Optimal guidance and control for investigating aircraft noise-impact reduction

NASA TP c.1

NASA Technical Paper 1237

Optimal Guidance and Control

for Investigating Aircraft

Noise-Impact Reduction

Elwdod C. Stewart and Thomas M. Carson ' I

, MAY 1978 ,

NASA

TECH LIBRARY KAFB, NM

I llllll 1 1 1 1 1 II 11111 lllll lllll I I l l 1 Ill

01134480

NASA Technical Paper 12 37

Optimal Guidance and Control

for Investigating Aircraft

Noise-Impact Reduction

Elwood C. Stewart and Thomas M. Carson Aines Research Center

Mofett Field, Cctliforpzia

National Aeronautics and Space Administration Scientific and Technical Information Office i TABLE OF CONTENTS Page

NOTATION . . . . . . . . . . . . . . . . . . . . . iii

SUMMARY . . . . . . . . . . . . . . . . . . . . . 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . 1 .

GENERAL STRUCTURE . . . . . . . . . . . . . . . . . . 3

GUIDANCE GENERATOR . . . . . . . . . . . . . . . .

. 4

Introduction . . . . . . . . . . . . . . . . .

. 4

Case 1 . Lateral and Normal Accelerations . . . . . . .

. 6

Lateral equations (Type A trajectory: arc with free end

points ) . . . . . . . . . . . . . . . . . 6

Normal equations (Type B trajectory: arc with constrained end points) . . . . . . . . . . . . . .

. 11

Lateral turn radius . . . . . . . . . . . . .

. 14

Case 2 . Lateral Acceleration Only . . . . . . . . .

. 15

Case 3 . Normal Acceleration Only . . . . . . . . . .

. 15

Case 4 . Longitudinal Acceleration Only (Type C Trajectory: Longitudinal Acceleration with Fixed End Points) .

. 1 9

Case 5 . Lateral and Longitudinal Accelerations . . . . .

. 1 9

Case 6 . Normal and Longitudinal Accelerations . . . . .

. 20

Case 7 . Lateral. Normal. and Longitudinal Acceleration . .

. 21

AIRCRAFT CONTROL GENERATOR . . . . . . . . . . . . . . . 22

Class A Aircraft: Thrust Angle Fixed . . . . . . . . . 24

Class B Aircraft: Thrust Angle Variable . . . . . . . . 25

Class B1 aircraft: aerodynamics independent of engine

parameter . . . . . . . . . . . . . . . . 26

Class B2 aircraft: aerodynamics dependent on engine

parameter . . . . . . . . . . . . . . . . 27

EXAMPLES . . . . . . . . . . . . . . . . . . . . . 28

CONCLUDING REMARKS . . . . . . . . . . . . . . . . . 33

REFERENCES . . . . . . . . . . . . . . . . . . . . 35

FIGURES . . . . . . . . . . . . . . . . . . . . . 37

i I . . . . .

NOTATION a t o t a l a c c e l e r a t i o n , m / s e c 2

a l o n g i t u d i n a l a c c e l e r a t i o n , m/ sec2

V a normal a c c e l e r a t i o n , m / s e c 2 Y lateral a c c e l e r a t i o n , m / s e c 2 all, d r a g c o e f f i c i e n t CD l i f t c o e f f i c i e n t CL d i - si ( s e e f i g . 2) c B D aerodynamic d r a g , N ground-track d i s t a n c e between way p o i n t s d i L aerodynamic l i f t , N d i r e c t i o n c o s i n e of i t h ground-track segment, R i i t h ground-track segment l e n g t h , m mi R t u r n r a d i u s i n ground t r a c k , m ground-track d i s t a n c e from l a s t way p o i n t t o n e x t t u r n ( s e e f i g . 2 ) si

T t h r u s t , N

V l o n g i t u d i n a l v e l o c i t y , m/sec W weight of v e h i c l e , N X p o s i t i o n v e c t o r i n h o r i z o n t a l p l a n e , X displacement i n x d i r e c t i o n of l o c a l e a r t h - f i x e d c o o r d i n a t e system, m displacement i n y d i r e c t i o n of l o c a l e a r t h - f i x e d c o o r d i n a t e system, m Y

p o s i t i o n v e c t o r i n v e r t i c a l p l a n e , p\, m

Z displacement i n z d i r e c t i o n of l o c a l e a r t h - f i x e d c o o r d i n a t e system, m Az change i n z as a r e s u l t of v e r t i c a l a c c e l e r a t i o n ( s e e f i g . 3 ) iii

I, I. I I - 1 1 1 . 1 , .,...,..-... .- . .. ~

a n g l e of a t t a c k , deg f l i g h t p a t h a n g l e , a n g l e between v e l o c i t y v e c t o r and h o r i z o n t a l p l a n e , deg d i s t a n c e a l o n g t h e t r a j e c t o r y , m f l a p d e f l e c t i o n , deg t h r u s t a n g l e , between body a x i s and t h r u s t v e c t o r , deg change i n i n c l i n a t i o n a n g l e , deg ( s e e f i g . 4 ) change i n heading of j t h curved segment, deg r a d i u s of f l i g h t p a t h i n v e r t i c a l p l a n e , m d i s t a n c e a l o n g ground t r a c k , m r o l l a n g l e , a n g l e about t h e v e l o c i t y v e c t o r , deg heading a n g l e , a n g l e between a r e f e r e n c e d i r e c t i o n i n h o r i z o n t a l p l a n e and p r o j e c t i o n s of V i n h o r i z o n t a l p l a n e , deg Note: I t a l i c s f o r t h e symbol denote q u a n t i t i e s d e f i n e d i n a v e r t i c a l p l a n e and are analogous t o t h o s e n o n i t a l i c i z e d letters i n t h e ground-track p l a n e .

i v I OPTIMAL GUIDANCE AND CONTROL FOR INVESTIGATING AIRCRAFT NOISE-IMPACT IZEDUCTION Elwood C. S t e w a r t and Thomas M. Carson A m e s Research Center S U M M A R Y A s p a r t of a NASA program t o i n v e s t i g a t e t e c h n i c a l approaches f o r i n c r e a s - i n g t h e t e r m i n a l area e f f e c t i v e n e s s of advanced s h o r t - h a u l a i r c r a f t , t h i s r e p o r t i s concerned w i t h a methodology f o r i n v e s t i g a t i n g t h e r e d u c t i o n of com- munity n o i s e impact. There are a number of computer programs a v a i l a b l e f o r t h e g e n e r a t i o n of n o i s e f o o t p r i n t s , a l l of which r e q u i r e v a r i o u s i n p u t d a t a t h a t are u s u a l l y u n a v a i l a b l e . This r e p o r t i s concerned w i t h t h e development of two models t o provide such d a t a : a guidance g e n e r a t o r and an a i r c r a f t con- t r o l g e n e r a t o r s u i t a b l e f o r v a r i o u s c u r r e n t and advanced t y p e s of a i r c r a f t .

The guidance g e n e r a t o r produces t h e commanded p a t h i n f o r m a t i o n from i n p u t s chosen by an o p e r a t o r from a g r a p h i c scope d i s p l a y of a land-use map of t h e t e r m i n a l area. The guidance g e n e r a t o r a l s o produces smoothing a t t h e junc- t i o n s of s t r a i g h t - l i n e p a t h s . The a i r c r a f t c o n t r o l g e n e r a t o r determines t h e o p t i m a l s e t of t h e a v a i l a b l e c o n t r o l s such t h a t t h e a i r c r a f t w i l l f o l l o w t h e commanded p a t h . The s o l u t i o n s f o r t h e c o n t r o l f u n c t i o n s are g i v e n and shown t o be dependent on t h e class of a i r c r a f t t o be c o n s i d e r e d , t h a t is, whether t h e t h r u s t v e c t o r i s r o t a t a b l e and whether t h e t h r u s t v e c t o r a f f e c t s t h e aerodynamic f o r c e s . For t h e c l a s s of a i r c r a f t p o s s e s s i n g a r o t a t a b l e t h r u s t v e c t o r , t h e s o l u t i o n i s redundant; t h i s redundancy i s removed by t h e addi- t i o n a l c o n d i t i o n t h a t t h e n o i s e impact be minimized. I n f o r m a t i o n from b o t h t h e guidance g e n e r a t o r and t h e a i r c r a f t c o n t r o l g e n e r a t o r i s used by t h e f o o t - p r i n t program t o c o n s t r u c t t h e n o i s e f o o t p r i n t .

The complete package of models p r o v i d e s a u s e f u l methodology f o r con- s t r u c t i n g an i n t e r a c t i v e g r a p h i c s t o o l f o r r a p i d l y a s s e s s i n g t h e e f f e c t s of d i f f e r e n t a i r c r a f t t e c h n o l o g i e s , f l i g h t p a t h s , a i r c r a f t mixes, and o t h e r vari- a b l e s , and f o r t h e minimization of n o i s e impact. These e f f e c t s are i l l u s - t r a t e d , u t i l i z i n g an e x i s t i n g f o o t p r i n t program, by s e v e r a l examples which c o n s i s t of t h r e e broad c l a s s e s of CTOL and V/STOL a i r c r a f t .

INTRODUCTION NASA h a s been engaged i n a program t o i n v e s t i g a t e t e c h n i c a l approaches f o r i n c r e a s i n g t h e t e r m i n a l area e f f e c t i v e n e s s of advanced s h o r t - h a u l a i r c r a f t .

The program i s i n t e n d e d t o p r o v i d e guidance t o NASA's advanced a i r c r a f t tech- nology program. The p r i n c i p a l purpose of t h e program i s t o i n v e s t i g a t e t h e i m p a c t of v a r i o u s t e c h n i c a l a l t e r n a t i v e s f o r i n c r e a s i n g t h e e f f e c t i v e n e s s of s h o r t - h a u l a i r c r a f t i n terms of a i r p o r t a i r s i d e c a p a c i t y and d e l a y , community n o i s e , a i r p o l l u t i o n , f u e l consumption, and r i d e q u a l i t y . The t e c h n i c a l a l t e r n a t i v e s i n c l u d e v a r i o u s advanced a i r c r a f t , t h e i r g r e a t e r o p e r a t i o n a l bounds and range of f l i g h t p a t h s , and a v i o n i c s .

T h i s r e p o r t i s concerned w i t h t h e e f f e c t s of a i r c r a f t n o i s e . Because of p u b l i c concern w i t h environmental i s s u e s , t h e importance of t h e e f f e c t s of a i r c r a f t n o i s e has i n c r e a s e d s i g n i f i c a n t l y i n r e c e n t y e a r s . W e are concerned h e r e w i t h d e v e l o p i n g a methodology f o r examining t h e t r a d e - o f f s between tech- n i c a l a l t e r n a t i v e s and community n o i s e impact.

A key t o o l i n t h e assessment o f n o i s e impact has been t h e development o f n o i s e f o o t p r i n t models and programs. Common t o a l l t h e s e models and programs, however, i s t h e need t o g e n e r a t e r e l i a b l e i n p u t i n f o r m a t i o n so as t o r e n d e r t h e r e s u l t s more r e a l i s t i c and t o enhance t h e i r u s e i n examining t h e many v a r i a b l e s and a l t e r n a t i v e s a s s o c i a t e d w i t h n o i s e r e d u c t i o n .

The i n p u t i n f o r m a t i o n i s of two d i s t i n c t t y p e s . F i r s t , a n o i s e f o o t p r i n t program must be s u p p l i e d w i t h a d e s c r i p t i o n of t h e a i r c r a f t p a t h . For t h r e e - dimensional p a t h o p t i m i z a t i o n purposes, t h e d e s i r e d a i r c r a f t p a t h i s b e s t s u p p l i e d by a guidance g e n e r a t o r which u t i l i z e s b a s i c i n f o r m a t i o n from land- use maps. I n t h i s way, t h e guidance g e n e r a t o r g e n e r a t e s d e s i r e d o r o p t i m a l p a t h s which are d i r e c t e d over areas t h a t are less s e n s i t i v e t o n o i s e . The guidance g e n e r a t o r a l s o g e n e r a t e s smoothing, t h a t is, smooth commanded p a t h s a t t h e j u n c t i o n s of s t r a i g h t - l i n e segments s o as t o avoid u n r e a l i s t i c a l l y high a c c e l e r a t i o n s .

Second, t h e n o i s e - f o o t p r i n t program must b e s u p p l i e d w i t h a i r c r a f t con- t r o l f u n c t i o n i n f o r m a t i o n i n o r d e r t o determine t h e n o i s e f o o t p r i n t based on s t o r e d i n f o r m a t i o n i n t h e f o o t p r i n t program. How much of t h e c o n t r o l informa- t i o n i s r e q u i r e d depends on t h e p a r t i c u l a r f o o t p r i n t program. Most programs i n h e r e n t l y assume i s o t r o p i c n o i s e c h a r a c t e r i s t i c s , and f o r them only one of t h e a i r c r a f t c o n t r o l s , t h e engine parameter (such as e n g i n e t h r u s t , t h r o t t l e s e t t i n g , e t c . ) i s r e q u i r e d . A more s o p h i s t i c a t e d f o o t p r i n t program t h a t could account f o r a n i s o t r o p i c n o i s e c h a r a c t e r i s t i c s would r e q u i r e a l l t h e a i r c r a f t c o n t r o l f u n c t i o n s - t h e engine parameter, t h e d i r e c t i o n a l c o n t r o l of t h e a c t i v e t h r u s t i n g f o r c e s , bank a n g l e , and a n g l e of a t t a c k . I n any case, it i s n e c e s s a r y t o relate t h e a i r c r a f t c o n t r o l f u n c t i o n s to t h e p a t h b e i n g flown as s p e c i f i e d by t h e guidance g e n e r a t o r . This i s b e s t accomplished by u t i l i z i n g t h e a i r c r a f t e q u a t i o n s of motion t o d e r i v e an a i r c r a f t c o n t r o l g e n e r a t o r t h a t determines t h e o p t i m a l s e t of t h e a v a i l a b l e c o n t r o l s from which t h o s e r e q u i r e d by t h e f o o t p r i n t program are taken.

I n t h i s r e p o r t , w e d e r i v e t h e models f o r g e n e r a t i n g t h e i n p u t i n f o r m a t i o n r e q u i r e d by n o i s e f o o t p r i n t models o r programs. The two models are a guidance g e n e r a t o r and an a i r c r a f t c o n t r o l g e n e r a t o r s u i t a b l e f o r v a r i o u s t y p e s of cur- r e n t and advanced a i r c r a f t . These two models w i l l be made compatible w i t h a p a r t i c u l a r n o i s e f o o t p r i n t program, widely used i n Government and i n d u s t r y , t h a t w a s developed s e v e r a l y e a r s ago by S e r e n d i p i t y f o r t h e Department of T r a n s p o r t a t i o n ( r e f s . 1 and 2 ) . The models could b e used w i t h a more g e n e r a l f o o t p r i n t program t a k i n g i n t o account a n i s o t r o p i c n o i s e s i n c e a l l t h e r e q u i r e d i n f o r m a t i o n i s a v a i l a b l e from t h e models developed h e r e .

