u i t t r a c k i n g

s y s te m i s g iv e n i n F ig u r e 2 - 3 . A s p e c i a l c a s e o f p u r s u i t

t r a c k i n g i s p r e c o g n i t i v e t r a c k i n g . The te r m " p r e c o g n i t i v e "

i s e m p lo y e d t o d e n o te t h e s i t u a t i o n i n w h ic h t h e f u t u r e v a lu e s

o f t h e i n p u t s i g n a l a r e v i s i b l e o r know n. D r iv in g a v e h i c l e

o r o p e r a t i n g a c r a n e a r e e x a m p le s o f p r e c o g n i t i v e t r a c k i n g .

2 . 3 . P h y s i o l o g i c a l A s p e c ts o f t h e Human O p e r a t o r

W. F . F i e l d i n g h a s su m m a riz e d some o f t h e m ore i m p o r t a n t

hum an c h a r a c t e r i s t i c s a n d f a c t o r s w h ic h a f f e c t th e m .

(1 ) Common o p e r a t o r r e a c t i o n s t o s t i m u l u s c a n b e d i v id e d

i n t o t h r e e m a jo r c a t e g o r i e s :

5

Compensatory tracking Pursuit tracking

2 -LA

C (control): FixedT (target): Movable C- T: Error

2 - IB

C(control): MovableT (target): MovableC -----T : shows d ifferen ce

in r e la tiv e locations

Figure 2-1 I llu s tr a t io n o f Coaipenaatory and pursuit tracking d isp lays.

A compensatory tracking d isp lay shows only the d ifferen ce (error) between the target T and the contro lled element C. A pursuit display shows the location (or other value represented) of both the target and the controlled element.

6

HUMANOPERATOR

CONTROLLED

Gjg

FEEDBACKELEMENT

Uhere: G^(i) ■ Ruaan Operator Transfer function6 p (i) ■ Controlled Eleaent Transfer function

r * aystea inpute ■ ayatea errora ■ Huaan operator outputc ■ ayatea output

■ observed ayatea output. •

Figure 3-3 The V isual Compensatory Tracking Syatea

Otiplay

G p IT)

where: CuCT) “ Huean Operator transfer functionG (T) - Controlled eleaent transfer function

Pr ■ ayaten inputm ■ hunan operator outputc ■ system outputc* “ observed system output

Figure 2-3 The Visual Pursuit System

a ) R e f la x a c t i o n : a lm o s t i n v o l u n t a r y w i t h o n ly a few m i l l i s e c o n d s b e tw e e n s t i m u l u s a n d r e a c t i o n

b ) N orm al r e a c t i o n : a c t i o n c a r r i e d o u t v o l u n t a r i l y a n d r e q u i r i n g a lm o s t t o t a l c o n c e n t r a t i o n upon s t i m u l u s .

c ) A u to m a t ic r e a c t i o n : f a s t v o l u n t a r y r e a c t i o n a c h i e v a b l e a f t e r much p r a c t i c e .

( 2 ) A s t i m u l u s m u s t e x c e e d a c e r t a i n t h r e s h o l d v a lu e

b e f o r e t h e s e n s o r y c e l l s w i l l t r a n s m i t an im p u l s e .

( 3 ) The hum an e y e h a s a h i g h s e n s i t i v i t y a n d c a n d e t e c t

m is a l ig n m e n t a s s m a l l a s f i v e s e c o n d s o f a r c a t t h e e y e u n d e r

s u i t a b l e c o n d i t i o n s .

( 4 ) C o n t r a s t b e tw e e n t a r g e t an d b a c k g ro u n d i s v e ry

i m p o r t a n t i n t a r g e t d e t e c t i o n a n d a l ig n m e n t i n t r a c k i n g .

( 5 ) R e a c t io n t i m e , t h e t im e d e la y b e tw e e n r e c e i p t o f an

a p p r o p r i a t e s t i m u l u s a n d i n i t i a t i o n o f r e a c t i o n , c a n b e

a p p ro x im a te d '* b y :

T - 0 .2 7 ln (N + 1 ) ( 2 . 3 - 1 )

w h e re : N = N um ber o f e q u i - p r o b a b l e s i g n a l sT = R e a c t io n t im e i n s e c o n d s .

T h is h a s b e e n r e p o r t e d a s a g o o d a p p r o x im a t io n , b u t i n some

c a s e s r e a c t i o n t im e o f a s l i t t l e a s .1 s e c o n d s h a s b e e n

o b s e r v e d . A v a lu e o f .1 5 s e c o n d s f o r u s u a l r e a c t i o n tim e i s

com m only u s e d , w hen m o d e lin g a hum an o p e r a t o r p e r f o r m in g a

t r a c k i n g t a s k .h

( 6 ) T he f o l lo w in g f a c t o r s a f f e c t human r e a c t i o n t im e s :

a ) R e a c t io n t im e v a r i e s w i t h t h e t y p e o f s t i m u l u s , r a n g i n g fro m .7 s e c o n d s f o r p a i n , t o a b o u t .19 s e c o n d s f o r l i g h t a n d s o u n d .

b ) R e a c t io n t im e v a r i e s w i th s e x , , m a le s h a v in g a p p r o x im a te ly .1 s e c o n d s h o r t e r r e a c t i o n t im e f o r so u n d a n d l i g h t t h a n f e m a le s .

9

c ) R e a c t io n t im e v a r i e s w i t h a g e . The v a r i a t i o n o f r e a c t i o n t im e f o r l i g h t a n d so u n d i s g iv e n i n T a b le 1 .

d ) R e a c t io n t im e c a n b e r e d u c e d b y p r a c t i c e a p p r o x i m a te ly 10 p e r c e n t .

( 7 ) V o lu n ta r y c o r r e c t i v e m ovem ent s u c h a s o c c u r s i n

t r a c k i n g a p p e a r s t o h a v e tw o c o m p o n e n ts :

a ) The f i r s t , a r a p i d j e r k t o i n i t i a t e t h e m o tio n .

b ) The s e c o n d , a f t e r t h i s i n i t i a l d i s c o n t i n u i t y , a c o n t in u o u s t r a c k i n g m o tio n i s a c h ie v e d .

( 8 ) P r a c t i c e a n d l e a r n i n g e n a b le a human o p e r a t o r t o

r e d u c e h i s r e a c t i o n t im e t o w i t h i n a c c e p t a b l e l i m i t s an d to

i n c r e a s e h i s a c c u r a c y by l e a r n i n g t o p r e d i c t f u t u r e e v e n t s .

F ig u r e 2-** show s t h e e f f e c t s o f t r a i n i n g o n t r a c k i n g q u a l i t y .

Upon r e c o g n i t i o n o f a r e g u l a r p a t t e r n , t h e o p e r a t o r u s u a l l y

s w i t c h e s f ro m a c t i v e t r a c k i n g t o w h a t i s c a l l e d " m o n i t o r i n g , "

t h u s r e d u c i n g h i s w ork l o a d w h i l e a c c o m p l i s h in g s u p e r i o r

c o n t r o l . I n t h i s c a s e , t h e hum an o p e r a t o r becom es a p r e c o g n i

t i v e s i g n a l g e n e r a t o r a n d t h u s n u l l i f i e s t h e t im e l a g .

I n a d d i t i o n t o t h e c h a r a c t e r i s t i c s su m m arize d b y F i e l d i n g ,C

F o g e l s u g g e s t s t h e e x i s t e n c e o f an a c q u i s i t i o n mode w h ic h i s

t r i g g e r e d b y an y e r r o r o r e r r o r r a t e o u t s i d e t h e t r a c k i n g

mode d o m a in . I n t h i s mode o f o p e r a t i o n t h e o p e r a t o r t r i e s to7 8 9m in im iz e t h e e r r o r a s r a p i d l y a s p o s s i b l e . N a d l e r , * *

s t u d y i n g arm m o tio n c o n d i t i o n s o f maximum s t r e s s , o b s e r v e d

v e l o c i t y a n d l im e h i s t o r i e s a lm o s t i d e n t i c a l w i th t h o s e o f

C o s t e l l o 's ^ ® t im e - o p t i m a l s e c o n d - o r d e r c o n t r o l l e r i n r e s p o n s e

t o s t e p i n p u t s . A n o th e r s y s te m w h ic h s u g g e s t s t h e e x i s t e n c e

10

The Variation

* i«

5

10

20

30

40

50

55

60

TABLE 1

o f Reaction Tla» with Age

Reaction Tlaie

0.40 aec.

0.30 sec .

0.20 aec.

0.22 sec .

0.25 sec .

0.28 sec.

0.35 sec .

0.50 sec .

11

Before Training

A fter Training

Target

Operator Aeaponee _

Figure 2-4 The E ffecta Training on Tracking Performance

12

o f a n a c q u i s i t i o n n o d e i s t h e c o n t r o l s y s te m w h ic h e n a b l e s u s11 12t o * move o u r e y e s a n d f o l lo w a m o v in g t a r g e t . The i n i t i a l

s c a n n in g m o v em en ts , c a l l e d s o c c a d i c e y e m o v e m e n ts , c a n b e

c h a r a c t e r i z e d b y v e ry r a p i d a c c e l e r a t i o n a n d d e c e l e r a t i o n (u p

t o 4 0 /0 0 0 d e g r e e s / s e c ^ ) a n d a p e a k v e l o c i t y d u r i n g t h e m o tio n

a s h ig h a s 400 t o 600 d e g r e e s / s e c . I n a d d i t i o n t o t h i s r a p i d

m ovem en t, t h e r e i s v e r y l i t t l e o v e r s h o o t o f t h e t a r g e t .

I t i s a l s o p e r t i n e n t t o l i s t s e v e r a l d e s ig n c h a r a c t e r i s

t i c s o f h a n d w h e e l c o n t r o l s .

( 1 ) The optim um d i a m e t e r f o r a h a n d w h e e l w i t h n i n e t y

in c h -p o u n d s o f f u n c t i o n a l t o r q u e i s 10 i n c h e s . ***

( 2 ) F o r op tim um a c c u r a c y i n o n e -h a n d e d t r a c k i n g , i t w as

fo u n d t h a t t h e h a n d w h e e l r a d i u s s h o u ld b e 4 .5 i n c h e s f o r s lo w

t u r n i n g r a t e s , b u t a t h i g h t u r n i n g r a t e s t h e a d v a n ta g e

s h i f t e d t o a 2 .2 5 in c h h a n d w h e e l .

(3 ) The f o r c e s t h a t c a n b e o v e rc o m e a t d i f f e r e n t s p e e d s

o f r o t a t i o n f o r a t h r e e - i n c h h a n d w h e e l i s show n i n F ig u r e

2 - 5 . 16

( 4 ) The hum an o p e r a t o r s h o u ld n o t b e r e q u i r e d t o p e r fo rm

a t maximum c a p a c i t y f o r a n y g r e a t l e n g t h o f t i m e ; t h e r e f o r e ,1 7i t i s recom m endedx t h a t t h e f o l l o w i n g maximum a l l o w a b le

f o r c e s —43 p o u n d s f o r a 1 0 " d i a m e t e r h a n d w h e e l a n d 12 p o u n d s

f o r an 1 8 ” d i a m e t e r h an d w h ee l)—b e u s e d w hen d e s i g n in g c o n t r o l

s y s t e m s .

13

3 f t

30

I% 38oV8£ 1-0

8040

Revolution* per Minute

Figure 2-5: Range of individual d ifference* using * handvtiedl toovercone forces at d ifferen t rotation speed*. The shaded area represents the range o f Individual d ifferen ces , fron the noxious of the weakest subject to the naxlnun of the strongest.

14

2 . 4 . M a th e m a t ic a l R e p r e s e n t a t i o n o f t h e Human O p e r a t o r

P r e s e n t l y m any m e th o d s a r e e m p lo y e d t o m o d e l t h e p e r f o r m

a n c e s o f a human o p e r a t o r e n g a g e d i n c o m p e n s a to ry t r a c k i n g .

T h e s e m ode1b r a n g e i n c o m p le x i ty fro m l i n e a r t im e i n v a r i e n t

m o d e ls t o t h e m ore s o p h i s t i c a t e d o n e s u s i n g s t o c h a s t i c a n d

a d a p t i v e a p p r o a c h e s . The f o l lo w in g i s a d e s c r i p t i o n o f som e

o f t h e m o d e ls p r e s e n t e d t o d a t e .

2 .4 1 . L i n e a r C o n s ta n t C o e f f i c i e n t M o d e ls . E a r l y a t t e m p t s

a t m o d e lin g t h e hum an o p e r a t o r r e s u l t e d i n t h i s r e p r e s e n t a t i o n

by a g a in p l u s a t im e d e l a y :

G j^ S ) = K e x p ( - T S ) • ( 2 . 4 - 1 )

T hen i n 1947 A. T u s t in ,* ® w o rk in g t o im p ro v e th e a im in g o f

t a n k g u n s , a d d e d a d e r i v a t i v e a n d i n t e g r a l mode t o t h e m o d e l ,

r e s u l t i n g i n t h e f o l l o w in g e q u a t io n :

G ^ S ) = e x p ( - T S ) ( A S + B + C /S ). ( 2 . 4 - 2 )

T h is c o n c e p t h a s b e e n e x p a n d e d , u n t i l p r e s e n t l y a com m onlyr I Q O Q

u s e d hum an o p e r a t o r t r a n s f e r f u n c t i o n * h a s t h e f o l lo w in g

fo rm :

G ^(S ) = K ’ e x P < - T S > ( T lS + 1 ) ( 2 . 4 - 3 )(T n S + 1 )(T £ S + 1 )

w h e re

X = g a in

T s r e a c t i o n t im a

= l e a d c o m p e n s a tio n t im e c o n s t a n t

*n = n e u r o m u s c u la r l a g t im e c o n s t a n t

= l a g c o m p e n s a tio n t im e c o n s t a n t

The o p e r a t o r seem s t o b e a b l e t o a d j u s t t h e p a r a m e te r s

w i t h i n l i m i t s t o a c h ie v e b e t t e r c o n t r o l o f t h e s y s te m . I t h a s

a l s o b e e n s h o w n ^ t h a t i f o n e a t t e m p t s t o u s e th e m ore g e n e r a l

t r a n s f e r f u n c t i o n i t r e d u c e s t o

GjjCS) = K (T n S + D ( T i 2 s + 1 ) • • • ( Tl n s + ^ e x p ( - T S ) Sm(T2 i S + 1 ) ( T 2 2S + 1 ) . . . C T 2 p S + 1 )

W here (m + p ) > n (2.**-*♦)

t h e fo rm o f e q u a t i o n ( 2 .U - 3 ) w i t h n e g l i g i b l e c o n t r i b u t i o n s

f ro m h i g h e r o r d e r t e r m s .

B e c a u se t h e o p e r a t o r i s n o n l i n e a r , h i s a c t i n g c a n n o t be

c o m p le te ly d e s c r i b e d by a l i n e a r m o d e l. To c o m p e n sa te f o r1

t h i s , t h e t r a n s f e r f u n c t i o n p r e s e n t e d i n e q u a t io n (2 .> t-3 ) can

b e m o d i f i e d by a d d in g a re m n a n t te r m N ( t ) . A b lo c k d ia g ra m

o f s u c h a m o d e l i s show n i n F ig u r e 2 - 6 .

M ost t e c h n iq u e s em p lo y ed t o d e te r m in e t h e p a r a m e te r s

a p p e a r in g i n t h e q u a s i - l i n e a r m odel a r e a fo rm o f l i n e a r i z a t i o n

i n w h ic h t h e re m n a n t i s m in im iz e d . The hum an o p e r a t o r i s

p l a c e d i n a c o n t r o l lo o p a n d r e q u i r e d t o t r a c k t h e i n p u t t o

t h e b e s t o f h i s a b i l i t y . H is i n p u t a n d o u t p u t a r e r e c o r d e d

Huaan Operator

R iT )

i Controlled Eleaent

c mLinear Model

P Gp(T)Output

Gf(T)Feedback Eleaent

Figure 2-6 Describing Function Block Dlagraa

f o r a n a n a l y s i s . I f a s i n u s o i d a l i n p u t o f d i f f e r e n t f r e q u e n

c i e s i s u s e d , b o d e p l o t s c a n b e u s e d t o d e te r m in e th e m o d e l.

I n t h e c a s e w h e re a s t a t i o n a r y ran d o m s i g n a l i s u s e d f o r t h e

i n p u t , t im e s e r i e s a n a l y s i s c a n b e e m p lo y e d . F i n a l l y , a c t u a l

t r a c k i n g d a t a c a n b e u s e d i f a n o p t i m i z a t i o n p r o c e d u r e i s

u s e d t o s e l e c t a s e t o f p a r a m e te r s w h ic h m in im iz e s a c o s t

f u n c t i o n b a s e d on t h e o u t p u t o f t h e o p e r a t o r a n d th e m o d e l.

The i n t e g r a l o f s q u a r e d e r r o r ( I S E ) :

i s a com m only u s e d c o s t f u n c t i o n . T h is t e c h n iq u e i s i l l u s

t r a t e d i n F ig u r e 2 - 7 .

T he " l i n e a r c o r r e l a t i o n " ( r a t i o o f l i n e a r l y c o r r e l a t e d

o u t p u t t o a c t u a l o u t p u t ) i s a b o v e o .9 f o r lo w f r e q u e n c y i n p u t s .

T he c o r r e l a t i o n d e c r e a s e s a s t h e t a s k c o m p le x i ty i n c r e a s e s an d

a s t h e b a n d w id th o f t h e i n p u t i n c r e a s e s .

2 .U 2 . S a m p le -D a ta M o d e ls . T h e re e x i s t s som e d i f f e r e n c e s

o f o p i n i o n r e g a r d i n g t h e way i n w h ic h a man c a r r i e s o u t a

t r a c k i n g t a s k , a s t o w h e th e r h e o p e r a t e s a s a c o n t in u o u s o r a s

IS E = f t f (CH( t ) - CM( t ) ) 2 d t t o

(2 .* * -5 )

w h e re = r e c o r d e d hum an o u t p u t

CM( t ) = m odel o u t p u t

t o = i n i t i a l t im e

t f = f i n a l t im e

18

RECORDED

am

OPTIMIZATION

A c re : (^ ( t ) * Recorded hunen outputCg ( t ) * Linear nodal outputM(t) ■ i i u n t L d ifferen t between two outputs Coat “ Coat function

The o p tin lsa tlo n procedure adjusts model parameter based on cost functions iron previous runs to minimise the rasmant.

Figure 2-7 Typical letu p for Remnant Minimization

19

a d l a o r a t a p e r f o r m e r . Some o f t h e e x p e r i m e n t a l e v id e n c e w h ic h

te n d e t o i n d i c a t e d i s a r e t e b e h a v i o r a r e t h e f a c t s t h a t t h e

t im e d e l a y o f a n o p e r a t o r c a n b e r e p r e s e n t e d b y a sa m p le h o ld

o p e r a t i o n a n d t h a t a human o p e r a t o r i s u n a b le t o r e s p o n d t o

t h e s e c o n d o f tw o c l o s e l y - s p a c e d s t i m u l i . B eck ey 2 3 * 2 4 * 25

t h o r o u g h ly e x a m in e d t h e e a r l i e r s a m p le d - d a ta m o d e ls a n d p r o

p o s e d a s im p le m o d e l o f h i s ow n. I n t h e l a t t e r o f t h e r e f e r

e n c e d a r t i c l e s , B eck ey p r o p o s e d a s y s t e m a t i c t e c h n iq u e f o r

e s t i m a t i n g t h e s a m p lin g f r e q u e n c y . E ven s o , t h e s y n t h e s i s o f

s a m p le d - d a ta m o d e ls o f hum an c o n t r o l l e r s i s d e f i c i e n t i n t h e

a r e a o f p a r a m e te r e s t i m a t i o n . The m o d el i s i l l u s t r a t e d i n

F ig u r e 2 - 8 .

2 . i ( 3 . A d a p t iv e M anual C o n t r o l ■ F ig u r e 2 .9 i l l u s t r a t e s

t h e f o u r m a jo r c a t e g o r i e s o f a d a p t i v e c o n t r o l . 2 ® T h e se c a t e -

g o r i e s a r e :

( 1 ) I n p u t a d a p t io n a n d p r e d i c t i o n w h ic h c o n s i d e r s t h e

a b i l i t y o f t h e hum an o p e r a t o r t o r e c o g n i z e r e p e a t e d i n p u t

p a t t e r n s a n d t r a c k t h i s i n a n o p e n lo o p m a n n e r;

( 2 ) C o n t r o l e le m e n t a d a p t a t i o n , w h ic h r e f e r s t o t h e

a b i l i t y o f a hum an t o c h a n g e h i s c o n t r o l s t r a t e g y due t o

c h a n g in g s y s te m d y n a m ic s ;

( 3 ) T ask a d a p t a t i o n , w h ic h i n c l u d e s t h e o p t i m i z a t i o n o f

t h e c o n t r o l lo o p b a s e d o n some p r e d e t e r m in e d c o s t f u n c t i o n ;

( 4 ) P rogram m ed a d a p t a t i o n o r o p e n lo o p a d a p t i v e c o n t r o l ,

w h ic h e n c o m p a s s e s t h e c h a n g e i n c o n t r o l s t r a t e g y b a s e d on

VisualReference

Input

Htaun Operator Model

Display

Figure 2-8 Sanpled-data Model o f Hunan Operator in Conpensatory Tracking

INPUTADAPTATION

Disturbance

ADAPTIVE MANUAL

CONTROLLER

ControlObjeotlves^ TASK

ADAPTATION ADAPTATION

la v lr i n t i lS ta tu

n d Tine

Fugure 2-9 Major Adaptive Functions In Manual Control

22

environment a l s ta tu s , suoh as wind v e lo c ity , road co n d itio n s,

and other s itu a t io n s which tha oparator has learned to com

pensate.

A good example o f th is approach to system mods l in g i s

the adaptive nodal o f tfents^? which i s a lin e a r model o f

f ix ed fora;

GhtS) - Ko . e ~ T S (AoS + 1)____________ (2 .4 -5 )

(TnS + 1) (TAs + 1 )

where Ko ***<* \> ar* autom atically adjusted by a s te e p e s t

descent procedure.

2 .4 4 . Multi-Modal Models. L. 6 . Fogsl28 in h is te x t ,

B iotechnologyi Concepts and A p p lica tion s, su ggests th a t i t

might be p o ss ib le to subdivide the human operation in to a

mutually ex c lu s iv e s e t o f modes, each corresponding to a par

t ic u la r region on the phase plane formed by the error s ig n a l

and i t s d e r iv a tiv e . Such an approach should r e s u lt in lower

order tran sfer functions fo r the in d iv id u a l modes and s t i l l

provide good tracking behavior when applied seq u en tia lly as

a group. He su ggests th a t the fo llow in g s e t o f modes might

work (although he did not pursue th is approach any fu r th er ):

Mode 1 . Reaction; No motor output.

Mode 2 . A cq u isition ; Mode which i s tr iggered by any error and/or error ra te o u tsid e the tracking mode domain. The purpose i s to rapid ly minimize erro r . This may cause overshoot.

Mode 3 . Tracking; The in te n t i s to zero the error .

23

Mode 4 . S y n c h ro n is m : T h is i s t r i g g e r e d by p e r c e p t i o no f som e i n v a r i e n t w ave fo rm i n t h e i n p u t s i g n a l . The o p e r a t o r a c t s a s an i n d e p e n d e n t g e n e r a t o r d u r i n g t h e t r a c k i n g m ech an ism s an d m o n i to r s t h e i n p u t s i g n a l .

Mode 5 . S te a d y S t a t e T re m o r: O n ly when i n t r e m o r d o m a in .

Mode 6 . R e a s s u r a n c e : A r t i f i o a l l y t r i g g e r e d b y t h e hum ano p e r a t o r i n o r d e r t o d e te r m in e d e f e n s i v e s v s te m q u a l i t y b y f e e d b a c k ; i t c a n o c c u r i n e i t h e r t h e t r e m o r o r t r a c k i n g s t a t e .

L i t t l e w ork h a s b e e n d o n e i n t h i s a r e a , w i t h t h e e x c e p t i o n

o f th e tw o mode m o d e ls w h ic h h a v e j o i n e d a s u r g e m o d e l t o a

c o n s t a n t c o e f f i c i e n t t r a c k i n g m o d e l. A good e x am p le o f t h i s 29 30i s C o s t e l l o ' s * s u r g e m o d el w h ic h a l lo w s d i s c o n t i n u o u s a s

w e l l a s c o n t in u o u s i n p u t s .

I n t h e s u r g e m o d e l , C o s t e l l o h a s u s e d t h e t r a d i t i o n a l

s e c o n d - o r d e r c o n s t a n t c o e f f i c i e n t m o d e l:

® h(S) = K e - T S (1 + Tl S ) ^(1 + STn ) (1 - STX)

w h e re TI s = hum an o p e r a t o r l a g t im e c o n s t a n t

Tx = 0 .5 = hum an o p e r a t o r l e a d t im e c o n s t a n t

Tn = 0 .2 = n e u ro m u s c u la r l a g t im e c o n s t a n t

T = 0 .1 5 = d e la y t im e

c o u p le d w i th a n o p t im a l m in im a l t im e f o r c i n g f u n c t i o n c a l c u

l a t e d w i th r e s p e c t t o s e c o n d - o r d e r l im b d y n a m ic s , t h i s f o r c i n g

f u n c t i o n d r i v e s t h e e r r o r a n d i t s d e r i v a t i v e t o t h e o r i g i n

b e f o r e a d d i t i o n a l c o r r e c t i o n i s m ad e . T he s u r g e m odel i s

e m p lo y e d when t h e f o l lo w in g i n e q u a l i t y e x i s t s :

24

1 -0 * ® o< t) ♦ 0 . 0 1 . - e 0 (T ) 0 .2 S ( 2 .4 - 7 )

The s u r g e m odel o f C o s t e l l o I s show n i n F ig u r e 2 - 1 0 .

2 .4 5 . A p p l i c a t i o n o f O p t i m i z a t io n T e c h n iq u e s I n t h e

H an -M ach in e S y s te m . I n g e n e r a l , m o s t o f t h e a p p l i c a t i o n s o f

o p t i m i z a t i o n i n m a n u a l o o n t r o l s y s te m s h a v e b e e n c e n t e r e d a ro u n d

d e te r m in in g m odel p a r a m e te r s w h ic h p r o v id e a " b e s t f i t " human

o p e r a t o r t r a n s f e r f u n c t i o n . A n o th e r a p p l i c a t i o n o f o p t i m i z a t i o n

c a n b e fo u n d i n t h e a r e a o f f i l t e r i n g a n d p r e d i c t i o n o f i n p u t

s i g n a l s . A t h i r d a r e a o f o p t im a l c o n t r o l a p p l i c a t i o n i s t h a t

o f d e te r m in in g s w i t c h i n g c u r v e s f o r o p t im a l f o r o i n g f u n c t i o n s .

T h is t e c h n iq u e w as u s e d i n d e v e lo p in g t h e s u r g e m o d e l.

2 .4 6 . O th e r M o d e ls . As s t a t e d b e f o r e , n o d e f i n i t e

m o d e l o f t h e hum an o p e r a t o r e x i s t s . I n s t e a d , t h e r e i s a v a r i e t y

o f m o d e ls , e a c h s te m m in g f ro m v ie w in g t h e o p e r a t o r f ro m a d i f

f e r e n t p o i n t o f v ie w . Some o f t h e o t h e r m o d e lin g t e c h n i q u e s ,

n o t p r e v i o u s l y m e n t io n e d , i n c l u d e n o n l i n e a r m o d e l in g , s t o c -

a s t i c m o d e l in g , a n d m u l t i p l e r e g r e s s i o n .

r (t )a<»)

Display

Contactor:Opoaa When fx(t-r) 0Closed Otherwise

Sugge-ModeModal

e ^ t )

ModalSelector

«2(t)

Continuous- Mode Model

c t ( t )

C j( t)

Figure 2-10 C oste llo 's Dual-Mode Model

SJUt

CHAPTER I I I

TRACKING SYSTEM DESCRIPTION

3 .1 . I n t r o d u c t i o n

T he m an -m ac h in e s y s te m s i m u la t e d c o n s i s t s o f l i n e - o f -

s i g h t t r a c k i n g by tw o hum an o p e r a t o r s w i th o n e a c t i n g to

r e d u c e t h e a z im u th e r r o r , w h i l e t h e o t h e r a c t s t o r e d u c e th e

e l e v a t i o n e r r o r . The c o n t r o l l e d e l e m e n t , an a n t i a i r c r a f t g u n ,

i s a p p ro x im a te d by a n o n l i n e a r s e c o n d - o r d e r m o d e l, a n d t h e

e r r o r s i g n a l i s r e p r e s e n t e d b y t h e a n g u l a r m is a l ig n m e n t a s

v ie w e d t h r o u g h t h e g u n 's s i g h t i n g s y s te m . I n a d d i t i o n t o

m o d e lin g t h e g u n an d d e v e lo p in g a c o o r d i n a t e s y s te m w h ic h

f a c i l i t a t e s t h e c a l c u l a t i o n o f a im e r r o r s , t h e gun p a r a m e te r s

a n d t a r g e t f l i g h t p a th s a r e d e s c r i b e d i n t h i s c h a p t e r .

3 .2 . The Gun M odel

P r o b a b ly t h e s i m p l e s t d y n am ic m o d el t h a t c a n b e u s e d t o

r e p r e s e n t t h e gun s y s te m i s t o c o n s i d e r i t a s a t w o - t o r t i o n a l

s y s te m , a s show n i n F ig u r e 3 - 1 . T hen t h e m o tio n o f t h e gun

in a z im u th a n d e l e v a t i o n may b e r e p r e s e n t e d b y :

26

0 AXIS OF ROTATION

TOP VIEW

e AXIS OF ROTATION

SIDE VIEW

FIGURE » GUN MODEL

w h e re

*z a a n g u l a r gun p o s i t i o n i n e l e v a t i o n p la n e

0g = a n g u l a r gun p o s i t i o n i n a z im u th p l a n e

= to r q u e a p p l i e d t o move gun i n e l e v a t i o n p la n e

= to r q u e a p p l i e d t o move gun i n a z im u th p l a n e

= m om ent o f i n e r t i a a b o u t t h e ^ a x i s o f r o t a t i o n

J f = m oment o f i n e r t i a a b o u t t h e 9 a x i s o f r o t a t i o n

a d r a g ( f r i c t i o n ) c o e f f i c i e n t i n a z im u th m ovem ent

B f a d r a g ( f r i c t i o n ) c o e f f i c i e n t i n e l e v a t i o n m om ent.

The e q u a t i o n s f o r t h e gun s i m u l a t i o n w e re o b t a i n e d by

a p p ly i n g t h e hum an o p e r a t o r o u t p u t , t o r q u e , t o a m e c h a n ic a l ly

s t i f f (K a o) r o t a t i o n a l s y s te m w h ic h c a n b e r e p r e s e n t e d b y :

J d 2W + B dW + KW a T ( t ) d t

( 3 . 2 - 3 )

w h e re

W a a n g u l a r d i s p l a c e m e n t

K a s t i f f n e s s c o e f f i c i e n t .

The e v a l u a t i o n o f t h e s e gu n p a r a m e te r s i s s t r a i g h t

f o r w a r d , s i n c e g o o d e s t i m a t i o n s f o r maximum a n g u l a r v e l o c i t y31an d a n g u l a r a c c e l e r a t i o n f o r t h i s gun a r e a v a i l a b l e a n d th e

29

maximum t o r q u e t h a t an a v e r a g e hum an o p e r a t o r w i l l e x e r t on

a h a n d w h e e l i s a p p r o x im a te ly 10 f t - l b s . S in c e t h e maximum

a c c e l e r a t i o n w i l l o c c u r when t h e gun i s m oved f ro m a r e s t p o s i

t i o n , w h e re ( d ^ g / d t ) i s z e r o , t h e moment o f i n e r t i a i s e a s i l y

d e te r m in e d fro m E q u a t io n 3 .1 - 1 a s :

I n a s i m i l a r m a n n e r , t h e d r a g - c o e f f i c i e n t , , may be

c a l c u l a t e d e a s i l y s i n c e t h e maximum v e l o c i t y w i l l o c c u r a s

t h e gun i s t u r n i n g w i th z e r o a c c e l e r a t i o n . T h u s ,

t h e s y s te m h a s r e l a t i v e l y p o o r s t a b i l i t y . T h is w i l l b e d i s

c u s s e d f u r t h e r i n S e c t i o n 3.H a n d j u s t i f i c a t i o n f o r th e v a lu e s

t h a t w e re u s e d i n t h e s i m u l a t i o n a r e g iv e n .

