Configuration management and automatic control of an augmentor wing aircraft with vectored thrust

APPENDIX A AWJSRA ENGINE AND AERODYNAMIC FORCE MODELS AND ELEVATOR TRIM FUNCTION The e n g i n e and aerodynamic f o r c e models used i n t h e c a l c u l a t i o n s of t h i s p a p e r are d e s c r i b e d i n d e t a i l . Both g r a p h s and t a b u l a t e d d a t a are g i v e n f o r t h e v a r i o u s f u n c t i o n s r e q u i r e d f o r t h e models (TH, TC, ;E, CD, CL). I n a d d i t i o n , e l e v a t o r t r i m v a l u e s and e n g i n e f u e l f l o w d a t a used i n t h e F o r c e Trimmap are i n c l u d e d .

\ F u n c t i o n e v a l u a t i o n i s c a r r i e d o u t by i n t e r p o l a t i o n of t h e t a b u l a t e d d a t a u s i n g s i m p l e , i n v e r t i b l e i n t e r p o l a t i o n f o r m u l a s g i v e n i n appendix B. To minimize s t o r a g e and computation time r e q u i r e m e n t s , t a b l e s i z e s have been a p p r o x i m a t e l y minimized w i t h i n t h e a c c u r a c y r e q u i r e m e n t t h a t t h e o r i g i n a l d a t a , based on t h e o r y , wind-tunnel tests, and f l i g h t tests, b e reproduced by t h e f o r c e model t o a n a c c u r a c y of 0.01 g. O t h e r classes of d a t a - f i t t i n g f u n c t i o n s ( s p l i n e f u n c t i o n s and o r t h o g o n a l polynomials) w e r e i n v e s t i g a t e d f o r c o m p u t a t i o n a l e f f i c i e n c y ; however, t h e y w e r e d i f f i c u l t t o i n v e r t and r e q u i r e d as many p a r a m e t e r s as t h e t a b u l a t e d d a t a method i n o r d e r t o o b t a i n +he r e q u i r e d a c c u r a c y . I n a d d i t i o n , t a b u l a t e d models p e r m i t s t r a i g h t f o r w a r d c o n t r o l over t h e maximum e r r o r i n t h e model w h i l e a n a l y t i c a l f u n c t i o n models, whose p a r a m e t e r s are computed v i a l e a s t s q u a r e s t e c h n i q u e s , are designed t o minimize a v e r a g e e r r o r s . The t a b l e i n t e r p o l a t i o n method i s a p p l i e d c o n s i s - t e n t l y throughout t h i s paper and w a s found t o b e s i m p l e and r o u t i n e l y a p p l i - c a b l e i n a l l p h a s e s of t h e work.

Engine Model The AWJSRA is powered by two twin-spool j e t e n g i n e s ( R o l l s Royce Spey M K 801-SF e n g i n e s ) . Some air i s d u c t e d o f f t h e f i r s t compressor and e x h a u s t e d through t h e augmentor f l a p s t o o b t a i n l i f t augmentation; t h e air is a l s o d u c t e d t o t h e a i l e r o n s f o r boundary-layer c o n t r o l . The remaining a i r p a s s e s through t h e e n g i n e n o r m a l l y b u t i s e x h a u s t e d through movable n o z z l e s t o p r o v i d e a j e t t h r u s t v e c t o r which can b e v a r i e d i n d i r e c t i o n over 90". The e n g i n e n o z z l e s and f o r c e s are i l l u s t r a t e d i n f i g u r e 29.

The d i r e c t e n g i n e f o r c e s are t h e t h r u s t from t h e h o t e x h a u s t and t h e r a m d r a g due t o momentum rate of t h e i n l e t a i r . These f o r c e s , r e f e r r e d t o s t a b i l i t y axes, are I n a d d i t i o n , t h e e f f e c t of t h e d u c t e d a i r used f o r l i f t augmentation is i n c l u d e d i n t h e aerodynamic d e s c r i p t i o n through t h e dependence of l i f t and d r a g c o e f f i c i e n t s on t h e "cold t h r u s t c o e f f i c i e n t " L C c E - J QSw where Tc i s t h e e q u i v a l e n t t h r u s t o b t a i n e d from i s e n t r o p i c expansion of t h e compressed d u c t e d a i r t o ambient a t m o s p h e r i c c o n d i t i o n s .

Three p a r a m e t e r s (TH, Tc, &E) i n e q u a t i o n s (Al) and (A2) remain t o b e d e f i n e d . These are f u n c t i o n s of a i r s p e e d , a t m o s p h e r i c p r e s s u r e and tempera- t u r e , and e n g i n e rpm. P l o t s and t a b l e s are g i v e n i n f i g u r e 30 and i n t a b l e 1 f o r t h e l l c o r r e c t e d l ' e n g i n e p a r a m e t e r s 6 6 where ~ , 6 = t e m p e r a t u r e and p r e s s u r e r a t i o s , PSSL,TSSL = s t a n d a r d m i d - l a t i t u d e sea l e v e l a t m o s p h e r i c p r e s s u r e , t e m p e r a t u r e (10.13 N / c m 2 , 288.15" K ) P1,T1 = e n g i n e i n l e t s t a g n a t i o n c o n d i t i o n s These d a t a d e r i v e d from e n g i n e t e s t s t a n d d a t a from t h e m a n u f a c t u r e r and from company r e p o r t s from t h e a i r c r a f t m o d i f i c a t i o n c o n t r a c t o r , i n c l u d e t h e e f f e c t s of power b l e e d and d u c t i n g l o s s e s . The r a t i o s T , 6 are c o n v e n i e n t s i m i l a r - i t y p a r a m e t e r s i n t r o d u c e d s o t h a t e n g i n e t h r u s t and m a s s f l o w a t a l l atmo- s p h e r i c c o n d i t i o n s c a n b e c a l c u l a t e d from t h o s e a t t h e r e f e r e n c e c o n d i t i o n s , T = 6 = 1. The independent v a r i a b l e , NH, i s t h e c o c k p i t tachometer d i a l read- i n g ("power s e t t i n g " ) which, f o r t h i s i n s t a l l a t i o n , i s rpmll21.3. Engine o p e r a t i o n a l l i m i t s on power s e t t i n g are I d l e power NI = 59.3 Maximum c o n t i n u o u s power Nplc = 95.8 (A3 1 Normal T.O. power NTO = 98.5 Emergency power NE = 101.4 J Such l i m i t s o c c u r f o r most j e t e n g i n e i n s t a l l a t i o n s ; t h e y are s e l e c t e d t o y i e l d s p e c i f i e d time i n t e r v a l s between r e q u i r e d major e n g i n e o v e r h a u l s and are marked by d e t e n t s on t h e t h r o t t l e c o n t r o l s .

The t h r o t t l e lever a n g l e , which is used t o c o n t r o l e n g i n e o u t p u t , i s d i r e c t l y r e l a t e d t o power s e t t i n g € o r t h e AWJSRA ( f i g . 30). Engine f u e l f l o w i s a l s o g i v e n i n f i g u r e 30 and t a b l e 1.

The domain o v e r which t h e s e d a t a d e f i n e t h e e n g i n e model 0 G V G 250 k n o t s 0 N H / 6 G 1 0 3 . 5 s u f f i c e s t o c o v e r the r a n g e of a i r s p e e d s of interest f o r t h e AWJSRA and opera- t i o n a t o r below maximum c o n t i n u o u s power f o r t e m p e r a t u r e s above -12' C .