I

The complete package, when implemented, p r o v i d e s a u s e f u l i n t e r a c t i v e g r a p h i c s o p t i m i z a t i o n t o o l f o r r a p i d l y a s s e s s i n g t h e e f f e c t s o f d i f f e r e n t air- c r a f t t e c h n o l o g i e s , f l i g h t p a t h s , a i r c r a f t mixes, and o t h e r v a r i a b l e s , and f o r minimizing of n o i s e e f f e c t s . S e v e r a l examples are given t o i l l u s t r a t e a p p l i - c a t i o n s of t h e methodology t o broad c a t e g o r i e s of advanced a i r c r a f t .

GENERAL STRUCTURE The g e n e r a l s t r u c t u r e o f a system i n c o r p o r a t i n g a n o i s e f o o t p r i n t model, t h e guidance g e n e r a t o r , and t h e a i r c r a f t - c o n t r o l system model are shown i n s k e t c h ( a ) .

STORED INFORYATION COMMANDED TRAJECTORY FOOTPRINT GENERATOR I I ACTUAL TRAJECTORY Sketch ( a ) The guidance g e n e r a t o r produces t h e d e s i r e d t r a j e c t o r y and t h e c o n t r o l system t h e n a t t e m p t s t o make t h e a i r c r a f t f o l l o w t h a t t r a j e c t o r y . The f i n a l element i s t h e f o o t p r i n t program which r e q u i r e s t h r e e k i n d s of i n f o r m a t i o n : (1) s t o r e d i n f o r m a t i o n d e s c r i b i n g t h e n o i s e c h a r a c t e r i s t i c s of t h e a i r c r a f t ; f o r t h e f o o t p r i n t program used h e r e t h i s s t o r e d i n f o r m a t i o n c o n s i s t s of t h e n o i s e c h a r a c t e r i s t i c s expressed as a f u n c t i o n of t h e s l a n t range and t h e engine n o i s e parameter; ( 2 ) i n p u t i n f o r m a t i o n d e s c r i b i n g t h e a i r c r a f t trajec- t o r y ; and (3) i n p u t i n f o r m a t i o n d e f i n i n g t h e v a r i a t i o n of t h e a i r c r a f t con- t r o l s d u r i n g t h e f l i g h t ; f o r t h e f o o t p r i n t program t o b e used h e r e , only t h e engine parameter is u t i l i z e d .

It w i l l be d e s i r a b l e t o modify t h e s t r u c t u r e from t h a t j u s t d i s c u s s e d t o accommodate t h e requirements of t h e p a r t i c u l a r n o i s e f o o t p r i n t program t o be used, and t o a c h i e v e c e r t a i n s i m p l i f i c a t i o n s . I n t h i s r e g a r d , t h e r e are t h r e e c o n s i d e r a t i o n s .

F i r s t , t h e f o o t p r i n t program r e q u i r e s t h e t r a j e c t o r y t o b e parameterized i n t e r m s of segment l e n g t h s , p a r t s of c i r c l e s , and i n c l i n a t i o n a n g l e s .

Second, s i n c e t h e a i r c r a f t can f o l l o w t h e commanded t r a j e c t o r y e x a c t l y ( i n t h e absence of c o n t r o l e r r o r s ) , t h e system s t r u c t u r e shown above can be s i m p l i f i e d as shown i n s k e t c h (b) on t h e f o l l o w i n g page.

Third, as a r e s u l t of t h e modified s t r u c t u r e , i t is n e c e s s a r y t o g e n e r a t e t h e o p t i m a l a i r c r a f t c o n t r o l s based on t h e d e s i r e d t r a j e c t o r y , as i n d i c a t e d i n t h e s k e t c h by t h e a i r c r a f t c o n t r o l g e n e r a t o r . The s i t u a t i o n shown i s f o r t h e f o o t p r i n t program t o b e used h e r e i n which only one o f t h e c o n t r o l s , t h e engine parameter T is used. For a more g e n e r a l program, a l l t h e c o n t r o l s I.

TRAJECTORY I N FORMATION

-f GENERATOR GUIDANCE 1 7 FOOTPRINT

I

PARAMETER

‘ I

qy-e] GENERATOR CONTROLS

Sketch (b) are a v a i l a b l e f o r use. A s w i l l be s e e n later, t h e a d d i t i o n a l c o n t r o l s c o n s i s t of t h e d i r e c t i o n a l c o n t r o l of t h e a c t i v e t h r u s t i n g f o r c e s 11, bank a n g l e $, and a n g l e of a t t a c k a. T h i s i n f o r m a t i o n , i n c o n j u n c t i o n w i t h t h e t r a j e c t o r y i n f o r m a t i o n , p r o v i d e s a l l t h e i n f o r m a t i o n t h a t would be r e q u i r e d by any f o o t - p r i n t program.

GUIDANCE GENERATOR I n t r o d u c t i o n The purpose of t h e guidance g e n e r a t o r i s t o g e n e r a t e i n f o r m a t i o n about t h e d e s i r e d p a t h of t h e a i r c r a f t . I n t h i s s e c t i o n w e w i l l c o n s i d e r how t o g e n e r a t e t h i s i n f o r m a t i o n from a v a i l a b l e d a t a and how t o g e n e r a t e i t i n a form t h a t w i l l b e compatible w i t h t h e e x i s t i n g f o o t p r i n t program.

F i r s t , c o n s i d e r t h e i n p u t i n f o r m a t i o n a v a i l a b l e t o t h e guidance gener- a t o r . The d e t e r m i n a t i o n of t h e d e s i r e d o r o p t i m a l ground t r a c k i n f o r m a t i o n f o r t h e purpose of minimizing n o i s e e f f e c t s must n e c e s s a r i l y s t a r t w i t h land- use maps of t h e a i r p o r t and t h e s u r r o u n d i n g community. This w i l l permit t h e s e l e c t i o n of c a n d i d a t e f l i g h t p a t h s t h a t are d i r e c t e d away from noise- s e n s i t i v e areas. The most convenient way i n which t o d i s p l a y such maps is on a g r a p h i c scope d i s p l a y s i n c e t h i s a c h i e v e s good man-machine i n t e r a c t i o n f o r t h e n o i s e minimization p r o c e s s . I n t h i s form, t h e g r a p h i c s d i s p l a y r e a d i l y i n c o r p o r a t e s p r o v i s i o n f o r t h e experimenter t o s p e c i f y c e r t a i n t r i a l way- p o i n t s t h a t are t h e s t a r t and end p o i n t s of s t r a i g h t - l i n e segments of t h e ground t r a c k . The c o o r d i n a t e s of t h e s e way-points are r e a d i l y s t o r e d by t h e computer and hence are a v a i l a b l e as i n p u t s t o t h e guidance g e n e r a t o r . To completely d e f i n e t h e p a t h of t h e a i r c r a f t i t i s a l s o n e c e s s a r y t o s p e c i f y t h e v e l o c i t y and i n c l i n a t i o n a n g l e a l o n g each segment. And f i n a l l y , i t w i l l be n e c e s s a r y t o s p e c i f y t h e t o l e r a b l e a c c e l e r a t i o n l e v e l s t o e n a b l e t h e genera- t i o n of s u i t a b l y smoothed t r a n s i t i o n s between s t r a i g h t - l i n e segments, as w i l l be d i s c u s s e d s h o r t l y .

Next, c o n s i d e r t h e form of t h e o u t p u t of t h e guidance g e n e r a t o r . S i n c e t h e guidance g e n e r a t o r o u t p u t w i l l b e used d i r e c t l y by t h e e x i s t i n g f o o t p r i n t program which is t o be l e f t unchanged, t h i s o u t p u t must be of a form s u i t a b l e f o r t h e f o o t p r i n t program. The l a t t e r r e q u i r e s t h a t t h e f l i g h t p a t h be d e s c r i b e d by a sequence of segments i n which each segment i s d e f i n e d by t h e f o l l o w i n g f o u r v a r i a b l e s : % o r ek k t h segment l e n g t h i n h o r i z o n t a l p l a n e , m y o r k t h subtended t u r n a n g l e , deg i n c l i n a t i o n a n g l e , climb o r d e s c e n t , f o r k t h segment, deg 'k e n g i n e n o i s e parameter f o r t h e k t h segment, p e r c e n t of maximum Tk t u r n r a d i u s of k t h segment, m Rk 0 = s t r a i g h t

+ = clockwise

- - - counterclockwise The p a r a m e t e r s , mk, ek, and Rk, need t o be provided by t h e guidance g e n e r a t o r , t h e Yk i s s p e c i f i e d a s an i n p u t , and t h e Tk i s t o be determined by t h e a i r c r a f t c o n t r o l g e n e r a t o r .

An i m p o r t a n t c o n s i d e r a t i o n i n t h e g e n e r a t i o n of t h e commanded t r a j e c t o r y i s t h e need t o g e n e r a t e a smooth t r a n s i t i o n between s t r a i g h t - l i n e segments so as t o avoid u n r e a l i s t i c s i t u a t i o n s a t t h e j u n c t i o n of s t r a i g h t - l i n e segments.

T h i s w i l l be done i n t h e g e n e r a l three-dimensional c a s e .

The t r a n s i t i o n between s t r a i g h t - l i n e segments i s c h a r a c t e r i z e d by v a r i o u s combinations of l a t e r a l , normal, and l o n g i t u d i n a l a c c e l e r a t i o n s . There are seven p o s s i b l e combinations of motion on t h e t r a n s i t i o n segments as enumerated by t h e following: 0 P O # O 0 0 # O

0 #o 0

#o 0 0

#O 0 # O

#o 0

# O

# O Po $0

For each of t h e s e c a s e s , i t w i l l b e n e c e s s a r y t o g e n e r a t e t h e i n f o r m a t i o n r e q u i r e d by t h e f o o t p r i n t program. However, many of t h e c a s e s are similar and t h e r e s u l t s f o r some may be c o n s t r u c t e d by combining o t h e r cases, as w i l l b e s e e n i n later developments. B a s i c a l l y , t h e r e are t h r e e t y p e s of t r a n s i t i o n t r a j e c t o r i e s which need t o b e c o n s i d e r e d : Type A: arcs w i t h f r e e end p o i n t s Type B: arcs w i t h c o n s t r a i n e d end p o i n t s Type C: l o n g i t u d i n a l a c c e l e r a t i o n w i t h f i x e d end p o i n t s Rather t h a n d i s c u s s each of t h e s e t y p e s i n g e n e r a l and t h e n a p p l y them t o each of t h e seven p o s s i b l e cases, i t w i l l b e more meaningful t o d i s c u s s them i n t h e c o n t e x t of t h e a p p r o p r i a t e case. I n p a r t i c u l a r , t h e correspondence w i l l b e as f o l l o w s : Type t r a j e c t o r y I Case A 1 (lateral e q u a t f o n s ) 3 3 1 (normal e q u a t i o n s ) C 4 The remainder of t h e seven p o s s i b l e c a s e s can b e c o n s t r u c t e d from t h e r e s u l t s f o r t h e s e b a s i c t r a j e c t o r y t y p e s .

To c l a r i f y t h e i n t e r r e l a t i o n of t h e v a r i o u s c a s e s , a flow graph of t h e computations t o be d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s i s shown i n f i g u r e 1.

The r e c t a n g l e s c o n t a i n t h e v a r i a b l e s t h a t are t o be computed a c c o r d i n g t o t h e a p p r o p r i a t e e q u a t i o n s t o follow. The i n d i c a t e d tests are made t o determine whether $, y, and V are r e q u i r e d t o change. Each p o s s i b i l i t y , r e p r e s e n t i n g one of t h e seven c a s e s , l e a d s t o a d i f f e r e n t p a t h on t h e flow graph. Dotted l i n e s i n d i c a t e a flow of i n f o r m a t i o n i n both d i r e c t i o n s between r e c t a n g l e s i n a manner made clear by t h e a p p r o p r i a t e e q u a t i o n s . The o u t p u t of t h e guidance g e n e r a t o r i s i n d i c a t e d by t h e seven p o s s i b l e o u t p u t s and by a list of t h e o u t p u t v a r i a b l e s .

Case 1. Lateral and Normal A c c e l e r a t i o n s In t h i s s e c t i o n t h e development of t h e e q u a t i o n s n e c e s s a r y t o g e n e r a t e t h e r e q u i r e d o u t p u t s from t h e a v a i l a b l e i n p u t s w i l l be considered. The gen- e r a l c a s e , i n which t h e a i r c r a f t is t u r n i n g and t h e i n c l i n a t i o n a n g l e i s changing, w i l l be c o n s i d e r e d h e r e , and s p e c i a l cases w i l l be d e a l t w i t h later.

Reference should be made t o f i g u r e 1 f o r t h e flow graph c l a r i f y i n g t h e compu- t a t i o n s t o be d i s c u s s e d .

LateraZ equations (Type A trajectory: m e with f r e e end p o i n t s ) - L e t s e l e c t e d on the g r a p h i c map d i s p l a y of t h e h o r i z o n t a l t h e p o i n t s which are where p l a n e be numbered by X g , x i , . . . xk . . .

c o o r d i n a t e s of t h e ground-track and t h e components Xk and Yk r e p r e s e n t t h e F i g u r e 2 i l l u s t r a t e s t h e g e n e r a l s i t u a t i o n of t h e three-dimensional d i s p l a y .

t r a j e c t o r y which i s based on t h e t h r e e chosen p o i n t s The G, X B , and %.

i t h and k t h segments are s t r a i g h t - l i n e segments and t h e j t h segment i s a c i r c u l a r segment which forms t h e smooth t r a n s i t i o n between t h e i t h and k t h segments.

Now c o n s i d e r how t h e q u a n t i t i e s m i and q , r e q u i r e d f o r t h e f o o t p r i n t program, can b e determined from t h e t h r e e g e n e r a l p o i n t s shown i n f i g u r e 2.