3 .3 C a l c u l a t i o n o f A lig n m e n t E r r o r s

The e q u a t i o n s u s e d t o d e f i n e t h e a n g u l a r m is a l ig n m e n t a s

v ie w e d th r o u g h t h e g u n 's s i g h t i n g s y s te m a r e d e v e lo p e d i n t h i s

s e c t i o n . S t a r t i n g w i t h t h e a im in g p o i n t , l o c a t e d in t h e

i n e r t i a l c o o r d i n a t e s y s te m , w h e re t h e gun i s l o c a t e d a t t h e

max

10 ft-lbs. 17057“ Raa/sec 10 f t - l b - s e c ( 3 .2 - 5 )

H o w ev er, i f t h e s e v a lu e s a r e u s e d i n c o n j u n c t i o n w i th32th e l i n e a r o p e r a t o r p a r a m e t e r s , a s p r o p o s e d by C o s t e l l o ,

30

o r i g i n , t h e g u n ’ s a n g u l a r a l ig n m e n t c a n b e d e s c r i b e d fro m an

a im in g v e c t o r p o s e d th r o u g h t h e tw o p o i n t s ( s e e F ig u r e 3 - 2 ) .

T he e q u a t i o n s f o r t h e g u n 's a n g u l a r p o s i t i o n a r e :

w h e re

0 = e l e v a t i o n a n g le

6 = a z im u th a n g le

( x , y , z ) = l o c a t i o n o f a im p o i n t i n i n e r t i a l c o o r d i n a t es y s te m .

H o w ev er, i n t h e c o m p e n s a to ry t r a c k i n g s y s te m th e o p e r a t o r

d o e s n o t u s e t h e a z im u th a n d e l e v a t i o n a n g l e s ; i n s t e a d , h e v ie w s

t h e t a r g e t th r o u g h t h e g u n 's s i g h t i n g s y s te m ( s e e F ig u r e 3 -3 )

a n d t r a c k s b a s e d on t h e a l ig n m e n t e r r o r d e te r m in e d b y com

p a r i n g t h e t a r g e t t o t h e c r o s s e d - h a i r s i n t h e g u n 's s i g h t .

T h is s i g h t i n g e r r o r c a n b e e x p r e s s e d i n te rm s o f an a n g u l a r

m is a l ig n m e n t , by d e f i n i n g a gun c o o r d i n a t e s y s te m w h ic h h a s i t s

p o s i t i v e Xg a x i s c o i n c i d e n t w i th t h e a im in g v e c t o r o f t h e a n t i

a i r c r a f t g u n . The e q u a t i o n s r e l a t i n g t h e t a r g e t p o s i t i o n i n

t h e i n e r t i a l c o o r d i n a t e s y s te m an d i t s p o s i t i o n i n t h e gun

c o o r d i n a t e s y s te m c a n b e o b t a i n e d b y r o t a t i n g t h e i n e r t i a l

c o o r d i n a t e s y s te m th r o u g h a n g le s ^ a n d 0 . T h e se e q u a t io n s

a n d t h e i r d e r i v a t i v e s a r e p r e s e n t e d i n A p p e n d ix A. B a se d on

( 3 .3 - 1 )

a n d 6 ( 3 .3 - 2 )

AIM POINT

GUN LOCATION

e(+)

FIGURE 3-2GUN POSITION IN INERTIAL COORDINATE SYSTEM

c+

PATH OF PU N E WITH GUN HILO STHI (LEFT ID RIGHT TRACKING)

TARGET AT TIME T

H AZIMUTH ERROR

(+) ELEVATION ERROR AIM POINT

FIGURE 3-3 SITE PATTERN GUN COORDINATE SYSTEM

FIGURE 3-4ANGLE MISALIGNMENT GUN COORDINATE SYSTEM

33

t h e gun c o o r d i n a t e p o s i t i o n o f t h e t a r g e t , t h e a n g u l a r m is

a l ig n m e n t e r r o r s c a n be c a l c u l a t e d a s show n b e lo w :

The a l ig n m e n t e r r o r s E l a n d E2 a r e i l l u s t r a t e d I n F ig u r e 3 - 4 .3 .4 . Gun P a r a m e te r D e te r m in a t io n

I n t h i s s e c t i o n t h e v a lu e s f o r t h e moment o f i n e r t i a

an d t h e dam ping c o e f f i c i e n t a r e d i s c u s s e d . The v a lu e s u s e d

i n S tu d y 1 h a v e b e e n c h o se n i n s u c h a way a s t o y i e l d a s t a b l e

b u t r e a l i s t i c t r a c k i n g s y s te m .

Some i n d i c a t i o n o f t h e g e n e r a l n a t u r e o f e a c h c o n t r o l

lo o p may b e o b t a i n e d b y n e g l e c t i n g t h e t r a n s f o r m a t i o n m a t r ix

( t h u s m ak ing t h e s y s te m l i n e a r a n d n o n - i n t e r a c t i n g ) an d

f o r m u la t i n g a s i n g l e c o n t r o l l o o p , a s show n i n F ig u r e 3 -5 . The

hum an o p e r a t o r h a s b e e n r e p r e s e n t e d b y t h e t r a d i t i o n a l s e c o n d -

o r d e r c o n s t a n t c o e f f i c i e n t m o d e l:

( 3 . 3 - 3 )

an d

w h e re

( 3 .3 - 4 )

= e l e v a t i o n e r r o r

E j = a z im u th e r r o r

( x ff, y , z ) = l o c a t i o n o f t a r g e t i n gun c o o r d i n a t e s y s te m 6 © S .

G^CS) = K<TLS- + 1 ) e x p ( - r S )

Tn S + 1 ) (T jS + 1 )( 3 .4 - 1 )

\ / ■Ny E(s) K(Tt s + I) T(s) i / j

(Desired v . J (Error) Tfi s + l)(T t s + 1) (Torque)S(S + B/J)

Aim Point) .

Gun S y i t «

Figure 3-5 Sim plified Tracking System

35

w h e re

*■3 H II 10

Tn = .2

n .5

F o r t h i s a n a l y s i s t h e t im e d e la y o f t h e hum an o p e r a t o r w as

a l s o n e g le c t e d } a l th o u g h i t c o u ld h a v e b e e n a p p ro x im a te d by

a P ade a p p r o x im a t io n .

From t h e b l o c k d ia g ra m , o n e c a n f o r m u la te a r o o t l o c u s

d ia g ra m i n o r d e r t o i n v e s t i g a t e s y s te m s t a b i l i t y . Such a

d ia g ra m i s show n i n F ig u r e 3 - 6 , w i th t h e v a lu e s o f t h e p a r a

m e te r s i n d i c a t e d .

As mAy'bte s e e n ' i n F ig u r e 3 - 6 , t h e s y s te m w i l l q u i c k l y go

u n s t a b l e a s t h e v a lu e o f t h e g a in c o n s t a n t , K, i s i n c r e a s e d .

The p r e d o m in a n t p a i r o f r o o t l o c i q u i c k l y l e a v e t h e r e a l a x i s

o f t h e c o m p lex p l a n e a n d c r o s s t h e im a g in a r y a x i s i n t o t h e

r i g h t - h a n d p l a n e , i n d i c a t i n g an u n s t a b l e s y s te m .

When c e r t a i n s y s te m p a r a m e te r s a r e c h a n g e d , a s i n d i c a t e d

i n F ig u r e 3 - 7 , much b e t t e r s y s te m p e r fo r m a n c e i s o b s e r v e d .

H e re t h e p r e d o m in a n t l o c i c r o s s t h e im a g in a r y a x i s i n t o th e

r i g h t - h a n d p la n e a t a much h i g h e r g a in c o n s t a n t . F o r t h i s

r e a s o n , th e l a t t e r s e t o f p a r a m e te r s w as u s e d i n t h e f i r s t s e t

o f o p e r a t o r s t u d i e s .

T h e se p a r a m e te r s n e c e s s i t a t e d t h e i n c l u s i o n o f l i m i t e r s

i n t h e g u n m o d e l s o t h a t maximum a c c e l e r a t i o n a n d s le w r a t e s

AXIS

I 10. 0I 2 0 . 0

1 3 0 . 0

. U A L 4 AXIS-4 •»

Figure 3 -0 Root L o cu s D iagram o f S i m p l i f i e d Tr a c k i ng S ystem - 1

uO)

r

0.50.21.0

16.05 .0

f O

Figure 3*7 Root Locus Diagram of S im plified Tracking System-II

u■si

38

w o u ld n o t be e x c e e d e d . T h e r e f o r e , o n e c h a n n e l o f t h e gun

s y s te m ( a z im u th ) a s f i n a l l y m o d e le d i s show n i n F ig u r e 3 -8 .

The o t h e r c h a n n e l i n S tu d y 1 i s i d e n t i c a l and t h e r e f o r e i s

n o t show n .

I n a d d i t i o n t o v e r i f y i n g t h e hum an o p e r a t o r m odel w i th

t h e s y s te m d e v e lo p e d i n t h i s s e c t i o n , tw o a d d i t i o n a l s y s te m s

w e re a v a i l a b l e .^ 3 A l l t h r e e o f t h e gun s y s te m s u t i l i z e th e

m odel i l l u s t r a t e d i n F ig u r e 3 - 8 . The n o n l i n e a r d i f f e r e n t i a l

e q u a t i o n s u s e d t o d e s c r i b e t h e s y s te m a r e :

® L im = % <T# ( t > - lA im >

an d

^ = z l (T, ( t ) - . ) ( 3 .4 - 3 )Lim * * L i m

w h e re

0 L im = l i m i t e d e l e v a t i o n a n g u l a r v e l o c i t y —0 L im = l i m i t e d e l e v a t i o n a n g u l a r a c c e l e r a t i o n 0

8 Lim = l i m i t e d a z im u th a n g u l a r v e l o c i t y 00

0 Lim = l i m i t e d a z im u th a n g u l a r a c c e l e r a t i o n

T ^ ( t ) = e l e v a t i o n t o r q u e

T p ( t ) = a z im u th to r q u e

= e l e v a t i o n d am p in g c o e f f i c i e n t

= a z im u th d am p in g c o e f f i c i e n t

= e l e v a t i o n moment o f i n e r t i a

= a z im u th moment o f i n e r t i a

w i th t h e o n ly d i f f e r e n c e i n t h e e q u a t io n s b e in g t h e v a lu e s o f

* Moment o f in c r taBg ■ Drag c o e ff ic ie n t

a ■ Laplace transform variable

Figure 3-8 Gun Model (Azimuth Channel) with Limiters

<40

t h e dam p ing c o e f f i c i e n t s a n d m om ents o f i n e r t i a . The gun

p a r a m e te r s f o r t h e t h r e e s y s te m s a r e g iv e n i n T a b le 2 .

3 .5 . I n p u t S i g n a l D e s c r i p t i o n

T he i n p u t s i g n a l s u s e d i n t h i s s t u d y a r e made up o f t h e

t r a j e c t o r i e s o f a i r c r a f t f l y i n g a t low a l t i t u d e s a n d h ig h

s p e e d s . The p o s i t i o n o f an a i r c r a f t , e n t e r e d i n t o t h e s im u

l a t i o n a s a f u n c t i o n o f t i m e , a lo n g w i th t h e a n g u l a r p o s i t i o n

o f th e g u n , i s u s e d t o c a l c u l a t e t h e a im in g e r r o r d e s c r i b e d

i n S e c t i o n 3 .3 , I t i s t h i s a im in g e r r o r w h ic h i s f e d b a c k t o

t h e o p e r a t o r .

T h e re a r e tw o t y p e s o f f l i g h t p a th s u s e d i n t h i s s t u d y .

The f i r s t ty p e c o n s i s t s o f s t r a i g h t an d l e v e l f l y - b y s . T h e se

f l i g h t p a th s a r e d e s c r i b e d i n T a b le 3 . The s e c o n d ty p e o f

f l i g h t p a th s u s e d i n t h i s s tu d y a r e f a r d i f f e r e n t fro m th e

low l e v e l f l y - b y f l i g h t s . The f l i g h t p a t h s a r e c h a r a c t e r i z e d

by s h a r p t u r n s , d iv e s a s s t e e p a s 6 0 ° a n d f a s t c l i m b s , p u l l i n g

a s h ig h a s 5 g * s on t h e a i r c r a f t . T h ese f l i g h t p a th s a r e

i n d i c a t e d i n T a b le <4. I n b o th c a s e s t h e f l i g h t p a t h s w ere

a l i g n e d i n t h e i n e r t i a l c o o r d i n a t e s y s te m s u c h t h a t th e minimum

d i s t a n c e t o t h e g u n , a t c r o s s o v e r , o c c u r r e d a f t e r 20 s e c o n d s o f

f l i g h t .

A b e t t e r r e p r e s e n t a t i o n o f t h e m an e u v e rs may b e g a in e d

fro m F ig u r e s 3 - 9 , 3 - 1 0 , 3 - 1 1 , a n d 3 - 1 2 , w h ic h p r e s e n t a p e r

s p e c t i v e p r o j e c t i o n o f f l i g h t p a t h s 1 0 2 , 1 , 3 , a n d 5 r e s p e c t

i v e l y . The r e m a in d e r o f t h e f l i g h t p a t h p r o j e c t i o n s p lu s a

d e s c r i p t i o n o f t h e p ro g ra m u s e d t o d raw t h e p e r s p e c t i v e

41

TABLE 2

Gun P a r iM tin

TernsStudy#1

Values for Study #2

Study#3 Units

B* 16 25.09 30.21 ft- lb -se c /r a d

"8 16 16.29 25.21 ft- lb -se c /r a d

J0 5 .247 .8007 f t - lb - s e c 2

Je 5 .1634 3.088 f t - lb - s e c 2

^L1b .384 .384 .384 rad/sec

*Lin .349 .349 .349o

rad/sec

*11Lin 1.065 1.065 1.065 rsd/aec

V .349 .349 .3492

rad/sec

42

TABLE 3

A ircraft Maneuvers Type I

F ligh t Path V elocity Plsplacenent A ltitude( f t /e e c ) ( f t ) ( f t )

101 760 600 375

102 760 1800 375

103 760 2700 375

43

TABLE 4

A ircraft Maneuver Type II

EMjfr* Pft.ft* Plcplaca—at( f t )

I 1500

2 1500

3 1500

4 1500

5 1500

6 1500

7 1500

8 1500

9 1500

10 1500

11 3000

12 3000

13 3000

14 3000

15 3000

16 3000

17 3000

18 3000

19 3000

20 3000

Dive Anale Pullup Maneuver

5 3

15 3

30 3

45 3

60 3

5 5

15 5

30 5

45 5

60 5

5 3

15 3

30 3

45 3

60 3

5 5

15 5

30 5

45 5

60 5

Figure 3 - i Perspective Projection of F light Path do. 102

10000' -I z

5000* -

-2000

Projection onto X-Y Plane

Figure 3-10 Perspective Projection o f F light Path Ho. 1

10000

5000

1000

2000Projection onto X-Y Plane

Figure 3-11 Perspective Projection o f F ligh t Path No. 5

10000

5000'

1000


Figure 3-12 Perspective Projection o f Fight Path No. 10

48

p r o j e c t i o n s c a n b e fo u n d i n A p p e n d ix B. The c o m p u te r p ro g ra m

i s l i s t e d i n A p p e n d ix C.

3 .6 . S e n s i t i v i t y A n a ly s i s o f t h e Gun S y s te m s

The w o rk i n t h i s s e c t i o n f u r t h e r d e m o n s t r a te s t h e s t a b i l i t y

o f t h e t h r e e t r a c k i n g s y s te m s s t u d i e d i n t h i s p a p e r . Due t o

th e n o n l i n e a r i t i e s o f t h e hum an o p e r a t o r m odel ( S e c t i o n 5 . 2 - 2 ) ,

t h e g un m odel ( S e c t i o n 3 .2 ) a n d t h e e r r o r d i s p l a y s y s te m

( S e c t i o n 3 . 3 ) , a n a l y t i c a l m e th o d s o f s t a b i l i t y a n a l y s i s h a v e

b e e n e l i m i n a t e d , t h u s l e a v i n g d i r e c t s i m u l a t i o n a s a p o s s i b l e

m eans f o r s t u d y i n g s y s te m s t a b i l i t y .

S e v e r a l f a c t o r s w h ic h h e lp t o s i m p l i f y t h e s tu d y a r e :

1 . A l l t h r e e gun s y s te m s s t u d i e d u t i l i z e t h e sam e

m o d e l; o n ly t h e v a lu e s o f B a n d J c h a n g e .

2 . The e l e v a t i o n c h a n n e l i n a l l t h r e e s y s te m s i s

h i g h l y o v e rd a m p e d , t h u s m ak in g i t u n n e c e s s a r y t o

s t u d y t h e e l e v a t i o n c h a n n e l .

3 . A l l t h r e e s y s te m s u s e t h e sam e d i s p l a y s y s te m .

4 . The p a r a m e te r s i n t h e o p e r a t o r m odel d e v e lo p e d i n

S e c t i o n 5 . 2 - 2 , w i th t h e e x c e p t i o n o f P 3 w h ic h i s s e t

e q u a l t o B , a r e i d e n t i c a l f o r th e t h r e e t r a c k i n g

s y s t e m s .

P o i n t s f ro m t h e p l a n e fo rm e d b y p l o t t i n g B v e r s u s J

w e re s e l e c t e d , t h u s c r e a t i n g d i f f e r e n t gun c o n f i g u r a t i o n s .

T h e se s y s te m s w e re t h e n t e s t e d b y c a l c u l a t i n g t h e a l ig n m e n t

e r r o r f o r tw o a i r c r a f t e n c o u n t e r s : F l i g h t P a th s 101 a n d 1 0 2 .

T he d i g i t a l s i m u l a t i o n p r e s e n t e d i n C h a p te r 5 w as u s e d a n d

i n b o th e n c o u n t e r s t h e d e t e r m i n a t i o n o f s y s te m s t a b i l i t y w as

made b y v i s u a l l y o b s e r v in g p l o t s o f t h e t r a c k i n g e r r o r g e n e r a t e d

d u r in g t h e s i m u l a t i o n . A s y s te m w as c o n s i d e r e d t o b e u n s t a b l e

when e i t h e r t h e t r a c k i n g e r r o r g rew e x p o n e n t i a l l y o r s u s t a i n e d

o s c i l l a t i o n s o c c u r r e d . The p l o t s fro m tw o u n s t a b l e s y s te m s

a r e shown i n F ig u r e 3 -1 3 .

R e s u l t s o f t h e s tu d y a r e g iv e n i n F ig u r e 3 -1 4 , A l l t h r e e

s y s te m s s t u d i e d i n t h i s p a p e r f a l l w e l l i n t o t h e s t a b l e r e g i o n .

The f i g u r e a l s o show s t h a t t h e r e g i o n o f u n s t a b i l i t y i s l a r g e r

f o r F l i g h t P a th 1 0 1 .

•iqsi

sun

OMi

jo|

••A

in3

••uo

d«»y

SK

u

>4U

e n n o ft

Em

m

M

OS

51

Region of In s ta b ility Couraa 102

Region In ln a ta b lllty Couraa 101

Study 1

Study 2

Study 3

B Damping C oeffic ien t ( f t - lb - s e c )

Figure 3-14 Reaulta o f S e n s it iv ity A nalysis

CHAPTER IV

REAL TIME SIMULATION OF THE TRACKING SYSTEM

4 . 1 . I n t r o d u c t i o n

E a r l y i n t h e s t u d y , i t becam e n e c e s s a r y t o d e v e lo p a

r e a l t im e s i m u l a t i o n o f t h e t r a c k i n g e n c o u n t e r i n w h ic h

a c t u a l hum an o p e r a t o r s w e re p e r f o r m in g c o m p e n s a to ry t r a c k i n g .

D a ta t h u s g e n e r a t e d w as u s e d t o d e v e lo p t h e p a r a m e te r s a n d t o

v e r i f y t h e hum an o p e r a t o r m o d e l , w h ic h i s d e s c r i b e d i n

C h a p te r 5 . T h is c h a p t e r d o c u m e n ts t h e h y b r i d c o m p u te r p ro g ra m

t h a t w as u s e d i n o b t a i n i n g t h i s t r a c k i n g d a t a .

4 . 2 . T he H y b r id C o m p u te r

T he h y b r i d c o m p u te r a t L .S .U . c o n s i s t s o f a S c i e n t i f i c

D a ta S y s te m s S igm a 5 d i g i t a l c o m p u te r w i th an E l e c t r o n i c

A s s o c i a t e s I n c o r p o r a t e d (E A I) 680 a n a lo g c o m p u te r a n d an EAI

69 3 i n t e r f a c e . The S igm a 5 d i g i t a l c o m p u te r h a rd w a re c o n

s i s t s o f :

1 . A c e n t r a l p r o c e s s o r c o n t a i n i n g a r i t h m e t i c an d c o n t r o l

l o g i c h a rd w a re f o r a w id e r a n g e o f o p e r a t i o n s .

2 . M a g n e tic c o r e memory - 2 0 ,0 0 0 w o rd s o f h ig h s p e e d

(7 5 0 n a n o s e c o n d s ) c o r e m em ory. A w o rd c o n s i s t s o f

32 d a t a b i t s a n d o n e p a r i t y b i t .

52

53

3 . M u l t i - l e v e l i n t e r r u p t s t r u c t u r e — e x t e r n a l i n t e r r u p t

c o n t r o l (6 l e v e l s ) w i th m a s k in g f o r d y n am ic p r i o r i t y

a s s ig n m e n t .

4 . P e r i p h e r a l i n p u t / o u t p u t e q u ip m e n t

a . o n e d i s k ( 3 / 4 m i l l i o n b y t e s )

b . c a r d r e a d e r (4 0 0 c . p . m . )

c . l i n e p r i n t e r (800 l . p . m . )

d . p a p e r t a p e r e a d e r (5 0 0 c . p .m . ) an d p u n ch 500 c . p .m . )

e . o p e r a t o r ' s c o n s o le w i th r e g i s t e r d i s p l a y s a n d o n - l i n e t y p e w r i t e r

The a n a lo g h a rd w a re c o n s i s t s o f :

1 . An a n a lo g t o d i g i t a l c o n v e r t e r (ADCS) w i t h a 24

c h a n n e l m u l t i p l e x e r .

2 . T w elve m u l t i p l y i n g d i g i t a l t o a n a lo g c o n v e r t e r s

(MDACS)

3. E ig h t s e n s e l i n e s

4 . S i x t e e n c o n t r o l l i n e s

5 . T h i r t y l o g i c g a t e s

6 . T w en ty g e n e r a l p u rp o s e f l i p f l o p s

7 . P e r i p h e r a l i n p u t / o u t p u t e q u ip m e n t

a . Two x - y p l o t t e r s

b . One 8 c h a n n e l r e c o r d e r

c . Two 4 c h a n n e l o s c i l l o s c o p e s

A lth o u g h b o t h t h e a n a lo g a n d t h e d i g i t a l c o m p u te rs a r e e a s y

t o u s e , i t i s t h e h y b r i d l i b r a r y , a s e t o f F o r t r a n s u b r o u t i n e s ,

t h a t p e r m i t s e f f i c i e n t u t i l i z a t i o n o f t h e s y s te m .

54

4 . 3 . T he H y b r id P ro g ra m s

The r e a l t im e s i m u l a t i o n p ro g ra m o f t h e t r a c k i n g s y s te m

w i t h a n a c t u a l hum an o p e r a t o r i n t h e lo o p i s p i c t u r e d i n

F ig u r e 4 - 1 . D e t a i l e d d ia g ra m s o f t h e a n a lo g an d d i g i t a l p r o

g ram s a r e g iv e n i n F i g u r e s 4 -2 a n d 4 -3 r e s p e c t i v e l y .

F ig u r e 4 - 1 i n d i c a t e s t h e o v e r a l l p r o c e s s a s th e d i g i t a l

c o m p u te r s a m p le s , v i a t h e i n t e r f a c e , t h e gun p o s i t i o n ( a s

d e te r m in e d b y 9 g a n d ^ g ) a n d t im e . I t t h e n l o c a t e s t h e

t a r g e t p o s i t i o n c o r r e s p o n d in g t o t h e t im e i n s t a n t ( t h e f l i g h t

p a t h i s s t o r e d i n t h e d i g i t a l c o m p u te r ) a n d c a l c u l a t e s t h e

a n g le o f m is a l ig n m e n t i n a z im u th a n d e l e v a t i o n . The tw o

e r r o r s a r e f e d v i a t h e i n t e r f a c e b a c k t o t h e a n a lo g a n d d i s

p la y e d o n o s c i l l o s c o p e s . The tw o hum an o p e r a t o r s c o n t i n u o u s l y

m o n i to r t h e e r r o r a n d a p p ly c o r r e c t i v e t o r q u e i n p u t s v i a

h a n d s e t p o t s o n t h e a n a lo g . The to r q u e f o r e a c h c h a n n e l i s

i n p u t e d to a gun m o d e l , a s d e s c r i b e d i n t h e p r e v io u s c h a p t e r ,

p rog ram m ed o n t h e a n a lo g c o m p u te r . A s im p le d ia g ra m o f t h e

a n a lo g gun m o d e l i s show n in F ig u r e 4 - 4 , w h e re a s t h e s c a l e d

m o d el a n d t h e l o g i c c i r c u i t u s e d t o l i m i t t h e v e l o c i t y i n t e

g r a t o r a r e l o c a t e d i n A p p e n d ix D. The a n a lo g c o m p u te r

i n t e g r a t e s t h e s e e q u a t i o n s i n r e a l t im e to p r o d u c e an a n g u la r

gun p o s i t i o n , 9 g a n d ^ g . The l a t t e r i s sa m p le d b y t h e d i g i t a l

c o m p u te r , h e n c e t h e lo o p i s c l o s e d .

F ig u r e 4 -2 g i v e s d e t a i l s o f t h e a n a lo g c o m p u te r p a t c h i n g

d ia g ra m . A t t h e s t a r t o f e a c h r u n , t h e o p e r a t o r s m a n ip u la te

p o t s Q4 a n d Q7 t o d i r e c t t h e g u n s to w a rd t h e t a r g e t , t h u s

z e r o i n g t h e i n i t i a l e r r o r s . T h is i s n e c e s s a r y t o a v o id any

55

d

Tg(#|

Z * 5 2 r ~

| Elevation J

Human O perators

Scope.Oitplaye

Aeeel<‘irotionf

Angular Velocity (#>

1 'Angular

t<Position

— 1Rotation and

Angular Misalignment

Targe!Position

z r z — j

TimeGeneration

Tg(+)

i tAccelor

i totiont )

Angular Velocity 4

AngulorU

Position

| Azimuth |

FIGURE 4-1 HYBRID SIMULATION: OPERATOR IN LOOP

—Hal

I Humon | pijnoi Camputar{ Oparater t

Display I

Misalign -Scop*

I HumanI Oparatar

Hsf.-

L inaS10 M s a s /j^

Manastabla

FIGURE 4-2 HYBRID SIMULATION OF COMPENSATORY TRACKING

57

c S ta rt

Sat up ana

< 9 -Icolculota oim arrortl

Read ond write flight data

..... — J -| Define crossover time

Error> MoxlmuErorr« Maximum

ILoad error on scope]

Enter scale factor

1Coiculate field dimension

S to re

P 1 — — — 4S w itc h (l)*f> ( Store torque error angular accelerationPick flight cose

and disptament

Switch (4)* I

NoI Calculate sta rt time

IAnalog in 'lC‘ and 'NS'

Read analog : T 6 $ Torque

: iCalculate target position

Coll ro ta te (Target in gun coordinate )

I

Store >T Store

Yes | Rood NGO|

_________ 2.3f Write output matrix |

Switch (4 )- I

No

Punch tope . Torque e r to tf

NGO>2

FIGURE 4 -3 DIGITAL FLOW CHART ( Stop )

58

LfWIK1/J

I S L _ ^ T ) - J

B/J

Figure 4*4 Analog SInulacion of Gun

The Logic signal (LSW) l i a i t s ln tegra ta l #2 by opening Che e lectron ic swlCch when the outpuC reaches one o f i t s Lim its, The switchremains c losed u n t il the angular acceleration (?LIj|) change d irection thus moving sway froei I t s lim ited value. A Logic c ir c u it diagramfor sign a l LSW can be found in appendix D.

59

a c q u i s i t i o n p ro b le m s a t t h e s t a r t o f th e r u n . As t im e i s

g e n e r a t e d , t h e t a r g e t b e g in s t o move an d t h e t r a c k i n g e r r o r

i s d i s p l a y e d on t h e s c o p e . The o p e r a t o r s m a n ip u la te p o t s Q2

an d Q 9, f o r e l e v a t i o n a n d a z im u th r e s p e c t i v e l y , t o i n p u t

t o r q u e t o t h e gun m o d e l. T h is g e n e r a t e s a n a c c e l e r a t i o n o f

t h e gun w h ic h i s i n t e g r a t e d t o o b t a i n v e l o c i t y a n d p o s i t i o n .

Gun p o s i t i o n i s sa m p le d b y t h e d i g i t a l c o m p u te r a t t h e

r a t e o f 100 t i m e s / s e c v i a t h e i n t e r f a c e . I t a l s o sa m p le s

t im e an d c a l c u l a t e s t h e c o r r e s p o n d in g p o s i t i o n o f th e t a r g e t

a t t h a t i n s t a n t . F ig u r e U-3 g i v e s d e t a i l s o f t h e d i g i t a l

p ro g ra m . The a n a lo g i s s e t up fro m t h e d i g i t a l b y c a l l i n g a

nu m b er o f h y b r i d s u b r o u t i n e s . F l i g h t p a th s a r e r e a d i n t o t h e

d i g i t a l fro m c a r d s o r g e n e r a t e d i n t e r n a l l y .

When a c a s e i s t o b e r u n , t h e s t a r t i n g p o i n t o n th e

t im e a x i s i s c a l c u l a t e d fro m c r o s s o v e r a n d th e a n a lo g i s p u t

i n t o t h e i n i t i a l c o n d i t i o n m ode, The d i g i t a l c o m p u te r th e n

s a m p le s t im e an d t h e i n i t i a l gun a n g le s ( a n d ^ j ) . T a r g e t

p o s i t i o n s , c o r r e s p o n d in g t o t h e t im e i n s t a n t , a r e c a l c u l a t e d

o r i n t e r p o l a t e d fro m s t o r e d a r r a y s . A s u b r o u t i n e ( R o t a t e ) i s

c a l l e d t o t r a n s f o r m t h e t a r g e t p o s i t i o n t o t h e gun c o o r d i n a t e

s y s te m . Upon r e t u r n fro m t h i s s u b r o u t i n e , t h e i n i t i a l a im in g

e r r o r s a r e c a l c u l a t e d a n d s u p p l i e d t o t h e a n a lo g f o r d i s p l a y .

E ac h o p e r a t o r t h e n z e r o s h i s e r r o r b y a d j u s t i n g th e c o r r e s

p o n d in g h a n d s e t p o t . The d i g i t a l c y c l e s a lo o p a w a i t in g t h e

a n a lo g t o o p e r a t e . T h is i s a c h ie v e d b y c o n t i n u a l l y c o m p a r in g

t h e v a lu e o f t im e on t h e a n a lo g w i th t h e t im e f o r t h e n e x t

60

s t o r a g e p o i n t . I n t e r v a l s f o r s t o r a g e o f d a t a a r e s e t a t 0 .2

s e c . As lo n g a s t h e s a m p le d t im e i s l e s s t h a n th e n e x t

s t o r a g e t im e , t h e lo o p i n d i c a t e d o n F ig u r e 4 -3 by e n t r y p o i n t

11 w i l l b e e x e c u te d c o n t i n u o u s l y .

When a l l i s r e a d y , an o p e r a t o r p u t s t h e a n a lo g in th e

o p e r a t e m o d e , t h u s g e n e r a t i n g t im e an d m ov ing t h e t a r g e t a c r o s s

t h e s c r e e n . A t c r o s s o v e r , i t w i l l a lw a y s b e c r o s s i n g th e

c e n t e r l i n e o f t h e s c o p e . A t t h e s p e c i f i e d i n t e r v a l s th e

d i g i t a l s t o r e s t im e an d t h e c o r r e s p o n d in g t r a c k i n g d a t a

( t o r q u e , a n g u l a r a c c e l e r a t i o n , gun p o s i t i o n an d e r r o r i n b o th

a x i s ) .