The maximum s t e a d y t h r u s t a v a i l a b l e i n f l i g h t o c c u r s a t maximum c o n t i n u o u s power, Nmc; i t d e c r e a s e s w i t h a l t i t u d e and t e m p e r a t u r e and i n c r e a s e s w i t h a i r s p e e d , as shown i n f i g u r e 31. The maximum c o l d t h r u s t c o e f f i c i e n t a l s o o c c u r s a t Nmc. It varies p r i n c i p a l l y w i t h a i r s p e e d ( f i g . 3 1 ( c ) ) and i s independent of p r e s s u r e a l t i t u d e . A s seen, CJ i s r e s t r i c t e d t o low v a l u e s a t c r u i s e speeds (160 k n o t s ) and t h e r e i s l i t t l e e f f e c t of e n g i n e power on a e r o - dynamic c o e f f i c i e n t s a t h i g h e r speeds. Much h i g h e r v a l u e s of CJ are a v a i l - a b l e a t l a n d i n g s p e e d s (65 k n o t s ) and a t low s p e e d s g e n e r a l l y as d e s i r e d f o r t h e l i f t augmentation system d e s i g n .

Aerodynamic F o r c e s The aerodynamic f o r c e s i n s t a b i l i t y axes are w r i t t e n where CL, C, are t h e u s u a l l i f t and d r a g c o e f f i c i e n t s and Sw i s t h e wing area, 8 0 . 4 m 2 .

The aerodynamic d e s c r i p t i o n of t h e AWJSRA used h e r e is d e r i v e d from r e f e r e n c e 4 , where wing-body l i f t and d r a g c o e f f i c i e n t s are t a b u l a t e d as f u n c t i o n s of t h r e e v a r i a b l e s , {a, 6 f , CJ), o v e r t h e domain cx E [-10.5",27.5"] 6 f E [5.6",72"]

(A5 1

cJ E

The d a t a of r e f e r e n c e 4 w e r e based l a r g e l y on wind-tunnel d a t a , b u t more a c c u r a t e d a t a have s i n c e been o b t a i n e d from f l i g h t tests ( r e f . 6) and are i n c l u d e d i n t h e model of t h i s paper. I n a d d i t i o n , t h e c o e f f i c i e n t s have been c o r r e c t e d f o r t h e symmetric deployment of a i l e r o n w i t h f l a p t h a t i s b u i l t i n t o t h e f l a p mechanism, and f o r t h e t a i l l i f t which i s r e q u i r e d t o t r i m t h e a i r c r a f t . Ground e f f e c t s , which occur a t a l t i t u d e s below 1 6 m are n o t i n c l u d e d i n t h e model.

The t a i l l i f t c o r r e c t i o n i s computed from - 1

- - - - [Icc% i- aLCLm - aDcD,]

a T H e r e , C M ~ is t h e wing-body p i t c h i n g moment, c is t h e mean aerodynamic c o r d , and are t h e moment arms f o r t a i l l i f t and wing-body aT, a L , and a D l i f t and d r a g f o r c e s ( r e f . 4 ) . The t a i l l i f t c o r r e c t i o n c o n t a i n s o n l y func- t i o n s of {a, 6 f , CJ} and can b e i n c l u d e d i n t h e l i f t c o e f f i c i e n t w i t h o u t i n c r e a s i n g t h e number of independent v a r i a b l e s . V a r i a t i o n s i n c . g . l o c a t i o n w i t h a i r c r a f t weight and t h e p i t c h i n g moment due t o e n g i n e t h r u s t have o n l y n e g l i g i b l e e f f e c t on t h e t a i l l i f t and are n e g l e c t e d .

The r e s u l t i n g aerodynamic model is t a b u l a t e d i n t a b l e 2 and p i c t u r e d i n f i g u r e 32 as l i f t - d r a g p o l a r s f o r f o u r f l a p s e t t i n g s , w i t h l i n e s of c o n s t a n t ct and CJ mapped on each p l o t . During approach and l a n d i n g t h e f l a p s e t t i n g p r o g r e s s e s s l o w l y from 5.6" a t c r u i s e t o 65" a t l a n d i n g . A t c r u i s e , e n g i n e power i s seen t o have l i t t l e e f f e c t on l i f t and r e d u c e s t h e aerodynamic d r a g by e x h a u s t i n g p r e s s u r i z e d a i r rearward through t h e f l a p . I n any c a s e , t h e r e i s l i t t l e a v a i l a b l e r a n g e of CJ a t c r u i s e s p e e d s and c o r r e s p o n d i n g l y l i t t l e e f f e c t of e n g i n e power s o t h a t t h e r e s u l t i n g model a t c r u i s e c l o s e l y resembles t h a t of a c o n v e n t i o n a l a i r c r a f t and h a s a maximum CL of a b o u t 2 . A t low s p e e d s and maximum f l a p t h e a v a i l a b l e range of CJ i s much g r e a t e r and i t s e f f e c t s on l i f t c o e f f i c i e n t i s pronounced w i t h maximum i n t h e r a n g e of 4 CL t o 5.

The purpose of a c o n v e n t i o n a l f l a p i s t o i n c r e a s e t h e maximum CL a v a i l - a b l e from t h e wing i n o r d e r t o p e r m i t lower l a n d i n g speeds. The s p e c i a l f l a p of t h e augmentor wing h a s t h e s a m e purpose b u t o b t a i n s a h i g h e r than a C L ~ ~ ~ c o n v e n t i o n a l f l a p i n o r d e r t o o b t a i n s u f f i c i e n t l y low l a n d i n g s p e e d s f o r s h o r t f i e l d l e n g t h l a n d i n g s . Thus, t h e p o l a r p l o t s i n f i g u r e 3 2 a l t h o u g h roughly s i m i l a r t o those f o r c o n v e n t i o n a l a i r c r a f t , r e q u i r e a n a d d i t i o n a l p a r a m e t e r , C J , t o a c c o u n t f o r e n g i n e power e f f e c t s ; they a l s o e x h i b i t l a r g e n o n l i n e a r v a r i a t i o n s w i t h f l a p s e t t i n g . A s a r e s u l t , t h e d a t a r e q u i r e d t o d e f i n e t h e aerodynamic f o r c e model i s g r e a t l y i n c r e a s e d compared t o t h e s i n g l e l i f t - d r a g p o l a r r e q u i r e d f o r CTOL a i r c r a f t .

E l e v a t o r Trimmap I n t h e a n a l y s i s of t h e trim e q u a t i o n i n t h i s paper t h e c o n t r o l v a r i a b l e s are t a k e n a s of which ( 6 f , 6 t , v ) are d i r e c t l y c o n t r o l l e d by t h e a u t o p i l o t through s e r v o s , and ct is c o n t r o l l e d by commands t o t h e e l e v a t o r s e r v o . T h e r e f o r e t h e rela- t i o n s h i p 6e(Z) i s n e c e s s a r y f o r t h e implementation of t h e a u t o p i l o t and con- - s t r a i n t s on e l e v a t o r u s a g e w i l l c o r r e s p o n d i n g l y l i m i t t h e u s a b l e r a n g e of u.