I n g e n e r a l , t h e two p o i n t s X, and X B , connected by t h e i t h l i n e , w i l l be

i

1 s e p a r a t e d by a d i s t a n c e d i and t h e l i n e connecting t h e s e two p o i n t s w i l l have a d i r e c t i o n c o s i n e R i g i v e n by R i = ( 3 ) Furthermore, any p o i n t X on t h e i t h l i n e connecting X , and XB i s given by

x = x, + S k i

( 4 ) where s i s t h e d i s t a n c e between X , and X. I n p a r t i c u l a r , f o r t h e tangency

p o i n t s Xp and % shown i n f i g u r e 2 , w e have

X p = X, + s i R i = XB - (di - si)Ri

(5) The c e n t e r of t h e c i r c u l a r arc X, i s given i n t e r m s of Xp and $ by a r e l a t i o n s i m i l a r t o e q u a t i o n ( 4 ) :

Xc = 5 + RRP

( 7 ) where R i s t h e r a d i u s of t h e c i r c u l a r arc and R p and R q are t h e d i r e c t i o n c o s i n e s of t h e r a d i i between Xp and Xc, and X and Xc, r e s p e c t i v e l y . The q d e t e r m i n a t i o n of t h e s e t h r e e q u a n t i t i e s i s d e f e r r e d t e m p o r a r i l y . Now, by from e q u a t i o n s (5)

e q u a t i n g ( 7 ) and (8) and s u b s t i t u t i n g f o r Xp and %

and ( 6 ) w e o b t a i n A(::)= D where A = ( t i -Rk)

D = dilli + R(R

4 - RP)

I

The s o l u t i o n i s I.

and t h e s e are t h e q u a n t i t i e s n e c e s s a r y t o determine t h e segments r e q u i r e d f o r t h e f o o t p r i n t program. That is, t h e i t h and k t h s t r a i g h t - l i n e segments shown i n f i g u r e 2 are m k =

‘k - ‘6

The sequence of c a l c u l a t i o n s i n e q u a t i o n s (11)-(13) needs some c l a r i f i c a - t i o n . A t t h e start, when t h e a i r c r a f t i s t a k i n g o f f , ca = 0, s o t h a t s i from e q u a t i o n (11) i s a l l t h a t i s needed i n e q u a t i o n (12) t o determine m i .

For t h e n e x t segment, q i n e q u a t i o n ( 1 3 ) , t h e s k comes from a second a p p l i c a t i o n o f e q u a t i o n (11) u s i n g t h r e e new p o i n t s s t a r t i n g w i t h Xg w h i l e t h e C B i s known from t h e f i r s t a p p l i c a t i o n of e q u a t i o n (11). I n s i m i l a r f a s h i o n t h e p r o c e s s c o n t i n u e s . On t h e l a s t segment, w e must have s k = dk s o t h a t

% = dk - c B ( l a s t segment) (14)

A s f o r t h e j t h segment, b o t h t h e subtended a n g l e €lj and t h e r a d i u s R are r e q u i r e d f o r t h e f o o t p r i n t program and f o r t h e p r e c e d i n g e q u a t i o n (11).

The subtended a n g l e i s given by t h e fundamental r e l a t i o n The f o o t p r i n t program does n o t r e c o g n i z e a s i g n on t h i s a n g l e s o t h a t t h e i n p u t t o t h e f o o t p r i n t program i s T

e j = ]cos-1 R

R I

P 9 The s i g n i n f o r m a t i o n i s accounted f o r by s p e c i f y i n g R as > O o r <O depending on whether e j w a s clockwise o r counterclockwise. The manner i n which t h i s i s done w i l l be d i s c u s s e d s h o r t l y .

The magnitude of t h e r a d i u s must be determined by t h e a c c e l e r a t i o n t o be allowed. The component of t h e v e l o c i t y i n t h e ground-track h o r i z o n t a l p l a n e f o r t h e j t h segment i s V j cos Yj. Hence, t h e l a t e r a l a c c e l e r a t i o n a$ i s g i v e n by (Vj c o s Y . ) 2 J = a,,, = V . $ COS Y R J j R should be based on t h e t o t a l a c c e l e r a t i o n , r a t h e r than determined from t h i s e q u a t i o n ; f o r t h i s r e a s o n t h e d e t e r m i n a t i o n of R i s d e f e r r e d t e m p o r a r i l y .

It s t i l l remains t o determine t h e e q u a t i o n s f o r R p and R q . It would appear t h a t because of t h e p e r p e n d i c u l a r i t y of t h e r a d i u s and t h e i t h and k t h segments, t h e c o n d i t i o n s

a

would be s u f f i c i e n t t o d e f i n e Rp and R q . However, t h i s i s n o t t h e c a s e . I n t h e approach t o be used h e r e , use w i l l be made of s k e t c h ( c ) .

Sketch (c) W e w i l l determine whether t h e e x t e n s i o n of l i n e i l i e s t o t h e l e f t o r r i g h t of l i n e k . L e t t h e l i n e i be extended beyond p o i n t XB. Any p o i n t on t h i s e x t e n s i o n a d i s t a n c e s from X, is given by

x = x, + S k i

( 2 0 ) o r L e t t i n g x = xy, we have The only unknowns h e r e are y and s and t h e s o l u t i o n f o r y i s xv - x, Y = Y , + a B i i Now i t i s c l e a r from t h e geometry of t h e s i t u a t i o n , r e c a l l i n g from equa- t i o n (3) t h a t ai E c o s 6i ( s e e s k e t c h ( c ) ) , t h a t yy > y 3 l e f t t u r n ai > 0 and yy < y =$ r i g h t t u r n

yr < y * l e f t t u r n

ai < 0 and yy > y =j r i g h t t u r n A l e f t t u r n i m p l i e s t h a t R l i e s on t h e l e f t of segment i w h i l e a r i g h t t u r n i m p l i e s t h a t R l i e s on t h e r i g h t of segment i. Define t h e components of R p and Rq by Now f o r a l e f t t u r n w e have I n a similar way, t h e components of R q are r e l a t e d t o t h e segment k. Thus, we have For a r i g h t t u r n , t h e r e l a t i o n s f o r R p are The components of Rq are r e l a t e d i n similar f a s h i o n t o t h e segment k. Thus 1 0 The p r e c e d i n g r e s u l t s are condensed by t h e f o l l o w i n g :

R p =

R q The o r t h o g o n a l i t y r e q u i r e m e n t s of e q u a t i o n s (18) and (19) are e v i d e n t .

Normal equations (!Type B trajectory: arc with constrained end p o i n t s ) - Consider now t h e motion and normal a c c e l e r a t i o n r e q u i r e d t o change t h e i n c l i - n a t i o n a n g l e . For t h i s purpose t h e new diagram shown i n f i g u r e 3 w i l l be c o n v e n i e n t . Here t h e a t t i t u d e i s d e p i c t e d as a f u n c t i o n of t h e d i s t a n c e CT t r a v e r s e d i n t h e ground t r a c k , and t h e p o i n t corresponding t o X, i n f i g u r e 2 i s d e f i n e d by t o r e f l e c t t h e correspondence. Three such p o i n t s are shown i n f i g u r e 3 . I f t h e normal a c c e l e r a t i o n o c c u r s between t h e s a m e p o i n t s Xp and as f o r t h e l a t e r a l a c c e l e r a t i o n , i t i s clear t h a t t h e t r a n s i t i o n t r a j e c t o r y due t o normal a c c e l e r a t i o n i s one w i t h f i x e d end p o i n t s . Thus, t h i s s i t u a t i o n i s an example of a type B t r a j e c t o r y w i t h c o n s t r a i n e d end p o i n t s .

F i r s t , l e t us determine t h e t r a n s i t i o n t r a j e c t o r y between t h e segments.

The i t h segment h a s t h e d i r e c t i o n c o s i n e z i g i v e n by

where t h e Y i v a l u e i s known, t h a t is, s p e c i f i e d ; s i m i l a r l y f o r zk. It i s i m p o r t a n t t o n o t e h e r e t h a t i n t h e e q u a t i o n s t o f o l l o w , y i s t a k e n t o be p o s i t i v e f o r a n g l e s above t h e h o r i z o n t a l and n e g a t i v e below. Thus, t h e range of i n t e r e s t i s - ~ / 2 < y < ~ / 2 . The p o i n t Zp i s given by i n s p e c t i o n of f i g u r e 3 as I where Az i s t h e i n c r e a s e i n z a l o n g segment j . The c u r v a t u r e r e q u i r e d i n t h e t r a n s i t i o n can b e found by l o c a t i n g t h e c e n t e r Zc of t h e c i r c u l a r arc as follows: are t h e d i r e c t i o n c o s i n e s of t h e r a d i i shown i n f i g u r e 3 , where zp and 2 , and are r e p r e s e n t e d by (30)

zP = ( 1 1 )

The d e t e r m i n a t i o n of t h e d i r e c t i o n c o s i n e s w i l l b e d e f e r r e d t e m p o r a r i l y .

Equating e q u a t i o n s ( 2 8 ) and (29) and u t i l i z i n g e q u a t i o n s (26) and (27) w e o b t a i n The s o l u t i o n f o r t h e unknowns p and Az i s R8 J P =

<P - %

and t h e s e w i l l be used in t h e f o l l o w i n g . It i s of i n t e r e s t t o n o t e t h a t i n t h e l i m i t as yk -+ Y i , Az = R 8 j t a n Y i , and by i n s p e c t i o n of f i g u r e 3 , t h i s can be s e e n t o be the. c o r r e c t l i m i t i n g v a l u e .

The r a d i u s p j u s t determined can now be used t o determine t h e normal a c c e l e r a t i o n on t h e j t h segment: 1 2

F

where e q u a t i o n ( 3 2 ) h a s been u t i l i z e d t o e l i m i n a t e p . T h i s e q u a t i o n w i l l be used i n t h e n e x t s e c t i o n t o determine t h e l a t e r a l t u r n r a d i u s .

The v a l u e of Az determined by e q u a t i o n ( 3 3 ) is used t o e s t a b l i s h a v a l u e f o r t h e i n c l i n a t i o n a n g l e d u r i n g t h e t r a n s i t i o n between s t r a i g h t - l i n e segments. T h i s m a t t e r w a r r a n t s f u r t h e r d i s c u s s i o n . When t h e i n c l i n a t i o n a n g l e s on c o n s e c u t i v e s t r a i g h t - l i n e segments are d i f f e r e n t , t h a t i s , Y i # Yk, t h e i n c l i n a t i o n a n g l e Yj on t h e c i r c u l a r connecting segment w i l l v a r y i n a continuous manner between Y i and Yk. However, t h e n o i s e f o o t p r i n t program only a c c e p t s c o n s t a n t Y v a l u e s . Consequently, i n o r d e r t o m e e t t h i s requirement of t h e f o o t p r i n t program; i t i s n e c e s s a r y t o determine an appro- p r i a t e c o n s t a n t v a l u e f o r Y j . This should be done s o t h a t t h e end p o i n t 2 q i s t h e same v a l u e t h a t would be o b t a i n e d from t h e c i r c u l a r segment. Thus,

Az ‘IP - ‘I9

t a n Y = - = ( 3 5 ) j R e j SP - C q where e q u a t i o n ( 3 3 ) h a s been used t o o b t a i n t h e e x p r e s s i o n on t h e r i g h t s i d e .

It i s s t i l l n e c e s s a r y t o determine t h e d i r e c t i o n c o s i n e s Z p and I,, which have been t a c i t l y assumed t o be known i n t h e p r e c e e d i n g development.

Since p i n f i g u r e 3 i s p e r p e n d i c u l a r t o b o t h t h e i t h and k t h segment, w e have T-

Z i L = o

( 3 6 ) P m

I I Z = o

( 3 7 ) k q Using e q u a t i o n (25) i n t h e above e q u a t i o n s y i e l d s The o r t h o g o n a l i t y r e l a t i o n is n o t s u f f i c i e n t t o determine t h e s i g n s , and more i n f o r m a t i o n i s needed.

Consider now t h e d e t e r m i n a t i o n of t h e s i g n s of t h e d i r e c t i o n c o s i n e s and 2,. From geometric c o n s i d e r a t i o n s i n f i g u r e 3 and t h e way i n which Y is d e f i n e d , it is clear t h a t lies below t h e p a t h Yi 7 Yk * p l i e s above t h e p a t h

Yi < Yk * p

as t h e a i r c r a f t proceeds a l o n g t h e t r a j e c t o r y . Thus f o r Yi > Yk, I n similar f a s h i o n , - nq - -<k For t h e o t h e r c a s e i n which Y i < Yk, t h e components of 1, are

sp = c o s ( Y i + ; ) = - s i n Y i = -qi

and t h e components of Z q are The e q u a t i o n s i n t h i s paragraph are v a l i d f o r - 7 ~ 1 2 < Y < 7 ~ 1 2 . The range of Y i s r e s t r i c t e d because f o r l a r g e a n g l e s n e a r 271 t h e above sequence of e q u a t i o n s b r e a k s down, and because [ - ~ r / 2 , ~ / 2 ] i s t h e range of p r a c t i c a l i n t e r e s t .

The f o r e g o i n g can be summarized by t h e f o l l o w i n g where - n / 2 < Y < 7 ~ 1 2 . The o r t h o g o n a l i t y r e q u i r e m e n t s of e q u a t i o n s ( 3 6 ) and ( 3 7 ) are c l e a r l y s a t i s f i e d .

Lateral turn radius- The d e t e r m i n a t i o n of t h e l a t e r a l t u r n r a d i u s R has been d e f e r r e d u n t i l now s o t h a t i t could b e based on t h e t o t a l a c c e l e r a t i o n .

i s g i v e n by combining e q u a t i o n s (17) and ( 3 4 ) : The t o t a l a c c e l e r a t i o n a, where t h e j s u b s c r i p t h a s been s u p p r e s s e d because t h e v e l o c i t y i s c o n s t a n t on segments i, j , and k. Equation ( 4 2 ) can b e s o l v e d f o r R f o r a s p e c i f i e d v a l u e of ao, and t h e r e s u l t i s S i n c e Y j v a r i e s from Yi t o Yk corresponding to t h e ends of t h e j t h seg- ment, t h e maximum a c c e l e r a t i o n w i l l occur a t e i t h e r end, and a maximum opera- t i o n is r e q u i r e d as shown i n e q u a t i o n ( 4 5 ) below. F u r t h e r , t h e f o o t p r i n t program r e q u i r e s a s i g n convention on R t o i n d i c a t e clockwise o r counter- clockwise t u r n i n g . The c o r r e c t s i g n i s o b t a i n e d from t h e following: s g n ( a i ) (y - y,) > 0 * clockwise * R > 0 ( 4 4 ) sgn(cri)(y - y,) < 0 * counterclockwise R < 0

I

as can be seen from e q u a t i o n ( 2 2 ) and t h e d i s c u s s i o n l e a d i n g up t o it.. Thus, This r e s u l t determines t h e l a t e r a l t u r n r a d i u s such t h a t t h e t o t a l a c c e l e r a - t i o n w i l l n o t exceed a s p e c i f i e d v a l u e . T h i s r e s u l t i s a l s o needed t o com- p l e t e t h e d e t e r m i n a t i o n of t h e normal e q u a t i o n s which have been dependent on R.

