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NASA-TP-1237 · Optimal guidance and control for investigating aircraft noise-impact reduction

NASA (NTRS) · 1978

Open the PDFPublic domain · NASA (NTRS)Technical Reports

Overview

A methodology for investigating the reduction of community noise impact is reported. This report is concerned with the development of two models to provide data: a guidance generator and an aircraft control generator suitable for various current and advanced types of aircraft. The guidance…

Pages
·
59

Key points

  • The report presents a methodology for investigating the reduction of community noise impact from aircraft.
  • It describes the development of two models: a guidance generator and an aircraft control generator.
  • The guidance generator creates commanded path information based on inputs from a land-use map of the terminal area.
  • The aircraft control generator determines the optimal set of controls to ensure the aircraft follows the commanded path.
  • The complete package of models aids in assessing the effects of different aircraft technologies and flight paths on noise impact.
Frequently asked questions
What is the purpose of the guidance generator?

The guidance generator produces commanded path information based on inputs chosen by an operator from a land-use map.

How does the aircraft control generator function?

The aircraft control generator determines the optimal set of controls required for the aircraft to follow the commanded path.

What are the two models developed in this report?

The two models developed are a guidance generator and an aircraft control generator suitable for various types of aircraft.

Why is noise impact reduction important?

Noise impact reduction is important due to public concern with environmental issues and the increasing significance of aircraft noise effects.

What does the complete package of models provide?

The complete package provides a useful methodology for constructing an interactive graphics tool to assess the effects of different aircraft technologies and minimize noise impact.

Document

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

-l-@/7 1 I I u

-1.6 4 9

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

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Doc number
·
NASA-TP-1237
Publisher
·
NASA (NTRS)
Year
·
1978
Pages
·
59
File size
·
2.0 MB
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