A r u n i s t e r m i n a t e d w hen t im e e x c e e d s TSTOP, t y p i c a l l y

40 s e c o n d s . A t t h e e n d o f a r u n , an i n t e g e r v a r i a b l e "NGO"

i s e n t e r e d v i a t h e t e l e t y p e , o f f e r i n g a d d e d f l e x i b i l i t y i n

t h e fo rm o f s e v e r a l o p t i o n s . "NGO" may b e 1 , 2 , 3 , o r 4

a c c o r d i n g t o w h e th e r t h e p ro g ra m i s t o b e t e r m i n a t e d w i th o r

w i t h o u t t h e o u t p u t f ro m t h e l a s t r u n , o r t o b e c o n t in u e d w i th

o r w i t h o u t t h e o u t p u t . I f a n o u t p u t i s d e s i r e d , a s w i tc h i s

u s e d t o d e c id e i t s fo rm , w h ic h c a n be a p r i n t o u t o n ly on th e

l i n e p r i n t e r , o r p r i n t o u t a n d a p u n c h e d t a p e f o r f u r t h e r

p r o c e s s i n g i n t o c a r d s .

As m e n t io n e d a b o v e , t h e p ro g ra m o p e r a t e s i n r e a l t im e ,

a t y p i c a l ru n t a k i n g a b o u t o n e m in u te ( s e t t i n g up t im e + 40

s e c . o f o p e r a t i o n ) . To o u t p u t t h e d a t a on p a p e r t a p e i s ,

h o w e v e r , t h e b o t t l e n e c k , a s i t t a k e s a b o u t 15 m in u te s p e r r u n .

D e t a i l s o f th e o u t p u t r o u t i n e a r e g iv e n i n A p p e n d ix E .

P r e l i m i n a r y r e s u l t s w i t h t h i s r e a l t im e t r a c k i n g s y s te m

w i th a c t u a l o p e r a t o r s d o in g th e t r a c k i n g a r e show n i n

F ig u r e 4 - 5 . I n t h i s f i g u r e we s e e how t h e p e r fo rm a n c e o f an

o p e r a t o r im p ro v e s a s h i s e x p e r i e n c e i n c r e a s e s . I n t h i s s e r i e s

o f r u n s , t h e o p e r a t o r im p ro v e d h i s p e r fo r m a n c e fro m a maximum

t r a c k i n g e r r o r o f a p p r o x im a te ly 1 0 ° (1 7 7 m i l e s ) t o l e s s th a n

3 .5 ° (62 m i l e s ) a f t e r 27 r u n s . The m ean t r a c k i n g e r r o r i s

c o n s i d e r a b l y l e s s . T h ese r e s u l t s w e re o b t a i n e d w i th a s e t o f

gun p a r a m e te r s c o r r e s p o n d i n g t o t h o s e d e te r m in e d f o r S tu d y 1 .

F ig u r e s 4 -6 th r o u g h 4 -8 i l l u s t r a t e t r a c k i n g p e r fo rm a n c e

o f t h e sam e o p e r a t o r f o r s e v e r a l f l y - b y c o u r s e s . N o te p a r

t i c u l a r l y t h e d i f f i c u l t y e n c o u n te r e d b y t h e r e a l o p e r a t o r

t r a c k i n g an a i r c r a f t a t 900 f e e t ( t h e minimum d i s t a n c e a s

show n i n F ig u r e 4 t .6 ) . I t s h o u ld b e n o t e d t h a t t h e S tu d y 1

gun m o d e l w as u s e d i n e a c h o f t h e s e c a s e s .

D a ta o b t a i n e d f ro m r e a l t im e t r a c k i n g e n c o u n t e r s , w i th

hum an o p e r a t o r s i n t h e l o o p , w e re u s e d t o d e v e lo p th e p a r a

m e te r s a n d t o v a l i d a t e t h e hum an o p e r a t o r m o d e l. T h is w ork i s

d e s c r i b e d i n C h a p te r 5 .

(MM

toQ)

HOMO

HlfW

NZ

V

0 0 0 feet minimum distance390 knots375 foot attitude

Run number from start:

27

20 24 4 0TIME (Seconds)

FIGURE «-5 PERFORMANCE VERSUS EXPERIENCE

\

360

320

280

COURSE I (SoM U m ): 450 Knott/Hour 3751800 Fool Distoneo

COURSE 2 (D oM U nat: 450 Knote/Hoor 37S Fiat OwwHow 900 Foot Oirtooco

240

200

160

120

80

40

-40

-80, 20TME (Stconds)

24 4036

FIGURE 4-6 TRACKING PERFORMANCE O F ACTUAL OPERATOR Io*U1

AZIM

UTH

ERRO

R (M

ila)

160

120COURSE 3 (Solid Lina):

450 Knott/Hour 300 Ftat Bavotion 2700 foot Diatonea

COURSE 4 (Oothad Lint): 450 Knots/Hour 375 Faat Elavotion 2700 foot Dtetanca

80

40

-8024 40

TIME (Saconds)

FIGURE 4-7 TRACKING PERFORMANCE O F ACTUAL OPERATOR II

CTl

AZIM

UTH

ERRO

R (M

ils)

120 COURSE 5 (Solid Line): 550 Knots/Hour 375 Feet Elevation 1800 Foot Distance

COURSE 6 (Dashed Line): 350 Knots/Hour 375 Feet Elevation 900 Foot Distance

80

4 0 -

N _ /

-40

-80 ; 20 TIME (Seconds)

24 40

FIGURE 4-8 TRACKING PERFORMANCE O F ACTUAL OPERATOR m

a ttn

CHAPTER V

THE HUMAN OPERATOR MODEL

5 . 1 . I n t r o d u c t i o n

The m odel o f t h e hum an o p e r a t o r t h a t w as f i n a l l y i n c o r

p o r a t e d i n t h i s s t u d y w as s t r o n g l y i n f l u e n o e d b y t h e w ork o f

C o s t e l l o 3* who h a d a m e a s u re o f s u c c e s s s i m u l a t i n g o b s e r v e d

o p e r a t o r p e r fo r m a n c e w i t h a tw o-m ode m o d e l. A m u lt i -m o d e m o d e l

was a l s o a d a p te d f o r t h i s s t u d y c o n s i s t i n g o f a t r a c k i n g

mode a n d tw o a c q u i s i t i o n m o d es . The p a r t i c u l a r mode w h ic h i s

a c t i v e d e p e n d s u p o n t h e o b s e r v e d e r r o r a n d i t s d e r i v a t i v e ,

a s show n i n t h e p h a s e p l a n t p l o t , F ig u r e 5 - 1 . When t h e e r r o r

a n d i t s d e r i v a t i v e a r e w i t h i n t h e e n c l o s e d a r e a , t h e t r a c k i n g

m ode o f t h e o p e r a t o r i s a c t i v e , o t h e r w i s e o n e o f t h e a c q u i r e

m odes i s a c t i v e u n t i l t h e e r r o r a n d i t s d e r i v a t i v e a r e b r o u g h t

w i t h i n t h e e n c l o s e d r e g i o n . I n a d d i t i o n t o d e v e lo p in g t h e

hum an o p e r a t o r m o d e l , t h e d i g i t a l s i m u l a t i o n s o f t h e t r a c k i n g

s y s te m i s d i s c u s s e d i n t h i s c h a p t e r .

5 . 2 . T he T r a c k in g Mode

As p o i n t e d o u t i n t h e i n t r o d u c t i o n , t h e human o p e r a t o r

h a s b e e n r e p r e s e n t e d b y a m u lt i -m o d e m o d e l . T h is s e c t i o n

d e a l s w i t h t h e e v o l u t i o n o f t h e m o d e l 's t r a c k i n g m ode, fro m

i t s b e g in n i n g a s a l i n e a r m o d e l ( E q u a t io n 2 .4 - 3 ) t o i t s f i n a l

f o r m u l a t i o n a s a n o n l i n e a r a d a p t i v e m o d e l .

66

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5 .2 1 . The L i n e a r Human O p e r a t o r . W ith t h e s e l e c t i o n ,

show n i n S e c t i o n 3 .> t, o f t h e m om ent o f i n e r t i a , J , t h e d r a g

o o e f f i o i e n t , B , a n d t h e hum an o p e r a t o r t im e c o n s t a n t s , t h e

t o t a l s y s te m i s r e a d y t o b e e v a l u a t e d f o r t r a c k i n g c a p a b i l i t y

o n c e t h e s e l e c t i o n o f t h e hum an o p e r a t o r g a in c o n s t a n t . K,

i s m ade. S in c e t h e s y s te m now c o n s i s t s e s s e n t i a l l y o f a

f e e d b a c k c o n t r o l l o o p , i t w as d e c id e d t o s e l e c t K b a s e d upon

a common c o n t r o l lo o p t u n i n g c r i t e r i o n , t h e d e c a y r a t i o c r i

t e r i o n .

S e l e c t i o n o f a p a r a m e te r u s i n g a d e c a y r a t i o c r i t e r i o n

r e q u i r e s t h a t t h e p a r a m e te r s i n q u e s t i o n be s e t s u c h t h a t

e a c h s u c c e s s i v e p e a k o f t h e r e s p o n s e o f t h e s y s te m i s a c e r t a i n

f r a c t i o n o f t h e p r e v i o u s p e a k w hen t h e s e t p o i n t o f t h e s y s te m

h a s a s t e p i n p u t c h a n g e . T h is w as a c c o m p l is h e d by f i x i n g th e

t a r g e t i n a s t a t i o n a r y p o s i t i o n . T y p ic a l r e s u l t s a r e show n

i n F i g u r e s 5 -2 an d 5 -3 f o r v a l u e s o f K o f 5 a n d 20 f t - l b / m i l ,

r e s p e c t i v e l y . N o te t h a t w i th a v a lu e o f 5 , a r e l a t i v e l y u n d e r -

dam ped r e s p o n s e w as o b t a i n e d , w h i l e w i th t h e h i g h e r g a in t h e

r e s p o n s e w as much f a s t e r a n d t h e r e f o r e m ore o s c i l a t o r y . W ith

t h e l a t t e r r e s p o n s e , a d e c a y r a t i o o f a p p r o x im a te ly 1 /1 0 w as

o b s e r v e d . The r e s p o n s e t im e o f t h e s y s te m may b e f u r t h e r

i n c r e a s e d b y f u r t h e r i n c r e a s i n g t h e g a in r a t i o ; h o w e v e r , t h i s

w i l l c a u s e t h e s y s te m t o becom e e v e n m ore o s c i l l a t o r y an d

e v e n t u a l l y u n s t a b l e .

T h ese r e s u l t s a l s o i n d i c a t e t h e in a d e q u a c y o f t h e l i n e a r

hum an o p e r a t o r m odel i n t h e a c q u i s i t i o n o f a t a r g e t . T h is h a d

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b e e n p o i n t e d o u t b y s e v e r a l a u t h o r s , a s m e n tio n e d i n C h a p te r

2 .

B a se d upon t h e a b o v e r e s u l t s , t r a c k i n g r u n s w e re made

w i th t h e hum an o p e r a t o r g a in c o n s t a n t o f b e tw e e n 10 a n d 30 f t -

l b / m i l . I n o r d e r t o e v a l u a t e t h e p e r f o rm a n c e o f t h i s p a r t i

c u l a r hum an o p e r a t o r , a l e f t t o r i g h t l e v e l f l i g h t p a th w as

c h o se n w i th a n a i r c r a f t v e l o c i t y o f 450 k n o t s . The a l t i t u d e

w as 375 f e e t an d t h e f l i g h t p a th w as c h o s e n to c o r r e s p o n d to

c o u r s e 1 0 2 , show n i n F ig u r e B - 2 . The i n i t i a l p o s i t i o n o f t h e

a i r c r a f t waB s e l e c t e d t o g iv e 20 s e c o n d s t o c r o s s o v e r . I n

a d d i t i o n , s i n c e c o n c e rn w as e x p r e s s e d w i t h t h e t r a c k i n g

c a p a b i l i t y o f t h e hum an m o d e l, t h e gun w as i n i t i a l l y p o i n t e d

a t t h e a i r c r a f t . T h a t i s , i t w as a ssu m e d t h a t th e gun o p e r a t o r

h a d a c q u i r e d t h e t a r g e t . M ost o f t h e t r a c k i n g s t u d i e s r e p o r t e d

h e r e w e re e v a l u a t e d w i t h t h i s p a r t i c u l a r f l y - b y .

I t s h o u ld b e e m p h a s iz e d t h a t i n t h e s e r u n s an a t t e m p t

w as b e in g m ade t o f i n d a m o d el t h a t w o u ld p e r fo rm i n a much

s u p e r i o r m an n e r t o o b s e r v e d t r a c k i n g b y a t r a i n e d o p e r a t o r ,

a s o b t a i n e d fro m t h e h y b r i d s t u d y . I n t h i s w ay , t h e p a r a m e te r s

o f t h e m odel t o f i t t h e o b s e r v e d d a t a c o u ld b e r e g r e s s e d .

T h e r e f o r e , t h e c r i t e r i o n u s e d t o e v a l u a t e t h e p e r fo rm a n c e o f

t h e v a r i o u s m o d e ls t e s t e d w as t h e m in im iz a t io n o f t h e i n t e g r a l

o f th e s q u a r e o f t h e a z im u th e r r o r :

20ISE = / ( e # ) 2 d t ( 5 . 2 - 1 )

- 2 0

t

71

R e s u l t s w i th t h e l i n e a r o p e r a t i o n a r e shown i n F ig u r e s

5 - 4 , 5 - 5 , an d 5 -6 f o r a g a in c o n s t a n t , K, o f 1 0 , 3 0 , a n d MO,

r e s p e c t i v e l y . I n e a c h c a s e , t h e a z im u th t r a c k i n g e r r o r i s

p l o t t e d a g a i n s t t im e t o c r o s s o v e r . E l e v a t i o n e r r o r was

g e n e r a l l y o b s e r v e d t o be a s m a l l e r e r r o r t h a n a z im u th s i n c e

f o r t h e s e f l y - b y ' s e l e v a t i o n t r a c k i n g i s a much s i m p le r t a s k .

T h e r e f o r e , f o r s i m p l i c i t y , o n ly a z im u th e r r o r i s r e p o r t e d .

T h ese r e s u l t s , when co m p ared t o th o s e o f t h e a c t u a l

hum an o p e r a t o r a s show n i n F ig u r e 4 - 6 , i n d i c a t e t h a t t r a c k i n g

w i th t h e l i n e a r human o p e r a t o r m o d el was v e r y p o o r , w i th

maximum e r r o r s n e a r c r o s s o v e r r a n g in g fro m 750 m i l s (K = 10)

t o 200 m i l s (K = 4 0 ) . (H e re a p o s i t i v e e r r o r i n d i c a t e s t h e

gun l a g g in g th e t a r g e t . ) Any a t t e m p t t o s i g n i f i c a n t l y r e d u c e

t h i s p e a k e r r o r by f u r t h e r i n c r e a s i n g t h e g a in c o n s t a n t

r e s u l t e d i n i n s t a b i l i t y a f t e r c r o s s o v e r w hen t h e a i r c r a f t i s

d e c e l e r a t i n g p a s t t h e gun s i g h t .

When o n e c o n s i d e r s t h a t t h e s e r e s u l t s w e re o b t a i n e d

a f t e r c o n s i d e r a b l e a d ju s tm e n t i n a n o t h e r p a r a m e te r o f th e

gun s y s te m ( s e e S e c t i o n 3 . 4 ) i n o r d e r t o im p ro v e t h e o v e r a l l

s y s te m s t a b i l i t y , i t w as o b v io u s t h a t c o n s i d e r a b l e m o d i f i c a

t i o n w o u ld h a v e t o b e made t o t h e hum an o p e r a t o r m odel i n

o r d e r t o o b t a i n r e a s o n a b l e r e s u l t s .

A n a ly s i s o f t h e p e r fo rm a n c e , i n d i c a t e d t h a t t h e o p e r a t o r

was n o t s u f f i c i e n t l y e x c i t e d d u r in g t h e f l y - b y . T h a t i s , a s

t h e a i r c r a f t a c c e l e r a t e d p a s t t h e g u n s i g h t , t h e i n c r e a s i n g

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Figure 5-5 Tricking v lth Linear Operator - I I

n

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Figure 5-6 Tracking with Linear Operator - III

75

l a g o f t h e gun w as c a u s e d b y i n s u f f i c i e n t to r q u e o u t p u t o f

t h e o p e r a t o r , n o t by th e a c c e l e r a t i o n o f th e a i r c r a f t e x c e e d in g

t h e maximum s le w r a t e o f t h e g u n .

S in c e t h e d e la y t im e and t im e c o n s t a n t s o f t h e human

o p e r a t o r m odel h a v e b e e n s u b s t a n t i a t e d by c o n s i d e r a b l e d a t a ,

i t w as d e c id e d t o a l t e r t h e s e o n ly a s a l a s t r e s o r t . T h e r e

f o r e , i t becam e a p p a r e n t t h a t t h e human o p e r a t o r g a in c o n s t a n t

m u s t b e made a f u n c t i o n o f t h e o b s e r v e d e r r o r . T h is l e d t o

t h e u s e o f t h e n o n l i n e a r human o p e r a t o r m o d e l.

5 . 2 2 . The N o n l in e a r O p e r a t o r . The i n c o r p o r a t i o n o f

th e g a in c o n s t a n t o f t h e hum an o p e r a t o r a s a f u n c t i o n o f

e r r o r r e s u l t s i n t h e human o p e r a t o r m o d el b eco m in g n o n l i n e a r .

The r e s u l t i n g e x p r e s s i o n w as o f t h e fo rm :

Gp(S) = (20 + |e #| p ) - ( T LS + 1 ) e~TS ( 5 . 2 - 2 )

(Tn S + I H T j S + 1)

w h e re t h e t e r m i n b r a c k e t s r e p r e s e n t s t h e v a r i a b l e g a i n ; e

i s t h e o b s e r v e d a z im u th e r r o r i n m i l s ; an d p i s a c o n s t a n t

t o be s e l e c t e d . The o t h e r c o n s t a n t s a r e a s p r e v i o u s l y

d e f i n e d . I n a d d i t i o n , i t w as fo u n d n e c e s s a r y t o l i m i t t h e

v a r i a b l e K t o a maximum o f 100 f t - l b / r a d i a n i n o r d e r to

im p ro v e p e r f o r m a n c e .

R e s u l t s o b s e r v e d w i t h t h i s m o d e l , f o r t h e sam e f l i g h t

p a t h , a r e show n i n F ig u r e s 5 - 7 , 5 - 8 , an d 5 -9 f o r v a r y in g

v a l u e s o f t h e p a r a m e te r s p . I t i s o b s e r v e d t h a t t h e s y s te m

t m c k j n g or mncmFYH1TM MN-UNGRN MM W WOCL r-i.o

RLTiM * 0 .0stpobl T T ft of crnon

q . . . OltNUTH

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Figure 5-7 Tracking with Non-Linear Operator - I

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Figure 5-8 Trucking with Non-Linear Operator - I I

S j y / X ^ X

Z.Sat •Z ? o- Ja g n M c x iN c v M ' t a n r r

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79

q u i c k l y w e n t u n s t a b l e w i th v a lu e s o f p o f 2 .0 a n d 1 .5 w h ile

m a rk e d ly im p ro v e d t r a c k i n g w as o b t a i n e d w i th a p o f 1 , 0 . In

t h e l a t t e r c a s e , w h i le t h e gun s t i l l l a g s t h e t a r g e t , t h e

maximum l a g n e a r c r o s s o v e r i s l e s s t h a n 80 m i l s . N o te th e

o s c i l l a t i o n s i n t h e t r a c k i n g a f t e r c r o s s o v e r . T h is i n d i c a t e d

t h e u n s t a b l e n a t u r e o f t h e t r a c k i n g a f t e r c r o s s o v e r and

s u g g e s t e d a d i f f e r e n t v a lu e o f p t o be u s e d b e f o r e c r o s s o v e r

a n d a f t e r c r o s s o v e r .

T h ese r e s u l t s i n d i c a t e d t h a t t h e hum an o p e r a t o r m o d e l,

a s s u g g e s t e d i n t h e l i t e r a t u r e , c o u ld n o t p e r f o r m a d e q u a te ly

b a s e d upon t r a c k i n g e r r o r f e e d b a c k . E i t h e r t h e p a r a m e te r s

o f t h e e x i s t i n g m o d el o r t h e b a s i c n a t u r e o f t h e o p e r a t o r

m o d el s h o u ld b e a l t e r e d .

I t seem ed l i k e l y t h a t an o p e r a t o r t r a c k i n g a t a r g e t

w o u ld be a w a re o f t h e a n g u l a r v e l o c i t y o f t h e a i r c r a f t r e l a

t i v e t o h i s p o s i t i o n a n d t h e r e f o r e w o u ld b a s e som e o f h i s

o u t p u t on t h i s k n o w le d g e . C o n s e q u e n t ly , t h e m odel w as a l t e r e d ,

a s show n i n F ig u r e 5 - 1 0 .

As show n i n F ig u r e 5 - 1 0 , t h e hum an o p e r a t o r s e n s e s th e

a n g u l a r a c c e l e r a t i o n o f t h e a i r c r a f t p a s t h i s p o s i t i o n . T h is

v e l o c i t y i s t h e n m u l t i p l i e d by a c o n s t a n t , a , ( t o b e

d e te r m in e d ) an d a d d e d t o t h e o u t p u t t o r q u e o f t h e human o p e r a

t o r m odel ( t h e n o n l i n e a r o n e w as r e t a i n e d ) . The human o p e r a t o r

m o d e l th u s a c t s a s a t r i m t o z e r o t h e o b s e r v e d e r r o r . A lso

n o t e t h a t th e t o t a l t o r q u e i s s t i l l d e la y e d b y 0 .1 5 s e c o n d .

R e s u l t s f o r v a lu e s o f a o f 10 a n d 20 f t - l b - s e c / r a d i a n a n d f o r

(Torque) 0

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A ircraft P osition & VelocityTarget A ircraft

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Figure 5*10 Control System with Huasn Operator Sensing A ircraft Velocity

81

v a r io u 6 v a lu e s o f t h e p p a r a m e te r a r e shown i n F ig u r e s 5 -1 1

th r o u g h 5 - 1 5 ,

R e a s o n a b ly good t r a c k i n g w as o b t a i n e d w i th a e q u a l t o

10 a n d p e q u a l t o 1 . 2 , a s show n i n F ig u r e 5 -1 1 . F u r t h e r

i n c r e a s e s i n t h e v a lu e o f p w i th t h i s sam e a l e d t o i n s t a

b i l i t y p a s t c r o s s o v e r , a s show n in F ig u r e s 5 -1 2 an d 5 -1 3 .

W ith a v a lu e o f a e q u a l t o 2 0 , t h e gun l e d t h e t a r g e t

t h r o u g h o u t m o st o f t h e e n c o u n t e r , a s show n i n F ig u r e 5 -1 4 .

Once a g a i n , i n c r e a i s in g t h e v a lu e o f p w i t h t h e sam e v a lu e o f

a l e d t o i n s t a b i l i t y a f t e r c r o s s o v e r ; h o w e v e r , t h e maximum

e r r o r a t c r o s s o v e r w as c o n s i d e r a b l y r e d u c e d . T h is i s show n

i n F ig u r e 5 -1 5 .

T h e se r e s u l t s s u g g e s t e d t h a t a d i f f e r e n t v a lu e o f p

b e f o r e a n d a f t e r c r o s s o v e r , w i th t h e v a lu e a f t e r c r o s s o v e r

b e in g c o n s i d e r a b l y s m a l l e r t h a n b e f o r e c r o s s o v e r , w o u ld c o n

s i d e r a b l y r e d u c e t h e t r a c k i n g e r r o r . Such w as t h e c a s e , a s

show n i n F ig u r e 5 - 1 6 . B e fo re c r o s s o v e r th e r e s u l t s a r e

i d e n t i c a l t o t h o s e o f F ig u r e 5 - 1 2 . H o w ev er, a f t e r c r o s s o v e r

t h e p p a r a m e t e r w as c h a n g e d 1 / 1 .5 o r 2 / 3 , r e s u l t i n g i n

s t a b i l i t y th r o u g h t h e e n t i r e t r a c k i n g s e q u e n c e .

One o f t h e p o s s i b l e d ra w b a c k s t h a t was f o r e s e e n in

u s i n g a p a r a m e te r s u c h a s p w as t h a t an op tim um v a lu e f o r i t

w o u ld p r o b a b ly b e h i g h l y d e p e n d e n t upon t h e p a r t i c u l a r c o u r s e

t h e a i r c r a f t was f l y i n g p a s t t h e gun p o s i t i o n . S e v e r a l w ays

w e re i n v e s t i g a t e d f o r a l t e r i n g t h e g a in e x p r e s s i o n o f t h e

hum an o p e r a t o r m odel o f E q u a t io n ( S . 2 - 2 ) . The m o s t s u c c e s s

f u l w as :

fi

ii

WW

.

IHft

»I

timouni w mmvrrHITH WS-UWW1 NUMW MOO.

f*l.«

stool TTft or m ang . . . nZIfUTN

"-io.oo - I S . DO - 1 0 .0 0 -fc.O O o'. 0 0 s'.0 0 1 0 .0 0 11.00 zb.ooTIHE TO CROSSOVER (SECONOS)

Figure 5-11 Trucking with Nan-Linear Plus Feed Forwerd Operator

TJMEXJNB dr MKMrr mm MBN-UNBMI MUMfW

r - i .snurm-io.

5THOOL TTfC d r CRRdO a . . . R1MUTH

'- 1 0 .0 0 - I S . 0 0 - 1 0 .0 0 - S . f - ----- ------ 757•to. 00 -d .O O 0.00 S.00 10.00TIME TO CROSSOVER (SECONDS)

oo co.oo

Figure 5*12 Tracking with Non-Linear Plus Feed Forward Operator - I I

-«.(

8l

ig-m

FiJot

W (h

M ..

...

timcxiw or RinaMrrWITH NBM.INCMI HUM

P*C.O HLfHA-10.

T t or D HtlfVTH

e

10.00 4.00 __ __TINE TO CROSSOVER (SECQNOS)

11.00 s. oe *o

Figure 5-13 Tracking with Non-Linear Pin# Feed Forward Operator - II I

O 1

o

TfMotiMc or mnawrr KITH MON-UMCMI HUMM

P-1.29THB0L TTPC V (I

B . . . ttlM IT H

fc.oo oTbo iTooTIME TO CROSSOVEB (SECONDS)

1 3 .0 0- 20.00 1 5 .0 0 10.00 10.00 20.00

Figure 5-14 Tracking with Mon-Linear Plue Feed Forward Operator - IV

e

timckinb or mncrnrT w:h mm-unom mmw

r-c.sRLPm-CO. '

X

AC

MOL TTPt O r Ob . . . m nm

■ is. o o - i o . o o - k . o o ik.ooTIME TO CROSSOVER (SECONOS)

10.00

Figure 5-15 Tracking with Non-Linear Plus Feed Forward Operator - V

cc

TIMOtlMO BP BIROMPT N1TH NON-UNCim MUWM MOO.

P - I .S . 1 /1 .5 SLfHB-lO.

« 8*u

STNBBL TTPt o r CWWI a . . • BZIWTN

-k.oo o'.oo s', oo ib.oo 15.00TIHE TO CROSSOVER (SECONOS)

£ 0 .0 0 - 1 5 .0 0 10.00

X

Figure 5-10 Tracking with Non-Linear Plus Feed Forward Operator •

88

GpCS) = (20 ♦ ! « . • W# /B > |).< T 1.S + l ) e - T S ( 5 . 2 - 3 )

<TnS + l > * ( T i s + l )

H e re t h e g a in c o n s t a n t , t h e e x p r e s s i o n i n b r a c k e t s o f

E q u a t io n ( 5 . 2 - 3 ) , w as o n c e a g a i n m ade a f u n c t i o n o f t h e e r r o r ,

b u t , i n a d d i t i o n , w as i n c r e a s e d i n p r o p o r t i o n t o t h e a n g u la r

v e l o c i t y o f t h e a i r c r a f t p a s t t h e g u n . A g a in , t h e maximum

v a lu e o f K w as l i m i t e d t o ± 100 f t - l b / r a d i a n . O f c o u r s e , t h e

o p tim u m p r o p o r t i o n a l i t y f a c t o r B w i l l a l s o be d e p e n d e n t upon

t h e c o u r s e . H o w ev er, i t w o u ld seem to b e l e s s s e n s i t i v e t h a n

a p a r a m e t e r s u c h a s p . The r e s u l t s w i t h t h i s m o d el a r e show n

i n F ig u r e 5 -1 7 f o r a v a lu e B o f 0 .1 f t - l b - s e e / r a d i a n 3 .

As s t a t e d e a r l i e r , t h i s p r e l i m i n a r y w ork w i th th e human

o p e r a t o r f u n c t i o n h a d b e e n d i r e c t e d to w a rd o b t a i n i n g a m odel

t h a t w o u ld t r a c k b e t t e r th a n o b s e r v e d d a t a s o t h a t t h e p a r a

m e te r s o f t h e hum an o p e r a t o r m odel w e re s u b j e c t e d t o a s i x -35p a r a m e te r o p t i m i z a t i o n u s i n g P o w e l l 's n o n d e r i v a t i v e m e th o d .

The hum an o p e r a t o r m odel c h o s e n f o r o p t i m i z a t i o n i s g iv e n b y

E q u a t io n ( 5 . 2 - 2 ) . The p a r a m e te r s o p t im iz e d w e re t h e t im e

c o n s t a n t s T j , T ^ , a n d Tn , an d t h e p a r a m e te r s p ( b e f o r e c r o s s o v e r ) ,

p ( a f t e r c r o s s o v e r ) , a n d a . T he o b j e c t i v e f u n c t i o n m in im iz e d

was IS E , a s g iv e n by E q u a t io n ( 5 . 2 - 1 ) . The op tim um p a r a m e te r s

w ere fo u n d t o b e :

Tn = 0 .1 9 1 1 s e c

Tl = 0 . 7 2 2 s e c

t

cc

TiwcxjNB or m ncm nWITH NM-UNCMI MUM

KT-21 * IEmN/BI 0*3.1

CO.

'* . 0 0 4 . 0 0 ffw * j» TIHE TO CIWSSCVEH (SECONOS)

i k . o o t b . o o10.00«o

Figure 5*17 Tracking with Non-Unear Plua Feed Forward Operator - VII

90

T i a 0 .5 6 8 s e c

« = 1 4 .5 4 1 1 s e c

p ( b e f o r e c r o s s o v e r ) = 2 .2 5 0 3

p ( a f t e r c r o s s o v e r ) » 1 .9 1 8 1

R e s u l t s w i th t h e s e p a r a m e te r s may be s e e n i n F ig u r e

5 - 1 8 . E x c e l l e n t t r a c k i n g r e s u l t e d b o th b e f o r e an d a f t e r c r o s s

o v e r w i th a maximum a z im u th e r r o r o f a b o u t 15 m i l s j u s t p r i o r

t o c r o s s o v e r .

H o w ev e r, when t r a c k i n g w as a t t e m p te d w i th a d i f f e r e n t

c o u r s e , s u c h a s o n e c o r r e s p o n d in g t o t h e c o u r s e 1 0 1 , v e ry p o o r

t r a c k i n g r e s u l t e d . T h is i s a d i r e c t r e s u l t o f o v e r t u n i n g th e

s y s te m f o r o n e p a r t i c u l a r c o u r s e . A new m odel h a d t o be

d e v e lo p e d w h ic h w o u ld g iv e s a t i s f a c t o r y t r a c k i n g p e r fo rm a n c e

w h i l e u s i n g o n ly o n e s e t o f p a r a m e te r s f o r v a r i o u s f l i g h t

p a t h s . The m o d e l may b e r e p r e s e n t e d b y t h e b lo c k d ia g ra m o f

F ig u r e 5 - 1 9 .

R e f e r e n c e t o F ig u r e 5 -1 9 i n d i c a t e s t h a t t h e m odel s e n s e s

t h r e e p a r a m e t e r s :

E = t h e t r a c k i n g e r r o r

t = t h e d e r i v a t i v e o f t h e t r a c k i n g e r r o r

a n d W = t h e a n g u l a r v e l o c i t y o f t h e a i r c r a f t

The m o d el c o m b in e s t h e s e i n p u t s d y n a m ic a l ly t o p ro d u c e an o u t

p u t t o r q u e t o t h e g u n . The m o d el i t s e l f c o n t a i n s s i x p a r a

m e te r s :

T = hum an o p e r a t o r d e a d t im e = 0 .1 5 s e c .