Case 2 . Lateral A c c e l e r a t i o n Only T h i s i s a t r i v i a l s p e c i a l c a s e of Case 1, and t h e e q u a t i o n s given t h e r e a r e s u f f i c i e n t . I n Case 2 , t h e r e i s no change i n t h e i n c l i n a t i o n a n g l e between c o n s e c u t i v e segments, and t h i s i n d u d e s s p i r a l t y p e motion. The flow graph of t h e computations involved h e r e are shown i n f i g u r e 1.

Case 3 . Normal A c c e l e r a t i o n Only The guidance e q u a t i o n s f o r t h e case i n which o n l y normal a c c e l e r a t i o n i s r e q u i r e d on t h e t r a n s i t i o n between s t r a i g h t - l i n e segments are considered h e r e .

This i n c l u d e s t h e p l a n a r s o - c a l l e d r'two-segmentlr p a t h s . The t r a j e c t o r y involved i s Type A , t h a t i s , an arc w i t h f r e e end p o i n t s . For t h i s r e a s o n , 1 5 I t h e r e s u l t s f o r this case are n o t o b t a i n a b l e from t h e normal e q u a t i o n s f o r Case 1 s i n c e t h e normal e q u a t i o n s t h e r e w e r e f o r a Type-B t r a j e c t o r y , an a r c w i t h f i x e d end p o i n t s . That i s , t h e j u n c t i o n p o i n t s of t h e t r a n s i t i o n segment f o r Case 1 w e r e determined by t h e l a t e r a l motion and t h e normal motion w a s " f i t " t o t h e s e j u n c t i o n p o i n t s . I n t h e p r e s e n t case t h e l a t e r a l motion is a b s e n t s o t h a t t h e j u n c t i o n p o i n t s f o r t h e normal e q u a t i o n s are f r e e . S i n c e t h e p r e s e n t case i s a Type-A t r a j e c t o r y , it i s b a s i c a l l y t h e s a m e as f o r t h e l a t e r a l e q u a t i o n s of Case 1, and t h e r e s u l t s t h e r e are immediately a p p l i c a b l e .

However, t h e r e are minor d i f f e r e n c e s , and i n o r d e r t o avoid c o n f u s i o n , t h e e q u a t i o n s and diagram of t h e s i t u a t i o n w i l l b e given. Reference should b e made t o f i g u r e 1 f o r a flow graph of t h e computations.

Consider t h e g e n e r a t i o n of t h e segment l e n g t h s r e q u i r e d - b y t h e nois: f o o t p r i n t program. For t h e case b e i n g c o n s i d e r e d , we have I ) = 0 and Y # 0 a t t h e j u n c t i o n between s t r a i g h t - l i n e segments. T h i s s i t u a t i o n i s d e p i c t e d i n f i g u r e 4 . S i n c e t h e e q u a t i o n s f o r t h i s c a s e w i l l be similar t o t h o s e devel- oped earlier f o r t h e lateral case, t h e d i s t a n c e v a r i a b l e s have been i t a l i c i z e d t o d i s t i n g u i s h them from t h o s e shown i n f i g u r e 2. The j u n c t i o n p o i n t s Zp and Zq are given by

zp = z , + sizi = Z B - (di - Si)Zi

where si and CB are t h e q u a n t i t i e s t o be determined. The c e n t e r of t h e cir- c u l a r arc Zc i s g i v e n i n t e r m s of Z and Zq by P 2, = zp + pz,

zc = zq + p z q

( 4 9 ) where p i s t h e r a d i u s of t h e c i r c u l a r arc, and 2 , and Z q are t h e d i r e c t i o n c o s i n e s of t h e two r a d i i shown i n f i g u r e 4 . Equating e q u a t i o n s ( 4 8 ) and ( 4 9 ) and u t i l i z i n g e q u a t i o n s ( 4 6 ) and ( 4 7 ) we o b t a i n BC;) = E where

E = diZi + p(Zq - Zp)

Of t h e v a r i o u s q u a n t i t i e s involved h e r e , are s t i l l given by equa-

z i and Zk

are g i v e n by e q u a t i o n s ( 4 0 ) and (41), and d i i s given t i o n (25), zp and 2, by 1 6 where d i is d e f i n e d i n e q u a t i o n ( 2 ) . The remaining v a r i a b l e , t h e r a d i u s p , must be based s o l e l y on t h e normal e q u a t i o n s s i n c e t h e r e i s now no l a t e r a l motion f o r t h i s c a s e . For a given choice of normal a c c e l e r a t i o n , ay, p t h e j t h segment i s g i v e n by

vj

=Ti

The s o l u t i o n of e q u a t i o n ( 5 0 ) i s and t h e s e are t h e q u a n t i t i e s t h a t are needed i n o r d e r t o determine t h e segment l e n g t h s r e q u i r e d by t h e f o o t p r i n t program. The l e n g t h s of t h e i t h and k t h s t r a i g h t - l i n e segments are t h e n given by

mi = si - ea

( 5 4 ) and they a r e determined s e q u e n t i a l l y i n c o n j u n c t i o n w i t h e q u a t i o n ( 5 3 ) much as h a s been d e s c r i b e d e a r l i e r i n c o n n e c t i o n w i t h e q u a t i o n s (12) and (13). Since t h e f o o t p r i n t program r e q u i r e s segments i n t h e h o r i z o n t a l p l a n e , t h e i t h and k t h segments t o be used i n t h e f o o t p r i n t program are g i v e n by - ( 5 6 ) mi - mi c o s yi = mici The e q u a t i o n s s i m p l i f y on t h e last segment where t h e Y ' s on c o n s e c u t i v e seg- ments are t h e s a m e . Thus i f t h e segment a f t e r t h e k t h segment h a s t h e s a m e i n c l i n a t i o n a n g l e , it i s c l e a r from f i g u r e 2 t h a t e q u a t i o n ( 5 3 ) i s simply Then e q u a t i o n ( 5 5 ) becomes and e q u a t i o n (57) becomes = dk - CB C O S Yk (58) Here t h e dk i s known from e q u a t i o n (2), and t h e eB h a s a l r e a d y been d e t e r - mined. The l e n g t h of t h e j t h segment i s , by o b s e r v a t i o n of f i g u r e 4 , This completes t h e d e t e r m i n a t i o n of t h e segment l e n g t h s m i , m j , and m k r e q u i r e d by t h e n o i s e f o o t p r i n t program.

The i n c l i n a t i o n a n g l e s Y i , Y j , and Yk f o r t h e i t h , j t h , and k t h seg- ments are a l s o r e q u i r e d by t h e n o i s e f o o t p r i n t program. The v a l u e s f o r Y i and Yk are known because they w e r e chosen, and t h e v a l u e of Y j v a r i e s con- t i n u o u s l y between t h e s e v a l u e s . S i n c e a c o n s t a n t v a l u e i s r e q u i r e d by t h e n o i s e f o o t p r i n t program, a s u i t a b l e v a l u e which g i v e s t h e c o r r e c t end p o i n t Zq i s

e ( s i n yi + s i n yk)

n i + nk

B -~ - ~- t a n Y = (60)

j "j <i + <k

This e q u a t i o n can be shown t o reduce t o t h e r e s u l t The remaining v a r i a b l e nceded from t h e guidance g e n e r a t o r i s t h e r a d i u s R which f o r t h i s c a s e w i t h J, = 0 i s This c o n d i t i o n can b e sensed as i n d i c a t e d i n f i g u r e 1 by determining i f con- s e c u t i v e d i r e c t i o n c o s i n e s , R i and Rk, are t h e s a m e ( o r w i t h i n some s m a l l v a l u e E ) . Thus, I R i - Rkl < E * R = O (62) Another v a r i a b l e , u s e f u l f o r l a t e r purposes, i s t h e a n g l e 9 which is given by

cos e = 2 T Z

( 6 3 ) P C l -I From e q u a t i o n ( 6 3 ) i t can b e shown, as e x p e c t e d , t h a t Case 4 . L o n g i t u d i n a l A c c e l e r a t i o n Only (Type C T r a j e c t o r y : L o n g i t u d i n a l A c c e l e r a t i o n w i t h Fixed End P o i n t s ) Case 4 t y p i f i e s t h e Type-C t r a j e c t o r y i n which t h e p a t h i s a s t r a i g h t l i n e i n s p a c e w i t h l o n g i t u d i n a l a c c e l e r a t i o n between f i x e d end p o i n t s . The f i x e d end p o i n t s a r e a r e s u l t of s p e c i f y i n g t h e p o i n t s on a land-use map of t h e t e r m i n a l area between which t h e l o n g i t u d i n a l a c c e l e r a t i o n is t o occur. A s k e t c h of t h e s i t u a t i o n is shown i n f i g u r e 5 , where i s t h e segment on d j which t h e a c c e l e r a t i o n o c c u r s . A flow graph of t h e computations is shown i n f i g u r e 1.

O f t h e v a r i a b l e s t o b e provided t o t h e f o o t p r i n t program, t h e segment l e n g t h s , d i , d j , and dk are g i v e n by e q u a t i o n ( 2 ) . The v a r i a b l e Y j i s con- s t a n t a t t h e chosen v a l u e , and s i n c e t h e r e i s no l a t e r a l motion, R = 0. This l a t t e r c o n d i t i o n can b e sensed by e q u a t i o n ( 6 2 ) .

The a c c e l e r a t i o n between t h e f i x e d end p o i n t s needs t o b e determined t o e n s u r e i t s r e a s o n a b l e n e s s and t o e n a b l e curved t r a n s i t i o n t r a j e c t o r i e s t o b e determined f o r o t h e r c a s e s . For purposes h e r e , w e w i l l estimate t h e a c c e l e r a - t i o n by d e t e r m i n i n g t h e c o n s t a n t v a l u e of a c c e l e r a t i o n r e q u i r e d t o m e e t t h e boundary c o n d i t i o n s , t h a t i s , t h e i n i t i a l and f i n a l v e l o c i t i e s , and t h e d i s - t a n c e t r a v e r s e d . A n a l y s i s shows t h a t a p o i n t m a s s moving i n a s t r a i g h t l i n e between two p o i n t s s e p a r a t e d by a d i s t a n c e A s t a r t i n g w i t h v e l o c i t y V i and ending w i t h v e l o c i t y vk r e q u i r e s a c o n s t a n t v a l u e of a c c e l e r a t i o n given by To apply t h i s t o t h e p r e s e n t c a s e , w e need o n l y t o n o t e from f i g u r e 5 t h a t A = d j / c o s Y . Thus, Case 5 . Lateral and L o n g i t u d i n a l A c c e l e r a t i o n s I n Case 5 , t h e l a t e r a l motion i s c o n s i d e r e d f i r s t and t h e l o n g i t u d i n a l motion i s " f i t " w i t h i n t h e a l l o w a b l e s p a c e . Thus, it i s a combination of a Type-A t r a j e c t o r y f o r t h e l a t e r a l motion and a Type-C t r a j e c t o r y f o r t h e lon- g i t u d i n a l motion. The lateral e q u a t i o n s are g e n e r a l l y t h e s a m e as f o r t h e lateral e q u a t i o n s developed i n Case 1. However, t h e e q u a t i o n f o r d e t e r m i n i n g t h e t u r n r a d i u s w i l l b e modified as shown below. The l o n g i t u d i n a l e q u a t i o n s are t h e same as developed i n Case 4 . A flow graph c l a r i f y i n g t h e computations i s shown i n figure 1.

The l a t e r a l t u r n r a d i u s i s based on t h e t o t a l a c c e l e r a t i o n c o n s i s t i n g of t h e lateral and l o n g i t u d i n a l components and i s g i v e n by

ao2 = a+2 + aV2

where w e have used e q u a t i o n s (17) and ( 6 5 ) . I n t h i s case t h e d i s t a n c e A can b e e x p r e s s e d i n t e r m s of t h e l a t e r a l t u r n r a d i u s R by R 8 j A =--- cos y Using t h e above r e l a t i o n s h i p , w e can then s o l v e e q u a t i o n (67) f o r t h e l a t e r a l t u r n r a d i u s . I n a d d i t i o n , s i n c e t h e v e l o c i t y varies from V i t o V j , i t w i l l be d e s i r a b l e t o e n s u r e t h a t t h e d e s i r e d v a l u e of t o t a l a c c e l e r a t i o n a, i s - n o t exceeded. F u r t h e r , t h e s i g n of R i s g i v e n by e q u a t i o n ( 4 4 ) . Combining t h e s e i d e a s , one o b t a i n s Case 6. Normal and L o n g i t u d i n a l A c c e l e r a t i o n s Case 6 i s s i m i l a r t o Case 5 e x c e p t t h a t t h e l a t e r a l motion i s r e p l a c e d by t h e normal motion. Thus t h i s c a s e is a l s o a combination of a Type-A t r a j e c - t o r y f o r t h e normal motion and a Type-C t r a j e c t o r y f o r t h e l o n g i t u d i n a l motion.

The normal e q u a t i o n s a r e obtained from Case 3 , b u t t h e e q u a t i o n s f o r t h e r a d i u s must be modified as shown below. The l o n g i t u d i n a l e q u a t i o n s are t h e s a m e as developed i n Case 4 . A flow graph of t h e computations r e q u i r e d f o r t h i s c a s e a r e summarized i n f i g u r e 1.

The r a d i u s p i n t h i s c a s e i s determined from t h e t o t a l a c c e l e r a t i o n a o 2 = a 2 + a Y V where w e have used e q u a t i o n s (52) and (65). Now t h e d i s t a n c e A is A = 0 0 (71) Combining t h e l a t t e r two e q u a t i o n s and s o l v i n g f o r p , we o b t a i n

1 [ + ( vk2 - vi2r]1-’2

p = m a x - Vn 20 n = i , k Case 7. Lateral, Normal, and L o n g i t u d i n a l A c c e l e r a t i o n Case 7 combines t h e motion due t o a c c e l e r a t i o n s i n a l l t h r e e a x e s . Thus i t i s a combination of Types A, B , and C t r a j e c t o r i e s . The lateral and normal e q u a t i o n s are g e n e r a l l y t h e s a m e a s f o r Case 1. However, t h e t u r n r a d i u s w i l l b e modified as shown below t o account f o r a l l t h e components of a c c e l e r a t i o n .