The e l e v a t o r a f f e c t s t h e a i r c r a f t f o r c e s and moments through t h e t a i l l i f t , which i s modeled by t h e l i n e a r r e l a t i o n

LT = LTo + L6e6e

w i t h e l e v a t o r travel l i m i t e d t o 6 e E [-25°,150] and E l e v a t o r s t a l l can b e n e g l e c t e d s i n c e i t h a s n o t been observed f o r a n g l e s of a t t a c k below wing s t a l l . I n a d d i t i o n , r e d u c t i o n o f dynamic p r e s s u r e a t t h e t a i l due t o t h e wing wake p r e s e n c e is a l s o n e g l i g i b l e b e c a u s e of t h e T - t a i l d e s i g n of t h e a i r c r a f t . i s t h e t a i l l i f t a t z e r o e l e v a t o r a n g l e and c a n LT, b e w r i t t e n as where

AaT = cx - E + LT - a0

T E = t a i l downwash a n g l e iT = t a i l i n c i d e n c e a = a n g l e of a t t a c k f o r which t a i l l i f t i s z e r o O T ~ C L ~ = - C

LTa a@

The downwash a n g l e i s g e n e r a t e d from a complex model ( r e f . 4 ) whose H e r e , C ~ T is t h e wing a n g l e of a t t a c k independent v a r i a b l e s are { S f , C ~ T , CJ}.

delayed by t h e t i m e i t t a k e s f o r a i r t o r e a c h t h e t a i l from t h e wing: a = a ( t - At) T aT A t = - vA where aT i s t h e d i s t a n c e between wing and t a i l c e n t e r of p r e s s u r e . For s t e a d y f l i g h t CXT and a are e q u a l . Values o f t h e remaining p a r a m e t e r s of t h e t a i l l i f t model, C L ~ ~ , . . ., aOT, are g i v e n i n r e f e r e n c e 4 .

5 2 The e l e v a t o r i s used t o c o n t r o l t h e p i t c h i n g moment b a l a n c e , g i v e n by are d i m e n s i o n l e s s e n g i n e f o r c e s H e r e , CDR, CT and are t h e moment arms of t h e e n g i n e and t a i l l i f t f o r c e s .

aI, a E , aT These moment a r m s are computed from a nominal c e n t e r of g r a v i t y l o c a t i o n .

Changes i n t h i s l o c a t i o n w i t h a i r c r a f t weight are s m a l l and can be n e g l e c t e d i n computing t h e t a i l l i f t and e l e v a t o r s e t t i n g f o r trim. Equation (A9) c a n b e s o l v e d f o r be and t h e r e s u l t i n g terms a r r a n g e d i n t h r e e p a r t s t o s e p a r a t e t h e e f f e c t s of dynamic r e a c t i o n s , e n g i n e f o r c e s , and aerodynamic f o r c e s and moments .

6 e = beD + 6 e E + &eA

6 e D = '

CLT, sW - -

6 e A = - (ccMWB + a c

A,T

LWB - a D C D ~ ~ ) CL6esTaT c~~~

I n t h e p r e s e n t a n a l y s i s t h e e l e v a t o r s e t t i n g f o r s t e a d y o r v e r y n e a r l y s t e a d y f l i g h t c o n d i t i o n s a l o n g t h e nominal p a t h is of i n t e r e s t so t h a t t h e dynamic t e r m can be n e g l e c t e d and t h e n t h e e l e v a t o r s e t t i n g f o r t r i m i s

6 e T = beA(a,df,CJ) + 6 e E

( A l l ) The p r i n c i p a l t e r m i n e q u a t i o n ( A l l ) i s b e A , which i s a f u n c t i o n of o n l y {ci,Gf,Cj} and can b e c a l c u l a t e d and t a b u l a t e d over t h e s a m e g r i d f o r which 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 g i v e n . The r e s u l t s are shown i n f i g u r e 33 and t a b l e 3 . Ground e f f e c t s (which are of a s i g n i f i c a n t s i z e ) are o m i t t e d from 6 e A .

The p h y s i c a l l i m i t s of e l e v a t o r t r a v e l (eq. (A8)) l i m i t t h e r a n g e of f o r which t r i m is p o s s i b l e . Under dynamic c o n d i t i o n s , t h e e l e v a t o r is a l s o used t r a n s i e n t l y t o c o n t r o l a t t i t u d e and t h e r e f o r e t h e allowed trim s e t t i n g s must b e f u r t h e r r e s t r i c t e d w i t h i n e q u a t i o n (A8) t o p r o v i d e a margin f o r t h e a t t i t u d e c o n t r o l f u n c t i o n . For t h e AWJSRA d e s i g n t h i s margin w a s t a k e n s i m p l y as 8" and it w a s found t h a t t h e c o n s t r a i n t s on e l e v a t o r usage had a l m o s t no i n f l u e n c e i n e s t a b l i s h i n g t h e b o u n d a r i e s of t h e f l i g h t envelope o r t h e r a n g e of a c c e p t a b l e c o n f i g u r a t i o n s a t any f l i g h t c o n d i t i o n w i t h i n t h e envelope.

APPENDIX B SOLUTION OF THE TRIM EQUATIONS S o l u t i o n of t h e a i r c r a f t t r i m e q u a t i o n s f o r two c o n t r o l v a r i a b l e s is r e q u i r e d e x t e n s i v e l y i n computing t h e c o n f i g u r a t i o n s c h e d u l e and is proposed as p a r t of t h e Trimmap element of t h e f l i g h t - c o n t r o l l o g i c ( f i g s . 1, 2 , and l o ) . These e q u a t i o n s are n o n l i n e a r and are s o l v e d n u m e r i c a l l y i n t h e case of t h e AWJSRA; t h i s i s expected t o b e t h e case f o r p o w e r e d - l i f t a i r c r a f t gener- a l l y s o t h a t e f f i c i e n t g e n e r a l approaches t o t h e s o l u t i o n of such e q u a t i o n s are of i n t e r e s t i n t h e c o n t r o l system d e s i g n .

An e x h a u s t i v e g r i d s e a r c h p r o c e d u r e w a s used i n t h i s work and is d e s c r i b e d i n t h i s appendix. T h i s method is c o n c e p t u a l l y s i m p l e ; d a t a d e f i n i n g t h e aerodynamic and e n g i n e f o r c e s a r e assumed g i v e n as t a b l e s of v a l u e s on a g r i d of p o i n t s c o v e r i n g t h e domain of i n t e r e s t . T a b l e i n t e r p o l a t i o n i s used t o p r o v i d e a p i e c e w i s e l i n e a r model on each p i e c e of t h e g r i d and t h e s e l i n e a r e q u a t i o n s are s o l v e d p i e c e by p i e c e u n t i l t h e p i e c e c o n t a i n i n g t h e s o l u t i o n i s found o r t h e g r i d i s e x h a u s t e d . The e x i s t e n c e of a s o l u t i o n is determined i n a f i n i t e number of s t e p s , and i t s uniqueness is determined from p r o p e r t i e s of t h e model d a t a .

A l t e r n a t i v e methods based on Newtonian i t e r a t i o n w e r e a l s o i n v e s t i g a t e d .