T j = m a jo r hum an t im e c o n s t a n t = 1 .0 s e c .

o

tmckjw op ftinmnrM1TN NON-U NOW HUM

mvmctoberr

•10. -o - S T o c 0T00 sToo T*ME Tl CROSSOVER (SECONOS)

:s.oo 10.00

Figure 5-1SOptlmua Tracking with Non-Linear Plue Feed Forward Operator * Course 102

(D ttoy)

(Aircraft ongulor vilocity)

(T1S+l)(TilS + l )( Torque)

FIGURE 5-1* TRACKING MODEL O F HUMAN OPERATOR

93

Tn = n e u ro m u s c u la r l a g = 0 .2 s e c .

p l» p2 t p 3= e m p i r i c a l c o n s t a n t s

S in c e T , T j a n d Tn a r e c o n s t a n t s t h a t h a v e b e e n e v a l u a t e d i n

p r e v i o u s w o rk 3®*3? t h e i r v a l i d i t y h a s b e e n w e l l d o c u

m e n te d , t h e s e w e re n o t a l t e r e d a n d t h e m odel w as tu n e d b y

a d j u s t i n g o n ly pj_, P2 , a n d P3.

T h is w as a c c o m p l is h e d i n t h e f o l l o w i n g m a n n e r . The

h y b r i d c o m p u te r s i m u l a t i o n o f t h e s y s te m , a s d e s c r i b e d in

C h a p te r 4 , w as u t i l i z e d e x t e n s i v e l y i n t a k i n g t r a c k i n g d a t a .

B a s i c a l l y t h e o n ly d i f f e r e n c e b e tw e e n t h i s m odel an d t h e a l l

d i g i t a l s i m u l a t i o n i s t h e f a c t t h a t o n e u s e s an a c t u a l human

o p e r a t o r w h i l e t h e o t h e r u s e s t h e m o d e l o f t h e hum an d e s c r i b e d

a b o v e . T h e r e f o r e , d a t a w as t a k e n on t h e h y b r i d c o m p u te r m odel

a n d u s e d t o s e l e c t t h e p a r a m e t e r s o f t h e hum an o p e r a t o r m o d e l.

The d e t a i l e d p r o c e d u r e i s d e s c r i b e d b e lo w .

F i r s t , p a r a m e te r s t o u s e i n t h e gun m o d el ( s e e S e c t i o n

3 .4 ) o f b o th s i m u l a t i o n s w e re s e l e c t e d a s

iT# = = 5 .0 s l u g - f t 2

B# = = 1 6 .0 f t - l b - s e c

A d e s c r i p t i o n o f t h e f l i g h t p a t h s s e l e c t e d c a n b e fo u n d i n

T a b le 3 o f S e c t i o n 3 .5 . T hey c o n s i s t e d o f a l e v e l f l y - b y a t

450 k n o t s a n d 375 f t . a . g . l . a n d a t m inim um d i s p l a c e m e n t s

f ro m t h e gun o f 600 f t . ( c o u r s e 1 0 1 ) , 1800 f t . ( c o u r s e 1 0 2 ) ,

an d 2700 f t . ( c o u r s e 1 0 3 ) .

94

A t o t a l o f f o u r t e e n r u n e w e re m ade w i th t h e s e f l i g h t

p a t h s a n d t h e a b o v e i n d i c a t e d gun p a r a m e te r s on th e h y b r i d

c o m p u te r w i t h a c t u a l o p e r a t o r s t r a c k i n g t h e t a r g e t a s d i s

p la y e d o n e l e c t r o n i c s c o p e s . T he a v e r a g e t r a c k i n g p e r fo rm a n c e

f o r t h e t h r e e f l i g h t p a t h s a r e show n i n F ig u r e s S - 2 0 , 5 - 2 1 ,

a n d 5&22.

The d i g i t a l m o d e l o f t h e human o p e r a t o r w as tu n e d by

r u n n in g t h e m odel on t h e i d e n t i c a l c a s e an d m in im iz in g th e

d i f f e r e n c e b e tw e e n t h e a v e r a g e t r a c k i n g d a t a an d t h e o u t p u t o f

t h e a l l d i g i t a l m o d e l. B e c a u se o f t h e l a r g e e r r o r s o c c u r in g

i n t h e c a s e w i t h 600 f t . d i s p l a c e m e n t , i t w as fo u n d t h i s c a s e

w as f a r o u tw e ig h in g t h e r u n s a t 1800 f t . a n d 2 700 f t . S in c e

t h e l a t t e r tw o c a s e s a r e c o n s i d e r e d t h e m ore i m p o r t a n t , t h e

600 f t . c a s e w as d i s r e g a r d e d i n d e te r m in in g t h e s e p a r a m e t e r s .

I t w as fo u n d t h a t t h e s e t o f p a r a m e t e r s t h a t g a v e t h e b e s t

f i t w e re :

P i = 1 .8 3 f t - l b - s e c / r a d i a n

P2 = 110 r a d i a n s 3/ f t - l b - s e c

P 3 = 1 1 .4 1 f t - l b - s e c / r a d i a n

C o m p ariso n o f t h e hum an o p e r a t o r m odel w i th t h e a v e r a g e

t r a c k i n g p e r fo rm a n c e a r e show n i n F ig u r e s 5 -2 3 , 5 - 2 4 , 5 - 2 5 , an d

5 - 2 6 . T h e se f i g u r e s i n d i c a t e t h a t w h i l e t h e m odel a d e q u a te ly

r e p r o d u c e s a v e r a g e p e r fo rm a n c e b e f o r e c r o s s o v e r , i t i s h i g h l y

s u p e r i o r a f t e r c r o s s o v e r . T h is s u g g e s t e d t h a t tw o s e t s o f

hum an p a r a m e te r s m ig h t b e u s e d , o n e s e t p r i o r t o c r o s s o v e r

a n d t h e o t h e r s e t a f t e r c r o s s o v e r . T h is c a u s e s no l o s s o f

Azi

mut

h Er

ror

(mil

s)

95

F lig h t Path 101 Cun l’araiNft.ors

J ■ 3B « 16

40 800

30

- 1400

200

— 200-10

-20 —400

-30Note change

of sca le

-20 -10 20

Time to Crossover (Seconds)

Figure t-aoa Average Human O perator T racking Performance - I

Ele

vatio

n Er

ror

(mil

s)

96

F lig h t Path 101 Gun Param eters

J m 5B ■* 16

100

50

100

-200

-250

-20 -10 0 10 20


F igure S*i*b Average Human O perator Tracking Performance - I

Azi

mut

h Er

ror

(mil

s)F l ig h t Path 102 Gun Paraiceters

-20

-40

-60 -20 -10 10Time to Crossover (Seconds)

Figure s-2ia Average Hunan Operator Tracking Perfornance - I I

Kle

vatio

n Er

ror

(mil

s)

F lig h t Path 102 Gun Parameters

J - 5

-10

-20 -10 0 1C 20Time to Crossover (Seconds) W

00

Figure S - » l b Average Human O perator Tracking Performance - I I

Axl

auth

Er

ror

(mil

t)

t

n ig h t Path 103 Gun Parameters

10

-20

-10 0-20 10 20Time to Crossover (Seconda)

«Figure M U Average Hunan Operator Tracking Performance - 1X1

f-;l«

?vat

ion

Krro

r (m

fla)

F lig h t Path 103 Gur. Parameters

J = 5B = 16

10

-20 -10 0 10 20*9

Time to Crossover (Secoads)

Figure Average Human Operator Tracking Performance - I I I

100

Acl

nuth

Er

ror

(mll

a)

Flight Path 102• Aslauth Operator60

60

20

20

-40

-60-10 0 10-20 20

Tine to Crossover (Seconds)Figure •-*» Performnce of Huaen Operator Model - I

101

Elev

atio

n Er

ror

(all

s)Flight Path 102Elevation Operator

Hybrid Kesults

SSP Model

-20

0 10-10 20-20Tine to Crossover (Seconds)

Figure *-*« Performance of Hasan Operator Model - I I

i

Azi

mut

h Er

ror

(ini

Is)

F light Bath 103Azimuth Operator

Hybrid R esults

SSP Model

-60-20 -10 0 10 20


Figure »-** Performance o f Human Operator Model - I I I

103

Kit!

vat

Lon

Erro

r (m

lla)

F light Path 103Elevation Operator

-10 Hybrid R esu lts

SSP Model

-2010-20 0 10 20

Time to Crossover ( Seconds)

F igure S-Ii Performance of Human O perator Model - IV

104

105

g e n e r a l i t y i n t h e m o d e l a n d c a n b e j u s t i f i e d on t h e p r e m is e

t h a t a t r a i n e d o p e r a t o r d o e s f o l l o w a d i f f e r e n t s t r a t e g y in

t r a c k i n g a t a r g e t b e f o r e a n d a f t e r c r o s s o v e r . B e fo re c r o s s

o v e r , h e i s c o n c e rn e d w i t h k e e p in g up w i t h t h e a c c e l e r a t i n g

t a r g e t , w h i l e a f t e r c r o s s o v e r , i f h e l a g s t h e t a r g e t h e

r e a l i z e s t h e d e a c c e l e r a t i n g t a r g e t w i l l f l y i n t o h i s r e t i c l e

i f h e s im p ly m a i n t a i n s t h e sam e gun m ovem ent.

T h is p r o c e d u r e w as f o l lo w e d a n d r e s u l t e d i n a d o u b le

s e t o f p a r a m e t e r s (D S P ). The op tim um p a r a m e te r s a r e i n d i

c a t e d i n T a b le 5 .

R e s u l t s w i t h t h e s e p a r a m e te r s a r e show n i n F ig u r e s 5 - 2 7 ,

5 - 2 8 , 5 - 2 9 , a n d 5 -3 0 f o r t h e l e v e l f l i g h t p a th s 102 a n d 1 0 3 .

I t i s s e e n t h a t t h i s DSP m o d el p r o d u c e s e x c e l l e n t d u p l i c a t i o n

o f t h e a v e r a g e t r a c k i n g d a t a .

F u r t h e r s t u d i e s w i t h t h i s p a r t i c u l a r hum an o p e r a t o r

m odel i n d i c a t e d t h a t t h e p a r a m e t e r Pg i s s t r o n g l y d e p e n d e n t

upon t h e v a lu e o f t h e d r a g c o e f f i c i e n t , B , i n t h e gun m o d e l,

a n d b e s t r e s u l t s a r e o b t a i n e d when t h e tw o a r e n u m e r i c a l ly

e q u a l . I n t h e s t u d y b e in g d i s c u s s e d h e r e , t h e v a lu e o f B

u s e d w as 1 6 .0 f t - l b - s e c , a n d o p t i m i z a t i o n p r o c e d u r e s fo u n d

v a l u e s o f Pg ( i n t h e DSP m o d e l) o f 1 5 .0 a n d 1 7 .1 4 f t - l b - s e c .

I n s u b s e q u e n t w o rk , s t u d i e s 2 an d 3 , t h e v a lu e o f t h e human

o p e r a t o r p a r a m e te r s w e re a s i n d i c a t e d i n T a b le 5 w i th t h e

e x c e p t i o n o f Pg w h ic h w as a d j u s t e d t o b e n u m e r i c a l ly e q u a l t o

t h e gun d r a g c o e f f i c i e n t , B . R e s u l t s w i t h t h e p a r a m e te r s f o r

S t u d i e s 2 a n d 3 a r e i n d i c a t e d i n F i g u r e s 5 - 3 1 , 5 - 3 2 , 5 - 3 3 , a n d

5 - 3 4 .

106

ZABLE 5

-'*4 » * »j&jkjdAev‘fo'tlauM FdrSttatare for as? ' Huaari O perator .Model

Before Crotiovar A fter Crossover

P f t- lb -se c /ra d ia n 2 .4 2 .4

P2 radian3/ f t - lb - s e c 4848.0 8.23 x 105

P3 ft- lb -se c /ra d ia n 15.0 17.14

Aalm

uth

Erro

r (n

ils)

I

Flight Path 102Aaiauth Operator

60

•20

- - Hybrid teau lta

- 4 0

-60-10 10-20 0 20

Time to Croaaovar (Seeonda)Figure f-tr Performance of OOP Human Operator 'iodel - 1

107

F light Path 102Elevation Operator t

Hybrid R esu lts

DSP Model

W

-10 0 10 20 Time to Crossover (Seconds)

Figure 1 11 Performance of DSP Human Operator Model - II

Azi

mut

h K

rror

(mil

s)

F light Path 103Azimuth Operator

Hybrid R esu lts

DSP Model-40

-20 -10 0 10 20Time to Crossover (Seconds)

Figure Performance of DSP Human O perator Model - I I I

109

Ele

vatio

n Er

ror

(ml

U)

Flight Path 103Elevation Operator

10

0

Hybrid R esultsV- /

DSP Model

10-20 10 0 10 20

Tine to Crossover (Seconds)

Figure S*S§ Performance o f DSP Human O perator Model - IV

110

Flight Psth 102Asiauth Operator

40

hOI*uhi

M<

-10

-20-20 -10

Tiae to Crossover (Seconds)Figure l - l u Mesa Tracking Error froa Model £or l y i t i s 3-1

I

111

Ele

vatio

n Er

ror

(Mila

)

Flight Path 102Elevation Ooerator

i-

5

0

-5

-20 -10 0 10 20

Time to Crossover (seconds)

Figure Mean Tracking E rro r from Model fo r System 3-1

112

A?.li

;nith

Er

ror

(mil

s)

Flight Path 103Azimuth Operator

20

10

0

-10

-20 -10 )00 20

Time to Crossover (seconds)

Figure s-sia Mean Tracking Error from Model for System 3 -II

113

Flight Fath 103Elevation Operator

Tin* to Crossover (seconds)

Figure S^skb Mean Tracking E rror from Model fo r System 3 - I I

Azi

mut

h Er

ror

(mH

n)Flight Path 102Azimuth Operator

15

10

5

0

5

-20 - 1 0 100 20

Time to Crossover (seconds)Figure Mean Tracking Error from Model for System 2-1

115

Klc

v.it

io

n Er

ror

(mil

s)

FIight Path 102Elevation Operator

-10 10-20

Time to' Crossover (seconds)Figure Mean Tracking Error fron Model for System 2-1

Azi

mut

h Er

ror

(mil

s)F’ ight Path 103Azimuth Operator

5

0

5

0-20 10 20

Time to Crossover (seconds)Figure 5-34M Kean Tracking Error from Model for System 2 -II

Ele

vatio

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ror

(mild

)

F light Path 103Elevation Operator

5

0

5

-20 -10 0 10 20Tine to Crossover (seconds)

Figure S-J^Kean Tracking Error from Model for Systen 2 -II 118

119

5 .3 . V e r i f i c a t i o n o f t h e T r a c k in g Mode M bdel

The s e v e r e s t t e s t o f a lm o s t a n y c o m p u te r s i m u l a t i o n i s

t h e d e t e r m i n a t i o n a s t o w h e th e r t h e s i m u l a t i o n c a n p r e d i c t

r e s u l t s f o r a s e t o f c o n d i t i o n s q u i t e d i s s i m i l a r t o t h o s e f o r

w h ic h t h e p a r a m e te r s i n t h e m o d el w e re t u n e d . D u r in g th e

c o u r s e o f t h i s r e s e a r c h , much d a t a w as r e c o r d e d on t h e t r a c k i n g

p e r fo rm a n c e o f hum an o p e r a t o r s , u s i n g t h e h y b r i d c o m p u te r .

T h e se r u n s i n c o r p o r a t e d t h e gun p a r a m e te r s o f S t u d i e s 2 an d 3 ,

b u t , o f c o u r s e , h a d a c t u a l o p e r a t o r s d o in g t h e t r a c k i n g .

T he f l i g h t p a th s u s e d i n t h i s s t u d y w e re , h o w e v e r , f a r

d i f f e r e n t fro m t h e low l e v e l f l y - b y f ro m w h ic h t h e human

o p e r a t o r m o d e l p a r a m e te r s w e re d e te r m in e d . T h e se f l i g h t p a th s

( a i r c r a f t m a n e u v e rs ty p e I I ) a r e d e s c r i b e d i n S e c t i o n 3 .5 and

i n A p p e n d ix B .

A c o m p a r is o n o f t h e p e r f o rm a n c e o f t h e m o d el v e r s u s th e

a v e r a g e t r a c k i n g p e r fo rm a n c e o f t h e o p e r a t o r s u s i n g th e

h y b r i d may b e s e e n i n F ig u r e s 5 -3 5 th r o u g h 5 -5 2 . F ig u r e s

5 -3 5 th r o u g h 5 -4 4 co m pare p e r f o r m a n c e s u s i n g t h e S tu d y 2 gun

m o d e l , w h i l e F ig u r e s 5 -4 5 th r o u g h 5 -5 2 co m p are S tu d y 3 gun

m odel p e r f o r m a n c e s . T h ese c a s e s w e re c h o s e n f o r p r e s e n t a t i o n

s i n c e e a c h a v e r a g e p e r fo rm a n c e r e p r e s e n t s t h e r e s u l t s o f a t

l e a s t f o u r d u p l i c a t e r u n s .

As c a n b e s e e n fro m t h e s e f i g u r e s , t h e m odel d o e s an

e x c e l l e n t jo b o f p r e d i c t i n g t h e m ean t r a c k i n g e r r o r f o r

f l i g h t p a t h s g r e a t l y d i f f e r e n t f ro m t h o s e f o r w h ic h t h e p a r a

m e te r s o f t h e m o d e l w e re t u n e d . T h is l e a d s o n e t o h a v e a

FLIGHT PATH. « 1. STUDY.,# 2

O.XKT

FIG

s .o r

n .

* - PREDICTED HERN ERRQRa PREDICTED HERN ERROR®

n .LS.OO 2 6 .QQT1HE I3EC!

3 1 .0 0 ;.oa ' 7 . 0 0 a .o o

5-35 AZIMUTH ERROR VS TIME

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dFLIGHT PATH n 1 STUDY « 2eO -

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« - AVERAGED HUMAN ERROR — PREDICTED MEAN ERRORn

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FIG 5-36 ELEVATION ERROR 75 TIME-

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FLIGHT PATH # 5 STUDY # 2

a .a o 5 .0 0

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F I G . 5 - 37 AZIMUTH ERRDR VS TIME

122

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R

* - AVERAGED HUMAN ERROR _ PREOICTEO MEAN ERROR

ik .Q Q 2 0 0 0TIME (SEC)

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F I G . 5 - 3 8 ELEVATION ERROR VS TIME 123

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J.OO 5 ,0 0 10.00

* - PREDICTED MEAN ERROR PREDICTED HERN ERROR

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F I G . 5 - 3 9 AZIMUTH ERROR VS TIME

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o .

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F I G . 5-40 ELEVATION ERROR VS TIME

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« - AVERAGED HUHRN ERROR — PREOICTEO HEAK ERROR

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F I G . 5-41 AZIMUTH ERROR VS TIME

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— •*- AVERAGED HUHRN ERROR PREDICTED HERN ERROR

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F I G . 5-4 6 ELEVATION ERROR VS TIME

8 F L I G H T PATH * 10S t

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F I G . 5-4 8 ELEVATION ERROR VS TIME

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F I G . 5-50 ELEVATION ERROR VS TIME

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F I G . 5-52 .ELEVATION ERROR VS TIME 137

138

g r e a t d e a l o f o o n f id e n e e i n t h e a b i l i t y o f t h i s m odel t o p r e

d i c t t h e t r a c k i n g e r r o r s f o r a lm o s t a n y f l i g h t m a n e u v e r .

5 . 4 . The A c q u i s i t i o n M ode. The o t h e r c h a r a c t e r i s t i c

mode i n c o r p o r a t e d i n t o t h e hum an o p e r a t o r m o d e l w as t h e a c q u i

s i t i o n m ode. D u r in g t h i s p h a s e o f o p e r a t i o n t h e o p e r a t o r

r e a c t s r a p i d l y t o l a r g e d i s t u r b a n c e s ( i . e . , e r r o r s i g n a l s o u t

s i d e t h e n o rm a l t r a c k i n g dom ain a s d e f i n e d b y F ig u r e 5 - 1 ) b y

a p p ly i n g a maximum d r i v i n g f o r c e t o t h e a p p a r a t u s b e in g c o n

t r o l l e d . I n o t h e r w o rd s , t h e hum an o p e r a t o r r e a c t s t o l a r g e

e r r o r s i n a m a n n e r s i m i l a r t o a b a n g -b a n g c o n t r o l l e r . I t i s

t h i s t y p e o f human c o n t r o l a c t i o n t h a t h a s b e e n d e v e lo p e d i n

S e c t i o n 5 .4 1 .

Due t o t h e s e v e r e n o n l i n e a r i t i e s i n t h e s y s te m n e a r

c r o s s o v e r t h e s w i t c h in g e q u a t i o n s becam e d i v e r g e n t , t h u s m ak in g

i t n e c e s s a r y t o m o d ify t h e m o d e l . T h is m o d i f i c a t i o n i s d e s

c r i b e d i n S e c t i o n 5 .4 2 .

5 .4 1 . D e r i v a t i o n o f S w i tc h in g T im e s . O nce t h e n o n l i n e a r

gun m o d el w as l i n e a r i z e d ( s e e A p p e n d ix 6 ) u s i n g a t r a n s f e r

f u n c t i o n o f t h e fo rm :

# ( s ) a K ( 5 .4 1 - 1 )"TTTaT s TR - 4 1 )—

w h e re u ( s ) = a p p l i e d o p e r a t o r t o r q u e

f ( s ) = a z im u th a n g le

K = gun t r a n s f e r f u n c t i o n g a in

T a gun t im e c o n s t a n t

139

a s e t o f a l g e b r a i c e q u a t i o n s w e re d e v e lo p e d f ro m w h ic h t h e

s w i t c h in g t im e s f o r t a r g e t a c q u i s i t i o n w e re c a l c u l a t e d .

G iv en t h e gun p o s i t i o n , when t h e a c q u i s i t i o n mode i s

e v o k e d a n d a i r c r a f t f l i g h t p a t h d a t a a lo n g w i th t h e t r a n s f e r

f u n c t i o n , e q u a t io n 5 . < t l - l , E q u a t io n 5 .U 1 -1 c a n b e r e w r i t t e n

a s ;

T # + f = K U ( t ) (5 .* * 1 -2 )

s u b j e c t t o t h e f o l lo w in g c o n s t r a i n t o n to r q u e

M 4 U ( t ) £ - M ( S . m - 3 )

w h e re M = l i m i t on to r q u e

a n d b o u n d ry c o n d i t i o n s

g o = $ ( t o )• * a t t = t fl| o ~ 9 ( t o ) °. (5 .W .-U )• f = # ( t f )

a t t = t f# f = f ( t f )

The p ro b le m now becom es a s e c o n d o r d e r , l i n e a r t im e o p t im a l

c o n t r o l p ro b le m w i th a c o n s t r a i n e d m a n ip u la te d v a r i a b l e .

The s o l u t i o n o f t h i s p ro b le m f o r t 0 = 0 i s

M o * t 4 t B

U Ct) = -H f o r t s 4 t ( 5 .H 1 -5 )

0 t w t f

w h e re t g = s w i t c h in g t im e

t f = c o m p le t io n o f a c q u i s i t i o n .

140

E u q a t io n 5 .4 1 - 2 c a n b e s o l v e d f o r t h e c a s e w h e re

o 4 t 4 t 8 a n d u C t) = M

| = TCK-M - $0 ) Exp C - t /T ) + K -M -t + (# 0 + T (# c - K-M))

( 5 .4 1 - 6 )

• »# = (# o - K-M) ex p C - t /T ) + K-M ( 5 .4 1 - 7 )

a n d w h e re t s 4. t « i t f w i th u ( t ) = -M

e| = - t / O f + K ‘M \ e x p C - t /T ) - K*M*t + ( 8 f + K .M »tf

e x p C - t f / T ) / + T + ( 5 .4 1 - 8 )

* / *

• =/ t f » y»M \p y p C - t /T ) - K - M . ( 5 .4 1 - 9 )V, ex p C-tf/T)/

To o b t a i n e q u a t i o n s f o r t s a n d t f t h e v a lu e s f o r 0 g iv e n b y

e q u a t i o n s 5 .4 1 - 6 a n d 5 .4 1 - 8 w h e re t = t s a r e s e t e q u a l t o

e a c h o t h e r , l i k e w i s e f o r 0 . T h is y i e l d s tw o e q u a t i o n s a n d

tw o unknow ns ( t s a n d t f ) , w h ic h upon r e a r r a n g e m e n t a r e :

.(. +. k .H ) exp- < t s / T ) - i - --------------— - ------------------------- , ----------------------------( 5 .4 1 - 1 0 )e x p C - ( 2 t s + _ 1 _ ( $ - + TC8 - 8 f ) ) / T )

K-M ° T © T

= 2K-M + (8 ^ - k -M) e x p ( - t s / T )

a n d

t f a 2 t s + _ 1 _ ( # Q - | f + T C |0 - # f ) ) . ( 5 .4 1 - 1 1 )K.'M

141

The s o l u t i o n o f e q u a t io n s S . 4 1 -1 0 a n d 5 .4 1 - 1 1 i n v o lv e s

a n e s t e d I n t e r a c t i v e p r o c e d u r e b e c a u s e e q u a t i o n 5 .4 1 -1 0 c a n n o t

b e s o l v e d e x p l i c i t l y f o r t fl a n d a v a lu e f o r t f m u s t b e a ssu m ed

b e f o r e t h e f i n a l , c o n d i t i o n s c a n b e c a l c u l a t e d . The f i n a l c o n d i -*

t i o n f o r # a n d & c a n b e c a l c u l a t e d s i n c e t h e a im v e c t o r

( F ig u r e 3 -2 ) a n d t h e v e c t o r fro m t h e gun p o s i t i o n t o t h e t a r g e t

m u s t b e c o i n c i d e n t . T hus o n c e h a v in g a ssu m e d t f , t h e t im e o f

a c q u i s i t i o n , a n d k n o w in g t h e f l i g h t p a t h , ^ f a n d c a n be c a l

c u l a t e d fro m t h e f o l l o w in g e q u a t i o n s :

® f 5 t a n " 1 ( Yt f \A Xt f /

( 5 .4 1 - 1 2 )© f = X .> - Y . . )

* t f 2 + ^ f 2

5 .4 2 . A c q u i s i t i o n a t C r o s s o v e r . As n o te d i n t h e i n t r o

d u c t i o n o f t h i s s e c t i o n , t h e a c q u i s i t i o n mode w as m o d if ie d n e a r

c r o s s o v e r d u e t o d iv e r g e n c e o f t h e s w i t c h in g e q u a t i o n s . The

p r im a r y f a c t o r s c o n t r i b u t i n g t o t h e i n s t a b i l i t y a r e t h e s e v e r e

n o n l i n e a r a l l t i e s o f t h e f l i g h t p a t h , w hen d e s c r i b e d i n s p h e r o id

c o o r d i n a t e s , a n d t h e b r e a k down o f t h e l i n e a r i z e d gun m odel

n e a r c r o s s o v e r .

The m o d i f i c a t i o n , t e l e o l o g i c a l i n n a t u r e , c o n s i s t s o f

a p p ly i n g maximum t o r q u e , f o r c i n g t h e g un t o l e a d t h e p l a n e ,

t h e n a p p ly i n g o n ly a s m a l l b r e a k i n g t o r q u e u n t i l t h e p l a n e f l i e s

142

i n t o t h e p r o p e r s i t e a l i g n m e n t . T he i d e a f o r t h e l e a d m o d i

f i c a t i o n i a fo u n d e d l a r g e l y o n e x p e r i e n c e g a in e d w h i l e t r a c k i n g

t a r g e t s u s i n g t h e h y b r i d s i m u l a t o r .

I n t h e r e g i o n o f c r o s s o v e r , c r o s s o v e r i s t h e p o i n t o f

m inimum d i s t a n c e b e tw e e n gun a n d p l a n e , t h e a n g u la r v e l o c i t y

o f t h e p l a n e e x c e e d s t h e maximum s le w r a t e o f t h e g u n . The

b e a t t r a c k i n g s t r a t e g y a t r a i n e d o p e r a t o r c a n u s e i s t o a p p ly

maximum to r q u e o n t h e gun u n t i l h e r e a c q u i r e s t h e t a r g e t .

F i e l d i n g 2 r e f e r s t o a n o p e r a t o r i n t h i s s i t u a t i o n a s a p r a c o g

n i t i v e s i g n a l g e n e r a t o r .

The l e a d mode i s in v o k e d w hen t h e t a r g e t i s n o t i n t h e

t r a c k i n g d o m ain ( F i g . 5 -1 ) an d n e a r c r o s s o v e r ( l e s s t h a n

2 5 0 0 ' f ro m s i t e ) . I f t h e t a r g e t i s n o t n e a r c r o s s o v e r t h e

a c q u i r e mode i s u s e d . A lth o u g h n o t u s e d i n t h i s w o rk , a

b e t t e r m eans o f i n v o k in g t h e l e a d mode w o u ld b e t o t e s t t h e

a n g u l a r v e l o c i t y o f t h e t a r g e t i n s t e a d o f t h e n e a r n e s s t e s t .

T h is c r i t e r i o n w o u ld b e b e t t e r b e c a u s e i t i s t h e a n g u la r

v e l o c i t y o f t h e t a r g e t e x c e e d in g t h e gun s le w r a t e t h a t l e a d s

t o t h e d iv e r g e n c e o f t h e s w i t c h i n g e q u a t i o n s . A f lo w c h a r t

o f t h e m u l t i - m o d a l t r a c k i n g m o d e l i s show n i n F ig u r e 5 - 5 3 .

5 . 5 . V e r i f i c a t i o n o f t h e A c q u i s i t i o n Mode M o d e l.

P r o b a b ly t h e b e s t way t o ju d g e t h e e f f e c t i v e n e s s o f t h e a c q u i

s i t i o n mode i s f i r s t t o r e v ie w t h e n e e d f o r i t s a d d i t i o n t o

t h e o p e r a t o r m o d a l a n d t h e n s e e how w e l l i t f u l f i l l s t h e

r e q u i r e m e n t s . T he a c q u i s i t i o n mode w as a d d e d t o :

143

1 . P r o v id e a n a a n s o f q u i o k l y a c q u i r i n g a now t a r g e t ,

2 . A llo w t r a c k i n g o f p l a n e s t h a t p a s s v e r y c l o s e t o t h e

o p e r a t o r , a n d

3 . A llo w t r a c k i n g o f i n p u t s i g n a l s w h ic h may c o n t a i n

d i s c o n t i n u i t i e s .

i t s h o u ld b e n o t e d t h a t im p ro v e m e n ts i n t h e n o n - l i n e a r t r a c k i n g

m o d e l r e d u c e d t h e im p o r ta n c e o f t h e s e c o n d r e q u i r e m e n t to w a rd

t h e e n d o f t h i s r e s e a r c h .

A c o m p a r is o n o f t h e p e r f o r m a n c e o f t h e d u a l-m o d e m odel

v e r s u s t h e s in g le - m o d e ( t r a c k i n g o n ly ) m o d e l f o r a c q u i s i t i o n

o f t a r g e t s u s i n g t h e s e t s o f g u n p a r a m e te r s may b e s e e n i n

F i g u r e s 5 -5 4 (A a n d B) th r o u g h 5 -5 6 (A a n d B ) . The c o m p a r i

s o n s w e re m ade f o r t a r g e t s w i t h 400 a n d 1000 m i l l s o f a l i g n

m e n t e r r o r . A P e r fo rm a n c e I n d e x ( t h e i n t e g r a l o f t h e e r r o r

s q u a r e d 'I S A ' o v e r t h e f i r s t s e v e n s e c o n d s ) i s a l s o g i v e n .

As c a n b e s e e n fro m t h e s e f i g u r e s , t h e d u a l-m o d e m o d e l

a c q u i r e s t h e t a r g e t r a p i d l y w i t h l i t t l e o s c i l l a t i o n , e v e n

when t h e i n i t i a l e r r o r i s v e r y l a r g e ; w h e r e a s , t h e s i n g l e -

mode m o d e l w as s l o w e r t o r e s p o n d f o r s m a l l i n i t i a l e r r o r s an d

o v e r - s h o t t h e t a r g e t f o r t h e l a r g e e r r o r s .

F i g u r e s 5 7 -6 0 show t h e r e s u l t s O f s i m u l a t i o n s a t c r o s s

o v e r w h e re t h e p l a n e cam e n e a r t h e g u n . H e re t h e a c q u i s i t i o n

m ode , w i t h t h e l e a d m o d i f i c a t i o n , p e r f o r m e d to o w e l l . I t

r e a c q u i r e d t h e t a r g e t a f t e r l o s i n g i t a t c r o s s o v e r w i t h o u t any

o v e r s h o o t . I n m o s t o f t h e d a t a s t u d i e d d u r i n g t h i s p r o j e c t .