The l o n g i t u d i n a l e q u a t i o n s come from Case 4 as i n d i c a t e d below. The flow graph of t h e computations f o r t h i s c a s e i s shown i n f i g u r e 1.

The t u r n r a d i u s R i s determined from t h e t o t a l a c c e l e r a t i o n which is

+ a 2 + aV2 ao2 =

(73) Y The f i r s t component is g i v e n i n t e r m s of R by e q u a t i o n (17) and t h e second component i s given by e q u a t i o n ( 3 4 ) . The t h i r d component i s given by equa- t i o n (65). However, i t is d i f f i c u l t t o determine t h e d i s t a n c e A i n t h a t e q u a t i o n because of t h e combined motions. W e w i l l be c o n s e r v a t i v e and t a k e t h e d i s t a n c e t r a v e r s e d due t o l a t e r a l a c c e l e r a t i o n . Thus t h e d i s t a n c e A i s given by e q u a t i o n ( 6 8 ) where t h e Y i s now Y j from e q u a t i o n ( 3 5 ) . That i s , R0 A = (74) cos Y j Hence w e have To s u i t a b l y l i m i t i t i s n e c e s s a r y t o know where t h e maximum o c c u r s on t h e a, j t h segment. I f V j and c o s Y j are g r e a t e s t a t t h e s a m e end o f t h e j t h segment, t h e maximum a c c e l e r a t i o n w i l l occur a t t h a f p o i n t . I f V j and cos Y j are n o t maximum a t t h e s a m e end of t h e j t h .segment, t h e maximum a c c e l e r a t i o n may t h e o r e t i c a l l y occur anywhere on t h e segment, and it i s d i f f i c u l t t o d e t e r - mine where. However, t h e maximum w i l l n e a r l y always occur where V j i s l a r g e s t s i n c e changes i n c o s Y j are i n h e r e n t l y v e r y s m a l l . Thus w e w i l l b a s e t h e t u r n i n g on t h e a c c e l e r a t i o n a t e i t h e r end of t h e j t h segment.

F u r t h e r , t h e s i g n convention of e q u a t i o n (44) w i l l b e r e q u i r e d . Thus w e have

I

AIRCRAFT CONTROL GENERATOR The g e n e r a t i o n of t h e o p t i m a l a i r c r a f t c o n t r o l s i s t h e s u b j e c t of t h i s s e c t i o n . Given t h e guidance i n f o r m a t i o n , i t i s d e s i r e d t o have t h e a i r c r a f t f o l l o w t h e commanded t r a j e c t o r y by making t h e p r o p e r c h o i c e of t h e a v a i l a b l e c o n t r o l s . One of t h e s e c o n t r o l s , t h e engine parameter i n f o r m a t i o n , w i l l b e r e q u i r e d by t h e n o i s e f o o t p r i n t program. The s o l u t i o n w i l l depend on t h e class of a i r c r a f t t o be c o n s i d e r e d , t h a t is, whether t h e t h r u s t v e c t o r i s r o t a t a b l e and whether t h e t h r u s t a f f e c t s t h e aerodynamic f o r c e s .

The a i r c r a f t e q u a t i o n s of motion w i l l b e needed as a s t a r t i n g p o i n t f o r d e t e r m i n i n g t h e c o n t r o l s . An examination of t h e e q u a t i o n s of motion i n v a r i o u s c o o r d i n a t e systems shows t h e v e l o c i t y c o o r d i n a t e system t o b e t h e most a p p r o p r i a t e . A set of o r t h o g o n a l c o o r d i n a t e s i s chosen t o be i n t h e d i r e c t i o n of t h e v e l o c i t y v e c t o r , p e r p e n d i c u l a r t o t h e v e l o c i t y v e c t o r i n a v e r t i c a l p l a n e , and p e r p e n d i c u l a r t o t h e v e l o c i t y v e c t o r i n a h o r i z o n t a l p l a n e . A diagram which d i s p l a y s t h e v a r i o u s f o r c e s involved i s shown i n f i g u r e 6 . The point-mass dynamic e q u a t i o n s are

mV = T c o s ( a + Q ) - m g s i n Y - D

(76)

mvi. = [L + T s i n ( a + rl) ] c o s 9 - m g cos Y

(77)

mv$ cos y = [L + T s i n ( a + r l ) ] s i n 4

(78) The k i n e m a t i c e q u a t i o n s are

> = v s i n Y

(79) The aerodynamic f o r c e s are given i n g e n e r a l by where t h e i n c l u s i o n of T r e p r e s e n t s t h e e f f e c t of t h r u s t on t h e aerodynamic f o r c e s . These e q u a t i o n s of motion have t h r e e state v a r i a b l e s , V, Y , and $, and f o u r c o n t r o l v a r i a b l e s , T, 17, 4 , and c1 which are t o b e determined. It i s worth n o t i n g t h a t t h e e q u a t i o n s i n t h i s c o o r d i n a t e system s e e m t o r e p r e s e n t a good b a l a n c e i n s i m p l i c i t y between t h e dynamic and k i n e m a t i c e q u a t i o n s . For example, i f t h e dynamic e q u a t i o n s are w r i t t e n i n an e a r t h - f i x e d c o o r d i n a t e system, t h e k i n e m a t i c e q u a t i o n s become s i m p l e r b u t t h e dynamic e q u a t i o n s are made more complicated, and a s o l u t i o n f o r t h e o p t i m a l c o n t r o l s i s d i f f i c u l t t o o b t a i n .

Now c o n s i d e r t h e s o l u t i o n of t h e e q u a t i o n s of motion f o r t h e c o n t r o l s .

Equations ( 7 6 ) t o ( 7 8 ) become, when s l i g h t l y r e a r r a n g e d ,

T cos(cr + n) = mg s i n Y + D + mi ( 8 4 )

m c o s Y + m~

g

L + T s i n ( a + 17) =

cos 4

mv$ c o s y - m(v c o s Y ) ’

L + T sin(c1 + rl) = -

R s i n 4 s i n 4 where i n t h e l a t t e r t h e f o l l o w i n g b a s i c r e l a t i o n s h i p f o r t h e l a t e r a l a c c e l e r a - t i o n a$ has been used: (v c o s Y ) ’ a+ = V$ cos y = R The c o n t r o l f o r 4 c a n be o b t a i n e d uniquely by e q u a t i n g e q u a t i o n s ( 8 5 ) and ( 8 6 ) : VIjJ cos Y - (V c o s Y ) 2 4 = tan-’ - - - tan- 1

g c o s y + v ; R(g c o s y + Vi.)

The c o n t r o l s T and c1 can be r e l a t e d by s o l v i n g e q u a t i o n s ( 8 4 ) and ( 8 5 ) , and t h e r e s u l t i s

~2 = (mg s i n Y + D + m i r > 2

+ (.. cos 4

cos + m~ - Lr

The two c o n t r o l s rl and a c a n b e r e l a t e d by means of e q u a t i o n ( 8 4 )

-1 m g s i n Y + D .+ mV

17 = c o s - - - - - T *) - L

( 9 0 )

(

These t h r e e e q u a t i o n s comprise t h e r e l a t i o n s h i p s t o be s a t i s f i e d f o r t h e f o u r c o n t r o l s .

It is now n e c e s s a r y t o examine t h e c o n d i t i o n s under which T i s c o n s t a n t s i n c e t h e f o o t p r i n t program w i l l a c c e p t only c o n s t a n t v a l u e s . S u f f i c i e n t con- d i t i o n s are Y = V = O (91)

and $ = c o n s t a n t )

Thus, t h e c o n t r o l e q u a t i o n s are

l l = cos-l(mg s i n Y + D

(94) I f t h e e n g i n e parameter T i s n o t c o n s t a n t , i t i s approximated h e r e by a l i n e a r v a r i a t i o n between t h e two c o n s t a n t v a l u e s of t h e p r e c e d i n g and follow- i n g segments. The f o o t p r i n t program c o n t a i n s t h i s c a p a b i l i t y . L e t T i and Tk be t h e c o n s t a n t v a l u e of the engine p a r a m e t e r on t h e i t h and k t h segments and l e t T j be t h e non-constant v a l u e of t h e e n g i n e parameter on t h e j t h segment. Then i f one sets T j = Tk t h e f o o t p r i n t program w i l l cause T j t o vary w i t h t h e d i s t a n c e 0 as where T h i s f u n c t i o n c a u s e s T j t o vary l i n e a r l y from T i t o Tk i n a f i x e d d i s - A b e t t e r procedure would be t o vary t a n c e of 304.8 m (1000 f t ) . over t h e T j a c t u a l computed d i s t a n c e between t h e i t h and k t h p o i n t s . This could be done from e q u a t i o n s g i v e n e a r l i e r , b u t would r e q u i r e m o d i f i c a t i o n t o t h e f o o t p r i n t program.

I f t h e c o n d i t i o n s f o r c o n s t a n t e n g i n e parameter T are m e t , i t w i l l b e n e c e s s a r y t o s o l v e e q u a t i o n s (92) t o (94). The s o l u t i o n of t h e s e e q u a t i o n s depends on t h e class of a i r c r a f t t o be c o n s i d e r e d and t h e r e are s e v e r a l p o s s i - b i l i t i e s . The t h r u s t v e c t o r a n g l e may o r may n o t be r o t a t a b l e and t h e t h r u s t v e c t o r may o r may n o t a f f e c t t h e aerodynamic f o r c e s involved. Thus i n t h e f o l l o w i n g s e c t i o n s , w e w i l l d i s c u s s t h r e e c l a s s e s of a i r c r a f t .

C l a s s A A i r c r a f t : T h r u s t Angle Fixed The Class A a i r c r a f t i s t y p i f i e d by t h e CTOL a i r c r a f t i n which t h e con- t r o l 11, t h e a n g l e of t h r u s t v e c t o r , is f i x e d w i t h r e s p e c t t o t h e a i r c r a f t .

For s i m p l i c i t y , 0 i s t a k e n t o be z e r o . An a i r c r a f t of t h i s class t y p i c a l of t h e c u r r e n t f l e e t i s d e s c r i b e d i n r e f e r e n c e 3 . F u r t h e r , t h e aerodynamic f o r c e s , l i f t , and d r a g are n o t a f f e c t e d by t h e engine t h r u s t s o t h a t t h e y are c h a r a c t e r i z e d by D(V,a) and L(V,a) f o r a f i x e d f l a p s e t t i n g . Thus t h e con- t r o l e q u a t i o n s (92) t o (94) become

m g s i n Y + D(V,a)

T = (97) cos a The f i r s t e q u a t i o n g i v e s cp based on i n f o r m a t i o n from t h e guidance g e n e r a t o r .

The second and t h i r d each e x p r e s s a r e l a t i o n s h i p between t h e two c o n t r o l s T and a, s o t h a t t h e r e i s a unique s o l u t i o n . W e w i l l n o t d e a l w i t h v a r i o u s e s o t e r i c approaches f o r s o l v i n g t h e s e e q u a t i o n s . Here w e w i l l simply s o l v e t h e l a t t e r two e q u a t i o n s s i m u l t a n e o u s l y by a one-dimensional s e a r c h o v e r a such t h a t e q u a t i o n s (96) and (97) are e q u a l . T h i s procedure w i l l be i l l u s - t r a t e d by a l a t e r example.

Class B A i r c r a f t : T h r u s t Angle V a r i a b l e For t h e Class B a i r c r a f t t h e a n g l e of t h e t h r u s t v e c t o r i s r o t a t a b l e , t h u s p r o v i d i n g a n o t h e r degree of c o n t r o l . I n t h i s case e q u a t i o n s (92) t o (94) form a redundant set of t h e form $ = f ( s t a t e s ) (98) cp T2 = f T ( c x , T , $ , s t a t e s ) (99) TI = f n ( a , T , s t a t e s ) (100) These e q u a t i o n s c o n t a i n f o u r c o n t r o l f u n c t i o n s , $, T , a , and and t h r e e s t a t e s , V , y , and $. The redundancy i s c l e a r because T and TI are u n i q u e l y determined from e q u a t i o n s (99) and (100) f o r any a r b i t r a r y a. To remove t h i s redundancy r e q u i r e s a n a d d i t i o n a l c o n d i t i o n . W e w i l l r e q u i r e t h e a d d i t i o n a l c o n d i t i o n t h a t t h e n o i s e be minimized by minimizing t h e engine t h r u s t T. For t h i s r e a s o n , i n t e r e s t c e n t e r s on e i t h e r e q u a t i o n (93) o r (99) i n o r d e r t o s a t i s f y t h e a d d i t i o n a l c o n d i t i o n . The o t h e r c o n t r o l e q u a t i o n s are n o t involved w i t h t h i s a d d i t i o n a l c o n d i t i o n because t h e d e t e r m i n a t i o n of t h e con- t r o l $ from e q u a t i o n (92) o r (98) i s . i n d e p e n d e n t of t h e o t h e r c o n t r o l v a r i a b l e s , and t h e d e t e r m i n a t i o n of q from e q u a t i o n (94) o r (100) f o l l o w s d i r e c t l y from t h e s o l u t i o n f o r t h e c o n t r o l s T and a. Thus, t h e a d d i t i o n a l c o n d i t i o n t o b e imposed is Min T: ~2 = f p ( a , T , $ , s t a t e s ) a s u b j e c t , of c o u r s e , t o any p a r t i c u l a r c o n s t r a i n t s on a and q, depending on t h e type of v e h i c l e . How t h i s e q u a t i o n i s s o l v e d depends on t h e aerodynamic c h a r a c t e r i s t i c s of t h e a i r c r a f t , ' t h a t i s , whether t h e aerodynamic f o r c e s are a f f e c t e d by t h e e n g i n e parameter, e i t h e r unavoidably o r purposely. Thus, t h e r e are two p r i n c i p a l t y p e s t o b e c o n s i d e r e d and two d i s t i n c t l y d i f f e r e n t approaches f o r a s o l u t i o n .

aerodynamics independent of engine parameter- For Class B 1 aircraft: c e r t a i n classes of V/STOL a i r c r a f t , t h e aerodynamic f o r c e s , l i f t , and d r a g , are n o t a f f e c t e d by t h e e n g i n e parameter T. These Class B 1 a i r c r a f t have been widely used i n v a r i o u s s t u d i e s such as t h a t of r e f e r e n c e 4 . Some l i f t - f a n a i r c r a f t are i n t h i s c a t e g o r y and many d e s i g n s t u d i e s have been made f o r t h e s e a i r c r a f t ( r e f s . 5 and 6 ) . The l i f t and d r a g f o r such v e h i c l e s are f u n c t i o n s of o n l y two v a r i a b l e s ; t h a t is, D(V,a) and L(V,a). The consequence of t h i s i s t h a t t h e minimization i n e q u a t i o n (101) becomes of t h e s i m p l e r form Min T: T~ = f T ( a , + , s t a t e s ) a If t h e aerodynamic f o r c e s are g i v e n i n t a b u l a r form, t h e minimization i n e q u a t i o n (102) can be solved by a simple one-dimensional numerical s e a r c h over a.