The classical Newton method i s s i m p l e and w e l l known b u t s u b j e c t t o a n a r r a y of d i f f i c u l t i e s i n p r a c t i c e a f f e c t i n g b o t h r e l i a b i l i t y and c o m p u t a t i o n a l r e q u i r e m e n t s . These d i f f i c u l t i e s m o t i v a t e t h e m o d i f i c a t i o n s found i n a v a r i e t y of quasi-Newton methods ( r e f . 13), such as t h e Newton-Powell a l g o r i t h m ( r e f . 1 4 ) . An a d v a n t a g e of i t e r a t i v e s e a r c h e s i s t h e i r g e n e r a l a p p l i c a b i l i t y ; t h e same i t e r a t i v e p r o c e d u r e a p p l i e s t o t h e s o l u t i o n f o r any p a i r of v a r i a b l e s i n t h e AWJSRA t r i m e q u a t i o n and t o s o l v i n g t h e trim e q u a t i o n f o r any powered- l i f t a i r c r a f t , i n c l u d i n g c a s e s where more t h a n two e q u a t i o n s i n two unknowns must b e s o l v e d , such as t h e tilt r o t o r a i r c r a f t which r e q u i r e s t h e s o l u t i o n of e i g h t e q u a t i o n s i n e i g h t unknowns. The Powell a l g o r i t h m r e q u i r e s l i t t l e more t h a n a u s e r - s u p p l i e d f u n c t i o n e v a l u a t i o n s u b r o u t i n e . I n a d d i t i o n , t h e a d a p t i v e s e a r c h of a n i t e r a t i v e scheme may converge i n fewer s t e p s on a d i s - t a n t s o l u t i o n t h a n t h e r i g i d g r i d s e a r c h p a t t e r n of t h e e x h a u s t i v e g r i d s e a r c h method, e s p e c i a l l y i f t h e a i r c r a f t model i s n o t s t r o n g l y n o n l i n e a r .

For f l i g h t c o n t r o l u s e , i m p o r t a n t p r o p e r t i e s of t h e t r i m s o l u t i o n algo- rithms are computation t i m e and r e l i a b i l i t y . The Powell a l g o r i t h m w a s t e s t e d w i t h t h e AWJSRA model a f t e r m o d i f i c a t i o n t o e n f o r c e c o n s t r a i n t s on t h e s e a r c h domain. It proved e f f i c i e n t w i t h convergence t o a s o l u t i o n i n a few s t e p s f o r a l l cases t e s t e d . However, methods of proving t h e convergence of such a l g o r i t h m s f o r a l l o p e r a t i o n a l c o n d i t i o n s , g i v e n t h e a i r c r a f t model, do n o t a p p e a r t o b e a v a i l a b l e . Consequently, t h e e x h a u s t i v e g r i d s e a r c h p r o c e d u r e w a s adopted f o r t h e p r e s e n t c o n t r o l system d e s i g n . Computation t i m e r e q u i r e - ments are p r o p o r t i o n a l t o t h e p r o d u c t of t h e t i m e r e q u i r e d f o r e a c h s t e p i n a s e a r c h and t h e number of s t e p s t o converge on a s o l u t i o n . An upperbound on computation t i m e is g i v e n by t h e case of s e a r c h i n g f o r a s o l u t i o n when none I exists. T h i s case w i l l o c c u r i n p r a c t i c e i n t h e a b s e n c e o f e x i s t e n c e condi- t i o n s t h a t can b e t e s t e d b e f o r e e n t e r i n g t h e a l g o r i t h m . When s o l u t i o n s do e x i s t , t h e convergence rate o r a v e r a g e number o f s t e p s t o converge is a prop- e r t y of interest.

For r e l i a b i l i t y , convergence of t h e a l g o r i t h m t o a s o l u t i o n , i f i t e x i s t s , and c o r r e c t s e l e c t i o n of a s i n g l e s o l u t i o n i f several e x i s t , are n e c e s s a r y p r o p e r t i e s . I n a d d i t i o n , such s o f t w a r e p r o p e r t i e s as i n s e n s i t i v i t y t o round- o f f e r r o r , and r i s k of programming e r r o r , which i n c r e a s e s w i t h t h e complexity of t h e a l g o r i t h m ' s s t r u c t u r e , are r e l e v a n t .

For convenience, t h e t r i m e q u a t i o n s f o r t h e AWJSRA are r e p e a t e d h e r e where TH(NH/&, 6 , V A ) , AE(NH/A, 6 1 , C D ( ~ , 6 f , CJ), C L ( ~ , 6 f , CJ) are scalar f u n c t i o n s of t h e c o n t r o l v a r i a b l e s o r of a u x i l i a r y f u n c t i o n s o f t h e c o n t r o l s and a r e g i v e n as t a b u l a t e d d a t a i n appendix A. A f o l l o w i n g s e c t i o n of t h i s appendix ( E x h a u s t i v e Grid S e a r c h S o l u t i o n Method) d e s c r i b e s e x h a u s t i v e g r i d s e a r c h a l g o r i t h m s f o r s o l v i n g e q u a t i o n ( B l ) f o r v a l u e s of t h e c o n t r o l p a i r s , { a , v} and { a , 6 f } , which y i e l d t h e commanded a p p l i e d s p e c i f i c f o r c e s , (A,,, AN^), g i v e n t h e v a l u e s of a l l remaining p a r a m e t e r s i n e q u a t i o n (Bl). S e a r c h e s o v e r o n e and two-dimensional g r i d s are used i n t h e s o l u t i o n s . These two a l g o r i t h m s a r e s u f f i c i e n t f o r t h e work d e s c r i b e d i n t h i s paper b u t , i n g e n e r a l , s o l u t i o n s o f e q u a t i o n ( B l ) f o r any p a i r of v a r i a b l e s among {Z, @, i i } a r e of i n t e r e s t i n t h e a n a l y s i s of v a r i o u s performance problems.

T a b l e i n t e r p o l a t i o n is r e q u i r e d e x t e n s i v e l y f o r f u n c t i o n e v a l u a t i o n i n t h e s e a l g o r i t h m s . I n t e r p o l a t i o n f o r m u l a s which are c o n t i n u o u s o v e r t h e t a b l e domain are g i v e n n e x t .

T a b l e I n t e r p o l a t i o n The e v a l u a t i o n of f u n c t i o n s , such as CL, CD, e t c . , by i n t e r p o l a t i o n of s t o r e d t a b u l a t e d v a l u e s i s used r e p e a t e d l y i n t h e t r i m s o l u t i o n a l g o r i t h m s .

I n t e r p o l a t i o n f o r m u l a s f o r f u n c t i o n s of t h r e e v a r i a b l e s are g i v e n below; t h o s e f o r f u n c t i o n s of one o r two v a r i a b l e s f o l l o w as s p e c i a l cases.

Assume t h a t a t a b l e of v a l u e s of t h e f u n c t i o n , f ( x , y , z ) , h a s been s t o r e d f o r a l l p o i n t s i n a g r i d c o v e r i n g t h e domain of i n t e r e s t ; t h a t i s , t h e set o f v a l u e s i = 1, 2 , . . ., Nx,

k = 1, . . ., Nz 1 (B2)

( f ( X i , y j , Z k ) ; j = 1, . . ., N y , T h i s t a b l e c o n t a i n s Nx NY N, v a l u e s and t h e g r i d s u b d i v i d e s t h e domain i n t o (Nx - 1) (NY - 1 ) (N, - 1) box-shaped p i e c e s .