144

COMPUTE PLAHB POSITION

COMPUTE ALIGNMENT BRROR |

OPERATOR IS INI AOQUIRB MOOB

YBS

NOOPERATOR IS IN LEAD MODE

IS ERROR WITHIN TRACKING DOMAIN

NOIS ERROR WITHIN TRACKING DOMAIN

YBS

IS PLANB NEAR CROSSOVER

+ NO&MPUTB SWITCHING

COMPUTBTORQUB

COMPUTBTORQUB

COMPUTBTORQUE

APPLY TORQUB TO TUB GUN AND CALCULATB

POSITION

FIGURB 5-53 FLOW DIAGRAM OF THE MULTI-MODAL TRACKING MODEL

SYSTEM No. 1

MAX. TORQUE

Difference Between Option*

I'm! and - multi-mode

and - tracking only

Model i s a Performance

— —— - Tracking only 1.068 X 10^— ■ i. Dual-node 5.141 X 10^

Optimum 4.851 X 10*

8 10 Tine (Sec)

FIGURE 54-A COMPARISON OF ACQUISITION PERFORMANCE

1000

500

SYSTEM NO. 1

Difference Between Optimum

•♦*/) and - multi-mode

and - tracking only

-500

-1000

MAX TORQUE

/


Tracking only 1.041 X 10®Dual-Mode 7.503 X 10sOptimum 7.435 X 10?

10Time (Sec)

FIGURE 54-B COMPARISON OF ACQUISITION PERFORMANCE

12a*

400

200

8uuM

-200 .

MAX. TORQUE

-400

/

SYSTEM NO. 2

D ifference Between Optimum

end - w lti-m ode

and - tracking only

Model Is a Performance

Tracking only Dual-Mode Optimum

1.026 X 105 4.91 X 104 4.837 X 104

- I142 4 6 8

Time (Sec)FIGURE 55-A COMPARISON OF ACQUISITION PEEFOBMANCE

10 12

147

1000 SYSTEM NO. 2

D ifference Between OptimumMAX. TORQUE

and - au ltl-node500

and - tracking only

500 Model la a Performance

9.6147.437.415

1000

Tine (Sec)FIGURE 55-B COMPARISON OF ACQUISITION PERFORMANCE 148

SYSTEM NO. 3

MAX. TORQUED ifference Between Opt in n

and - mult i-node

and - tracking only

Model la a Performance200

1.224 X 105Tracking only

5.620 X 10H 4.90 X 104

■400

Time (Sec)

FIGURE 56-A COMPARISON OF ACQUISITION PERFORMANCE

1000*

500 -

uuM-500

-1000 .

SYSTEM HO. 3

Difference Between Optimist

and - track in g only

VMAX. TORQUE


_ _ Tracking only

- Dual-Mode — . — Optimum

9.622 X 103

7.483 X 105 7.426 X 105

8Time (Sec)

FIGURE 56-B COMPARISON OF ACQUISITION PERFORMANCE

10 12 14

150

SYSTEM NO. 15 0 0 FT. FLY-BY COURSE

MODEL

TRACKING ONLY

DUAL*M OOE

i t MTIME (SCC1

FIGURE 5 * 5 7 p e r f o r m a n c e at c r o s s o v e r

SYSTEM N O -Zl o o FT. FlY-OY COURSE

MOO E l

TRACKING ONLY

D U A L -M O D E

FtSORfi 5 - 5 8 PERFORMANCE AT CROSSOVER

153 OHO XV 1O M V N O Q J014 I S - S 9 0 0 0 1 4

<9*S? 1MU

r \

r

i d o w - i v n a A1NO o n » mo v o x

1 9 0 0 M

3SOHOO 4 0 - /1 4 U4 ooS 1 -ON H31SAS

SYSTEM n o . 14 0 0 FT. FLV-W COURSE

MODEL

T R A C K I N G o i I L Y

D U A L-M O D E

F 1 6 U W S 'W O PttlfO AM A HCC AT CROSSOVER.

ERRO

R (M

ILS)

SYSTEM NO. I

INITIALLY I S M FT. FLY-BY

AFTER 10 SECONDS 2000 FT. FLY-RY

00

40

0

- 40

TIME (SEC)

FIGURE 5 -S A DUAL- MODEL TRACKING DISCONTINUOUS TARGET

155

f ItA

Oft

(MIL

S)

IYI1KM MO. B

INITIALLY i A FTg* U SECONDS 2340 FT. FLY-BY

SO

40

40

flftU R E M l DUAL-MODEL T S A C K lM DISCONTINUOUS TARGET 9ST

157

t h e o p e r a t o r h a d a te n d e n c y t o o v e r s h o o t t h e t a r g e t l i k e t h e

a I n g le - m o d e m o d e l p r e d i c t e d . H o w ev er, t h e r e w e re soma c a s e s

w h e re t h e hum an o p e r a t o r se em e d t o u s e a l e a d s t r a t e g y a n d

p e r f o r m e d i n a m an n e r s i m i l a r t o t h e a c q u i s i t i o n m o d e . I t

s h o u ld a l s o b e n o t e d t h a t t h e h y b r i d t r a c k i n g d a t a w as som ew hat

b i a s e d n e a r c r o s s o v e r b y t h e way t h e o p e r a t o r c o n t r o l l e d t h e

g u n . T he o p e r a t o r c o n t r o l l e d t h e h y b r i d gun by a p p ly in g

to r q u e w i t h a h a n d s e t p o t e n t i m e t e r . He h a d t o r e t u r n i t t o

t h e z e r o p o s i t i o n i n o r d e r t o t a k e a d v a n ta g e o f d a m p in g ,

w h e re an a c t u a l o p e r a t o r o n l y h a d t o s t o p a p p ly in g f o r c e t o

t h e g u n .

F i n a l l y , F i g u r e s 61 a n d 62 w e re i n c l u d e d t o i n d i c a t e how

t h e d u a l-m o d e m o d e l p e r f o r m s w hen t r a c k i n g a d i s c o n t i n u o u s

s i g n a l . S u ch d i s c o n t i n u i t i e s c a n o c c u r w hen t h e t a r g e t i s

h id d e n b y c l o u d s , t r e e s , e t c . , o r w hen t h e o p e r a t o r ' s a t t e n

t i o n s p a n i s i n t e r r u p t e d b y o t h e r s t i m u l i . T h e s e f i g u r e s

i n d i c a t e t h a t t h e m o d e l c a n e f f e c t i v e l y t r a c k a d i s c o n t i n u o u s

t a r g e t .

5 . 6 . The D i g i t a l S i m u l a t i o n . A l i s t i n g o f t h e d i g i t a l

s i m u l a t i o n s o f t h e t r a c k i n g s y s te m u s e d i n t h i s s tu d y i s g iv e n

i n A p p e n d ix H . T he s i m u l a t i o n i s m o d u la r i n n a t u r e a n d u s e s

4 t h o r d e r R u n g a -K u tta t o s o l v e a s e t o f s im u l t a n e o u s d i f f e r e n

t i a l e q u a t i o n s . A f lo w c h a r t o f t h e p ro g ra m i s g iv e n i n

F ig u r e 6 3 .

1 5 8

The fo llow in g i s • fu n ction a l d escr ip tion o f each o f

the subroutines used in the sim u la tion s.

MAIM PROGRAM - used to s e t up date for the ease to be run,

c a l l the equation s o lv e r , and c a l l the p lo t rou tin e .

RUNG A - Master schedule fo r the so lu tio n o f the s e t o f sim ul

taneous d if f e r e n t ia l equations.

DERIV - c a lle d by Runga. C alcu lates i n i t i a l con d ition s,

eva lu ates interm ediate va lu es, and term inates the so lu

t io n o f the s e t o f simultaneous d if fe r e n t ia l equations

d escrib in g the tracking system .

IODATA - output subroutine which i s c a lle d by Runga.

TARGET - used to en ter fli< £ it path data and c a lcu la te the

p o s it io n o f the plane as a function o f tim e.

AOQUIR - used to compute the sw itch ing times for the a cq u is i

t io n mode.

DELAY - s e ts up, s to r e s , and r e tr ie v e s values from the torque

h isto ry ta b le s . I t a lso contains the lead m odification

used by the acquire mode.

WMODB - used to determine which operator mode to be used,

c a lle d by DERIV.

ROTATE - used to perform the coordinate transform ation needed

to eva lu ate alignment erro r .

P L O T MA I N l— 1 R U N G A IODATAP ROGRAM

EVAL

R OT AT E DE R I V

A C Q U I R

T A R G E T

V I L

FIGURE 5 - « S F L O W CHART O F D I G I T A L S I M U L A T I O N 159

160

VEL - u s e d t o c a l c u l a t e t h e a n g u l a r v e l o c i t y o f t h e a i r c r a f t .

EVAL - u s e d t o e a l o u l a t e t h e a im e r r o r w h ic h i s f e d b a c k t o

t h e o p e r a t o r .

PLOT - l i n e - p r i n t e r r o u t i n e s c a l l e d by MAIN PROGRAM.

CHAPTER VI

EXAMINATION OF RESULTS AND CONCLUSIONS

6 . 1 . E x a m in a t io n o f R e s u l t s . R e v ie w in g t h e d a t a show s

t h a t t h e i n i t i a l o b j e c t i v e , t h e d e v e lo p m e n t o f a m odel o f a

w e l l t r a i n e d o p e r a t o r , h a s b e e n e s t a b l i s h e d . The f i g u r e s i n

S e c t i o n 5 .3 show a t a g l a n c e how w e l l t h e hum an o p e r a t o r m odel

u s e d i n c o n j u n c t i n w i th d i f f e r e n t n o n - l i n e a r a im in g d e v ic e s

a g r e e s w i t h e x p e r i m e n t a l d a t a . A ls o i t i s i m p o r t a n t t o p o i n t

o u t t h a t t h e hum an m odel p a r a m e t e r s u s e d t o p r e p a r e t h e s e

r e s u l t s a r e b a s e d on d a t a o b t a i n e d f ro m v e r y d i f f e r e n t t a r g e t

t r a j e c t o r i e s . T h is l e a d s o n e t o h a v e a g r e a t d e a l o f c o n f i

d e n c e i n t h e a b i l i t y o f t h i s m o d e l t o p r e d i c t t h e t r a c k i n g

e r r o r s f o r a lm o s t a n y f l i g h t m a n e u v e r .»

The d i f f i c u l t i e s e n c o u n te r e d e a r l y i n t h e s t u d y w i th

l i n e a i r c o n t in u o u s m o d e ls a n d t h e f i n a l r e s u l t s o b t a i n e d w i th

t h e d u a l - m o d a l m odel c l e a r l y p o i n t o u t t h e s u p e r i o r i t y o f

t h e new o p e r a t o r m o d el t o t h e c o n t in u o u s m o d e l . Due t o i t s

s t a b i l i t y , S e c t i o n 3 - 6 , a n d i t s a b i l i t y t o p r e d i c t t r a c k i n g

e r r o r s f o r v e r y d i f f e r e n t t a r g e t t r a j e c t o r i e s , t h e d u a l-m o d a l

m o d el a l s o show s p ro m is e o f s u p e r i o r i t y t o s t o c a s t i c m o d e ls .

A l th o u g h s t o c a s t i c m o d e ls p r o v id e good r e s u l t s , t h e y c a n n o t b e

u s e d t o t r a c k t a r g e t s w i th t r a j e c t o r i e s t h a t d e v i a t e v e r y much

f ro m t h o s e u s e d t o d e v e lo p t h e i r p a r a m e t e r s .

161

162

6 . 2 . C onclusions . T he r e s u l t s show d e a r l y t h s t t h e

d u a l-m o d a l m o d e l d o e s an e x c e l l e n t j o b o f p r e d i c t i n g m ean

t r a c k i n g e r r o r s f o r t h e s y s te m s t u d i e d . I t i s a l s o d e a r l y

s u p e r i o r t o t h e c o n t in u o u s m o d e l i n c h a r a c t e r i s i n g an o p e r a t o r ' s

r e s p o n s e s . I t s h o u ld a l s o b e n o t e d t h a t e v e n th o u g h t h e o p e r a

t o r m odel d e v e lo p e d i n t h i s p a p e r d o e s a n e x c e l l e n t jo b o v e r a

w id e r a n g e o f t a r g e t s , t h e f a c t t h a t t h e hum an o p e r a t o r a c t s

a s a v e r y a d a p t i v e c o n t r o l l e r a n d t h e f a c t t h a t t h e o p e r a t o r

m odel h a s b e e n d e s ig n e d c l o s e l y a ro u n d a p a r t i c u l a r a p p l i c a

t i o n i n d i c a t e s t h a t i t i s u n l i k e l y t h a t a u n i v e r s a l hum an

o p e r a t o r m o d e l w i l l b e d e v e lo p e d . I n s t e a d , u s e f u l o p e r a t o r

m o d e ls w i l l c o n t i n u e t o b e c l o s e l y t i e d t o t h e d e v ic e b e in g

c o n t r o l l e d .

F i n a l l y , t h i s r e s e a r c h s u p p o r t s t h e g ro w in g l i s t o f

u s e s f o r g e n e r a l p u r p o s e h y b r i d c o m p u ta t io n . W ith a l i t t l e

i n v e n t i v e n e s s i n t h a a r e a o f v i s u a l d i s p l a y s a n d i n p u t d e v i c e s ,

t h e h y b r i d c o m p u te r b eco m es a n in e x p e n s i v e a n d u s e f u l t o o l i n

t h e d e s ig n o f e x p e r i m e n t s , o p e r a t o r t r a i n i n g an d hum an f a c t o r s

s t u d i e s . As i n t h i s p r o j e c t , t h e h y b r i d c o m p u te r c a n b e u s e d

t o p r o v i d e u s e f u l d a t a a n d t o r e d u c e t h e num ber o f e x p e n s iv e

a n d so m e tim e s u s e l e s s e x p e r im e n t s n e e d e d t o d e v e lo p human

o p e r a t o r p a r a m e t e r s .

BIBLIOGRAPHY

BIBLIOGRAPHY

1 . M cC orm ick, E . J . Human F a c to r * E n g i n e e r i n g . 2nd e d .New Y o rk : M e G ra w - i i i i i , i n c . " 1 9 "U . p p . 2 4 8 -2 5 0 .

2 . F i e l d i n g , W. F . "Some C h a r a c t e r i s t i c s o f t h e HumanO p e r a to r a n d h i s M a th e m a t ic a l R e p r e s e n t a t i o n i n t h e T r a c k in g R o l e , " Royal Aircraft E s t a b l i s h m e n t , T e c h n ic a l R e p o r t N o. WE 3 8 . F a r e n n r o u g h , f c n g lin (A u g u s t , 1 9 8 3 ) .

3 . H ic k s , W. E . "Man a s a n E le m e n t i n a C o n t r o l S y s te m ,"M e d ic a l R e s e a r c h C o u n c i l , A p p l i e d P s y c h o lo g y U n i t ,APU 1 5 0 /5 1 , ( 1 9 5 1 ) .

4 . D r e y f u s s , H e n ry . T he M e a su re o f M an. New Y o rk : W h itn e yL i b r a r y o f D e s ig n s , 1 9 6 7 . p . i< t.

5 . M o rg an , C. T . , e t a l . Human E n g in e e r i n g G u id e t o E q u ip m en t D e s ig n . Nfew Y o rk : r f c G r a w - r t i l l . i n c . , 1 9 5 3 . p .- g'5 6 .

6 . F o g e l , L . J . B io te c h n o lo g y : C o n c e p ts a n d A p p l i c a t i o n s .E n g lew o o d C l i r r s , Mew J e r s e y : P r e n t i s s - f l a i l , I n c . ,1 9 6 3 . p . 2 4 3 .

7 . N a d l e r , G . , a n d J . G o ldm an . "T he U n o p a r ," T he J o u r n a lo f I n d u s t r i a l E n g i n e e r i n g » V o l . 9 , 1 9 5 8 . p p . s d - 6 5 .

8 . N a d l e r , G. a n d J . G o ldm an . " O p e r a to r P e r fo rm a n c eS t u d i e s : I , One P la n e M o tio n L e a r n in g , n T he J o u r n a lo f I n d u s t r i a l E n g i n e e r i n g , V 61. 9 , 1 9 5 8 . p p t i t l ? - i 9 7 .

9 . N a d l e r , G. a n d J . G oldm an. " O p e r a t o r P e r fo rm a n c eS t u d i e s : I I , L e a r n in g A n a l y s i s f o r T h re e P l a n e M o t io n s ,"The J o u r n a l o f I n d u s t r i a l E n g i n e e r i n g , V o l. 9 , 196 3 .

1 0 . C o s t e l l o , R . J . A M odel o f t h e W e ll T r a in e d Human O p e ra t o r : As a n E le m e n t i n an A u to m a t ic c o n t r o l s y s te m .P h .D . d i s s e r t a t i o n , M a d iso n , W is c o n s in : U n i v e r s i t y o fW is c o n s in , 1 9 6 6 . p p . 3 4 -4 2 .

164

165

1 1 . Young, L . R . and L . Stark. "Variable Feedback Experiment*T eeting a Sampled Data MBdel fo r E ye Tracking Movements," IEEE Transactions on Human Factors in E le c tr o n ic s , Vol. h f e U , No . 1 , September, 1963. pp. 38-51.

1 2 . M a n fre d , C. a n d J . H . M ilsu m . B i o a e d io a l E n g in e e r in gS y s te m s . New Y o rk : M c G ra w -H ill', I n c . " " 1 9 7 0 . p p . 1 - 2 0 .

1 3 . M cC orm iok, o p . c i t . . p p . 3 3 2 -3 4 3 .

1 4 . I b i d . , p p . 3 3 4 -3 3 5 .

1 5 . I b i d . , p p . 3 3 8 -3 3 9 .

1 6 . M cC orm ick , E . J . Human E n g i n e e r i n g . 1 s t e d . New Y o rk :M c G ra w -H ill, I n c . , 1 9 5 7 . p . 3b77

1 7 . D r e y f u s s , o p . c i t . , p . 8 .

1 8 . T u s t i n , A . "The N a tu r e o f t h e O p e r a t o r 's R e sp o n se i nM anual C o n t r b l a n d i t s I m p l i c a t i o n f o r C o n t r o l D e s ig n ," J o u r n a l o f t h e 1 i n s t i t u t i o n o f E l e c t r i c a l E n g in e e r s ,Vol. 94, Far* II-A, No. 2, 1947.-----------*------

1 9 . M cR uer, D. e t a l . "Human P i l o t D ynam ics i n C o m p e n sa to ryS y s te m s ," A i r F o rc e F l i g h t D ynam ics L a b o r a t o r y , W r i g h t - P a t t e r s o n A i r F o r c e B a s e , O h io , T e c h n ic a l R e p o r t N o. A FFD L -T R -65-1S , J u l y , 1 9 6 5 .

2 0 . B e k e y , G. A . " A d a p tiv e C o n t r o l S y s te m M o d e ls o f t h eHuman O p e r a t o r . " I n A d a p t iv e C o n t r o l S y s te m s , p p . 2 0 3 -2 1 9 , E d . b y F . P . C a r u th e r s a n d ft. L e v e n s t e i n ,New Y o rk , New Y o rk , T he M a c m illa n C om pany, 1 9 6 3 .

2 1 . I b i d . , p . 2 0 8 .

2 2 . E l k i n d , J . . I . " C h a r a c t e r i s t i c s o f S im p le M anual C o n t r o lS y s te m s ," M .I .T . L in c o ln L a b o r a to r y T e c h n ic a l R e p o r t N o . I l l , A p r i l , 1 9 5 6 .

2 3 . B e k e y , G. A . "T he Human O p e r a t o r a s a S a m p le -D a taS y s te m ," IRE T r a n s a c t i o n s Human F a c t o r s i n E l e c t r o n i c s , H F E -3 , S e p te m b e r , 1 9 6 2 . p p . 4 3 - 5 1 .

2 4 . B e k e y , " A d a p tiv e C o n t r o l , " o p . c i t . , p p . 2 0 9 -2 1 9 ,

2 5 . B e k e y , G. A . a n d C . B. N e a l . " I d e n t i f i c a t i o n o f S a m p lin gI n t e r v a l s i n S a m p le d -D a ta M o d e ls o f Human O p e r a t o r s , "IEEE T r a n s a c t i o n s o n M an-M ach ine S y s te m s , V o l . MMS-9,No. 4, December, 1966.

166

2 6 . Y oung , L . R . "On A d a p t iv e M anual C o n t r o l , " A u to m a tio na n d R em ote C o n t r o l , J u n e , 1 9 6 8 . p p . 2 9 2 -8 3 1 .

2 7 . W e r tz , H . J . " A d a p t iv e C o n t r o l o f S y s te m s C o n ta in in gt h e Human O p e r a t o r , " P h .D . D i s s e r t a t i o n , U n i v e r s i t y o f W is c o n s in , M a d iso n , W is o o n s ln , 1 9 6 2 .

2 8 . F o g e l , o p . c i t . , p p . 2 4 1 -2 4 5 .

2 9 . C o s t e l l o , R . 6 . "T he S u rg e M odel o f t h e W e ll T r a in e dHuman O p e r a t o r i n S im p le M an u al C o n t r o l , " IEEE T r a n s a c t i o n s o n M an-M achine S y s te m s , V o l . MMS-9, N o. 1 , M a rch , 1 9 6 9 . p p . 2 - 9 .

3 0 . C o s t e l l o , " D i s s e r t a t i o n , " o p . c i t . , p p . 3 5 -3 7 .

3 1 . P l a n c h a r d , J . A . a n d T . M. P e r k i n s . "D e v e lo p m e n t o f aHuman O p e r a t o r M odel f o r a T r a c k in g T a s k ," A i r F o rc e A rm am ent L a b o r a t o r y , E g l i n A i r F o r c e B a s e , F l o r i d a , T R -7 1 -2 0 , V o l. I l l , F e b r u a r y , 1 9 7 1 .

3 2 . C o s t e l l o , " IE E E ," o p . c i t . , p p . 9 - 1 7 .

3 3 . P l a n c h a r d , J . A . , T . M. P e r k i n s a n d J . B a r z i n l i ." D e te r m in a t io n o f T r a c k in g E r r o r s i n a n A n t i A i r c r a f t E n v i r o n m e n t ," A i r F o r c e A rm am ent L a b o r a t o r y , E g l i n A i r F o rc e B a s e , F l o r i d a , T e c h n ic a l R e p o r t , J u n e , 1 9 7 2 . p p . 4 2 - 5 5 .

3 4 . C o s t e l l o , " D i s s e r t a t i o n , " o p . c i t .

3 5 . P o w e l l , M. J . D. "An E f f i c i e n t M ethod f o r F in d in g t h eMinimum o f a F u n c t io n o f S e v e r a l V a r i a b l e s W ith o u t C a l c u l a t i n g D e r i v a t i v e s , " C o m p u te r J o u r n a l , V o l . 7 . p . 1 5 5 . -------------------------------

3 6 . P l a n c h a r d , J . , T . P e r k i n s a n d J . B a r z i n j i . "A R e s e a r c ha n d A n a l y s i s P ro g ra m o f S t u d i e s o n t h e A n a l y s i s o f W eapon E f f e c t i v e n e s s - P h a s e I I R e s u l t s , Volum e I I I , D e v e lo p m e n t o f a Two-Mode Human O p e r a t o r f o r a T r a c k in g T a s k ," AFATL, E g l i n A i r F o rc e B a s e , F l o r i d a , T R -1 1 0 , V o l. I l l , A u g u s t , 1 9 7 1 .

3 8 . B re w e r , J o h n . C la s s n o t e s p r e p a r e d f o r E n g in e e r in gG r a p h ic s 154 (C o m p u te r G r a p h i c s ) . F a l l , 1 9 7 1 .

APPENDICES

APPENDIX A

CALCULATION OF ALIGNMENT ERROR

A - l . D e v e lo p m e n t o f C o o r d in a te R o t a t i o n E q u a t io n

The e q u a t i o n s t h a t r e l a t e s a p o i n t i n t h e i n e r t i a l

c o o r d i n a t e s y s te m t o i t s c o r r e s p o n d in g p o s i t i o n i n t h e gun

c o o r d i n a t e s y s te m a r e d e v e lo p e d b e lo w . F i r s t r o t a t e t h e

i n e r t i a l c o o r d i n a t e s y s te m th r o u g h a n g l e & ( s e e F ig u r e A - l a ) .

Xg Cos e S i n ^ 0 x i

* 8= - S i n e Cos 9 0 y i ( A - l )

Zg 0 0 z1

T hen r o t a t e t h e t r a n s f o r m e d i n e r t i a l c o o r d i n a t e s y s te m th r o u g h

a n g le ( s e e F ig u r e A - l b ) .

y i

7I

Co 8 0 - S i n 0

0 1 0

S i n 0 0 C o s0

X

Y

Z

(A -2 )

By c o m b in in g th e a b o v e r e l a t i o n s h i p s , a p o i n t i n t h e gun

c o o r d i n a t e s y s te m c a n b e c a l c u l a t e d , g iv e n t h e i n e r t i a l c o o r

d i n a t e a n d t h e a n g l e s £ a n d 0 . T h is c o m b in e d t r a n s f o r m a t i o n

i s

168

169

Z. * ' Y'

X

(a) rotation through angla

(b) rotation through angle

Figure A -l Rotation o f In e r t ia l Coordinate'a System through Anglte $ end Q

170

g

o r

C o s # O - S i n ^ C o s # S i n # 0

0 1 0 - S i n # C o * # 0

Sln^ 0 Cos^ 0 0 1

(A -3)

X9

y~9

Z9

( c o a ^ • C o s # ) ( c o s # • S i n # ) - S i n ^

- S i n # C o s #

( S i n ^ , G o s s ) ( S i n # • S i n # ) C o * #

(A -4)

X

y

z

Xy

z

i

APPENDIX B

PERSPECTIVE PROJECTION

T he p ro g ra m u s e d t o d raw t h e p e r s p e c t i v e p r o j e c t i o n s

o f t h e a i r c r a f t f l i g h t p a t h s i s d o c u m e n te d i n t h i s s e c t i o n .

G e tam ,38 a p a c k a g e o f F o r t r a n s u b r o u t i n e s f o r t h e g e n e r a

t i o n , m a n i p u l a t i o n an d d i s p l a y o f g r a p h i c d a t a , fo rm s th e

h e a r t o f t h i s p ro g ra m .

B a s c i a l l y , t h e p ro g ra m t a k e s i n d a t a p o i n t s fro m a c u rv e

i n 3 - s p a c e , p l a c e s th e m i n t o a (>f x N) m a t r i x , t h e n t r a n s

fo rm s t h e d a t a p o i n t s i n t o a tw o - d im e n s io n a l p r o j e c t i o n

p l a n e . The t r a n s f o r m a t i o n i s c a r r i e d o u t b y m u l t i p l y i n g t h e

d a t a m a t r i x b y a t r a n s f o r m a t i o n m a t r i x T , w h ic h i s g e n e r a t e d

f ro m t h e l o c a t i o n o f t h e o b s e r v a t i o n p o i n t an d t h e p r o j e c t i o n

p l a n e . Once t h e p o i n t s on t h e c u rv e a r e t r a n s l a t e d i n t o t h e

p r o j e c t i o n p l a n e t h e y a r e s c a l e d t o f i t i n t o a p r e d e te r m in e d

w indow . I n a d d i t i o n t o t h e d a t a m a t r i x , a p l o t t e r p e n c o n

t r o l a r r a y i s c o n s t r u c t e d . The f i n a l p l o t i s d raw n b y u s in g

t h e C alcom p r o u t i n e p l o t . T he p l o t t e r i s m oved fro m d a t a

p o i n t t o d a t a p o i n t i n s t r a i g h t l i n e in c r e m e n t s w i th t h e p en

c o n t r o l a r r a y d e te r m in in g w h e th e r t h e p l o t t e r p e n i s down a n d

171

172

w r i t i n g o r u p . T he c o m p u te r p ro g ra m l a H a t e d i n A p p e n d ix C.

A f lo w d ia g ra m o f t h e p e r a p e c t i v e p r o j e c t i o n p ro g ra m

i a g iv e n i n F ig u r e B - l . The f o l lo w in g c o n t a i n s a b r i e f d e e -

c r i p t i o n o f eome o f t h e a u b r o u t i n e a u s e d .

S u b r o u t in e VALUE: T h is r o u t i n e r e a d a f ro m c a r d a t h e maximum

" v ie w in g w indow " s i z e , a x i a p a r a m e t e r s , a n d t h e d a t a p o i n t s

fro m t h e c u rv e t o b e p l o t t e d . Then i t s e t s up th e d raw

m a t r i x , DR1, t o i n c l u d e t h e a x e s , w i t h t i c m ark s a n d th e

c u r v e . F i n a l l y t h e l o c a t i o n o f t h e o b s e r v a t i o n p o i n t i s

c a l c u l a t e d i n s u b r o u t i n e VALUE.

S u b r o u t in e TRANSL; T h is r o u t i n e d e f i n e s t h e t r a n s f o r m a t i o n

m a t r ix T w h ic h w i l l b e u s e d t o c a u s e t r a n s l a t i o n i n a n y o r

a l l o f t h e 3 p r i n c i p a l d i r e c t i o n s .

S u b r o u t in e PROJ 1 : T h is i s t h e i n i t i a l s t e p i n f o rm in g a

p e r s p e c t i v e p r o j e c t i o n o f a n a r r a y . T h is s u b r o u t i n e d e f i n e s

t h e p r o j e c t i o n m a t r i x T w h ic h w i l l b e u s e d w i th TRANSF a n d

PROJ 2 t o c a u s e p e r s p e c t i v e p r o j e c t i o n i n t o t h e 1 , 2 p l a n e .

S u b r o u t in e TRANSF: T h is r o u t i n e p e r f o r m s t h e t r a n s f o r m a t i o n

d e f i n e d b y s u b r o u t i n e s TRANSL a n d PROJ 1 .

S u b r o u t in e PROJ 2 ; T h is r o u t i n e p e r f o r m s t h e f i n a l s t e p i n

p e r s p e c t i v e p r o j e c t i o n . I t " b a c k - p r o j e e t s " t h e H-D o r d i n a r y

c o o r d i n a t e s fo rm e d by c a l l i n g PROJ 1 a n d TRANSF i n t o th e

p i c t u r e p l a n e .

r BEGIN ^

I —

CALL VALUE

CALL TIANBL

CALL PftOJl

CALL TRAHF

fcg-— — i i

CALL PBOJ 2

CALL SCALE

/ \I PLOT CURVE 1

Figure B -l Projection Frograa Flow Chert

17H

S u b r o u t in e SCALE: T h is s u b r o u t i n e s c a l e s t h e 2 - d im e n s io n a l

d raw a r r a y s o t h a t I t w i l l f i t i n t o t h e " v ie w in g w in d o w .”

S u b r o u t in e PLOT: A C aloom p s u b r o u t i n e t h a t m oves t h e p l o t t e r

p e n fro m p o i n t 1 t o p o i n t 2 w i th a p e n c o n t r o l p a r a m e te r

d e te r m i n i n g w h e th e r t h e p e n i s down a n d w r i t i n g o r u p .

A 8 a m p le s e t o f t h e d a t a c a r d s i s g iv e n i n A p p e n d ix C.

The f o l lo w in g f i g u r e s a r e e x a m p le s o f f l i g h t p a t h s n o t show n

i n S e c t i o n 3 . 5 .