I f t h e aerodynamtc f o r c e s can b e approximated by a simple e x p r e s s i o n , t h e minimization can b e accomplished a n a l y t i c a l l y . The well-known Kuhn-Tucker problem i s t o Max f ( a ) s u b j e c t t o where a~ i s t h e maximum p e r m i t t e d v a l u e of a. Since w e want t o minimize T , w e set f ( a ) = -T2 where T2 is g i v e n by e q u a t i o n ( 9 3 ) . The approximations t o be used f o r l i f t and d r a g a r e Using t h e s e approximations, w e o b t a i n

f ( a ) = - ~ 2 = - s i n y + - P S V ~ ( D , + D ~ C X

- - pSV2(L, + L

(105)

= - ( a + b a 2 ) 2 - (c - d a ) 2 = - ( a + b a 2 ) 2 - (c - d a ) 2

I

where a , b y c y and d are f u n c t i o n s of Y , V, and 4 ; t h e f u n c t i o n s are I a p p a r e n t from t h e above e q u a t i o n . The i n e q u a l i t y c o n s t r a i n t can b e made i n t o ' a n e q u a l i t y c o n s t r a i n t by t h e s l a c k v a r i a b l e y:

a + y2 - aL = 0 (106)

Thus, t h e Lagrangian f o r t h i s problem is

L = - ( a + b a 2 ) 2 - ( c - d a ) 2 - X(a + y2 - a L )

(107) and t h e n e c e s s a r y c o n d i t i o n s f o r a minimum are

L, = -4b2a3 - 2(2ab + d 2 ) a + 2cd - X = 0

Ly = -2Xy = 0 (109)

LA = -(a + y2 - aL) = 0

From e q u a t i o n (109) it i s s e e n t h a t t h e r e are two p o s s i b l e c a n d i d a t e s f o r t h e optimum. E i t h e r X = 0 o r y = 0 , and i t i s n e c e s s a r y t o examine b o t h p o s s i b i l i t i e s t o see which w i l l l e a d t o t h e s o l u t i o n . If A = 0 , t h e n a i s n o t on t h e boundary and t h e optimum a is o b t a i n e d from t h e s o l u t i o n of t h e c u b i c e q u a t i o n i n a , e q u a t i o n (108) w i t h X = 0. On t h e o t h e r hand, i f y = 0 , t h e optimum a i s on t h e boundary. I n e i t h e r case t h e s o l u t i o n f o r a can b e used t o determine T from e q u a t i o n (105) o r ( 9 3 ) and t h e t h r u s t a n g l e q by e q u a t i o n (94). A l a t e r example w i l l i l l u s t r a t e t h i s d i s c u s s i o n .

Class B 2 aircraft: aerodynamics dependent on engine parameter- For many V/STOL a i r c r a f t t h e e f f e c t of t h e t h r u s t parameter on t h e aerodynamic f o r c e s cannot be n e g l e c t e d . Thus t h e l i f t and d r a g are f u n c t i o n s of t h r e e v a r i a b l e s : L(V,a,T) and D(V,a,T). There are many examples of Class B 2 a i r c r a f t , such as t h e e x t e r n a l l y blown f l a p , t h e augmentor wing, and t h e upper s u r f a c e blowing j e t f l a p a i r c r a f t ( r e f s . 7-10).

For t h i s class of a i r c r a f t , t h e t h r u s t i n g f o r c e i s a f u n c t i o n of a con- t r o l v a r i a b l e 8 , such as t h r o t t l e s e t t i n g and t h e aerodynamic l i f t and d r a g c o e f f i c i e n t s are f u n c t i o n s of two v a r i a b l e s f o r a f i x e d f l a p s e t t i n g : Generally, CL and CD w i l l b e e x p r e s s e d i n g r a p h i c form. The v a r i a b l e 0 i n fL and fD reflects t h e i n t e r a c t i o n of t h r u s t w i t h t h e aerodynamics, whether it i s unavoidably i n t r o d u c e d o r whether i t is p u r p o s e l y i n t r o d u c e d . The con- sequence of t h i s i n t e r a c t i o n i s t h a t t h e minimization i n e q u a t i o n (101) i s d i f f i c u l t because T a p p e a r s on b o t h s i d e s of t h e e q u a t i o n and because CL and CD c o n t a i n e d i n f T w i l l g e n e r a l l y b e i n g r a p h i c form.

The m i n i m i z a t i o n i n e q u a t i o n (101) can b e s o l v e d by a two-dimensional s e a r c h f o r t h e o p t i m a l v a l u e s of T and a. Although v a r i o u s s o p h i s t i c a t e d approaches f o r s o l v i n g e q u a t i o n (101) could be u t i l i z e d , h e r e we w i l l simply do t h e f o l l o w i n g . F i r s t , d e f i n e c 2 t o b e t h e s q u a r e of t h e e r r o r by which e q u a t i o n (99) o r (93) is n o t s a t i s f i e d , t h a t is, Second, examine t h e mapping Txcl -+ E by f l o o d i n g t h e Txa s p a c e . T h i r d , select t h e set (Txa: E = 0 1 , t h a t i s , t h e s e t s a t i s f y i n g t h e e q u a t i o n s of motion. And f o u r t h , c a r r y o u t t h e m i n i m i z a t i o n i n d i c a t e d i n e q u a t i o n (101) by s e l e c t i n g t h e member f o r which T is minimum. More c o n c i s e l y , MinfT: T E (T,a) 3 E: = 01 a I f d e s i r e d , t h e whole p r o c e s s can be c a r r i e d o u t d i r e c t l y on t h e computer s o t h a t r e s u l t s of i n t e r m e d i a t e s t e p s are n o t d i s p l a y e d .

EXAMPLES I n t h i s s e c t i o n we w i l l i l l u s t r a t e t h e r e s u l t s of a p p l y i n g t h e p r e c e d i n g method t o t h e e v a l u a t i o n of n o i s e impact a t a t y p i c a l a i r p o r t . F i g u r e 7 shows a n a c t u a l a i r p o r t , i n c l u d i n g t h e a i r p o r t boundary, t h e runway, and t h e s u r - rounding zoning p a t t e r n as i n d i c a t e d by t h e codes. Each of t h e i r r e g u l a r - shaped areas (even w i t h t h e same zoning) c o r r e s p o n d s t o a d i f f e r e n t l a n d v a l u e , w i t h t h e h i g h e r p r i c e d l a n d i n d i c a t e d by planned r e s i d e n t i a l zoning g e n e r a l l y l y i n g i n t h e s o u t h e r n p o r t i o n of t h e map. The n o i s e impact i s determined by superimposing a n o i s e f o o t p r i n t on t h e zoning p a t t e r n and by e v a l u a t i n g t h e t o t a l v a l u e of l a n d s u b j e c t e d t o u n a c c e p t a b l e n o i s e l e v e l s .

For t h i s purpose w e w i l l c o n s i d e r t h e g e n e r a t i o n of t h e t a k e o f f f o o t p r i n t s f o r t h e v a r i o u s t y p e s of a i r c r a f t . The s a m e e q u a t i o n s a r e v a l i d f o r l a n d i n g s .

F i r s t , w e w i l l c o n s i d e r a n a p p l i c a t i o n t o an a i r c r a f t of Class A , a r e p r e s e n t a t i v e c u r r e n t CTOL a i r c r a f t . For t h i s a i r c r a f t c l a s s t h e t h r u s t a n g l e i s f i x e d and t h e aerodynamic f o r c e s a r e f u n c t i o n s of o n l y v e l o c i t y and a n g l e of a t t a c k f o r a f i x e d f l a p p o s i t i o n . The d a t a needed h e r e f o r such a n a i r c r a f t were t a k e n from r e f e r e n c e 3 and are as f o l l o w s : W = 781,047 N (175,595 l b ) S = 144.93 m2 (1560 f t 2 )

CL = 0.60 + 0.1065

CD = 0.0845 + 1 . 1 3 6 ~ 1 0 - ~ a2

Tmax = 192,154 N (43,200 l b ) The n o i s e s o u r c e c h a r a c t e r i s t i c s , which are needed as s t o r e d i n f o r m a t i o n i n t h e f o o t p r i n t program, c o n s i s t of d a t a d e p i c t i n g EPNdB as a f u n c t i o n of two 2 8 v a r i a b l e s , t h e s l a n t range t o t h e o b s e r v e r and t h e magnitude of t h e engine parameter ( i . e . , t h r u s t l e v e l ) . R e p r e s e n t a t i v e d a t a f o r c u r r e n t CTOL a i r c r a f t are shown i n f i g u r e 8.

A f a c s i m i l e of t h e scope d i s p l a y of t h e a i r p o r t and surrounding zoning p a t t e r n s i s i l l u s t r a t e d i n f i g u r e 9. Shown h e r e i s t h e c o o r d i n a t e system s t a r t i n g a t t h e o r i g i n of t h e runway. Also shown on t h i s f i g u r e i s t h e f l i g h t p a t h p a s s i n g through t h e chosen way-points ( i n d i c a t e d by s m a l l c i r c l e s ) w i t h chosen i n c l i n a t i o n a n g l e s . The way-points are p u t i n d i r e c t l y on t h e scope d i s p l a y by t h e o p e r a t o r and sensed by t h e computer, w h i l e t h e i n c l i n a t i o n a n g l e s are p u t i n on a keyboard. With t h i s i n f o r m a t i o n t h e guidance g e n e r a t o r c a l c u l a t i o n s are made i n accordance w i t h t h e e q u a t i o n s f o r Case 3 and t h e flow diagram of f i g u r e 1. The computations made by t h e guidance g e n e r a t o r are s e n t d i r e c t l y t o t h e f o o t p r i n t program and are n o t normally made a v a i l a b l e t o t h e o p e r a t o r . However, f o r purposes of i l l u s t r a t i o n , t h e s e c a l c u l a t i o n s are shown i n columns 2, 3, and 4 of t h e f o l l o w i n g : Guidance g e n e r a t o r o u t p u t C o n t r o l g e n e r a t o r o u t p u t

Segment I

ILength, m . ( f t ) IY, deg I R , m (ft> I 1 100 0 0 0

1 ~ 1 1,.524.0- (5,000)

Varies -- 0 2 954.87 (3,132.8) 3.75 0 S e t t o 82.6

82.6 0 5.55 3 0 30,004.61 (98,440.3) 1 7.50 I

The a i r c r a f t c o n t r o l g e n e r a t o r c a l c u l a t i o n s a r e determined by t h e solu- t i o n of e q u a t i o n s (95) through (97). The s o l u t i o n of e q u a t i o n s ( 9 6 ) and (97) i n v o l v e s a one-dimensional s e a r c h f o r t h e v a l u e of a which s a t i s f i e s b o t h e q u a t i o n s . This i s i l l u s t r a t e d i n f i g u r e 10 f o r t h e t h i r d segment, t h e only one needed f o r t h i s simple t r a j e c t o r y . Here t h e v a r i a t i o n of e q u a t i o n s (96) and (97) w i t h a is shown, and i t can b e s e e n t h a t t h e r e are two s o l u t i o n s c l o s e t o g e t h e r . The d e s i r e d s o l u t i o n i s t h e one i n d i c a t e d because i t r e q u i r e s less t h r u s t and hence r e s u l t s i n less n o i s e . The c o n t r o l s f o r t h e t h r e e seg- ments are summarized i n t h e l a s t t h r e e columns of t h e above t a b l e .

The i n f o r m a t i o n s e n t t o t h e f o o t p r i n t program c o n s i s t s of columns 2 , 3, 4,and 5 of t h e above t a b l e . The o u t p u t of t h e f o o t p r i n t program i s g i v e n by t h e s i n g l e - e v e n t n o i s e contour i n f i g u r e 9. It i s s e e n t h a t t h e n o i s e impact on t h e s u r r o u n d i n g community is s u b s t a n t i a l . Mqreover, t h e l i m i t e d maneuver- a b i l i t y of t h i s a i r c r a f t r e l a t i v e t o t h e later V/STOL a i r c r a f t means t h a t l i t t l e can b e done t o a l l e v i a t e t h i s e f f e c t . For example, t h e ground-track corresponding t o a 0.1-g t u r n s t a r t i n g a t a n a l t i t u d e of 304.8 m (1000 f t ) i s shown by t h e d o t t e d l i n e i n f i g u r e 9; t h e corresponding contour i s n o t shown b u t i t i s clear t h a t t h e n o i s e impact would s t i l l b e s u b s t a n t i a l because of t h e l i m i t e d m a n e u v e r a b i l i t y of t h i s a i r c r a f t .

The second example is f o r Class B 1 a i r c r a f t which are t y p i f i e d by r o t a t - a b l e t h r u s t v e c t o r s and aerodynamic f o r c e s t h a t are independent of t h r u s t . A v e h i c l e t h a t f i t s t h i s c a t e g o r y approximately i s t h e l i f t - f a n 100-passenger commercial s h o r t - h a u l t r a n s p o r t s t u d i e d i n r e f e r e n c e s 5 and 6 . To avoid r e p e t i t i o n , w e w i l l i l l u s t r a t e . f o r t h i s example o n l y t h e a i r c r a f t c o n t r o l e q u a t i o n s . For t h i s purpose we w i l l take t h e f l i g h t p a t h of t h e a i r c r a f t and t h e guidance e q u a t i o n s t o b e t h e same as f o r t h e n e x t example, s i n c e t h e a i r - c r a f t f o r b o t h examples are h i g h l y maneuverable and can r e a d i l y f o l l o w t h e r e q u i r e d p a t h . T h i s p a t h (shown i n f i g . 11) h a s been chosen as a p o t e n t i a l c a n d i d a t e f o r minimizing t h e n o i s e e f f e c t on t a k e o f f because it l a r g e l y a v o i d s much of t h e r e s i d e n t i a l p r o p e r t y . It might b e noted t h a t t h i s p a t h does n o t r e q u i r e t h e p r e s e n t l i f t - f a n a i r c r a f t t o u t i l i z e i t s f u l l c a p a b i l i t y s i n c e t h e p a t h i s a STOL f l i g h t p a t h whereas t h e l i f t - f a n a i r c r a f t h a s v e r t i c a l f l i g h t c a p a b i l i t y .