The o r i g i n a l f u n c t i o n i s approximated as p i e c e w i s e l i n e a r by i n t e r p o l a t - i n g t h i s t a b l e . For a g i v e n p o i n t , ( x , y , z ) , i n 9 t h e t a b l e is i n t e r p o l - l a t e d i n two s t e p s as f o l l o w s . F i r s t , l o c a t e t h e g r i d p i e c e c o n t a i n i n g t h e p o i n t ( x , y , z ) ; t h a t i s , f i n d is e a s i l y done by several methods; t h e one used h e r e i s d e f i n e d later T h i s i n e q u a t i o n (B14) and is e f f i c i e n t i n real time s i m u l a t i o n and c o n t r o l .

Having found ( i , j , k) i t i s c o n v e n i e n t t o d e f i n e t h e f o l l o w i n g v e c t o r which l o c a t e s t h e p o i n t ( x , y , z ) r e l a t i v e t o ( x i , y j , zk) and n o r m a l i z e s t h e c o o r d i n a t e s T x - x

- i Y - Y j = ( P x , P y . P z ) T

i 'j+i - y j ' Z k+i - 'k i+ 1 - The c o o r d i n a t e s of p are each numbers i n [ 0 , 1). For b r e v i t y , t h e n o t a t i o n { f o o o , f l o 0 , . . ., f , , , ) i s adopted t o r e f e r t o t h e t a b u l a t e d v a l u e s of f a t t h e e i g h t c o r n e r s of t h e g r i d p i e c e : ( x i , Y j , z k ) , (xi+,, y j , Zk), . . ., ( x i + , , Y j + l ' 'k+l)' Second, t h e l i n e a r approximation of f is g i v e n from t h e u s u a l t r u n c a t e d T a y l o r series expansion of f a b o u t ( x i , y j , z k ) :

E ( ; ) = f o o o + p -

,-.

The approximation, f , is r e q u i r e d t o b e c o n t i n u o u s and t o match t h e t a b u l a t e d v a l u e s . However, t h e o r i g i n a l f u n c t i o n , f , is i n g e n e r a l n o n l i n e a r , and i t s v a l u e s a t t h e e i g h t c o r n e r s of t h e g r i d p i e c e w i l l n o t f a l l - i n a s i n g l e h y p e r p l a n e , s u c h A a s e q u a t i o n (B6a). Consequently, t o o b t a i n t h e d e s i r e d p r o p e r t i e s f o r f i t i s n e c e s s a r y t o d e f i n e h y p e r p l a n e s by c a l c u l a t i n g t h e - g r a d i e n t v e c t o r , Vf, from a d i f f e r e n t set of f o u r g r i d p o i n t s depending on - t h e l o c a t i o n of p w i t h i n t h e g r i d p i e c e s . T h i s i s i l l u s t r a t e d i n s k e t c h ( a ) f o r t h e s i m p l e r case of a f u n c t i o n of two v a r i a b l e s . The r e c t a n g u l a r domain i s d i v i d e d i n t o two t r i a n g u l a r p i e c e s and t h e g r a d i e n t c a l c u l a t e d from a d i f f e r e n t set of t h r e e g r i d p o i n t s i n each t r i a n g l e as n o t e d i n t h e s k e t c h .

The r e s u l t i n g f u n c t i o n is l i n e a r i n each t r i a n g l e and c o n t i n u o u s a t t h e i r common edge. I n t h r e e dimensions t h e l i n e a r a p p r o x i m a t i o n becomes more complex and r e q u i r e s s u b d i v i s i o n of t h e box-shaped domain i n t o s i x t e t r a h e d r a l p i e c e s ; approximate f o r m u l a s f o r t h e g r a d i e n t v e c t o r are: [of]; = P Y P X f (x. Y)

I f01 /

Sketch ( a ) . - L i n e a r i n t e r p o l a t i o n .

I n t e r p o l a t i o n f o r m u l a s o t h e r t h a n e q u a t i o n s ( B 6 ) are a l s o a v a i l a b l e and have d i f f e r e n t p r o p e r t i e s ; f o r example, s u b d i v i s i o n of t h e g r i d p i e c e as i n e q u a t i o n ( B 6 b ) is avoided by adding t e r m s i n c r o s s - p r o d u c t s of { p , , p y , p , } from t h e Taylor series, which y i e l d s

+ P x P y ( f l l 0 - f a l o - f l a a + f o o o )

+ f 1

+ PxPz(flOl - f o o l - f l O O

0 0 0

+ ~ y P z ( f 0 l l - - + fooo)

+ P , P y P z ( f l l l - foil - f , o l + f o o l - f , . l o + f o l a + floe - f o o o ) 037) T h i s f o r m u l a t i o n e x t e n d s t o more t h a n t h r e e v a r i a b l e s by r e t a i n i n g a l l t e r m s t o f i r s t o r d e r i n {px, p y , p z , . . . } from t h e g e n e r a l T a y l o r series expansion ( c f . r e f . 1 5 ) .

The l i n e a r a p p r o x i m a t i o n i n e q u a t i o n s (B6) works w e l l f o r f u n c t i o n s of two v a r i a b l e s , i n c l u d i n g t h e i n v e r s i o n of such f u n c t i o n s , b u t e q u a t i o n (B6b) i s d i f f i c u l t t o g e n e r a l i z e t o f o u r o r more v a r i a b l e s . The q u a s i - l i n e a r approximation of e q u a t i o n (B7) h a s worked e a s i l y f o r f u n c t i o n s of two o r t h r e e v a r i a b l e s , i n c l u d i n g t h e i r i n v e r s i o n . For some a p p l i c a t i o n s , it i s u s e f u l f o r t o have c o n t i n u o u s f i r s t d e r i v a t i v e s o v e r t h e t a b l e domain 9, b u t t h a t p r o p e r t y w a s n o t r e q u i r e d i n t h e p r e s e n t work; e q u a t i o n (B7) h a s d i s c o n - t i n u o u s d e r i v a t i v e s on t h e s u r f a c e s of t h e g r i d p i e c e s w h i l e e q u a t i o n s (B6) h a s d i s c o n t i n u o u s d e r i v a t i v e s a l o n g a l l i n t e r i o r boundary s u r f a c e s i m p l i e d by e q u a t i o n (B6b) as w e l l as on t h e g r i d p i e c e s u r f a c e s .

E x h a u s t i v e Grid Search S o l u t i o n Method SoZution f o r n o z z l e and angle of attaek- The problem i s t o s o l v e t h e t r i m e q u a t i o n s ( B l ) f o r { a , v } given t h e v a l u e s of a l l remaining v a r i a b l e s i n t h e e q u a t i o n s . It i s c o n v e n i e n t t o d e f i n e t h e q u a n t i t i e s I n e q u a t i o n s (B8), CD and CL are f u n c t i o n s of a and are assumed t a b u l a t e d on a g r i d of v a l u e s

{ a i , i = 1, . . ., N,)

where a~~ c o r r e s p o n d s t o s t a l l o r maximum CL. The t r i m e q u a t i o n s c a n b e r e a r r a n g e d t o i s o l a t e t h e n o z z l e a n g l q i n a s i n g l e e q u a t i o n by s u b s t i t u t i n g e q u a t i o n s (B8) i n t h e t r i m e q u a t i o n , which becomes

- T

E = TH(cos(a -I- v ) , s i n ( a + v ) )

and t h e n i t f o l l o w s t h a t m

v = a + tan-1 ( - ; )

- S i n c e E depends on a o n l y and c a n be t a b u l a t e d f o r t h e g r i d p o i n t s , { a i } , e q u a t i o n ( B 9 ) can b e s o l v e d n u m e r i c a l l y f o r a a f t e r which v i s c a l c u l a t e d from t h e c l o s e d form e x p r e s s i o n i n e q u a t i o n (B10).