500

1000

100

F n ja ctlo n onto X-Y PlMK 1000

Figure B -2 Perspective Projection o f F light Path Ho. 101

175

500

-X 1000

100‘-

1000

2000*

X « 3000Pcojeeckn onto X-Y Plane

Figure B-3 Perspective Projection o f F lig i t Path Ho. 103

17«

10000

5000

20001000


Figure B*4 Perspective Projection of F light Path No. 2

177

10000 '

5000' -

-X1000' --Y

2000 'P ro jec tio n onto X-Y Plane

Figure B-5 Perspective Projection of F light Path No. 3

178

T Z

10000

5000

1000


Figure B-6 Perspective Projection of F light Path Mo. 4

10000

50001 -X

4000'

• 20001

2000Projection ontoX-T Plane

Figure B~7 Perspective Projection of F ligh t Path Ho. 6

10000

5000

-4000

2000

X * 2000'P ro je c tio n onto X-T Plane

Figure B-8 Perspective Projection of F light Path No. 7

10000 '

5000*

2000*

-Y


Figure B-9 Perspective Projection of F light Path Ho. 8

182

10000'

5000

/ T:


Figure B“10 Perspective Projection of Flight Path Ho. 9

183

APPENDIX C

COMPUTER LISTING OF PERSPECTIVE

PROJECTION PROGRAM

/ / J O B O l JOB ( 1 0 0 ' 1 3 0 3 ' S V1 0 ) ' *40197 PERKINS**MS6LEVEL«1 / / S T E P I EXEC FORTGCLG«REGION«GO*10QK'TIM£*5 //FORT*SYSIN DO «

DIMENSION DRI( 9 0 0 « 4 l ' 1 C 1 ( 5 0 0 ) 'B U F F I10001 COMMON /B/MESS/C/XMIN'XMAX'YMIN'TMAX'ZMIN'ZMAX COMMON /PARM/1AXES16)*YTRAN. I T I C ( 3 ) . WINDOM( 2 )MESS * 1N1DIM * 500CALL PLOTS(BUFF'IQOO)

CC READ DATA' GENERATE AXES AND PEN CONTROLSC

CALL V A L U E (D R l ' IC l 'N l 'N lD IM )CC MOVES OBSERVER POINT PARALLEL TO V AXISC

CALL TRANSL(-YTRAN'-YTRAN*0*>CC CALCULATE DISTANCE FROM PICTURE PLANE TO OBSERVERC

Z I P « -(XMAX-XMtNI/(2**TAN(30*/57*29S7B>>M R IT E I6 '2 2 ) XMAX.XMIN

22 FORMAT(1X'2F10*2)CALL P R O J1 (Z IP )

CC PERFORMS MATRIX TRANSFERMATIONC

CALL TRANSF( DR1 • IC1»N1'N1DIM.DR1• I C 1 , N 1 ' N I O I M ' l ) CALL P R O J 2 (D R l 'N l 'N lD IM )CALL SCALE(DRl 'N l 'NIDIM)DO 100 1 * 1 *N1M R I T E ( 6 ' 2 0 0 ) O R 1 ( I ' l > ' D R 1 ( I ' 2 )

20C FORMAT!1X.2F20*3)100 CALL P L 0 T ( D R l ( l ' l ) ' D R l ( I ' 2 ) ' I C l ( i n

CALL P L O T ! 0 * .0 * *999)STOPEND

BLOCKDATACOMMON/ APT/T<4 . 4 > *TCOM(4 . 4 )DATAT/1 * . 4 * 0 * 0 . 1 * « 4 * 0 * 0 * W . 4 * 0 * 0 . 1 . /DATATCOM/1* . 4 * 0 * • 1 * . 4 * 0 * . 1 • . 4 * 0 * • 1 • /END

SUBROUTINE PROJ1 (E )COMMON/APT/T<4 . 4 ) .TCON < 4 .4 ) /B/MESS IF tM E 5 S * £ 0 * i )M R IT E (6 * 1 IE TC0M<3.3)*0*T C 0M (3*4)*1* /E CALL COMBIF (M E S S*EQ *I)«R ITE <6.2 )RETURN

1 FORMAT ( / / • ENTER P R O J I */ 3 X . 'P O S IT I O N CN NEC* Z AXIS I S * . 1F9*0«* FROM THE X*V PLANE*)

2 FORMAT ( • EXIT PROJI*)ENO

SUBROUTINE PROJ2 IDRN.N.NDIM)DIMENSION DRN (NDtM.4)COMMON/8/MESS IF(MESS*EQ*1) WRITE(6*1>DO 20 1 * 1 «NI F ( D R N ( I .4 ) * L T * * 0 0 1 ) D R N (I* 4 )* * 0 0 1 DO 20 J * 1 . 4

20 O R N I I . J ) * D R N ( I • J ) / D R N ( I *4)IF (M E S S *E O *l)«R IT E I6*2)RETURN

1 FORMAT!//* ENTER PROJ2• )2 FORMAT(* EXIT P R 0 J2 * I

END

186

SUBROUTINE TRANSF(DRt• IC1*Nl*N1D1N*DR2*IC2*N2*N2DIM*KEEP)OIMENSION SU8(4 ) *ORl(N1DIM*4)*DR2tN2DIM*4) • I C 1 ! N1DIM). IC2!N2DIM> COMMON/APT/T( 4 « 4 ) * TCOM( 4 * 4 ) /B/MESS IFIMESS*EQ*I) WRITE( 6 * 1 I N2*N1DO20K*l*NlIC2(K)*IC1<K>0 0 1 0 J * 1 * 4 S U B ! J ) *0*0 0 0 1 0 1 * 1 * 4

10 S U B C J ) * S U B t J ) 4 D R l f K * I ) « T ( I * J )0 0 6 0 L * 1*4

60 DR2fK*t.)*SUB<L)20 CONTINUE

IF!MESS*EQ* 1 )WRITE(6*2)IFIKEEP*EQ*11 6 0 TO SO 0 0 3 0 1 * 1 * 4 D 0 3 0 J* 1 * 4

30 T( I* J 1*0*00 0 4 0 1 * 1 * 4

40 T ( 1 * 1 1*1*IF(MESS*EO*11MRITEC6*31 GO TO 70

50 IF(MESS*EQ*1 ) MR1TEC6*4170 IFfMESS*EO*l) WRITEC6*S)

RETURN1 FORMAT!//* ENTER TRANSF*12 FORMAT(3X*'OBJECT HAS BEEN TRANSFORMED'>3 FORMAT!3X*'TRANSFORMATION MATRIX HAS BEEN RESET TO IDENTITY*)4 FO RMAT!3X* 'TRANSFORMATIO N MATRIX HAS BEEN KEPT FOR R E U S E * )5 FORMAT! • E X I T TR A N S F*)

END

SUBROUTINE TRANSL ( A A . B B . C C )C 0 M M 0 N / A P T / T ( 4 * 4 ) * T C O M ( 4 * 4 l / B / M E S S 1 F C M £ 5 S « E Q « I 1 M R I T E I 6 * I )11*1 1 2 * 2 1 3 * 3T C Q M ( 4 « 1 )*AA T C O M < 4 * 2 )* B B T C 0 M ( 4 . 3 ) = C C CALL COMBI F ( M £ S S » £ Q « 1 ) V R I T E | 6 * 2 ) A A * X 1 * B B * I 2 * C C * I 3 RETURNFORMAT< / / • ENTER TRANSL*)F O R M A T « 3 X . 'D E F I N E D TRANSLATION O F * .

1 3 ( T 2 6 * F 9 « 2 « * INCHES PARALLEL TO THE N O « * * I 2 * * A X I S ' / ) .2 • E X I T TRANSL*)

END

SUBROUTINE COMBCOMMON✓APT/T< 4 • 4 > • TCOMI4 • 4 )DIM ENSIO N S U B « 4 )DO 3 0 K * 1 . 4 OO 1 0 J * 1 * 4 S U B C J ) * C *DO 1 0 1 * 1 * 4S U B t J ) * S U B ( J ) + T ( K * I > * T C O M < I , J >DO 2 0 L * l * 4 T ( K * L ) * S U B C L >CONTINUE DO 4 0 1 * 1 * 4 DO 4 0 .1*1 • 4 T C O M ( I « J ) * 0 *0 0 5 0 1 * 1 * 4 TCOM (1 * 1 1 * 1 *RETURNEND

SUBROUTINE S C A L E ( O R I . N 1 . N 1 D I M )DIMENSION D R M N 1 0 I M . 4 )COMMON / B / M E S S / C / X M 1 N . X M A X . YMIN.YMAX.ZMIN.ZMAX COMMON / P A R M / I A X E S I 6 ) • Y T R A N » I T I C ( 3 ) . W I N D 0 W ( 2 )I F ! M E S S « E Q * I ) W R 1 T E I 6 * 1 0 1 )

101 FORMAT( 1 0 X . *ENTER SUBROUTINE S C A L E * )X M IN*D R1( 1 * 1 >XMAXsXMlN YMIN=DR1 ( 1 . 2 )YMAXsYMIN OO 1 0 0 I * 2 . N l XX * D R U M )YY * 0 R 1 < 1 . 2 )I F ( XX*LT*XMIN) XMIN-XX IF(XX*GT*XMAX) XMAX*XX IF!YY*LT«YM1N) YMIN*YV IF!YY«GT«YMAX) YMAX*YV

100 CONTINUEI F ( MESStEO*I) WRITE( 6 * 1 2 5 ) XMIN.XMAX.YMIN.YMAX

125 FORMAT(5X» *XMIN * • . E l 2 . A . • XMAX * * . E l 2*4* * YMIN ■ * . E 1 2 * 4 .1 *YMAX * * . E 1 2 « 4 )XDIF * XMAX - XMIN YDIF * YMAX - YMIN SFX * WINDOW!11/XDIF SFY a WINDOW!2 l /YOIF IFISFX«L£*SFY) GOTO 200 SF * SFY GOTO 300

200 SF * SFX 300 OO 500 I = 1.N1

OR 1 ( 1 * 1 ) = D R l ( I « t ) * S F S 0 9 O R K I . 2 ) * 0 R 1 ( I . 2 ) * S F

XMIN 3 XMIN • SF YMIN 3 YMIN * S F CALL P L 0 T ( - X M I N . - Y M l N . - 3 )I F ( M E S S . E Q . 1 ) W R I T E ! 6 . 1 0 3 ) ! J . D R 1 C J . I ) . O R I ! J . 2 )* O R l ! J . 3 ) •

189

1 J * 1 * N 1 >103 FORMAT!IX.#L 1ST ORI ARRAY**5X.* Ml • *3X* • OR1(Ml• 1 ) • . 6 X .

1 * O R l |N l « 2 ) * « 6 X * * O R l(N l* 3 1 " *SX*"IC1CN1) " / ! 2 0 X * I 5 * 3 E 1 S « S ) 1 IF<ME5S*EQ*1) VRITE<6*102)

1 0 2 FO RM AT!10X ** E X I T SUBROUTINE S C A L E " 1 RETURN ENO

SUBROUTINE V A L U E C D R 1 * I C 1 * N 1 . N I O I M l DIMENSION D R 1 < N 1 D I M * 4 ) * I C 1 I N 1 D I M )COMMON / B / M E S S / C / X M I N • XMAX *VN1N* YMAX *ZMIN»ZMAX COMMON / P A R M / I A X E S 1 6 1 • V T R A N * I T I C I 3 ) » M I N O O M ( 2 ) I F ! M E S S * E O « I ) V R I T E C 6 * 1 0 1 1

101 FORM AT!1 OX*"ENTER SUBROUTINE VALUE")N l * lI C 1 I N 1 ) * 3R E A D ! 5 * * ) « I N D O « ! l ) * t t l N O O V ! 2 )OO 1 0 1 * 1 * 3

10 R E A D I S * * ) I A X E S ! I ) * I T I C ! 1 )DO 2 0 I * 4 * 6

2 0 R E A D !5 * 4 ) I A X E S I I )1 0 0 R E A D ! 5 * 4 * E N D * 5 0 0 ) O R I ( N 1 * 3 ) * D R 1 ! N 1 * 1 ) * D R 1 I N I * 2 )

1 F ! N 1 « G T « N 1 D I N > GOTO 9 9 8 I F ! N 1 * E 0 * 1 ) GOTO I S O I C 1 I N 1 ) * 2 XX* D R K N 1 . 1 )YY= 0 R 1 ! N 1 * 2 )Z Z * 0 R K N 1 . 3 )I F ( X X * L T « X M I N ) XMIN * XX IF !X X*GT«XM AX) XMAX * XX I F ! YY*LT«YMIN) YMIN * YY IF!YY«GT*YMAX) YMAX * YY

190

150

200

500

501

103

650700750

I F ( Z Z * L . T * 2 M I N ) ZMIN * ZZIF(ZZ«GT«ZMAX) ZNAX « ZZGOTO 2 0 0XMIN * O R I ( N 1 • 1 1XMAX x XMINYMIN * O R K N 1 . 2 )YMAX * YMIN ZMIN x D R H N 1 . 3 )ZMAX * ZMIN N1 a N1 ♦ 1 GO TO 1 0 0 NSTOP * N1 - 1 NN x 2*N S T O P N * N10 0 5 0 1 N 1 x N .N N I C l f N l ) x I C 1 I N I —N S T O P 1 OR 1 ( N l • 1 ) x D R M N 1 - N S T 0 P . 1 )O R K N 1 . 2 ) x 0«D R K N 1 . 3 ) x ORI ( N l - N S T O P * 3 1 N l x NN ♦ II F ( M E S S *E Q «11 I I R I T E ( 6 , 1 0 3 > t J . D R I C J , n , D R i ( J , 2 ) . D R 1 C J , 3 > . I C I ( J > ,

1 J = 1 . N S T O P )FORMAT( I X , ' L I S T OR I A R R A Y * , 5 X , * N l • , 3 X , * D R 1 ( N l • 1 ) • , 6 X ,

1 * D R 1 ( N 1 , 2 ) * . 6 X , * 0 R 1 ( N I , 3 ) * , 5 X . * I C 1 ( N 1 ) • / ( 2 0 X , I 5 , 3 £ 1 5 * 5 » 1 1 0 1 ) D R 1 ( N 1 , 1 ) = 0 *D R K N l , 2 ) x 0 »D R 1 ( N 1 , 3 ) » 0 .I C 1 ( N l ) * 3 XD1F x XMAX - X M IN Y O IF x YMAX - YMIN Z D I F x ZMAX - ZMINT I C L x A M A X 1 ( X D I F , V O I F , Z D l F ) * 0 * 0 1 I F ( I A X E S f 1 ) « E O « 1 ) GOTO 1 0 0 0 I F I I A X E S ( 2 ) « E Q « 1 ) GOTO 2 0 0 0 I F ( I A X E S ( 3 ) * E Q « 11 GOTO 3 0 0 0 I F I I A X E S C 4 ) * £ Q * 1 ) GOTO 4 0 0 0

191

800 I F ( IAXESf 5 1*EQ»1> GOTO 5000 850 I F ( I AXES( 6 )«EQ«1) GOTO 6 0 0 0

GOTO 9600 1003 IF ( XMAX«LE«0*)GOTO 650

XLEN * 0*1100 Nl * Nl ♦ 1

1F(N1*GT«N1DIM) GOTO 998 XLEN = XLEN ♦ ITICC 1) O R l ( N l . l ) * XLEN DR 1 ( N1 • 2 ) > 0*O R K N 1 .3 ) a 0*IC1CN1) = 2 Nl > Nl ♦ IOR 1I N I • 1 ) > ORI(Nl —1*11 DR1CN1*2) * 0*D R K N 1 .3 ) a —TICL 1C1CN11 = 2 Nl * N l ♦ 1 DR1(N1«1>3DR1(N1-1«1>OR1CN1«2 >3 0*O R I ( N l * 3 ) 3 0*IC1(N1>* 3IF(XLEN«LT*XMAX> GOTO1100 Nl * Nl -f 1 D R M N 1 .1 )= 0 *O R l(N l ,2 > x O *ORI (N l *3 >3Q«I C K N l > 33 GOTO 650

2000 IF lY MAX.LE*^ I GO TO 700 YLEN 3 0*

1200 Nl 3N1+ 1IF(N1«GT*N1DIM> GOTO 998 YLEN 3 YLEN ♦ IT IC (2> D R l ( N l . l ) 3 0*O R 1IN 1 .2 ) x YLEN

toK)

D R U N 1 . 3 ) * 0*I C 1 ( N l ) * Z

N l *N1 +1D R 1 ( N 1 , 1 ) * - T I C LOR 1 ( N 1 * 2 ) * OR 1 ( N l — 1 *2 1D R K N 1 . 3 ) x C .I C 1 ( N l 1 s 2 Nl *N1 +1 0 R 1 (N 1 * 1 )« 0*D R K N 1 . 2 ) * O R I ( N l —1 * 2 )OR 1 (N l *9 ) a* 0*I C K N l ) = 3I F ( YLEN«LT«YMAX) GOTO 1200 Nl * Nl ♦ I DR1<N1'1>*0»O R l( N l .2 ) x O *DR1(N1«3)»0*I C K N l 1*3 GOTO TOO

3000 1F(ZMAX*LE«0«) GOTO 7 5 0 ZLEN » 0»

1300 Nl * Nl ♦ 1IF (N U G T .N ID IM ) GOTO 998 ZLEN * ZLEN ♦ ITICC3)O R I ( N l • 1 1 x 0*0 R 1 ( N 1 « 2 > * 0 *D R K N 1 . 3 ) * ZLEN I C l ( N l ) = 2 N 1*N 1 ♦ 1 OR 1( N 1 • 1 ) = T IC L OR K N 1 »2 ) * 0 .OR 1 ( N 1 • 3 ) * O R I ( N l - t * 3 ) I C K N l ) x 2 N l s N l H OR 1 ( N1 , 1 ) s 0*OR 1 ( N 1 * 2 ) s 0«

to

OR 1 ( N1 • 3 ) = D R K N l - 1 , 3 ) I C l ( N l ) = 3IF(ZLEN«LT«ZMAX) GOTO 1300 Nl * Nl + 1 DR1 ( N l • 1 ) * 0 *OR 1( N l * 2 ) 3 0«DRKN1«3>»0*I C K N l )=3 GOTO 75C

4000 IF<XMIN.GE*0*» GOTO 800 XLEN * 0*

1400 Nl * Nl + 1IF (N U G T .N ID IM ) GOTO 9 9 6 XLEN * XLEN - I T l C ( l )O R I ( N l *1) * XLEN 0 R K N 1 . 2 ) * 0 'O R K N 1 .3 ) » 0*I C K N l 1* 2Nl s Nl 4 1DR1( N l • 1 )*DR1(N1—1*1)O R I ( N l * 2 ) * 0*DRKN1 . 3 ) = - T I C L I C K N l ) * 2 Nl * Nl ♦ 10 R K N 1 . 1 ) * OR 1 (Nl —1 *1 ) O R K N 1 .2 ) * 0*D R K N 1 .3 ) = 0*I C K N l ) s 3IF<XLEN«GT«XM1N) GOTO 1400 Nl * Nl M 0 R K N 1 .1 )>0*DR1IN1 «2)*0*O R 1 ( N 1 , 3 ) = 0 *I C K N l >=3 GOTO 800

5000 1F(YMIN*GE*0*) GOTO 850

194

YLENsO*1 5 0 0 Nl * N l ♦ 1

! F ( N U G T « N 1 D I M ) GOTO 9 9 6 YLEN * YUEN - I T I C < 2 >Oft K N 1 • 1 ) * 0«Oft1 ( N 1 • 2 ) * YLEN Oft 1 ( N 1• 3 ) * 0«I C K N l ) = 2 N l * N l H D R l ( N l . l ) * -T IC L 0 R 1 I N 1 . 2 ) *D R1 ( N l —1 « 2 I O f t l ( N 1 . 3 ) » 0*I C l ( N l ) * 2 N l « N l 4 |OR K N 1 • 1 ) x 0*DR1(N1*2) * O R I (N l—1*2) O R I ( N l . 3 ) * 0 *I C K N l ) * 3IF(YLEN*GT«YMIN) G 0 T 0 1 5 0 0 N l * N l ♦ 1 D f t 1 ( N 1 » 1 ) * 0*D f t l ( N l » 2 ) x 0*D f t K N1 » 3 ) * 0* l C l ( N l ) = 3 GOTO 6 5 0

6 0 0 0 I F ( Z M I N * G E * 0 ) GOTO 9 0 0 0 ZLEN x 0*

1 6 0 0 Nl * N l 4 1I F ( N I . G T . N I O I M ) GOTO 9 9 8 ZLEN * ZLEN - I T I C ( 3 t D R K N 1 . 1 ) x 0*0 R K N 1 . 2 1 « 0*O R K N 1 .3 1 x ZLEN I C K N l ) x 2 Nl s N l 4 1 D R K N t . l ) x TI CL

MUl

I

0 R 1 1 N 1 . 2 ) = 0*DR l (Nl t3 ) =0R1IN| -1 .3>I C K N l ) * 2 N l * N l U D R I ( N l . l ) * 0 .0 R K N 1 .2 ) = 0*DR1(N1,3) * O R l l N l - 1 . 3 )I C K N l ) * 3I F ( Z L E N « G T « Z M I N ) GOTO 1 6 0 0 N l * N l 4 1 O R K N 1 * 1 ) * 0 *0 R K N 1 . 2 ) * 0 «D R K N 1 . 3 ) » 0*I C K N l ) = 3

9 0 0 0 DO 9 5 0 0 I = 1 >N1 9 5 0 0 O R K U 4 1 * 1 * 0

YTRAN * YMAX ♦ 1 * 5 GOTO 9 9 9

9 9 0 M R 1 T E I 6 . 1 0 4 )MESS * 1

1 0 4 F O R M A T ! I X * * * * * * * * ERROR * * * * « « N l HAS EXCEEDED N 1 D I M * )9 9 9 NSTR s NSTOP ♦ 1

I F ( M E S S * E O « l ) V R I T E I 6 . 1 0 3 M J . D R 1 I J . D . O R 1 ! J . 2 > . D R l f J . 3 ) . I C K J ) • 1J s N S T R # N 1 )

I F ! M E S S * E Q « 1 > M R I T E I 6 * 1 0 2 )1 0 2 F O R M A T ! 1 0 X .« E X I T SUBROUTINE VALUE*)

RETURNEND

196

/ / G O . P D D UNIT=TAPE«LA8EL»(»NU)•VOLsSER»TPLOT //G O ,S Y S IN OO *

9» 6*I 1000 1 500a 500 1 1000 01 500-6 1 1 1 * 4 5 -5 7 4 0 * 9 4 -5 3 7 1 * 0 0 -5 0 0 1 * 3 3 -4 6 3 1 * 9 2 -4 2 6 2 * 7 6 -3 8 9 3 * 8 8 -3 5 2 6 * 0 7 -3 1 6 0 * 7 0 -2 7 9 8 * 2 2 -2 4 3 9 * 2 5 -2 0 8 4 * 5 6 - 1 7 3 4 * 9 3 -1 3 9 1 * 1 0 -1 0 5 4 * 9 5

-7 3 2 * 3 2 - 4 2 6 * 0 2 -1 3 7 * 9 7

1 2 9*99 3 7 6 * 1 7 599*01 79 7 * 1 0

10053*399994*6199 36*049877*5198 1 9 * 0 297 6 0 * 5 69702*0196 3 9 * 3 595 64*4694 74*5293 6 9 * 0 892 5 0 * 7 491 1 7 * 3 689 70*0288 06*828 6 1 9 * 2 08 4 0 6 * 6 46 1 7 0 * 5 37912*417 6 3 3 * 9 57 3 3 6 * 9 77023*39

14 35*7814 67*471 4 99*921532*391564*871597*351 6 2 9 * 9 01 6 63*121692*941710*961715*451 7 0 6 * 3 91683*791 6 4 7 * 6 51599*521549*011 4 98*661448*4713 98*451346*5812 98*671249*32

197

APPENDIX D

THE GUN MODEL

'E L I. D e r i v a t i o n o f S c a l e d Gun E q u a t io n s

A s c a l e d a n a lo g f lo w d ia g ra m o f t h e a z im u th a n g u la r

d i s p l a c e m e n t f o r t h e a n t i - a i r c r a f t gun i s shown i n F ig u r e

D - l , w i t h a l i s t o f t h e g a in f a c t o r s g iv e n i n T a b le D - l .

N o te t h a t t h e g a in f a c t o r s f o r t h e a z im u th m o d el a r e d i f f e r

e n t f ro m t h o s e o f t h e e l e v a t i o n m o d e l, b e c a u s e o f t h e

d i f f e r e n c e s i n l i m i t s p l a c e d on t h e a n g u l a r a c c e l e r a t i o n a n d

t h e a n g u l a r v e l o c i t y . The s c a l i n g e q u a t i o n s a r e a s f o l lo w s :

F o r t h e a z im u th p l a n e :

L im i t Summer #1

# •( D - l )

w h e re l i m i t o n (P = * • 3H907 = t #23271 • 5

(D -2 )

I n t e g r a t o r #2

CD-3)

198

LIM

LSW

Figure D*1 Scaled Gun Model

199

200

TABLE D - l

SCALING TABLE OF THE GUN MODEL (AZIMUTH)

P ro b le m E s t im a te d C o m p u te rV a r ia b l eV a r ia b l e D e s c r i p t i o n Maximum

T Q (t - T ) F o o t p o u n d s 20 -T Q ( t)

tb R a d ia n 3 .1 4 2 t* - T O

* • d > L ± m R a d ia n /s e c o n d 1 .0 6 5 - 0 Lim^ " l.f fB S

0 Lim R a d ia n /s e c o n d 2 1 .5 Lim175”

201I n t e g r a t o r #3

[ T . t k r I s - * 1 5 9 2 ^ - ^ 5 - m31 (D -^ )“cPF

F o r t h e e l e v a t i o n p l a n e :

L im i t Summer #1

2 . 6 87 [ z ^ C t l ] . 2 . 267 [ ^ L i » l (D . 5 )

w h e re l i m i t on & = - * -^ ^ g '7 = " .2 3 2 7 (D -6 )


d~3F [ 4 m Q = - 1 -1,03f e i E] (D- 7)


(D -8 )

T he p o t s e t t i n g s f o r a l l t h r e e gun m o d e ls a r e g iv e n i n

T a b le D -3 .

D -2 . D ev e lo p m en t o f L o g ic S i g n a l f o r E l e c t r o n i c S w itc h

The l o g i c s i g n a l , LSW, show n i n F ig u r e D - l an d i n

F ig u r e D-2 i s u s e d t o c o n t r o l t h e i n p u t t o t h e Lim i n t e g r a t o r ,

s o t h a t t h e o u t p u t Lim i® l i m i t e d . I n i t i a l l y , a l i m i t sum m er

was e m p lo y e d i n t h e gun s i m u l a t i o n t o l i m i t L im i n s t e a d o f t h e

LowerLimit

+ ref

LL LSW

-re f

UpperLimit

Figure D-2 C ircuit D lagr« - Generation of the Logic Control Signal (LSW)

202

20 3

TABLE D-2

SCALING TABLE OF THE GUN MODEL (ELEVATION)

P ro b le m E s t im a te d C o m p u te rV a r ia b l e D e s c r i p t i o n Maximum V a r ia b l e

T Q (t - ) F o o t p o u n d s 20 - T 0 ( t )

&

& L im

■ •© L im

R a d ia n

R a d ia n /s e c o n d

2R a d i a n / s e c o n d

3.1H 2

.5

1 .5 &LimITT

e l e c t r o n i c s w i t c h . B u t t h i s a n a l o g m o de l a l l o w e d I n t e g r a t o r

#2 t o d e v e l o p v o l t a g e h i g h e r t h a n t h e d e s i r e d l i m i t e d v a l u e s

A l o g i c c i r c u i t d i a g r a m o f s i g n a l LSW i s shown i n F i g u r e D-2

a l o n g w i t h a d e s c r i p t i o n o f t h e l o g i c v a r i a b l e s u s e d in

T a b l e D-*t. A t r u t h t a b l e f o r t h e s w i t c h i n g s i g n a l c i r c u i t

i s g i v e n i n T a b le D -5 .

205

TABLE D-3

POT SETTINGS FOR THE ANALOG MODEL*

P o t Number AZIMUTH PLANE

S tu d y #1 S tu d y #2 S tu d y #3

1 .2 2 6 7 . 8345 .5 4 0 02 .1 0 6 7 .6290 .3 3 8 03 .3 0 0 0 .3000 .3 0 0 04 .1 5 9 2 .15 92 .1 5 9 2

P o t Number ELEVATION PLANE

S tu d y #1 S tu d y #2 S tu d y #3

1 .2 6 6 7 .4 3 2 0 .8 1 6 02 .2 2 6 7 .5800 .7 0 6 03 .1 4 0 3 .1 4 0 3 .1 4 0 34 . 3390 . 3390 .3390

NOTE T h e s e p o t n u m bers c o r r e s p o n d t o t h e p o t s i n F i g u r e D - l , n o t t o t h e p o t s i n t h e H y b r id s i m u l a t i o n d i a g r a m , F i g u r e 4 - 2 .

206

TABLE D-4

DESCRIPTION OF LOGIC SIGNAL

S i g n a l S t a t e D e s c r i p t i o n

L I 1 u p p e r l i m i t0 Lim u p p e r l i m i t

L2 1 Lj_m l o w e r l i m i t0 ^ im l o w e r l i m i t

L3 1 A n g u l a r a c c e l e r a t i o n n e g a t i v e0 A n g u l a r a c c e l e r a t i o n p o s i t i v e

L4 1 S y s te m m ov ing away from u p p e r l i m i t0 S y s te m a g a i n s t u p p e r l i m i t

L5 1 S y s te m m o v ing away f ro m lo w e r l i m i t0 S y s te m a g a i n s t l o w e r l i m i t

L6 1 S y s te m m ov ing away f ro m l i m i t s0 S y s te m a g a i n s t a l i m i t

L6H H o ld s i g n a l f o r f i x e d am ount o f t im ewhen L6 g o e s h i g h

LL 1 L i m i n s i d e c o n s t r a i n t0 Li.m a c o n s t r a i n t

LSW 1 E l e c t r o n i c s w i t c h c l o s e d0 E l e c t r o n i c s w i t c h o p e n

207

TABLE D-5

TRUTH TABLE FOR LOGIC SWITCHING SIGNAL*

L o g icS t a t e s L o g ic V a i r a b l e L o g ic F u n c t i o n s

L I L2 L3 L4 L5 L6 LL LSW

1 0 0 0 0 1 1 0 1

2 0 0 1 0 0 0 0 0

3 0 1 0 0 0 0 1 1

4 0 1 1 0 0 0 1 1

5 1 0 0 w i l l n o t o c c u rb o t h Lim v i o l a t e d

6 1 0 1

7 1 1 0 0 0 0 0 0

8 1 1 1 1 0 1 0 1

&N o te ( s e e F ig u r e D -2 )

APPENDIX E

PAPER TAPE CONVERSION PROGRAM

The j o b o f r e p r o d u c i n g t h e h y b r i d d a t a on p u n c h e d c a r d s

i s t h e t o p i c o f t h i s a p p e n d i x . S i n c e t h e h y b r i d c o m p u te r

o n l y h a s p a p e r t a p e a n d l i n e p r i n t e r o u t p u t d e v i c e s a n d t h e

U n i v e r s i t y C o m p u te r C e n t e r c a n n o t r e a d p a p e r t a p e , i t becam e

n e c e s s a r y t o u s e an IBM-49 p a p e r t a p e t o c a r d p u n c h f o r t h i s

d a t a t o c a r d p r o b le m . The p ro g ra m t o f o l l o w e n a b l e s t h e

h y b r i d c o m p u te r s p a p e r t a p e p u n c h , w h ic h n o r m a l l y p r o d u c e s 8

c h a n n e l EBCDIC t a p e , t o p r o d u c e an 8 c h a n n e l e v e n - p a r i t y BCD

t a p e i n an on l i n e m a n n e r . T h is BCD t a p e c an t h e n b e p r o

c e s s e d w i t h an IBM -49.

The h y b r i d d a t a f ro m a r u n i s s t o r e d i n t h e 8 by 250

w r i t e a r r a y , NOUT, a s d e s c r i b e d i n C h a p t e r 4 . T h is w r i t e

a r r a y , NOUT, i s t r a n s f e r r e d t o s u b r o u t i n e PPTIBM, w h ic h p e r

fo rm s t h e p a p e r t a p e p r o d u c t i o n . D a ta f ro m two t im e i n c r e

m e n t s , a t o t a l o f 16 v a l u e s , i s r e a r r a n g e d i n t o t h e 1X16

a r r a y , NDATA; s t a r t i n g w i t h t h e v a l u e NOUT ( 1 , 1 ) s t o r e d i n

NDATA ( 1 ) and f i n i s h i n g w i t h t h e v a l u e o f NOUT ( 8 , 1 + 1 ) b e i n g

s t o r e d i n NDATA ( 1 6 ) . T h i s d a t a w i l l e v e n t u a l l y b e p u n c h e d

on a s i n g l e 80 c h a r a c t e r c o m p u te r c a r d w i t h t h e f o r m a t o f

208

1 615 . T hen t h e n u m b ers s t o r e d i n NDATA a r e b r o k e n down i n t o

i n d i v i d u a l c h a r a c t e r s ( s i g n s , n u m e r a l s , a n d b l a n k s ) , c o n v e r t e d

from EBCDIC t o BCD by m eans o f a t a b l e l o o k - u p , a n d s t o r e d

i n t o t h e 1 x 80 a r r a y , NDUM. F i n a l l y , t h e a r r a y , NDUM, i s

t r a n s f e r r e d t o t h e p a p e r t a p e p u n c h f o r o u t p u t t i n g . T h i s i s

a c c o m p l i s h e d by u s i n g a s s e m b ly l a n g u a g e s t a t e m e n t s . I t i s

i n t e r e s t i n g t o p o i n t o u t t h a t t h i s i s e a s y t o do w i t h t h e

h y b r i d s y s te m b e c a u s e t h e c o m p i l e r a l l o w s t h e m ix in g o f

b o th a s s e m b ly l a n g u a g e a n d / o r F o r t r a n s t a t e m e n t s i n t h e same

p ro g ra m .