The a i r c r a f t c o n t r o l g e n e r a t o r u t i l i z e s t h e o u t p u t of t h e guidance gener- a t o r ( s e e r e s u l t s given i n t h e n e x t example) and t h e b a s i c a i r c r a f t d a t a t o determine t h e o p t i m a l c o n t r o l f u n c t i o n s . The a i r c r a f t c h a r a c t e r i s t i c s needed f o r t h e s i m p l i f i e d model h e r e are taken from r e f e r e n c e s 5 and 6 and are W = 5 6 1 , 7 8 2 N ( 1 2 6 , 3 0 0 l b ) S = 7 3 . 2 1 m2 ( 7 8 8 f t 2 )

CL = 0 . 9 4 + 0 . 1 0 1 7

Cn = 0 . 1 8 + 0 . 0 0 1 3 4 2 a2

L

Tmax - - 6 4 6 , 0 5 0 N ( 1 4 5 , 2 4 5 I b )

"L = 10" where t h e e x p r e s s i o n s f o r CL and CD have been f i t t o aerodynamic d a t a i n t h e above r e f e r e n c e s . The p e r m i s s i b l e magnitude of t h e a n g l e of a t t a c k a~ i s i n f l u e n c e d by several c o n s i d e r a t i o n s . F i r s t , v e r t i c a l g u s t c o n s i d e r a t i o n s r e q u i r e t h a t a n a s a f e t y margin b e provided t o i n s u r e t h a t t h e peak of t h e l i f t c u r v e n o t b e exceeded. Reference 11 i n d i c a t e s t h e magnitude of t h i s s a f e t y margin s h o u l d be sin-1(20/V)o. Second, t h e a n g l e o f a t t a c k should b e l i m i t e d f o r r e a s o n s of passenger comfort, t h a t i s , t o restrict t h e p i t c h a t t i t u d e of t h e a i r c r a f t . A reasonable l i m i t on a t o s a t i s f y b o t h consider- a t i o n s w i l l be t a k e n t o be ciL = 10". The s o l u t i o n of t h e c o n t r o l e q u a t i o n s i s r e q u i r e d f o r each segment of t h e f l i g h t and f o l l o w s p r e v i o u s d i s c u s s i o n .

For segment 3 , f o r example, t h e c o n t r o l + = 0. The d e t e r m i n a t i o n o f t h e con- t r o l a i n v o l v e s s a t i s f y i n g t h e n e c e s s a r y c o n d i t i o n s given by e q u a t i o n s (108) through (110). For t h i s purpose t h e Lagrangian is L = - ( 2 3 , 2 0 2 . 9 2 + 17:576 C X ~ ) ~ - ( 1 1 2 , 2 5 6 . 9 3 - 1 , 3 3 1 . 9 3 9 - A(a + y2 - aL) Now i f X = 0, Ly = 0 i s s a t i s f i e d . Then

L, = 1 2 3 5 . 6 4 a 3 + 5 . 1 7 9 3 8 ~ 1 0 ~ a - 2 . 9 9 0 3 9 ~ 1 0 ~ + A = 0

i s s a t i s f i e d by a = 4 1 . 1 3 " . But t h e n L A = -(a + y2 - 10) = 0 cannot b e

s a t i s f i e d . On t h e o t h e r hand, i f y = 0 , Ly = 0 i s s a t i s f i e d , and LA = 0 i s s a t i s f i e d by (y. on t h e boundary, t h a t i s , a = c i ~ = 10". F i n a l l y , La = 0 i s s a t i s f i e d f o r A = 2.46Ox1O8. Thus, a l l t h r e e n e c e s s a r y c o n d i t i o n s f o r t h e optimum are s a t i s f i e d w i t h a on t h e boundary. The o t h e r c o n t r o l s T and T I are then found u s i n g t h i s v a l u e of c1 and e q u a t i o n s ( 9 3 ) and ( 9 4 ) . The r e s u l t s f o r a l l t h e c o n t r o l s are summarized f o r a l l segments i n t h e following: CONTROL GENERATOR OUTPUT T , % Segment n , deg 0 1 0 100 Varies --

--

S e t t o 70.3

0 l o 3 70.3

Varies -- -- S e t t o 61.7 57.97

0 l o 5 , 7, 9, 11 61.7

6 , 8, 1 0 14.2 1 1 0 64.6 59.0

The t h i r d a p p l i c a t i o n i s t o a v e h i c l e of class B2 which i s c h a r a c t e r i z e d by a r o t a t a b l e t h r u s t v e c t o r and aerodynamic f o r c e s t h a t are s i g n i f i c a n t l y a f f e c t e d by t h r u s t . I n accordance w i t h p r e v i o u s d i s c u s s i o n , t h e c h a r a c t e r i s - t i c s of such v e h i c l e s are r e p r e s e n t e d i n s k e t c h (d) where 0 i s t h e primary engine parameter c o n t r o l l i n g t h r u s t . From an input- o u t p u t viewpoint, t h e i n t e r n a l s t r u c t u r e is unim- -1

p o r t a n t . However, i n o r d e r t o g e n e r a t e and i n t e r p r e t ' ; 1 1 T

t h e input-output c h a r a c t e r i s t i c s , i t i s n e c e s s a r y t o become concerned w i t h t h e i n t e r n a l s t r u c t u r e f o r a p a r t i c u l a r a i r c r a f t . It w i l l b e convenient f o r t h i s CL purpose t o use a r e s e a r c h v e r s i o n of t h e augmentor 01 I 1 CD wing a i r c r a f t because o f t h e ready a v a i l a b i l i t y of I I L - - - - - - t h e r e q u i r e d d a t a .

The augmentor wing a i r c r a f t i s provided w i t h Sketch (d) d i r e c t l i f t by means of t h e h o t e x h a u s t of t h e e n g i n e s which can be r o t a t e d downward from h o r i z o n t a l , c o v e r i n g a range of q from 0 t o 98". I n a d d i t i o n , t h e normal l i f t of t h e wings h a s been augmented by t h e a d d i t i o n of a c o l d a i r d u c t and f l a p r e d e s i g n t h a t e x h a u s t s t h e c o l d a i r a l o n g t h e wing. This r e s u l t s i n added l i f t due t o i n c r e a s e d c i r c u l a t i o n .

For such a v e h i c l e t h e i n t e r n a l s t r u c t u r e i s as shown i n s k e t c h ( e ) , where 0 i s t h e e n g i n e t h r o t t l e a n g l e , T i s t h e t h r u s t due t o t h e h o t e x h a u s t g a s , TC i s t h e c o l d t h r u s t , and CJ is t h e c o l d t h r u s t c o e f f i c i e n t d e f i n e d by CJ = TC/qS. The c h a r a c t e r i s t i c s of t h e u n l a b e l e d boxes w e r e o b t a i n e d from r e f e r e n c e 9 and are shown i n f i g u r e s 1 2 and 13.

I I f f I 1 -c, It is worth r e i t e r a t i n g t h a t t h e

' +CD

I aerodynamic c h a r a c t e r i s t i c s shown are I r e p r e s e n t a t i v e of t h i s class of air- c r a f t i n which t h r u s t i n t e r a c t s w i t h Sketch ( e ) aerodynamic f o r c e s . That is, t h e l i f t and d r a g c o e f f i c i e n t s are i n v a r i a b l y given as a f u n c t i o n of two v a r i a b l e s , t h e a n g l e of a t t a c k and an i n t e r a c t i o n parameter (such as CJ) r e l a t e d t o t h e t h r u s t ( s e e d a t a given i n r e f . 7 f o r a v a r i e t y of c o n f i g u r a t i o n s ) . Although t h e p h y s i c a l mechanism of t h i s i n t e r a c t i o n depends on t h e s p e c i f i c a i r c r a f t t y p e , t h e input-output char- acteristics are a l l d e s c r i b e d by similar k i n d s of d a t a .

The remaining c h a r a c t e r i s t i c s of t h e v e h i c l e under s t u d y are ( s e e r e f . 9 ) : W = 1 7 7 , 9 2 0 N ( 4 0 , 0 0 0 l b ) S = 80.36 m2 ( 8 6 5 f t 2 ) T , , = 6 1 , 6 0 5 N ( 1 3 , 8 5 0 l b ) F i n a l l y , t h e n o i s e c h a r a c t e r i s t i c s may v a r y depending on t h e a i r c r a f t t y p e and t h e l e v e l of n o i s e s u p p r e s s i o n technology i n c o r p o r a t e d i n t o t h e a i r c r a f t .

Since t h e s e n s i t i v i t y of n o i s e e f f e c t t o t h e l e v e l of n o i s e s u p p r e s s i o n i s of i n h e r e n t i n t e r e s t , w e w i l l t a k e t h e two d i f f e r e n t levels, A and B ( f i g . 1 4 ) t o i l l u s t r a t e t h e r e s u l t s . Level B is t a k e n from unpublished FAA d a t a which are approximately t h e s a m e as d a t a g i v e n i n r e f e r e n c e 7 , w h i l e level A h a s been i n c r e a s e d by 10 dB i n o r d e r t o i n v e s t i g a t e t h e s e n s i t i v i t y t o t h i s parameter.

The f l i g h t p a t h chosen f o r t h e a i r c r a f t t o f l y i s shown by t h e f a c s i m i l e of t h e scope d i s p l a y i n f i g u r e 11. The way-points, which are chosen d i r e c t l y on t h e scope d i s p l a y , are i n d i c a t e d by t h e s m a l l c i r c l e s , and t h e i n c l i n a t i o n a n g l e s , which are p u t i n through t h e keyboard, a r e a l s o i n d i c a t e d . With t h i s i n f o r m a t i o n , t h e guidance g e n e r a t o r c a l c u l a t i o n s are made i n accordance w i t h p r e v i o u s d i s c u s s i o n . Again, t h e r e s u l t s of t h e s e computations are normally s e n t d i r e c t l y t o t h e f o o t p r i n t program and are n o t a v a i l a b l e f o r i n s p e c t i o n .

However, t h e r e s u l t s are i l l u s t r a t e d i n t h e following.

GUIDANCE GENERATOR OUTPUT .~ Length - o r a n g l e ,

l------ Segment V , k n o t s

m ( f t ) o r deg I 1 0 0 Varies 1 9 4 . 9 ( 6 3 9 . 4 ) 2 4 . 7 5 0 70 218.2 ( 7 1 5 . 8 ) 3 9.5 0 7 0 1 9 6 . 5 ( 6 4 4 . 6 ) 4 9 . 5 0 Varies 914.4 ( 3 , 0 0 0 ) 5 9 . 5 0 100 239.6 ( 7 8 6 . 2 )

6 9.5 +1050.71 (+3447.2) 100 38. a"

7 9.5 0 100 2 5 4 . 3 ( 8 3 4 . 2 ) 8 9.. 5 +1050.71 (+3447.2) 100 27.4" 9 9 . 5 0 1 0 0 8 4 7 . 2 ( 2 , 7 7 9 . 4 ) 1 0 9.5 +1050.71 (+3447.2) 100 45.8" 9.5 0 6 0 3 7 . 4 ( 1 9 , 8 0 7 . 8 ) 11 100 . .

The i n f o r m a t i o n i n columns 2 , 3 , and 5 are s e n t t o t h e f o o t p r i n t program while t h e i n f o r m a t i o n i n columns 2 , 3 , and 4 are s e n t t o t h e a i r c r a f t c o n t r o l program.

The g e n e r a t i o n of t h e o p t i m a l c o n t r o l s f o l l o w s t h e t h e o r y p r e v i o u s l y d e s c r i b e d . The mapping Txa + E h a s been i n v e s t i g a t e d numerically and t h e 3 2 set o b t a i n e d f o r which E = 0 ( s e e eq. (111)). For i l l u s t r a t i o n , t h i s s e t is d i s p l a y e d i n f i g u r e 1 5 f o r segment 5 of t h e f l i g h t p a t h . A l s o shown i s t h e o t h e r c o n t r o l q. From t h i s f i g u r e , t h e removal of t h e redundancy by t h e min- i m i z a t i o n i n e q u a t i o n (101) s o as t o minimize t h e n o i s e amounts t o simply choosing t h e minimum T s u b j e c t t o c o n s t r a i n t s on c1 and n . It can be s e e n t h a t t h e t r u e minimum T o c c u r s a t e s s e n t i a l l y t h e boundary of t h e 11 con- s t r a i n t q -< 0 as shown. The computer program does n o t l o c a t e t h i s p o i n t p r e c i s e l y because o f t h e f i n i t e increments i n T used ( l i m i t e d t o about 80 l b ) . Thus, t h e computer program s o l u t i o n is s l i g h t l y i n e r r o r . Note t h a t t h i s e r r o r r e s u l t s i n a s i g n i f i c a n t d i f f e r e n c e i n 0 between t h e two p o i n t s .

However, t h i s is an academic p o i n t t h a t i s n o t o f g r e a t concern because t h e e n g i n e parameter T, which determines t h e n o i s e impact, i s e s s e n t i a l l y t h e s a m e f o r e i t h e r s o l u t i o n . A more s o p h i s t i c a t e d o p t i m i z a t i o n procedure could l o c a t e t h e t r u e minimum p r e c i s e l y , b u t t h i s w a s n o t considered n e c e s s a r y h e r e .

A summary of t h e r e s u l t s of c a l c u l a t i o n s f o r a l l t h e c o n t r o l s o v e r t h e e n t i r e f l i g h t p a t h i s shown i n t h e f o l l o w i n g . Of t h e c o n t r o l s shown, only t h e d a t a f o r T needs t o b e s e n t t o t h e f o o t p r i n t program.

CONTROL GENERATOR OUTPUT Segment T, % I 100 Varies S e t t o 55.3 3 55.3 Varies S e t t o 59.78 5 , 7 , 9, 1 1 59.78 6 , 8, 1 0 60.36 __ The o u t p u t of t h e f o o t p r i n t program i s d i s p l a y e d i n 1 6 by t h e s i n g l e - e v e n t 90 EPNdB c o n t o u r s . The two c o n t o u r s r e p r e s e n t t h e r e s u l t s f o r t h e two d i f f e r e n t l e v e l s of technology. For t h e l e v e l A technology, i t i s s e e n t h a t t h e n o i s e e f f e c t on t h e r e s i d e n t i a l p r o p e r t y i s f a i r l y small because of t h e c a p a b i l i t y of t h e a i r c r a f t t o f l y a curved p a t h t h a t a v o i d s most of t h e n o i s e - s e n s i t i v e areas. F u r t h e r , t h e s t e e p f l i g h t - p a t h c a p a b i l i t y c o n t r i b u t e s t o t h i s r e d u c t i o n . On t h e o t h e r hand, t h e l e v e l B technology i s s u f f i c i e n t t o c o n f i n e t h e n o i s e e f f e c t t o w i t h i n t h e a i r p o r t ' b o u n d a r y . Comparisons such as t h i s are of obvious use i n i n v e s t i g a t i n g t h e s e n s i t i v i t y of t h e a c t u a l n o i s e impact t o t h e n o i s e technology level of t h e a i r c r a f t .