The problem of s o l v i n g e q u a t i o n ( B 9 ) i s v i s u a l i z e d i n s k e t c h ( b ) , which shows t h e o p e r a t i n g p o i n t (Au , AN^) f o r which trim v a l u e s of { a , V I are C d e s i r e d , and t h e l i n e t r a c e d o u t by t h e aerodynamic f o r c e s o ! = o !

No!

AN Sketch ( b ) . - S o l u t i o n of e q u a t i o n ( B 8 ) .

as a i s v a r i e d over t h e domain [ a l , a ~ ~ ] . T h i s l i n e i s c l o s e l y r e l a t e d t o t h e CL - CD p o l a r s (appendix A ) ; it h a s a n e a r - v e r t i c a l s l o p e and AN i s s t r i c t l y i n c r e a s i n g i n a below t h e s t a l l a n g l e . The s o l u t i o n of equa- t i o n (B9) i s e q u i v a l e n t t o f i n d i n g a* such t h a t t h e d i s t a n c e from t h e l i n e t o t h e o p e r a t i n g p o i n t , IE(a*) I , matches t h e r a t i o , TH/W; t h i s i s v i s u a l i z e d i n t h e s k e t c h as t h e i n t e r s e c t i o n of t h e l i n e w i t h a circle of r a d i u s TH/W.

I n g e n e r a l , t h e r e may b e no s o l u t i o n ( t h e l i n e i s e n t i r e l y o u t s i d e o r i n s i d e t h e c i r c l e ) , one s o l u t i o n (one e n d p o i n t of t h e l i n e f a l l s i n s i d e t h e c i r c l e ) , o r two s o l u t i o n s a s i n t h e c a s e s k e t c h e d above. When two s o l u t i o n s o c c u r , t h e lower v a l u e of a is s e l e c t e d i n o r d e r t h a t t h e r e s u l t i n g n o z z l e a n g l e f a l l w i t h i n t h e hardware l i m i t s on t h e n o z z l e , [6", 104'1. The h i g h e r v a l u e of a c o r r e s p o n d s t o a n e g a t i v e v a l u e of EN and t h i s i m p l i e s a n e g a t i v e n o z z l e a n g l e and downward d i r e c t e d t h r u s t v e c t o r from e q u a t i o n ( B 1 0 ) i n a l m o s t a l l cases.

The numerical s o l u t i o n of e q u a t i o n (B9) is c o n s i d e r e d i n two p a r t s : f i r s t , a method i s d e f i n e d f o r t e s t i n g whether a g i v e n i n t e r v a l of t h e a - g r i d , s a y [ a i , a i + l ) , c o n t a i n s t h e s o l u t i o n ; and, second, a p r o c e d u r e f o r s e a r c h i n g t h e i n t e r v a l s of t h e g r i d is given.

The i n t e r v a l [ a i , ai+ ) can b e t e s t e d a s f o l ows . D e f i n e t h e n o t a t i o n a - a i - - a i p a - a

I

i+ 1 - Then E ( a ) i s g i v e n on [ a i , ai+l] by:

E ( a ) = Ei + AEipa

q u a d r a t i c i n p a T h i s i s s u b s t i t u t e d i n e q u a t i o n (B9) and t h e r e s u l t i n g s o l v e d :

E 2 ) / n E 2

While t h e q u a d r a t i c h a s two s o l u t i o n s f o r p g , o n l y t h e smaller s o l u t i o n , g i v e n i n e q u a t i o n (B13), need b e t e s t e d because t h e l a r g e r one c o r r e s p o n d s t o n o z z l e s e t t i n g s o u t s i d e t h e n o z z l e hardware l i m i t s . I f t h e c o n d i t i o n i s s a t i s f i e d by e q u a t i o n (B13) t h e n t h e s o l u t i o n o c c u r s i n t h e t e s t e d i n t e r - v a l , [ a i , ai+l) and is 6 1 Otherwise, t h e s o l u t i o n e i t h e r does n o t e x i s t o r o c c u r s i n some o t h e r i n t e r v a l of t h e a - g r i d .

Each i n t e r v a l of t h e a - g r i d is t e s t e d as d e s c r i b e d above u n t i l e i t h e r a s o l u t i o n i s found o r t h e g r i d is exhausted w i t h no s o l u t i o n i n s t e p s . N a - 1 While t h e g r i d c a n b e s e a r c h e d i n any o r d e r a minimum computation t i m e s e a r c h o r d e r i n g i s d e s i r e d f o r real t i m e c o n t r o l o r s i m u l a t i o n a p p l i c a t i o n s . I f a s o l u t i o n e x i s t s , computation t i m e i s p r o p o r t i o n a l t o t h e number of i n t e r v a l s t e s t e d b e f o r e f i n d i n g t h e s o l u t i o n and can b e minimized, on t h e a v e r a g e , by t e s t i n g i n t e r v a l s i n t h e o r d e r of t h e l i k e l i h o o d t h a t t h e y c o n t a i n t h e s o l u - t i o n . For f l i g h t c o n t r o l , s o l u t i o n s a r e r e q u i r e d e v e r y c o n t r o l c y c l e and t h e expected v a l u e of t h e s o l u t i o n c o r r e s p o n d s t o t h e nominal p a t h and i s e i t h e r c o n s t a n t o r s l o w l y v a r y i n g w i t h t i m e . Consequently, t h e most p r o b a b l e i n t e r - v a l i s t h e same i n t e r v a l i n which it w a s found f o r t h e p r e v i o u s c o n t r o l c y c l e and p r o b a b i l i t y d e c r e a s e s w i t h d i s t a n c e from t h i s i n t e r v a l . The c o r r e s p o n d i n g s e a r c h o r d e r i n g b e g i n s i n t h e s a m e i n t e r v a l , I, i n which t h e s o l u t i o n w a s p r e v i o u s l y found and expands t o n e i g h b o r i n g i n t e r v a l s o u t t o t h e ends of t h e g r i d ; t h a t is, i n t h e o r d e r {I f k l g e n e r a t e d a s k v a r i e s i n t h e e x p r e s s i o n . , I - 1 I - k k = l , . .

i = {

I + k k = l , . . . , N , - I - l

The maximum computation t i m e r e q u i r e d w i t h t h i s s e a r c h p r o c e d u r e o c c u r s when a l l i n t e r v a l s must b e s e a r c h e d and is p r o p o r t i o n a l t o N, - 1. T h i s worst-case time i s minimized by minimizing t h e number of g r i d p o i n t s i n t h e a - g r i d used i n t h e aerodynamic model w i t h i n t h e l i m i t s imposed by t h e d e s i r e d model a c c u r a c y .