A f lo w d i a g r a m o f t h e d a t a c o n v e r s i o n p r o c e s s i s g i v e n

i n F i g u r e E - l , w h i l e a l i s t i n g o f s u b r o u t i n e PPTIBM c an b e

fo u n d i n A p p e n d ix F .

APPENDIX F

COMPUTER LISTING OF HYBRID PROGRAM

u u

u o

JOB-o r t r a n h g o .l s .s *

DIMENSION PLANE!J>,GUN(3>,NUM(SC>, VALUE!5C»•ANGC2> .IT IME< 2 0 > ,CV( 9) .QEPI2) . SCALE ( 2 ) . TCMOSS ( 10 > • JOUTIB . 2C *: . 4 ) • 2K0UT( 9 .20 0 «LUUT(H* 20'*') ,M0UT(8.20O ,NOUT(8,20t >ECU I VALENCE (JOLT<1, 1 * I).KC1UT<1*1>),(JOUT(1 ,1,2),LOUT! 1 . 1 ) 1 I t(JOLT( 1•I,3)•MQUT(1*1 ))«(JOUT(1,1,4) ,NOUT< 1,1))DIMENSION nPATHI1G > *NCR0SS(1.>,X(5.1j0),Y(5.IO0>,Z(5,10Q>

PLANE (3) TARGET LOCATION VECTOR <FTJ 4GUN I 3) TARGET IN GUN SYSTEM (FT) *AZ IM AZIMUTH MISSALIGNMENT <RAD» 4ELt V ELEVATION MISSALIGNMENT (RAO) 4NUM( SC) POT NUMBER 4VALUE! S O POT VALUE 4THOA ELEVATION POSITION OF GUN 4PHI AZIMUTH POSITION OF GUN 4NP NUMBER OF POTS TO BE SET 4

SET UP ANALOG COMPUTER

23 READ!5*2) NPREAD I 5 • 2 ) ( NUM ( I ) , VALUE( 1 ) * 1= l .NP)4RITEIE.J>DO 6 I - l .NPCALL SPOTINUMd), VALUE ( I ) )CALL SACO( *P • ,NUM( I )}

c al l O W i) {RVALUE )WRITE (6. 4) \U*(I)» VALUE< I > .RVALUE

5 CONTINUECALL5AV01 'PC1 )AL T = 1 v .rad = ?• i a i s o / i R:•CVLPu = I : l BC ■»I SAVi_ = 1TERM = 3 9a 75PR TINT = ,25NPTS = T CRM/PR T INT+1*TSTOP = TERM ♦ PRTINT/2#C R£AD(5,410)NPATH,NCROSSWRITt< 6 , 4 1 1 ) (NPATHCI) .NCROSSII )* 1 = 1 * 10)D0413I=1,ICCTIM£(2*1— 1) = FLOAT( I )/2 - o

413 CONTINUE D04C3K=1,5READ! 5 * 4 0 1 ) ( Y ( K , n , Z ( K . I ) » X ( K , n , I = l , l C C )KK = KWR IT£(c * 1C 3) KKMRtTE(6.1G2HTIM£(2*I-l).X(K.I),Y<K,I),Z(K,I),|=l,lCC>Mss 1

004141=2,100IF ( Z < K » I )«LT,Z(K.M)) W = I

414 CONTINUEDZ=Z( K *M)—50C«DX=X(K,M)-150C.DY=Y(K,P)DO4151=1,100 X(K , I >=X(K,I )-OX Y(K, I > =Y ( K , I )-OY ZCK,I>=ZCK,I)-DZ

415 CONTINUE«RITEI6,103)KK* R1 T£ < 6 . 1 0 2 ) ( T 1 R E { 2 * 1 - 1 > , X ( K , I ) , Y ( K . l ) , Z ( K . I ) , 1 = 1 , 10C >

ft'"! CONTINUErcRossti)= 2 ■* size=s:SCALcIl) =*3 SC ALc(2) = *4 00412NRU 4 = 1,1 t K = N P A T X 4RUN )<x =n c r q s s <n r u n )*RirtC 1 . 2*315) K. «KX WR 1TC(6 ift2 C)K,KX CALL SAWo(*PC•)CALLSSCL(2 )AW 1TE( 112,599>PAUSECALL SSwTCHIA,JSKIP)IF(J3KIP *EO*1) GO TOA12 CALL SAKO(MC«)CALL STCO('NS*)XX =0.CIFCKX.E0.2 >XX=15?C*

77 CONTINUE J = 1TSTORE - 0,OCCC:CALL SSCL(l)INTRP=2 INTTT=3

li oo a i=i,7 E CALL CRAC(I,CV(I»I

PHI =CV(3)*3*142THCA =CV(2)*3.142 T=CV(1)*50 9IF( <TSTORE—T>#GT#3**PRTINT)GOT077 IF(T»GT»TIME(INTTT))INTRP=INTRP+1 INTTT = 2*INTRP -1PLANE( 1)=(X(K,INTRP)-XCK.INTRP-1))#IT-TIMEI INTTT-2) 1/1 TIME! INTTT)- 1 TIME!INTTT-2>)+X(K,INTRP-1 ) +XX

213

PL ANiL ( 2 ) -1 YIK. * INTRP)-Y(K, INTRP-1 ) > * IT-TI MEI I NTTT-2))/1TIMEI INTTT ) 1TIMEIIn TTT-2) )♦VIK,INTRP-1 )PLANE(J)=(ZIK,INTRP)-ZIK,INTRP-1) >*I T-TI MEC INTTT-2I)/C TIMEt INTTT) i T IME( I NTTT-2 » )*ZIK,INTRP-I I CALLROTA TE(OH I.THOA.PLANL•GUN)ANG<11 = —AT AN(GUN12) /GUNtl) )DIA - SCRTIGUNt1)4*2 ♦ SUNt2)**2)ANGI 2) = AT AN(GLNI J ) /DIA)DO ie I = 1.2IPtAQSIANGII) )»GT«ALT>ANGII) = SIGNIALT*ANGCI I >DcRII)= (ANGII)/RAD*17*7777)/SIZ£

16 IFIABSIOERI I 1 )«GT«SCALEII ) > UERl I ) =SIGNI SCALEI I) *OER( I ) )DO 9 I = 1*2

9 CALL L SDA(I * OERII))CALL TLDAIFIT*GT» TSTOREI GO TO 1?GOTO 11

lv JOUTI 1.J.ISAVE)=-20.*CV(5)*IGOCi JOUT ( 2. J. ISAVE)=-2G«*CVI4)*100 ) •JOUTI3.J.I SAVE)-ANG(1)*CMLRDJOUTI 4.J, ISAVE)=ANGI2)*CMLRDTIMEIJ) = TTHDOT=CVI6>»3*142PHIDOT =C VI7) *3*142JOUTI5.J .1SAVE)=PHI*1;0C»JOUT I 6 • J * I SAVE ) =THDA41 SGi3«JOUTI7.J.ISAVE)=PHIDOT*lOCC*JOUTIS.J.I SAVE)=THDOT*1COO*TSTORE = TSTORE ♦ PRT1NT 1FITSTORE.GT.TSTOP) GOTO 12 J = J + I GOTO I 1

12 CALL RSCL11)CALL SAM01*HO• )MR ITEI 1C2.25)

r> n o

n n

C M30=1 TERMINATE ***********************= 2 PS INT.PUNCH AND TERMINATE **********************= 3 PS INT PUNCH AND RUN AGAIN ***********************

Rr.AD( 1-.1,2? ) NGC IFIMGO.LQ.1>CALL RCLFCE WR1TZ(6.21 ;>MRITC(6.211)*R IT^{ £.212)WRITE<6.215)< J.IJOUTII.J.ISAVE).I=I.B),J=1.NPTS)WRITE< 1C2.*43)PAUSE

SWITCH 1 POSITION 1 SKIP TAPE ***********************SWITCH 2 POSITION C< SAVE AND RUN ***********************SWITCH 3--- POSITION 1 RERUN SAME CASE ***********************

CALL SSWTCH(1•I)IF( I.EQ.2)G0TQ43C CALL SSwTCHl3.1)IF CI . E C U 1)G0TG77 G0T0412

43. IF(1SAVE«EQ«4)GOT0429 CALL SSWTCHI2.1)IF(I•EQ« 2)GOTO432

42y CALL PPTIOM(KOUT.NPTS)IF( ISAVE.EQ.1)G0TO431 CALL PPTIBM(LOUT.NPTS)IF I ISAVC.EQ.2)GCT0431 CALL PPTI8M(MOUT.NPTS)IF!ISAVE.EG*3)GOTO*31 CALL PPTIBM(NOUT.NPTS)

431 ISAVE=C IF(NoO.EQ* 2 >CALL RELECE

432 ISAVI=ISAVE+1 412 CONTINUE

GO TG 4CJ2 FORMAT(I5.F1C.5)3 FORMAT(IHl« 5X.•POT ADDRESS* *5X«*DESIRED VALUE*.9X,•SET VALUE*.//)

215

* FC.RMAT(-jX. I 12,2F2'-„A)7 FORMAT C/ / S X , ‘ANALOG COMPUTER IS SET UP *** LETS oCJ'///)15 FORMAT*5X.F15,2. FI 5.3 .F15»1,F15.3,F15,1»2. FORMAT(II)25 FORMAT(SX,‘ENTER NOG*)99 FORMAT( 1H1 )

1 M FORMAT (AF 1 C * 2 >112 FORMAT(IH:,16X.‘TIMF* *15X*X»,19X*Y*,19X*Z*/17X*----•,14X*--- •17X*-

1— *•17X *---•/(AF20* 1 » >1^3 FORMAT(1H1,'DATA FOR FLIGHT CASE =*,I1C> lib FORMAT(5F 7# A )2 1*: FORMAT( 1H0, I4X, • TORQUE FT* LG/1 uOO * * 1AX *• ERROR MLS/10 * • 1 AX, • GUN P

10SITI0N RAO/IOOO**3X,‘ANGULAR ACCELARAT ION RAO/SEC/1000 * )211 FORMAT ( 1 5X * * • , 14X,-*-------- • , 1 AX, *---- -----

|-------------• , 3X , *------------------- — * )212 FORMAT( /,6X,A(RX,•AZIMUTH■,6X,•ELEVAT ION*)./1215 FORMAT(IX,(15*1CX*I5,10X,I5,10X,I5*ICX,15,1OX,15,10X,15,1CX,15,

1 6X.I5))315 FORMAT(215//1AO 1 FORMAT!1CX,3F10*1)4 1 J FORMAT(2C12)All FORMAT(25X,2110)A2C FORMAT( 1H1, 5X«*PATH = * , I2 , 5X, • 0 ISPLACEMENT CODE *12)AAC FORMAT < 5X »’CHECK SENSE SWITCHES*)999 FORMAT(1 OX,* TO SKIP SET SWITCH A a 1 • )

END

216

S U B R O U T I N E R O T A T E ( A . 0 • C O L O , C N E it )

******************* COORDINATE TRANSFORMATION SUBROUTINE ***♦! A GUN ANGLE PHI (k AD)H GUN ANGLE THEDA (RAD)ULD(3) VECTOR CONTAINING TARGET LOCATIONN£*(3) VECTOR CONTAINING TARGET LOCATION IN GUN SYSTEM

DIMENSION COLD(3)»CN£W(3)«C(3 *3) CA=COS(A)CQ=COS(B )SA=SIN(A)SU = SIN(B >C( 1* 1 > = CA*CB Ct 1.2) = CB* SA C(1.3) - -SB C(2.1 ) = -SA C(2.2) as CA C(2*3) = c.C(3.1> = SB*C A C(3•2 ) = SA* SB C(3.3) = ce DO 1Z 1=1.J SUM=„.DO 2G J=1.3

2: SUM=SUM+C(1.J )*COLD(J )1C CN£W(I)=SUM

RETURN END

suPMLuri vid ^PTir^(NAA.Nrs)DIMENSION NAA(e,25'>D I MENS ION KA(T).NCSI2C > ,NDATA( 16) .NDUMI80 ) • NB ( IC ) data nplus.^inus.nhlk.nav .NNNEL/azoO:c•:si j• bzc oo; j:■ a.,8z:C JCOC 1 c •

la z c o .r :? * * : « e z « c * : data k a/i . i ;' * i /data Ne/t z. • : ■',d7C‘:: : i ,azv0 jo;o: 2 .az: j: : i3*hz_v-C-wvA.

l h»Zv. c ; : . . l o . a z ; : . : i b . e z ^ j . r j j t; 7 . 8 Z O ‘. o ; < J O ' : 8 * 8 Z o ; ; c ; : i •>/

*** U5E« SUPPLIES

* NAA(I,J) an e p y n m a t r i x c o n t a i n i n g t h e d a t a t o be p u n c h e d o n t a ^ e I REPRESENTS THF 8 VARA6LES PHI.THEDA ETC*J REPRESENTS THE NUMBER OF TIME STEPS TO BE PUNCHEDTHE VARABLE J MUST BE AN EVEN NUMBER SINCE THE FORMAT FOR THE TAPE IS 1815) FOR EACH TIME STEP • IN OTHER WORDS YOU GET TWO TIME STEPS PER CARO

YOU GET J/2 CARDS AND J/2*B INCHES OF PAPERTAPE PER RUN GOOD LUCK WITH THE PROGRAM SEE YOU ABOUT THE 5TH OF JULY TOMMY PERKINS

w R I T C { 6 • 2 1 C )WRITEC6.211) wRITEIE.212)DO EC 1 N TP =s I.NTS.2 CALLSSWTCHIA,JSKIP)IF< JSKIP * EQ*1)GOT06Gl JTP = NTP + 1wRITc(6*215)NTP.(NAA(I.NTP),1=1,8)

218

v»9 I Tu ( fc . El 5 > JTP. (NAA( I , JTP) • 1 = 1 *cl>DO 6C? MPT - 1*~MMT = MPT + o NPATA(MPT) = NAA(MPT.NTP)

b ? 2 NDATA(WMT) = NAA(mpt,JTP)DU 7 I = 5.H: . «J = 1/3NG I GN = MINUS1F(NDATA( J) ,GE» . JNS1GN = NPLUS NDATA(J) = IABS(NOA T A(J ) )LLP = 1 - 4 DO A L = 1.3LL = LLP ♦ LNDUMM = NDA TA(J >/KA(L) ♦ INOATA(J) = NDATA(J) -(NDUMM-1)*KA(L>

8 NDUM(LL>= NB(NOUMM)NDUMM = NOATA(J) ♦ 1NOUM(I) = NB(NOUMM)LLL = 1 - 3DO 9 L = LLL.ILL = L - 11F(NDUM(LUN£#NB( 1 » > GOTO 1C

9 NDUM(LL) = N8LK NDUM([) = NBLK GOTO 7

i: NDUM(LL) =NSIGN 7 CONTINUE

s L I • 1 V

s LI.5s LI .6 3

toMVO

Lwi A 41.1 SL S • j L* . A

1CW. 1ln =:S I * • 3 A I * b A I * 1 A I * 6 Cl ,5 fiNE

NOOM * 1

vOLN', 1S A

tSss

7 : s N C S * 6 1

c

SS

A2 C 7 1 Sc

CALL VnhA TE INCS.dC 1 NCS(1) = NNN=L CALL KhATclNC S * 1)

t : i CQNTlNUF 12 KORMAKSX.Llie)le; f o r m a t (bx, u ; 1 lo FORMAT ( 5X.61 1-: )

2i: FORMAT!1H1.I AX*•TORQUE FT«L3/1C30• .1 AX , •FRROH MLS/1C• •1AX••GUN P iOStTlON RAD/1CCC•*3X,•ANGULAR ACCELARATIQN RAD/SEC/IvC'*)

211 FORMAT! 1 5X , • • , 1AX. • • , I 4 X . •----------1 • • 3X »-•----------------------------------- • >

212 FORMAT! /» 6X.A!RX ••AZIMUTH* »6X#*ELLVATIQN* )*/)215 FORMAT!lX.!I5,l\X.I5*l:x,IS.13X*15.i:X.I5.1CX.IS*KX.IS«irX.I5,

1 6X, 15 > )RETURNEND

220

LYMBCL LC, S I .oCDEF «IHA TEWLF FC69

i»' h A T F L * « 1 J * 15s T * . n L>GFFA I . 1: 1L « • 1 3 * 1 5L» * 1 3 *1 3S T » • 1 J HYTCALI, 1 FP TA I t 15 1B *15

FPT GEN.fl,24 X •DATA X • 30A

Ri-FF HLS 1-YTE RES 1

END

K>KJ

t -LC A Li ( GO ) . ( UDCb • 6 ) . ( T E M P , 4 J > • ( L I B *USEH* SYSTEM ) « ( F O R r • 1 6 J C ) /, ASSIGN (F.5,51), A S S I G N ( K „ 6 , L f l ) . V F C ' A S S I G N ( F „ 1 ' I • OC > • VFC;ass ion (f :i :?.u c ).v f c

i S S IGIJ ( F I ): -E. SIONI ( F v i i i ^ P A ' l ) « ti 1 N

- V 'j yf

145 a 2926 C« 5 11 >176215 C*32. *15923E « £ 20 3t> :^54: ,59541 a49J 45 *14*35. .,33951 ,4«S9 ; 292 *4

V

222

APPENDIX G

LINEARIZATION OF THE GUN MODEL

The p r o c e d u r e u s e d t o l i n e a r i z e t h e n o n - l i n e a r gun model

d e v e lo p e d i n S e c t i o n 3-4

■ *

© L I M = <T ( t ) - E1 © LIM> G - l

. . Jfi><j^LIM = _1 ( T ^ ( t ) - B 0 L I M ) G-2

*

i s p r e s e n t e d i n t h i s a p p e n d i x . W here t h e l i n e a r fo rm was

c h o s e n t o bet * *

T O + & = K#U ( t ) G-3

w h e re T a n d K a r e p a r a m e t e r s t o b e d e t e r m i n e d .

The t e c h n i q u e u s e d c o n s i s t e d o f d r i v i n g b o t h t h e n o n

l i n e a r m o del a n d t h e l i n e a r m o d e l w i t h t h e same f o r c i n g f u n c

t i o n , m e a s u r i n g t h e d e g r e e o f c o r r e l a t i o n b y c o m p u t in g t h e

i n t e g r a l o f t h e e r r o r s q u a r e d , a n d a d j u s t i n g t h e p a r a m e t e r s o f

t h e l i n e a r m odel CT a n d K) i n a s e a r c h p r o c e d u r e C p a te r n s e a r c h )

t o p r o v i d e t h e b e s t f i t . F i g u r e G - l i l l u s t r a t e s t h e p r o c e d u r e

u s e d .

F O R C I N GF U N C T I O N

N O N L I N E A R eM O D E L

1 >

c

L I N E A R 0

M O D E Lt -------------

E R R O R( E R R O R ) d t

/ A D J U [ L I N E \ T H E

P A T E R N S E A R C H

8 T P A R A M E T E R S IN THEA R M O O E L TO M I N I M I Z E E R R O R )

F I G U R E G- l F L O W D I A G R A M OF L I N E A R I Z A T I O N PROCEDURE

APPENDIX H

COMPUTER LISTING OF DIGITAL SIMULATION

MAIN PROGRAMDOUflLfc PRECISION Z ! 8 ) .ZNEW( 8 > • SUM(8 ) *FU N( 0 ) . KKC4 , 8 ) *

I T * T1 • DELTCCMMON I PR I NT .KOUNT • OELT • T. TI • TF IN • TI_OG IC * ISTP COMMON /AREA I / TI . TL • TN.TD«KP• SPJET• TH1LIM. TH2L1M.

1PH1LIM.PH2LIM,JTHDA.JPHI.BTHDA.BPHI.RAOEG.CMILRD COMMON /AREA4/PLTZ!2.8t).OUTPUT! 2 . 5 0 0 ) .KCCNT COMMON / PERK/Z1LIM .Z2LIM.GA IN.ALPHA. ADDER REAL JTHOA.JPht.KP RADEG = 2 . 4 3 . 1 4 1 5 9 3 / 3 6 0 .CMILRC = 1 0 1 8 .6 READ!5 . 2 )NRUN TH1LIM = TH1L IM4RA0EG TH2LIM = TH2LIM4RA0EG PM1LIM = PHILIM4RADEG PH2LIM = PH2LIM4RA0EG 00 1 1 = 1 .NRUNREAD!5 .1 0 0 0 ) Z1L1M.Z2LIM.GAIN.ALPHA*ADDER

1000 FORMAT!5F10.2)WRITE!6 . 1 0 0 1 ) Z1L1M.Z2LIM.GAIN.ALPHA• ADDER

1 0 0 1 FORMAT! lO X . 'A C Q U l R MODE PARAMETERS*/I OX*Z1L1M = • . F I S . S / 1 O X • • Z2LIM 1 = • . F I S . 5 / 1 OX. 'GAIN * • . F 1 5 . 5 / 1 OX.'ALPHA = • . F 1 5 . 5 / 1 OX. » ADDER = • 2 . F 1 5 . 5 / / )N * 8 ICASE = 1OELT = 0 . 0 5CALL RUNGAIZ.ZNEW.XK.SUM.FUN.N.ICASE)OO 10 J= 1.KCCNT

10 WRITE16.3) (OUTPUT(L.J ) , L = I . 2)K=0DC 4 J=1 . KCCNT. 5 K= K + lPLTZ! 1 . K)= OUTPUT!l.J)

4 PLTZ!2.K)= OUTPUT!2 . J )

22t>

WRITEC6.5) CPLT7!1.J ).PLTZ<2,J ).J= 1 .K )5 FORMAT{20X * 2F15*3)

CCCCCC CALCULATE COST FUNCTION CCCCCC CC USE THE ISE COST FUNCTION AND CC THE TRAPIZODIAL RULE FOR INTEGRATION C

TFIKAL * 30*N = TFINAL*10.+ 1.NM1 a N - 1COST = (OUTPUT(I•1)**2 + OUTPUTI 1»N)**2)/2.0 DO 11 J * 2.NM1

11 COST = COST ♦ OUTPUT! l . J 1**2 COST = COST♦DELT MRITE<6.12) NtCOST

12 FORMAT!10X,15.'POINTS MERE INTEGRATED TO YIELD COSTIISE) **.F15.S) C CCCCCC CCCCCC

CALL PLOT!K >1 CONTINUE2 FORMAT!13)3 FORMAT!10X.5F15.5)

STOPENO

roN1

s u b r o u t i n e t a r g e t * t 3 . ZZ)COMMON / AREA 1 / T I * T L » T N . T O . K P . S P J E T . T H I L I N . T H 2 L I M •

1 P H 1 L I M . P H 2 L I M . J T H O A . J P H I . B T H D A . B P H I . R A D E G . CMtLRD REAL J T H C A . J P I - I . K P DI MENSION ZZ( 6 )READ* S . 1 ) Xt.X2.TM

I FORMAT( 3 F 1 0 * 3 )WRITE* 6 . 2 ) X 1 . X 2 . T M

2 F O R M A T * 1 0 X . ' F L I G H T PATH DATA ---------- X QRG = • . F 1 0 . 3 / .1 3 5 X * * XF IN ss ' . F 10 • 3 / 3 5 X . • TCHN = * . F 1 0 . 3 )

XO = XIYO = 1 5 2 0 0 . 0 0 0 ZO= 3 7 5 . 0 0 XV - 0 . 0 0 0YV = - S P J E T * 6 0 6 7 . / 3 6 0 0 .ZV = 0 . 0 0 0 RETURNENTRY W H E R E * T 3 . Z Z )ZZ* 1 ) * XOI F I T 3 . G E . T M ) Z Z * 1) = X2

Z Z * 2 ) * YO ♦ YV*T3 Z Z ( 3 ) - ZO ZZ ( 4 ) * XV Z Z * 5 ) * VV Z Z ( 6 ) * ZV RETURN END

2?8

SU6R0UTINE ACQU1R! PHI*PHID.T1.TTS.TTF*ERROR•TQR. TF IN)COMMON/6LK1/KACC DIMENSION ZZ!6).ERROR!2.2>REAL K , K PFUNF(X) = 2.*X ♦ < 1 ./1<P*K ) )*! X 10-XlF + TAU* I X20-X2F) )f u n s t x j = <x 2 f +k p *k ) * e x p ! - x / t a u ) / e x p i - f u n f ! x > / t a u ) - 2 . * k p *k + i k p * k- IX20)*EXPC-X/TAU>CALCIIX.V) = AT AN2I V • X )CALC2(X* XO ,Y,YD) a IX*YD-Y4XD )/( X4X4Y*Y)RADEG a 2.*3.1A 159/360.KP = 1 . 7 5 3 3 TAU « 4 9 . 9 7 tCACO a KACQ + 1 I F I K A C O .G T . 1 0 ) GOTO 1 0 4 A = 1 . 0 0 0 0 0 I a 1TF a 1.1 + T1X10 a PHIX20 a PHIDEZERO a ERROR!1.I>K a 10.0IF!EZERO.GT.0.00) K a -10.TOR a KCALL WHERE!Tl.ZZ)EPS a 0 . 0 0 0 1P 10 a PHI/RAOEGP 2 0 a PHID/RAOEGX I I = CALC1! Z Z ! 1 ) • Z Z ! 2 ) ) /RADEGX 12 a CALC2 ! Z Z ! 1 ) . Z Z ! 4 ) . Z Z ! 2 ) . Z Z 1 5 ) ) /RADEGW R | T E ( 6 . 5 0 0 ) T l . P 1 0 . P 2 0 . XI I . X I 2

5 0 0 FORMAT 1 S X . • IN IT IAL CONOITIONS• / O X , • TIME* . 2 1 X . * G U N * . 3 7 X , * P L A N E * / 5 X , 1 FI 0 . 5 . 4 £ 2 0 . 6 )

102 CALL WHERE!TF *ZZ)X a ZZ! 1 ) bZ

Z

Y a Z Z ( 2 )X1F s CALC1< X .Y >X2F a C A L C 2 ( X , Z Z ( 4 ) « Y , Z Z ( 5 ) )TL = 0 . 0 1 TR a 8 . 0

7 3 FTL = FUNS<TL )FTR = Funs ( t r )I F ( FTL * FTR ) 2 1 * 9 9 , 4 0

21 TAPP a TL + ( T R - T L 1 / 2 . 0 FTAPP = FU N S(T A P P )I F ( A B S ( T R - T L ) * L T , 0 . 0 0 1 ) G Q T O 9 9 FTSQ a FTAPP * FTAPP IFC SORT( F T S Q ) - EPS ) 9 9 , 9 9 , 6 0

6 0 I F ( F T A P P * F T L ) 6 5 , 9 9 , 7 5 6 5 TR a TAPP

FTR a FTAPP GO TO 21

7 5 TL a TAPP FTL a FTAPP GO TO 21

4 0 I F ( A B S ( F T L ) , G T * A B S ( F T R ) ) TLaTL + 0 , 1 I F I A B S ( F T R ) , G E • A 0 S ( F T L ) ) TRaTR - 0 , 1 I F ( T R . L E . T L ) GOTO 11 GOTO 7 3

U TAPP * TL 9 9 TMAX1aTAPP

TMAX2 a FU NF(TM AXl)TS a TMAX1 ♦ Tl T T F I N a TMAX2 ♦ T l ♦ TMAXl OER a EZERO/RAOEG CALL « H E R E (T T F I N ,Z Z )X a 2 Z( 1 )Y a 2 Z ( 2 )X 1FO a CALC1I X, Y l /RADEGX2FC a C A L C 2 I X ,Z Z ( 4 ) * Y , Z Z ( 5 ) l /RADEG

*RlTEtb. 100) TUTS. TT FIN, OER.X 1FO.X2F0 100 FORMAT(5X « 6E15.5)

CHECK = ABS(T TFIN-TF)IF(CHECK.LE.0.01) GOTO 101 IF( I.GT.6) A = 0.6000 1 = 1 * 1IF(I.GT.ISO) GOTO 10*TF = TTF1N*A ♦ (A - 1.000>*TF GOTO 102

10 1 CONTINUEIFtTMAXl.LT.EPS) TMAX1 = O.COO lFtTMAX2.LT.EPS) TMAX2 = 0.000 TTS = TMAX1 TTF = TMAX2 GOTO 105

10* TF IN = 3.105 RETURN

ENC

s>

CCCCCccCCCCC

100

ccc

66 6

SUBROUTINE DEHIV(Z.FUN.N)DOUBLE PRECISION Z ( 8 ) • FUN( B >, T , T 1 .DEL TCOMMON IPRINT .KUUNT• O E L T , T , T l , T F I N « TLOG1C * 1 STPCOMMON / A R E A ! / T 1 . T L . T N . T D , K P • S P J E T . TH1LIM• TH2LIM.P H 1 L I M .P H 2 L I M ,J T H D A ,J P M I , BTHDA*BPHI, RAOEG• CM1L « 0 COMMON / P E R K / Z 1 L I M . Z 2 L I M , GAIN,ALPHA.ADDER REAL J T H D A , J P H . K PDIMENSION Z Z ; 6 ) .HUMAN!2 , 4 > ,Z 11 8 ) .ERROR( 2 . 2 ) .P L A N E ( 6 )DATA HUMAN/A.8 4 7 8 . 8 2 3 . 0 8 3 . 2 . 4 , 2 . 4 , 1 6 . • 1 6 . • 1 6 . • 1 6 . /TF IN = 1 . 0 T = 0 . 0 0 0 TLOGIC = 2 . 0 0 KT VPE = 2 KHG * 1GC = I . 0 / 1 T N 4 T I )G1 « T l ♦ TN IS T P a 1 0 0 0 NPT * 1 0 0CALL TARGET!TCROSS,ZZ)CALL CELAY( N P T )CALL tfHERE( 0 . 0 * Z Z )

CCCCCREAD IN FL IGHT DATA AND RETURN TCRQSS AND ZZ AT TIME EQUAL 0 . 0 0 0

CCCCCDO 1 0 0 J * 5 , 8 Z ( J ) * 0 . 0 0 Z ( 2 ) = 0 . 0 0 ZI 4) = 0 .00

IFIJLK IS LESS THAN 1 THE I N I T I A L CUN P O SITIO N I S CALCULATED

READ!5 . 6 6 6 ) JLK.A.E FORMAT( I 1 0 . 2 F 1 0 . 5 )

IF ! JLK «LT • 1 ) GOTO 667 Z(1) = A Z!3> a E GCTC 668

667 ZZX s ZZ(1 )ZZV = ZZ12)ZZZ = ZZ(3 )01 A 3 SCRT(ZZ{ IJAZZC1> ♦ ZZ!2 )*ZZ(2)>Z(l) = AT AN2iZZY.ZZX)Z! 3 ) * -ATAN2!ZZZ.D1A)

658 OTl * TFILIM/RADEG 0T2 = TH2LIM/RACEG OPIspHILIH/RADEG 0P2 3 PH2LIM/RADEG WRITE16.967) Z!ll.Z!3)

967 FORMAT!10X.*THE INITIAL POSITOON OF THE GUN IS AZIMUTH 3 «.F15.5. 15X.*ELEVAT ION s *.F15.5/)WRITE!6.501)BPHI. OP 1.TI.HUMAN! I•I).HUMAN!2•1)•1JPHI. 0P2.TL* HUMAN(1*2).HUMAN!2*2).BTHOA• OTI.TN,2HUMAN!1.3).HUMAN!2.3).JTHDA, 0T2.TO,HUMAN 11.4).HUMAN!2,4)WR ITE16.502 >T2 3 Ti m o o e = o t x o z = o .TQXV 3 0.CALL EVALlT2» Z,ERROR.PLANE )CALL WMOOE!ERROR. IMODE.T2 )IF! IMCOE.EQ.1 ) CALL SETUP!Z.T2.ERROR•TFtN.TXOZ)

502 FORMAT!27X.•|OEGREES)*.4 0X.*(MILLS)*/lX.*TIMc* • 9X»1*PHI•.8X.•PHIC*.9X.*THD*.8X.•THDD•.7X,•ERPHI•*6X••EROOTP* *7X.2*T CPHI *.7X**ERTHD*.6X.*ERD0TT*.7X.•TQTHD*//)

501 FORMAT(56X.•SYSTEM PARAMETERS**//85X. •BEFORE*.1 OX.•AFTER•/112X.* EPH1••FI 1,2.5X,*PHILIM••F 10.2•5X.*TI*.F10.2.5X.•0*. 2F13.2.5X.F10.2/12X.*JPHI•,F11.2.5X,*PH2LIM•,F10.2.5X.•TL* • 3F10.2.5X,•PPP'.FI 1.2.5X.F1C.2/12X.*B THDA *.F10.2,5X.•TH1LIM••