CONCLUDING REMARKS A few p o i n t s r e g a r d i n g t h i s paper are worth emphasizing. The methodology p r e s e n t e d h e r e a l l o w s complete f l e x i b i l i t y i n d e t e r m i n i n g t h e d e s i r e d f l i g h t p a t h of t h e a i r c r a f t . T h i s i s i m p o r t a n t because advanced V/STOL a i r c r a f t have much g r e a t e r maneuvering c a p a b i l i t y t h a n t h e c o n v e n t i o n a l c u r r e n t CTOL a i r c r a f t . I f t h e advantages of t h i s g r e a t e r m a n e u v e r a b i l i t y i n terms of i n c r e a s e d a i r s p a c e c a p a c i t y , reduced d e l a y s , reduced n o i s e , etc., are t o b e shown, t h e methodology must be f l e x i b l e enough t o b e a b l e t o account f o r t h e s e g r e a t e r c a p a b i l i t i e s . The methodology d i s c u s s e d h e r e i s f l e x i b l e enough t o p e r m i t t h e simultaneous c o n s i d e r a t i o n of maneuvers i n a l l t h r e e axes. Also, t h e t r a n s i t i o n s between s t r a i g h t - l i n e segments are uniquely d e f i n e d , t h a t is, t h e r e i s no n e c e s s i t y t o iterate between t h e t r a n s i t i o n f l i g h t p a t h s and t h e a c c e l e r a t i o n s .

There are a number of areas f o r f u r t h e r r e s e a r c h and development. F i r s t , s i n c e t h i s r e p o r t w a s r e s t r i c t e d t o t h e development of t h e e q u a t i o n s , f u r t h e r work should d e a l w i t h t h e implementation of t h e computer program of t h e models t o p r o v i d e t h e d e s i r e d on-line i n t e r a c t i v e c a p a b i l i t y . Second, f u r t h e r devel- opment of t h e f o o t p r i n t program t h a t w a s used could be made i n t h e f o l l o w i n g d i r e c t i o n s : 1. The speed of t h e program could b e i n c r e a s e d f o r b e t t e r man-machine i n t e r a c t i o n .

2 . The maximum number of segments allowed could be i n c r e a s e d from t h e p r e s e n t f o u r .

3. A r e l a t i v e l y minor change could be made t o enhance o n - l i n e o p e r a t i o n so as t o g i v e t h e o p e r a t o r c o n t r o l of t h e t e r m i n a t i o n of t h e contour computa- t i o n s r a t h e r t h a n a l l o w i n g t h e program i t s e l f t o t e r m i n a t e t h e computations, sometimes p r e m a t u r e l y .

4 . The program could be changed t o avoid t h e p o s s i b i l i t y of producing an i n c o r r e c t n o i s e l e v e l a t a given o b s e r v e r p o s i t i o n by b a s i n g t h e computations on t h e segment producing t h e g r e a t e s t n o i s e r a t h e r t h a n on t h e c l o s e s t segment.

5. The program could be improved t o i n c o r p o r a t e a n i s o t r o p i c n o i s e c h a r a c t e r i s t i c s .

F i n a l l y , t h e g e n e r a t i o n of t h e a i r c r a f t c o n t r o l f u n c t i o n s w a s n o t d e a l t w i t h a d e q u a t e l y . S o p h i s t i c a t e d o p t i m i z a t i o n methods could b e u t i l i z e d t o g r e a t advantage t o s e e k o u t t h e d e s i r e d s o l u t i o n s more e f f i c i e n t l y t h a n w a s done h e r e .

Ames Research Center N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Moffett F i e l d , C a l i f o r n i a 94035, November 1 7 , 1977 REFERENCES 1. A Study of t h e Magnitude o f T r a n s p o r t a t i o n Noise Generation and P o t e n t i a l Abatement. S e r e n d i p i t y , I n c . f o r Department of T r a n s p o r t a t i o n .

DOT-OS-A9-018, November 1970.

2. S a f e e r , Harvey B.; and W i l l i a m s , Louis J.: A i r p o r t Noise Exposure Con- t o u r Model User Manual. Department of T r a n s p o r t a t i o n , NO. OST-ONA 72-3, August 24, 1972.

3. Snyder, C. Thomas: Analog Study o f t h e L o n g i t u d i n a l Response of a Swept- Wing Transport A i r p l a n e t o Wind Shear and S u s t a i n e d Gusts During Landing Approach. NASA TN D-4477, A p r i l 1968.

4. Mehra, R. K . ; and Bryson, Jr., A. E . : Conjugate Gradient Methods w i t h a n A p p l i c a t i o n t o V/STOL F l i g h t - P a t h Optimization. J o u r n a l of A i r c r a f t , v o l . 6 , no. 2 , March-April 1969.

5. Addendum t o Conceptual Design of V/STOL Lift-Fan Commercial Short-Haul T r a n s p o r t s , 1980-1985. F i n a l Report, v o l . I V , The Boeing Company, March 5 , 1971.

6. Zabinsky, J. M . ; Minker, W . F.; Bohn, J. G . ; Derbyshire, T . ; Middlebrooks, J . E . ; Barron, J. P.; W i l l i a m s , B . ; and Miller, C . W . : V/STOL L i f t Fan Commercial Short-Haul T r a n s p o r t s , NASA CR-2437, p r e p a r e d by Boeing Commercial Airplane Company, March 1974.

7 . Study of Q u i e t Turbofan STOL A i r c r a f t f o r Short-Haul T r a n s p o r t a t i o n .

F i n a l Report, v o l . 11, A i r c r a f t . NASA CR-114607, prepared by Douglas A i r c r a f t Company, June 1973.

8. Quiet Turbofan STOL A i r c r a f t f o r Short-Haul T r a n s p o r t a t i o n . F i n a l Report, v o l . I. NASA CR-114612, prepared by Lockheed A i r c r a f t C o r p o r a t i o n , June 1 4 , 1973.

9. Rumsey, P. C . ; and S p i t z e r , R. E . : Simulator Model S p e c i f i c a t i o n f o r t h e Augmentor Wi'ng J e t STOL Research A i r c r a f t . NASA CR-114434, prepared by The Boeing Company, December 1971.

10. Quigley, H . C . ; I n n i s , R. C . ; and Grossmith, S . : A F l i g h t I n v e s t i g a t i o n of t h e STOL C h a r a c t e r i s t i c s of an Augmented J e t Flap STOL Research A i r c r a f t . NASA TM X-62,334, May 1974.

11. S c o t t , B. C . ; H y n e s , C . S . ; Martin, P. W . ; Bryder, R. B . : P r o g r e s s Toward Development of C i v i l Airworthiness C r i t e r i a f o r Powered-Lift A i r c r a f t . FAA-RD-76-100 and NASA T M X-73,124, p r e p a r e d f o r U . S . Dept.

of T r a n s p o r t a t i o n , May 1976.

I

CORRECT WAY POINT - INPUT R = O Figure 1.- Flow graph of guidance g e n e r a t o r computations.

kth segment ‘k .

jth F i g u r e 2.- I l l u s t r a t i o n f o r Case 1: Lateral and normal a c c e l e r a t i o n s .

. . .

i

\segment i I I

ua I,--- si -

F i g u r e 3 . - I l l u s t r a t i o n f o r Case 1: Normal t r a j e c t o r y w i t h f i x e d end p o i n t s .

Figure 4.- Illustration for Case 3: Normal acceleration only.

V = O

Figure 5.- Illustration for Case 4 : Longitudinal acceleration only.

V E RTI CA L T PLANE mg cos 7 PLANE PERPENDICULAR TO V Y I

I L + T sin L + n ) I sin

HORIZONTAL PLANE Figure 6 . - Diagram of forces.

I I 1538 m

- 1 "NZONED

................... ; : : : ............ (6030 f t )

- 1 PLANNED RESIDENTIAL

- 3 RESIDENTIAL

MANUFACTURING F i g u r e 7.- T y p i c a l a . i r p o r t and s u r r o u n d i n g community.

4 3 I - 100% THRUST - m -a

z

n - I 100 1000 10,000 SLANT RANGE, m Figure 8.- Noise characteristics for example CTOL a i r c r a f t .

F i g u r e 9.- F l i g h t p a t h and 90 EPNdB n o i s e f o o t p r i n t f o r example CTOL a i r c r a f t .

w-

+-

VJ CT I I - 1 00 DESIRED SOLUTION

L 1 . .-

70 1 for example CTOL aircraft.

T and 01 Figure 10.- Solution for controls

H

WAY 1 7,deg 1 V, knots

POINTS 1838 m (6030 ft) 1-2 0 Varies 2-3 4.75 70 I I

3-4 I 9.5 I 70

I Varies

F i g u r e 11.- F l i g h t p a t h f o r example STOL a i r c r a f t .

- 7000 30,000 HOT THRUST, T - 20,000

z

6 4000

I I -

+

- 10,000 - 5 10 15 20 25 30 35 40 THROTTLE ANGLE 8, deg Figure 12.- Hot and c o l d t h r u s t v e r s u s t h r o t t l e a n g l e f o r example j e t STOL a i r c r a f t .

- c -----

:I 0 1 -~

I '

.4 0 0 I -

-

0 -.4

-

U U w

8 -.a

c3

a

a -1.2

n

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I

-

LEVEL A ----LEVEL B a 110 W -100 UJ

5 90

UST Y Lu

>

! 80

v U Lu & w 7n

>

\ \ \ \ \

Lu 60 50

I

100 1000 10,000 SLANT RANGE, m Figure 14.- Noise c h a r a c t e r i s t i c s f o r example j e t STOL a i r c r a f t .

z

rc 0) Q)

s

-a k-.

o s

c n a -10

- 20

-30 -40 -50 50 -60 -2 -1 0 1 2 3 4 5 6 a, deg Figure 15.- Optimal c o n t r o l s f o r example powered-lift STOL a i r c r a f t of Class B 2 ; Segment 5.

Figure 16.- 90 EPNdB n o i s e f o o t p r i n t s f o r example STOL a i r c r a f t .

_ - ..

2. Government Accession No. 3. Recipient's Catalog No.

1. Report No.

NASA TP1237 5. Report Date 4. Title and Subtitle OPTIMAL GUIDANCE AND CONTROL FOR INVESTIGATING AIRCRAFT NOISE-IMPACT REDUCTION 8. Performing Organization Report No.

7. Author(s) Elwood C . S t e w a r t and Thomas M. Carson = _ . _ _ _ - 9. Performing Organization Name and Address NASA Ames Research C e n t e r 11. Contract or Grant No.

M o f f e t t F i e l d , C a l i f . 94035 -- 2. Sponsoring Agency Name and Address N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n 14. Sponsoring Agency Code Washington, D.C. 20546 ~ - 5. Supplementary Notes __ - _ ~ 6 Abstract A s p a r t of a NASA program t o i n v e s t i g a t e t e c h n i c a l approaches f o r i n c r e a s i n g t h e t e r m i n a l a r e a e f f e c t i v e n e s s o f advanced s h o r t - h a u l a i r c r a f t , t h i s r e p o r t is concerned w i t h a methodology f o r i n v e s t i g a t i n g t h e r e d u c t i o n of community n o i s e i m p a c t . There are a number of computer pro- grams a v a i l a b l e f o r t h e g e n e r a t i o n of n o i s e f o o t p r i n t s , a l l of which r e q u i r e v a r i o u s i n p u t d a t a t h a t a r e u s u a l l y u n a v a i l a b l e . T h i s r e p o r t is concerned w i t h t h e development of two models t o pro- v i d e such d a t a : a guidance g e n e r a t o r and an a i r c r a f t c o n t r o l g e n e r a t o r s u i t a b l e f o r v a r i o u s cur- r e n t and advanced t y p e s of a i r c r a f t . The guidance g e n e r a t o r produces t h e commanded p a t h informa- t i o n from i n p u t s chosen by a n o p e r a t o r from a g r a p h i c scope d i s p l a y of a land-use map of t h e t e r m i n a l a r e a . The guidance g e n e r a t o r a l s o produces smoothing a t t h e j u n c t i o n s of s t r a i g h t - l i n e p a t h s . The a i r c r a f t c o n t r o l g e n e r a t o r d e t e r m i n e s t h e o p t i m a l set of t h e a v a i l a b l e c o n t r o l s s u c h t h a t t h e a i r c r a f t w i l l f o l l o w t h e commanded p a t h . The s o l u t i o n s f o r t h e c o n t r o l f u n c t i o n s are g i v e n and shown t o be dependent on t h e class o f a i r c r a f t t o be c o n s i d e r e d , t h a t i s , whether t h e t h r u s t v e c t o r i s r o t a t a b l e and whether t h e t h r u s t v e c t o r a f f e c t s t h e aerodynamic f o r c e s . For t h e c l a s s of a i r c r a f t p o s s e s s i n g a r o t a t a b l e t h r u s t v e c t o r , t h e s o l u t i o n i s r e d u n d a n t ; t h i s redundancy i s removed by t h e a d d i t i o n a l c o n d i t i o n t h a t t h e n o i s e impact be minimized. I n f o r m a t i o n from b o t h t h e guidance g e n e r a t o r and t h e a i r c r a f t c o n t r o l g e n e r a t o r i s used by t h e f o o t p r i n t program t o c o n s t r u c t t h e n o i s e f o o t p r i n t .

The complete package of models p r o v i d e s a u s e f u l methodology f o r c o n s t r u c t i n g an i n t e r a c t i v e g r a p h i c s t o o l f o r r a p i d l y a s s e s s i n g t h e e f f e c t s of d i f f e r e n t a i r c r a f t t e c h n o l o g i e s , f l i g h t p a t h s , a i r c r a f t mixes, and o t h e r v a r i a b l e s , and r o r t h e m i n i m i z a t i o n of n o i s e impact. These e f f e c t s are i l l u s t r a t e d , u t i l i z i n g a n e x i s t i n g f o o t p r i n t program, by s e v e r a l examples which c o n s i s t of t h r e e broad c l a s s e s of CTOL and V/STOL a i r c r a f t .

~ . .

18. Distribution Statement 7. Key Words (Suggested by-Author(s) ) Noise-impact r e d u c t i o n Unlimited A i r c r a f t c o n t r o l A i r c r a f t guidance STAR Category - 08 NASA-Langley, 1978