SoZution for throttZe and angle of attack- The problem i s t o s o l v e t h e t r i m e q u a t i o n s f o r { a , 6 t ) g i v e n t h e v a l u e s of a l l remaining v a r i a b l e s i n t h e s e e q u a t i o n s . The a p p l i e d a c c e l e r a t i o n due t o e n g i n e and aerodynamic f o r c e s i s e x p r e s s e d as a f u n c t i o n of (a, Nc) by It i s c o n v e n i e n t t o Here, Nc d e n o t e s t h e c o r r e c t e d e n g i n e power, N H / & .

s o l v e f o r Nc i n p l a c e of t h r o t t l e s i n c e t h e e n g i n e o u t p u t i s t a b u l a t e d i n appendix A f o r t h i s parameter on t h e g r i d The t h r o t t l e s e t t i n g c o r r e s p o n d i n g t o any v a l u e of Nc i s then g i v e n from The problem i s now t o d e t e r m i n e (a*, N : ) such t h a t t h e commanded f i g u r e 30.

-

a c c e l e r a t i o n (Auc, AN,) o r , f o r b r e v i t y , a,, i s o b t a i n e d ; t h a t i s , s o l v e 6 2

-

a(a,Nc) = ZC The a p p l i e d a c c e l e r a t i o n , Z(a, Nc), c a n b e t a b u l a t e d f o r v a l u e s of (a, N,) i n t h e g r i d s used f o r t h e a i r c r a f t and e n g i n e models. T h i s y i e l d s t h e table

-

i = 1, . . ., N,,

j = 1, . . ., NNc 1 (B17)

{ Z i j - = a ( u i , N c j )

-

is g i v e n O n a g r i d - p i e c e of t h e domain, s a y [ a i , a i + l ] 8 [ N c j , Ncj+l], a from t h e l i n e a r i n t e r p o l a t i o n formula (B5) as where - N c - N C j PNc - N c j + l - N C j Using e q u a t i o n s (B6), t h e J a c o b i a n is g i v e n by t h e 2 x 2 matrices The problem of s o l v i n g e q u a t i o n ( B 1 6 ) i s v i s u a l i z e d i n s k e t c h ( c ) . Equa- t i o n s (B17) and (B18) map v a l u e s of ( a , Nc) i n t o v a l u e s of (Au, AN) and t h i s map i s shown i n t h e s k e t c h f o r t h e g r i d l i n e s of t h e t a b u l a t e d model. The mapping d i f f e r s depending on c o n f i g u r a t i o n ; i n t h e c o n v e n t i o n a l c o n f i g u r a t i o n and f o r CTOL a i r c r a f t g e n e r a l l y t h e e f f e c t s of t h e e n g i n e power and a n g l e of a t t a c k are n e a r l y o r t h o g o n a l and AN(a, N,) is n e a r l y independent of e n g i n e power. Consequently, t h e r e i s l i t t l e c o u p l i n g between t h e two e q u a t i o n s and t h e normal a c c e l e r a t i o n e q u a t i o n can be s o l v e d a p p r o x i m a t e l y f o r a*. However, i n t h e p o w e r e d - l i f t c o n f i g u r a t i o n , AN(", N,) depends s t r o n g l y on e n g i n e power, b o t h from t h r u s t v e c t o r i n g and l i f t augmentation, and a* c a n no l o n g e r b e s o l v e d s e p a r a t e l y from N g . N e v e r t h e l e s s , t h e s o l u t i o n of e q u a t i o n (B16), i f i t e x i s t s , is u n i q u e a t a l l f l i g h t c o n d i t i o n s and c o n f i g u r a t i o n s i n t h e d e s i g n domain based on t h e model p r o p e r t i e s : 1. AN(a, Nc) i s monotonic and s t r i c t l y i n c r e a s i n g i n a and N, 2. Au(a, Nc) i s monotonic and s t r i c t l y i n c r e a s i n g i n N, a l o n g any l i n e f o r which AN(", Nc) i s c o n s t a n t 6 3 OPERATING /POINT POWERED L I F T CONFIGURATION AN (6 F = 65". v = 80') OPERATING POINT CONVENTIONAL CON FIGURATION AN (6 F = 5.6'. u = 6")

Nc -

A" Sketch ( c ) . - A c c e l e r a t i o n map f o r ( a , ").

The numerical s o l u t i o n of e q u a t i o n (B16) i s c o n s i d e r e d i n two p a r t s ; f i r s t , a method of t e s t i n g whether a g i v e n g r i d p i e c e c o n t a i n s t h e s o l u t i o n i s g i v e n and t h e n a p r o c e d u r e f o r s e a r c h i n g t h e p i e c e s of t h e two-dimensional g r i d i s g i v e n .

A g r i d p i e c e , s a y [ a i , a i + l I C 3 [Ncj, N c j + l l , c a n b e t e s t e d t o d e t e r m i n e i f i t c o n t a i n s t h e s o l u t i o n by s o l v i n g e q u a t i o n ( B 1 8 ) T h i s s o l u t i o n i s c a l c u l a t e d once f o r each J a c o b i a n g i v e n i n e q u a t i o n (B19).

The r e s u l t i n e q u a t i o n (B20) g i v e s t h e s o l u t i o n of e q u a t i o n (B16) i f it s a t i s f i e s t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s o r depending on which of t h e two g r a d i e n t s i n e q u a t i o n (B19) w a s used. I f t h e s e are s a t i s f i e d , t h e n t h e s o l u t i o n i s Each p i e c e of t h e g r i d i s t e s t e d u n t i l e i t h e r a s o l u t i o n i s found o r t h e g r i d i s exhausted w i t h no s o l u t i o n i n (Na - 1) * ( N N ~ - 1) s t e p s . A search o r d e r i n g which is e f f i c i e n t i n real-time u s e i s d e f i n e d n e x t by e x t e n s i o n of e q u a t i o n (B14) t o two dimensions. T h i s o r d e r i n g tests g r i d p i e c e s approxi- m a t e l y i n t h e o r d e r of t h e i r p r o b a b i l i t y of c o n t a i n i n g t h e s o l u t i o n ; t h a t i s , i t b e g i n s a t t h e g r i d p i e c e (I, J) i n which t h e s o l u t i o n w a s found f o r t h e p r e c e d i n g c o n t r o l c y c l e , and expands o u t t o t h e ends of t h e two-dimensional g r i d , t e s t i n g groups of g r i d p i e c e s which frame t h e s t a r t i n g p i e c e (I, J) a t each s t e p . The g r i d p i e c e s i n e a c h frame are assumed t o b e e q u a l l y l i k e l y t o c o n t a i n t h e s o l u t i o n . The i n d i c e s f o r t h e g r i d p i e c e s i n t h e k t h such frame ( s e e s k e t c h ( a ) ) are enumerated i n e q u a t i o n ( B 1 2 ) f o r each of t h e f o u r s i d e s of t h e frame; g r i d p i e c e s and s i d e s which are o u t s i d e t h e model domain are d e l e t e d from t h e enumeration by t h e extremes p l a c e d on i, j, and k.

MODEL DOMAIN AND

/ GRIDLINES

(GRID PIECE I , J) k = 0 N C k = l k = 2 Sketch ( d ) . - Two-dimensional g r i d s e a r c h p a t t e r n .