4F10.2.SX*•TN* .FlO.2.5X,'POWP* *F|0.2•SX*F10•2/12X,•JTHDA*,F10.2. 55X «•TM2LIM* ♦F10.2.5X.*TD**Fl0*2*5X.*POWT* *F10.2.5X.F10.2///)RETURNENTRY DER(Z.FUN)

CCCCC NONLINEAR GUN MODEL CCCCCc cc cC Z ARRAY OF STATE VARIABLES CC Z(l) OH PHI AZIMUTH ANGLE OF GUN CC Z (2) ANGUt-AR VELOCITY OF GUN AZIMUTH CC Z(3) OR THEDA ELEVATION ANGULAR POSITION CC ZI4) ANGULAR VELOCITY ELEVATION CC CC CCCCCC CCCCC

T2 = TlCALL CLY0UT(T2«TRPHI,TRTHDA.Z) PHI = Z(l)DPHI = Z (2)THEOA * Z(3)DTHEDA s Z(4)IF(ABSIDPHI).GT.PH1LIM> GOTO 1 FUNI1) a Z(2)DOPhI s ITRPHI-ePHIAZt2))/JPHl IF!ABS(CDPHI>.GT*PH2LIM)GOTO 2 FUN(2) s DDPHI GOTO 3

1 FUN(l) s SIGNIPH1L IM.DPHI )FUN(2) = 0.0Z < 2 1 * FUNII)GOTO 3

2 FUN12) s S IGNIPH2L IM.DDRHI )3 CONTINUE

IF(THE0A.GT.0«0) GOTO 7 IF(ABS(OTHEOA ).GT.TH1LIM) GOTO 4

W.Z

FUN(J> a Z ( 4 )DOTHOA a (TRTFDA - BTH0A4Z(4))/JTHDA 1F( ABStCCTHDA ).CT .TH2LIM ) GOTO 5 FUM4) * DOTHCA GOTO 6

4 FUN13) = SIGN! T H u IM.OTHEDA)f u n (4) a o.oZ( 4 ) * FUN( 3)GOTO 6

5 FUN (4) a SIGN(TH2LIM •DDTHDA)6 CONTINUE

GOTO 67 Z(3> a o.O

Z( 4 > a 0.0 FUN(3> « 0.0 FUN14) a 0.0

8 CONTINUECCCCC NLOM HUMAN MODELCcc HUMANt1 , 3 ) NLDM OPERATOR PARAMETERSc ( 1 . 1 ) Q BEFORE TCROSSc ( 1 , 2 ) PPP BEFORE TCROSSc ( 1 . 3 ) POWP BEFORE TCROSSc ( 1 , 4 ) POWT BEFORE TCROSSc ( 2 * 1 ) 0 AFTER TCROSSc ( 2 , 2 ) PPP AFTER TCROSSc ( 2 , 3 ) POWP AFTER TCROSSc ( 2 , 4 ) POWT AFTER TCROSScc

T CROSS TIME TO CROSSOVER

cCCCCC

IFIKTYPE.EQ • 2 . AND.T2 •GT.TCROSS) KHO a 2CALL EVAL( T 2 , Z* ERROR• PLANE )

CCCCCccccCCCCcccccc

CCCCC 2 35

EPhI a ERROR( 1 * 1 ) + HUMAN!KH0.2)*ERROR(1*2)ETHD s ERROR( 2 . 1 ) + HUMAN( KHO, 2 )*ERROR(2 • 2 )KPPMI 3 20 .+ABS(EPHI*PLANEC 2 > ♦ 1 0 0 0 . /HUMAN!KHO.1 I > IFIKPPHI.GT . 1 0 0 . JKPPHI 3 ICO.KTHCA = 20 *♦ AES(ETHDAPLANE(S )A 1000./HUMAN(KHO.I )) IF(KTHDA.GT . 100. KTHDA s ICO.FUN(5) = Z ! 6 )FUN(6 I = (K PPH I*EPH I-Z(5) -Z(6>*G1>*G C F U M 7 J » Z ( B )FUN( 8 ) s (KTHDA*ETHO-Z(7)-Z(8)*G1)*GC RETURNENTRY OERFtN(Z)T2 3 TI F ( T 2 . G T . 4 0 . )TFIN s 3 . 0 IF (TFIN.GT.TLGGIC) GOTO 9 CALL. EV AL ! T 2 . Z . ERROR. PLANE )ATCPH1 3 PL ANE( 2 ) * HUMAN( KHO• 3 >ATOTHO 3 PLANE! 5 >*HUM AN<KHO*4 I TCPHI * —Z( 5 ) ATQPHI TQTHD s —Z (71 ♦ ATQTHOIF !A B S !T O P H I) .G T .1 0 .> T Q P H I= S IG N (1 0 . .T Q P H I)IF(ABS(TOTHO).GT.iO.ITQTHD 3 S IG N (1 0 . • TQTHD)CALL 0LYIN(T2.TOPHI.TQTHO)IF( I M 0 0 E .E 0 .2 ) GOTO 10 1F(T 2 .L E .T X Q Z ) GOTO 9 CALL WMODEIERROR.IM00E.T2)IF(1M 0CE.EQ.1 ) CALL SETUP(Z.T2.ERR0R.TFIN.TXQ2)I F ( I M 0 0 E . E 0 . 2 ) CALL LEAO(T2.ERROR. TXQY)GOTO 9

10 IFCT2.GE.TXQY ) CALL CHECK!T2 . ERROR. IMODE. TXQV.TXOZ) 1 RETURN

END

SUBROUTINE DELAY(NPT)DOUBLE PRECISION Z<8)COMMON /AREA 1 / T l . T L • TN,TD .K P . SPJET,THILIM , TH2LIM,

1PHILIM.PH2LIM*JTHDA,JPF 1, 8THDA,QPHI.RADEG,CMILRD COMMON /P E R K / Z 1LIM«Z2LIM.GAIN,ALPHA,ADDER DIMENSION STORE( 3 . 1 0 1 > • ERROR!2 • 2 ) .PLANE(6 )REAL JTHDA,JPHI.KP

CCCCC C CC TIME C NPT C TO C STORE C ST O R E ! l .X )C STCREI2.X)C STORE( 3 . X )C DTP C OTT C TP C TT C CCCCCC

NUM * 3 NPM = NPT - 1 NPP * NPT 4 |S T C R E U . t l = O.COO STORE! 2 . 1 I * 0 . 0 0 0 STORE!3 . I ) = O.COO STORE!1 . 2 ) « TO STCRE!2*2 I = 0 . 0 0 STORE I 3 , 2 I = 0 . 0 STO RE!I .NPP) = 1 0 0 0 0 . 0

NOMENCwAYURE CCCCCCC

TIME SUBROUTINE CALLED CNUMBER POINTS IN STORAGE ARRAY C LENGTH OF TIME OELAY CNAME OF STORAGE ARRAY CVALUES OF TIME CVALUES OF TORQUE AZIMUTH CVALUES OF TORQUE ELEVATION CCELAYEO TORQUE AZIMUTH CDELAYED TORQUE ELEVATION CINPUT TORQUE AZIMUTH CINPUT TORQUE ELEVATION C

Cc

CCCCC

c c c c

c c c c

7

BGKMOO = 0 *PTF = 0.PTFP = 0.ENOMOCZ Z Z Z Z - 2 1 L IM /1 C 1 8 .6 RETURNENTRY SETJPCZ ,PTB.ERROR* TFIN.TXQZ >PHI s Z i I )PHIC = 0 .CALL ACOUIRCPHI.PHID.PTB.PTS.PTF.ERROR»PTQt. TFIN)PTFP* PTF+ AOCER PT 02 = -PTOI TXQZ sPTFP

NOTE CCCCC

THIS SECTION ALLOWS THE OPERATOR TO INPUT C HIS OWN SWITCHING DATA CWHERE PTB s BEGINNING OF CONTROL SEQUENCE

PTS s SWITCH TORQUE PTF s END OF CONTROL SEQUENCE PTFPs RTF ♦ 0 . 5 P T Q ls FIRST TORQUE PTQ2s SECONO TORQUE

CCCCCC

W R IT E (6 .7 )P T B .P T S .P T F .P T Q I .P T Q 2 .P T F PFORMAT( 1 OX.• PTB s • , F 1 5 . 5 / 1 OX. «PTS * • *F15 . 5 / 1 OX.■PTF 3 » , F 1 5 . 5 / 1 O X.* PT GI3 * , F 1 5 . 5 / 1 0 X . • PTQ2= » . F I S . 5 /1 OX• 'P T F P * • . F l 5 . 5 )RETURNENTRY LEADC T2 .ERROR.TXQY)ERROLO 3 ERROR( 1 . 1 )ENOMOO = T2 + 0 . 0 5 TXQY 3 ENOMOD JACO 3 i NERN s l

BCZ

TOLEAO * - 1 0 .I F ( ERROR( 1 * 1 J . L T . O . > TOLEAO* 10.BGNMOD * T 2■RITE!6 , 5 0 0 ) T2 . ERROR!1 . I).TOLEAO

500 FORMAT!IOX,*BEGIN LEAD TRACKING TIME * * , F I 0 . 2 . » ERROR * * , F 1 0 , S , 1 •TORQUE * * . F 1 0 . 2 )

RETURNENTRY CHECK!T,ERROR,IMODE. TX0Y.TXO2>IFIN ERN .EQ.2) GOTO 309I F ! ERROR!1 , 1 )*ERROR!1 , 2 1 > 3 0 9 , 3 0 0 , 3 0 0

309 IF !JA C Q .E 0 ,5 > GOTO 307 E20LD * ERROR 1 1 , 2 )JACQ * 5NERN * 2

3 0 7 IF!ERROR!1 , 2 1 *E20LD>3 1 0 , 3 1 1 , 3 1 1 311 IF1ABS1ERRO R!1 ,2))«LE«0,01 ,A N D,A 8S(ERRO R!1 ,1) ) ,LE ,ZZZZZ)G O TO 3 1 0

TOLEAO * 0 .• R I T E ! 6 * 5 0 3 ) T,ERROR!1 , I ) , ERROR!1 . 2 ) .TOLEAO

5 0 3 FORMAT!10X,*HA—HA*, A F 1 0 .4 )300 ERROLD * ERROR! 1, 1)

EN0M0D * T + , 0 5 TXOV * EN0M0D GOTO 320

310 IMOOE * 0 W R IT E!6 .50 I> T .ERROR!1 , 1 )

501 FORMAT!1OX,*ENC LEAD TRACKING TIME * • . F I 0 . 2 . ‘ERROR * • . F 1 0 . S ) 320 CONTINUE

RETURNENTRY 0LYIN!TIME.TP.TT)IFINUM.GT.NPT ) GOTO 1 STORE!1,NUM) a TIME ♦ TO STORE!2.NUM) * TP STORE!3.NUM) * TT NUN 3 MUM ♦ 1 RETURN

1 00 2 J=1 . NPM I * J ♦ 1STORE!1•J ) = STORE( 1 * 1 )STORE!2*J) * STORE!2*1)

2 STORE!3 . J) * STORE!3 * I)STORE!1 . NPT) x TIME 4 TO ST0RE!2tNPT) * TP STORE!3.NPT) * TT RETURNENTRY OLYOUT! TIME.DTP.OTT• Z )DO 3 1 * 2 , NPP1F!TIME.LT.ST0RE!l.I)> GOTO 4

3 CONTINUE4 IF!I.GT.NUM.OR.I.EO.NPP) MRITE16.5ITIME

NI > I - IFT » ( TIME—STO RE!I*N I) I / ! S T O R E ! 1 • I )—S T O R E !I .N 1 1)OTT « ! S T 0 R E ! 3 . I I-STOREI3.Nl)»*FT 4 ST0REf3.NI)OTP *f STORE!2 . 1 ) —STORE!2.NI) )4FT 4 STORE!2.NI)TOOLO m OTPIF!TIME.GE.BGNMOD.ANO.TIME.LT.ENOMOD) GOTO 50 IF!TIME.GE*PTB.AN0.TIME.LT• PTF> GOTO 100 IF!TIME.GE.PTF.AND.TIME.LT.PTFP) GOTO 150 GOTO 200

50 OTP * TOLEAO GOTO 200

100 OTP « PTQ1IFIT1ME.GE.PTS) OTP » PTQ2 GOTO 200

150 CALL EVAL!TIME.2 . ERROR.PLANE>TORN » JPHI*PLANE!3) 4 BPHI4PLANE12I - GAIN*! ERROR! 1 . I ) 4 ALPHA*

1ERR0R!1 * 2 ) )IF(ABS(TORN).GT. 1 0 . ) TORN*SIGNf1 0 . . TORN)TACO * PTFP - TIMEXLAMDA * ! ADOER - TAOO)/ADDEROTP s XLAMDA*TORN 4 It-XLAMOA)*TOOLO

240

I F ( ABS( D T P ) * 6 T . 1 0 . )OTP a S IG N ( 1 0 . . D T P )WRITE<6.13) TORN.ERROR(I.1 > .ERROR( 1 . 2 > . TI ME

13 FORMAT( 10X * *TO*ERROR! 1 . I ) • ERROR( 1 * 2 ) * • IOX*4F15*5)2 0 0 CONTINUE

RETURN5 FORMAT( 5 X . • * * * * * ♦ * ♦ * ♦ ♦ * * * * ♦ • * * * ' . / / S K * • AT TIME * «

1 FI 0 . 5 . S X . ‘ INCORRECT VALUE FOUND BY SUBROUTINE D E L A Y '• 2 / / 5 X . • • * • ♦ ♦ * * * * • ♦ * * * ♦ • * * * * • / ✓ ✓ ✓ / )

ENO

241

SUBROUTINE VMOOEIERROR*I.T)COMMON /P E R K / Z I L I M .Z 2 L I N .G A I N .A L P H A .A D D E R DIPENSIO N E R R 0 R ( 2 . 2 ) . Z Z f 6 )DATA A 1 . A 2 . C M R / 2 5 . . 1 0 0 . . 1 0 1 8 . 6 /1 * 0VI * ABS( ERROR ( 1 . t > ) *CMR/Z 1LIM V2 * AB S(E R R O R !1 . 2 ) >*CMR/Z2LIN VALUE * VI ♦ V2 I P ( V A L U E . G E . 1 * 0 ) 1 * 1 I P ( I . E O . O ) GOTO 10 CALL V H E R E ( T .Z Z )D IS T * S O R T I Z Z I 1 ) * * 2 ♦ Z Z ( 2 ) * * 2 ) I F ( 0 1 S T . L E , 2 S 0 0 . ) 1 * 2 RETURN END

SUBROUTINE IOCATA!Z.N.ITIME>OOUBLE PRECISION Z<8 ) • T .T 1 .DELTCOMMON i p r i n t . ko unt . d e l t . t . t i . t f i n . t l o g i c . i s t pCOMMON /AREA1/ T I . TL.TN.TD.KP.SPJET.TH1LIM.TH2LIM•

IPHILIM.PH2LIM.JTHDA*JPHI• BTHDA.BPHI.RADE6.CMILRD COMMON /AREAA/PLTZt 2* 0 1 ) • OUTPUT( 2 * 5 0 0 ) .KCCNT REAL JTHDA*JPHI.KP01 MENS ION ZOUTI1 1 ) • ERROR!2 * 2 ) .PLANE!6)T2 * T K* 2I PRINT * IPRINT ♦ KIF ( ITIME*E0*2) GO TO 2KCCNT* 0KCCNT* KCCNT ♦ 1CALL OLVOJTIT2*OTP*OTT*Z)CALL EVAL(T2*Z*ERROR*PLANE)OUTPUT( 1 .KCCNT)* ERRORC1 . 1 )«CNILRO OUTPUT!2 . KCCNT)* ERROR!2 . 1 )*CMILRD RETURN END

SUBROUTINE RUNGA <Z . Z N E t t . X K . S U M . F U N . N . I C A S E l OOUBLE PRECISION Z C 8 ) « Z N E W < 8> .S U M |8 ) . F U N ! 0 > • XK< 4 . 8 ) *

I T . T t . D E L TCOPMON I P R 1 N T . K O U N T . O E L T . T . T I . T F I N . T L O G I C . I S T P REAL MM.NNDIMENSION CON (4 . 3 ) .CONK 4 ) . C0N2I 4)IF(ICASE.EQ.2) GO TO 415 N M 2 . / 3 .M M «l. /3 .R*NN4(PM>NN)/(4a*MM**2-2•♦MM)G1m pM-4• 4NN«* 2 4 5 . 4NM-2a 6 2 - 6 . 4PP4NN—4 .4PM-4.*NN+3.S»Il.-MM)4GI/(2.*MM*<MM-MM>*G2>TT*CI.-2.4MM)• ( t.-MM)4(l.-NN)/<NN4fNN-MM)4G2)P * l .C0N1( 1 >«(6.4MM*NN-2.4MM-2a*NN+t.) / ( I2.4MM*NN> C 0 N K 2 ) » ( 2 . 4 N N - t . ) / < I2.*MM*(NN>MM)4( l.-MM))CONIf3)»<2.*MN—I • ) / ( 12*4NN4(MM-NN)«(l.-NN)1 CONK4 1 * 0 2 / II2««1 l.-MM)*I l . - N N ) )GO TO 418

415 CONTINUE MM*1s/ 2 *N N » l . / 2 .P*1 •TT-1.+SQRTI0.5)R-0 .5 /TT S* Ia-TT CONK I )*1 a / 6 .C 0 N U 2 ) * ( 2 . - T T > /3 .CONI(3J«TT/3.CQNt( 4 ) * 1 « / 6 .

416 CONTINUE C 0 N 2 II)» 0 .C0N2I2 )*MM

244

© n ♦ © © ♦ ©M a o © (* « © «© O o Ul A « A oN in o v> o x o r t H N o n o o i i o x a f i n f t f t f t o n n o x o x X oz c Q C O X O > A A o A o o O > a X > o o o o o a o o x o x X am X X A r N <_ c z r c X r z z Z Z Z z z II II A

ma ♦ A > A © r H A ♦ A ♦ © h r z A r A A A A A © X * o A *u. A C M • o ♦ N in N 01 o A H Z * ♦ u u t o x « X © • ©

ImA o A o <- o o O N A N A o Z A N H o • • • * • • • A ♦ A c oN M A i n m A z c a O * 11 U M A to A A A A A

IT (A A O «. H c X r c X in ( . A m o O X A A A A A A A X N wN c N • It *11 N -4 A M « N N > A M N II N II • n N II O NZ X A A c A A • ♦ A A A A H < -1 « X X z X o X A • Am A • • Z • N o X • c * • > A A 1 z X • X • •mc . © z z * o X M z © A N (A T * u Za a t - Tl z A N Z 1 X CJ( . ♦ A C N X z m HA X • z A 0 m a H♦ o A A Im ■ •© A 0 A A • ■nc A r * z cz * n • zA c . O A •c A Z A za • A A

8 •z XA AA

A

£*Z

♦*t«

i 06

* oa

•l

*(t)

ZN

03

NN

*(C

)ZN

DD

T*T +DELT K0UNT*K0UNT+1I F ( K O U N T .E G .I P R I N T ) C A L L IOOATA<ZNEtf*N*2)CALL OERFINCZNEV)I F ( T F I N .G E .T L O G I C )R E T U R NIFtKOUNT « G £ • 1 S T P)RETURNGO TO 4 9 9END

246

SUBROUTINE RO TA TE !A .8 . COLD.CNEMJ DIMENSION C 0 L 0 I 3 I . C N E M I 3 I . C I 3 . 3 1 CA *CO S(A)C B - C O S I 8 )S A * S INI A )S B * S 1 N I B >C l 1 . 1 >-CA*CB C ( 1 . 2 ) * C B A S A C ( 1 • 3 >*—SB C I 2 . 1 ) * - S A C ( 2 » 2 I * C A C ( 2 . 3 I * 0 .C ( 3 * 1 )» C A * S B C I 3 . 2 ) * S A * S B C ! 3 . 3 1 * C B DO 10 I « t • 3 S U M « 0 .DO 2 0 J * 1 « 3

2 0 SUM*SUM*CI I* J 1AC0L0! J I 1 0 C N E M lD s S U M

RETURN ENOSUBROUTINE V E L l A . 8 . ADOT. BOOT • AC INE R.O XINE R.D X G }DIMENSION ACINERI 31 .DX I N E R t 3 1 . D X G ! 3 I . C D 0 T I 3 . 3 1 .X E D 0 T I 3 1 .CNEMI3* CA * C O S ! A )CB * C O SIB )SA ■ S I N ( A )SB = S I N I B )C D O T I l . l ) * -ADOTASAACS - 8D 0T «C A *SBCOOT!1 . 2 ) * -BOOTASA*SB - ADOT*CA*CBC D 0 T U . 3 ) * -BDOTACBC 0 0 T I 2 . 1 ) * -ACOTACAC D C T I 2 . 2 ) * -ACOTASACOOT!2 . 3 1 * 0 . 0 L*

?Z

COOT( 3 . I ) = BOCT*CA*CB - AOOT*SA*SBCOOK 3* 2 ) * BDOT*CB*SA +ADOT*SB*CACOOT( 3 * 3 ) * -BOOT*SBDO 2 I * 1 * 3SUM * 0 * 00 0 t J * 1 • 3

1 SUM « SUM ♦ COOK I . J ) * A C I N E R ( J )2 X E D C T ( I ) x SUM

CALL ROT AT E(A*B«DXINER«CNEH )0 0 3 K x | • 3

3 OXG(K) x XEOOT(K) + CNEMCK)RETURNEND

N>00

SUBROUTINE EVALlT.Z*ERROR*PLANE)01 MENS ION ERROR!2 . 2 1 . P L A N E ! 6 > . Z Z ! 6 >*A C I N E R !3 ) .

1 A C G U N ! 3 > .D X I N E R ! 3 > .D X G U N ! 3 )DOUBLE PRECISION Z C 8 )

CCCCC NOMENCLATURE

THE PURPOSE OF THIS SUBROUTINE I S TO CALCULATE THE ANGULAR ALIGNMENT ERRORS AND ANGULAR P O SITIO N OF THE TARGET

ERROR ARRAY OF ALIGNMENT ERRORSAND O ERIVIT IVES AZIMUTH ERROR RATE OF CHANGE OF AZIMUTH ELEVATION ERROR

ERROR!1 * 1 ) ERROR!1 . 2 ) E R R O R !2«1 ) E R R O R !2*2 I

PLANE

P L A N E !1 ) PLANE!2 ) P L ANE !3 ) P L A N E ! 4 ) PLANE!5 ) P LANE!6 )

CCCCC Cc c c c c c c c c

ERROR Cc

RATE OF CHANGE OF ELEVATION ERROR C ARRAY CONTAINING ANGULAR LOCATION C

OF THE TARGET AZIMUTH ANGLE AZIMUTH VELOCITY AZIMUTH ACCELERATION ELEVAT10Y ANGLE ELEVATION VELOCITY ELEVATION ACCELERATION

CCCC

Ccccccccc

CCCCCCALL WHEREIT.ZZI 0 0 1 I a 1 * 3 A C I N E R ! I ) * Z Z ! I ) 0 0 2 I = 1 . 3 0 X I N E R 1 I ) * Z Z ! I ♦ 3 )

PHI « 2 1 1 1 OPHI a 2 ( 2 )THEDA * 2 ( 3 >DTHEDA * Z ( 4 )CALL ROTATEIPHI,THED A,ACINER.ACGUN)Z1 * ACCUNI1 ) AACGUNtI> t A C G U N !2)*A C G U N (2)2 2 * SORT( 2 1 )2 3 « ACGUN!I>4ACGUN(11 ♦ ACGUN!3>*ACGUN!3)ERROR( I , | ) * -A T A N C A C C U N ( 2 ) / A C G U N ! 1 ) )ERROR( 2 , I ) * ATAN(ACGUN!3 ) /ACGUN( 1 ) )CALL V E L ( P H I • THEDA,DPHI,DTHEDA• ACINER, 0 X 1 NER. DXGUN) ERROR!1 , 2 > * ( A C G U N t 2 >*DXGUN!1 ) - ACGUN11 >*DXGUN(2 > > / Z 1 E R R 0 R ( 2 . 2 > * ( ACGUNI I >«OXGUN< 3 > - ACGUNt 3)*DXGUN! 1 ) ) / Z 3 Z1 * Z Z ! l ) * Z Z ( i > ♦ Z Z ( 2 ) * Z Z ( 2 )2 2 * SORT( 2 1 >Z 3 * Z Z ( I ) * Z Z ( 4 ) 4 Z Z ( 2 ) * Z Z ( 5 )PLANE( I ) * AT AN2( ZZ( 21 , 2 2 ( 1 ) 1PL ANE !2 1 * ( Z Z ( 1 > * Z Z ( 5 ) - Z Z ! 2 ) * 2 2 ( 4 ) J / 2 1PLANE!3 ) * 0 . 0PLANE!4 ) « - A T A N 2 ( Z Z ( 3 1 . Z 2 )PLANE( 5 ) * ( Z Z ! 3 ) 4 2 3 —Z Z ! 6 ) 4 2 1 > / ( Z 2 * ( 2 1 ♦ Z 2 ( 3 ) * Z 2 I 3 ) > )PLANE!6 ) * 0 . 0 0RETURNEND

250

SUBROUTINE PL OT (N PT)COMMON / AREA4/PLTZ! 2* 8 1 I .Q U T PU T ! 2 . 5 0 0 ) . KCCNT DIMENSION 8 0 U N D ( 8 1 ) . S C A L E ! 1 0 ) . D N U M ( 5 )DATA S C A L E / I . . 2 . , 5 . , 1 0 * . 2 3 . . 5 0 . . 7 5 . . I 0 0 . . 2 0 0 . . 5 0 0 . /OATA DASH, S T A R . P L A N K , X L I N E / * - * . « * • , • • * •DATA C N U M / « 0 * . M « . ' 2 * . ' 3 < . " 4 * /DO 7 I N K * 1 , 2 M R I T E ( 6 « 1 2 )PFAX ■ 0 . 0 0 0 0 1 I * I . N P TI F ! A B S I P L T Z ! I N K . I ) ) . G T .P M A X ) P M A X * A B S ( P L T Z < I N K .I >>

1 CONTINUE0 0 2 I * 1 . 1 0I F I P N A X . L E . S C A L E ! I > * 4 . 6 2 5 ) GOTO 3

2 CONTINUE1 * 10

3 RANGE « S C A L E ! I ) / 4 . 0 0 B IG * S C A L E ! I > * 4 . 6 2 5ISCALE » S C A L E ! I I ♦ 0 . 0 0 1JSCALE * ISCALE44KNT « tDO 7 J * 1 . 3 6SMALL * B IG — RANGEBOUND!1 ) * XLINEI F ! J . E 0 . 1 9 ) GOTO 8DO 4 K * 2 . NPT

4 BOUND!K) * BLANK5 0 0 6 K * I . N P T

I F 1 P L T Z ! I N K . K ) . L E .B I G .A N D .P L T Z ! 1 N K .K ) .C T .S M A L L ) B O U N D ! K ) * S T A R6 CONTINUE

KNT * KNT ♦ 1 I F ( K N T . E 0 . 4 ) GOTO 11 M R 1 T E 1 6 . 1 3 ) BOUND

7 B I G * SMALL

251

W R I T E < 6 . I 2 >GOTO 1 5 DO 9 K * 2 * 8 1 BOUNO(K) * DASH KK * 100 10 K * 1*8 It 20 0OUNOCK) * ONUM(KK)KK s KK M GOTO 5MR1TEC 6# 14 ) JSCALE* BOUND KNT * 0JSCALE * JSCALE - I SCALE GOTO 7 RETURN FORMAT C « l * )FORMAT(20X*81 At >F O R M A T ! 1 2 X * I 6 . 2 X . 8 1 A 1 ) END

MUt14

BLOCK OATAREAL JTH DA *JPH I.K PCOMMON / A R E A I / T I *T L *T N *T O *K P * SPJET *T H IL IM .T H 2L IM *

1PH1LIM*PH2LIM*JTHOA*JPHI*BTHOA«BPHI*RAOES*CMILRODATA T 1 *TL • TN • TO

1 / I . • 0 * 5 • 0 . 2 • 0 * 1 5 /DATA KP • S P J E T • THILIM •TH2LIM

1 / 2 0 . . 4 5 0 . • 2 2 * . 2 0 . /DATA PH1LIM •PH2LIM •JTHDA • JPH I

IOATA

1END

/ 6 1 • BTHDA

/ 1 6 .

• 2 0 *• BPHI• t 6 .

• 5 * 0

/

• 3 . 0

203 0 0 0 . 3 0 0 0 .

6 0 0 • 600* 5 0 .3 1 . 4 3 - . 0 2 4 4 9too. 1 0 0 0 . 1 0 .600* 600* 5 0 .5 1 . 4 3 - . 0 2 4 4 9

3000* 3 0 0 0 .500* 3 0 0 . .O10

5 1 . 4 3 - . 0 2 4 4 9100* 1000* 1 0 .500* 5 0 0 . 5 0 .s 1 . 4 5 - . 0 2 4 4 9

3000* 3 0 0 0 .900* 9 0 0 . 5 0 .

5 I . 4 3 - . 0 2 4 4 9too* 1 0 0 0 . 1 0 .9 0 0 . 9 0 0 . SO.

5 1 . 4 5 - . 0 2 4 4 93000* 3 0 0 0 .

1800* 1 8 0 0 . SO.

5 0 * 4 0 6 - • 0 2 4 4 91 0 0 * 1 0 0 0 * 1 0 * • 1 • 5taoo. 1 8 0 0 . 5 0 *5 0 * 4 0 6 - • 0 2 4 4 9

3 0 0 0 . 3 0 0 0 *1 6 0 0 . 1 8 0 0 * SO*5 1 * 1 0 4 - • 0 2 4 4 91 0 0 . 1 0 0 0 * 1 0 * • 1 • S1 8 0 0 . 1 8 0 0 * 5 0 .5 1 * 1 0 4 - . 0 2 4 4 9

roUl

255

VITA

Thomas M. P e r k i n s , J r . , w as b o m o n M arch 9 , 1945 i n

B a to n R ouge , L o u i s i a n a . He i s t h e s o n o f M r. a n d M rs .

Thomas M. P e r k i n s , S r . , o f C l i n t o n , L o u i s i a n a . He a t t e n d e d

e le m e n ta r y a n d s e c o n d a r y s c h o o l i n C l i n t o n , L o u i s i a n a , g r a d u a

t i n g i n J u n e , 1 9 6 3 .

H is u n d e r g r a d u a te w o rk w as d o n e a t L o u i s i a n a S t a t e

U n i v e r s i t y , g r a d u a t i n g w i t h a B a c h e lo r o f S c ie n c e D e g ree i n

c h e m ic a l e n g i n e e r i n g i n J u n e , 1 9 6 7 .

I n S e p te m b e r , 1 9 6 7 , h e m a r r i e d t h e fo rm e r B a r b a r a H a le y

o f B a to n R o u g e , L o u i s i a n a . On A p r i l 1 2 , 1 9 6 9 , h e becam e

t h e p ro u d f a t h e r o f a s o n , W il l ia m M ahoney . I n S e p te m b e r ,

1 9 6 7 , h e e n t e r e d L o u i s i a n a S t a t e U n i v e r s i t y 's g r a d u a t e s c h o o l

a s a f u l l - t i m e s t u d e n t . D u r in g t h i s t im e h e w as c o m m iss io n e d

th r o u g h ROTC a n d s e r v e d o n a c t i v e d u ty i n t h e U n i te d S t a t e s

A rm /. I n A u g u s t , 1 9 7 1 , h e r e c e i v e d a M a s te r o f S c ie n c e D eg ree

i n C h e m ic a l E n g i n e e r i n g . On F e b r u a r y 2 1 , 1 9 7 3 , h e becam e

t h e p ro u d f a t h e r o f a s e c o n d s o n , M ic h a e l Shaw n. C u r r e n t l y

h e i s e m p lo y e d by M e a su re x C o r p o r a t i o n , C u p e r t i n o , C a l i f o r n i a ,

i n t h e a r e a o f d i g i t a l c o n t r o l . He i s now a c a n d i d a t e f o r a

D o c to r o f P h i lo s o p h y i n t h e D e p a r tm e n t o f C h e m ic a l E n g in e e r

i n g .

EXAMINATION AND THESIS REPORT

Candidate:

Major Field:

Title of Thesis:

Thomas Mahoney P e rk in s , J r .

Chemical Engineering

A Math Model o f a H ell Trained Human Operator Performing A Tracking Task

Approved:

airman

f f Dean of the Graduate School

EXAMINING COMMITTEE:

S' /I,

Date of Examination:

March 1, 1974

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