(1 - k , j ) k < I j = jmin,jmin + 1, - * -3jmax

= max{l,J - k}; (1 + k , j ) k < N , - I jmin

= min{NNC - 1,J + k}

j m a x { ' i 9 j ) ) ( k ) =

(',J + k - 1) k < N - J

c1 i = imin,imin + 1, . . .,imax

( i , J - k) k < J . bin = m a x { l , I - k + 1)

imax = min{N, - 1,1 + k - 1)

I

The s e a r c h p r o c e e d s , t a k i n g k = 0, 1, 2 , . . ., u n t i l a s o l u t i o n i s found ( N N ~ - 1) g r i d p i e c e s have been o r t h e g r i d i s exhausted a f t e r (Na - 1) t e s t e d . T h i s s e a r c h o r d e r i n g i s a p p l i c a b l e t o t h e s o l u t i o n of any f u n c t i o n of two v a r i a b l e s .

.. .

APPENDIX C , CONTROL MARGIN AND CONFIGURATION SCHEDULE COMPUTATIONS Computation of C o n t r o l Margin C o n t r o l margin is a measure of t h e a c c e l e r a t i o n c a p a b i l i t y a v a i l a b l e t o r e g u l a t e p a t h e r r o r s by v a r y i n g t h e f e e d b a c k c o n t r o l s around t h e i r t r i m v a l u e s a s s o c i a t e d w i t h t h e nominal f l i g h t c o n d i t i o n . C o n t r o l margin qepends s t r o n g l y on c o n f i g u r a t i o n and a p p e a r s i n t h e c o n f i g u r a t i o n s c h e d u l e d e s i g n b o t h as a c o n s t r a i n t on a c c e p t a b l e c o n f i g u r a t i o n s and as a parameter t o qe o p t i m i z e d .

S e v e r a l d i f f e r e n t c o n t r o l margins w e r e d e f i n e d i n t h e t e x t a s s o c i a t e d , r e s p e c t i v e l y , w i t h t h e u s e , o f t h r e e c o n t r o l s f o r r e g u l a t i o n and w i t h v a r i o u s p a i r s of c o n t r o l s . The s u b s c r i p t s i n t h e n o t a t i o n i n d i c a t e t h e c o n t r o l s h e l d f i x e d i n each c a s e . C o n t r o l margin w a s d e f i n e d as f o l l o w s . The a c c e l e r a t i o n c a p a b i l i t y of t h e r e g u l a t o r , g i v e n & f , vE, P3 is dGf(vE,P) = { ( ~ ( V E Y P , ~ ) ,AN(VE,P,U)): (CI)

a € d R E G y G t E % E G ~ V ExREG 1

where a c c e l e r a t i o n s are g e n e r a t e d u s i n g t h e a i r c r a f t model ( e q . ( 9 ) ) . Next, d e f i n e t h e r e g i o n e n c l o s e d by e l l i p s e s w i t h 5-to-1 a x e s r a t i o s and c e n t e r e d a t Z The c o n t r o l margin i s then t h e l a r g e s t such e l l i p s e t h a t can be i n s c r i b e d C o n t r o l margin i s computed i n t h r e e s t e p s and t h e s e are o u t l i n e d below 1. G e n e r a t e t h e envelope of &Gf(vE, p) ( o r of d 6 f , v , d s f , b t , 4 S f , a ) 2. Determine i f (A,,, AN^) i s i n s i d e p) ( o r i n s i d e J % f , V , e t c . ) 3 . I f it i s , c a l c u l a t e CMgf(x, p) ( o r C M G ~ , , , e t c . ) The a c c e l e r a t i o n envelope i n s t e p 1 i s g e n e r a t e d a p p r o x i m a t e l y as a polygon. The end p o i n t s of t h e s i d e s of t h e polygon a r e computed as a set of p o i n t s {(b, AN)i} on t h e envelope c o r r e s p o n d i n g t o a sequence of v a l u e s of t h e c o n t r o l s { i i } t h a t form a g r i d c o v e r i n g t h e r e g u l a t o r regime. For convenience i n l a t e r c a l c u l a t i o n s , t h e s e p o i n t s are g e n e r a t e d i n a c l o c k w i s e manner around t h e envelope and t h e c o n t r o l s c u i ) must b e t a k e n i n a d e f i n i t e sequence i n o r d e r t o do so. The r e q u i r e d computations are t h e n 1. C o n s t r u c t g r i d s of v a l u e s of t h e c o n t r o l s , a , 6,, w, c o v e r i n g t h e i r i - 1 v i = Vmin + (‘ma, - v m i n ) n

J

2. G e n e r a t e and s t o r e v a l u e s of ( A u ( V ~ ,p , u t ) , AN(VE, p , u i ) ) i n t h e f o l l o w i n g o r d e r : The e n v e l o p e , d s f ( V ~ ,p ) , now c o n s i s t s of 6n p o i n t s { ( A u i , AN^), i = 1, . . ., 6n) w i t h s u c c e s s i v e p o i n t s connected by s t r a i g h t l i n e s . A t y p i c a l d 6 f i s shown i n f i g u r e 35. The e n v e l o p e s , d 6 f , b t , d 6 f , v , d 6 f , a , a r e g e n e r a t e d s i m i l a r l y e x c e p t t h a t a n o t h e r c o n t r o l i n a d d i t i o n t o f l a p i s h e l d f i x e d , and t h i s r e d u c e s t h e number of p o i n t s t o 4n.

A s i m p l e geometric c r i t e r i o n c a n b e used t o d e t e r m i n e whether t h e g i v e n o p e r a t i n g p o i n t (AU,, AN^) i s i n s i d e t h e envelope ( s t e p 2 of t h e c o m p u t a t i o n s ) .

L e t {pi, i = 1, . . ., 6 n ) r e f e r t o t h e p o i n t s g e n e r a t e d i n s t e p 1 t o d e f i n e

-

t h e f i g u r e , and l e t 0 i b e t h e a n g l e subtended a t t h e o p e r a t i n g p o i n t , pc,

by t h e d i r e c t e d l i n e segment j o i n i n g F i , p i + l ( s e e f i g . 36)

I

where and t h e n

pc E d IFF C O i = 27~

-

pc 4 d IFF c ei = o

-

Assuming pc i s i n d , t h e c o n t r o l margin c a n now b e c a l c u l a t e d . The

problem of c a l c u l a t i n g t h e l a r g e s t e l l i p s e of t h e form g i v e n i n e q u a t i o n ( C 2 ) , w i t h c e n t e r a t (Aut, AN,) which c a n be i n s c r i b e d i n t h e envelope, s i ' , gener- a t e d by t h e p o i n t s {(Aui, AN^)) is e q u i v a l e n t t o f i n d i n g t h e r a d i u s of t h e l a r g e s t c i r c l e c e n t e r e d a t (5Auc, AN^) which can b e i n s c r i b e d i n t h e e n v e l o p e .ai'', g e n e r a t e d by t h e p o i n t s {(5Aui, AN^)). For s i m p l i c i t y , t h e n o t a t i o n is' w i l l r e f e r t o p o i n t s (5Au, AN). The c o n t r o l margin i s c a l c u l a t e d as t h e smallest d i s t a n c e from t h e o p e r a t i n g p o i n t t o any one of t h e s i d e s of t h e polygon .d', as d e s c r i b e d n e x t .

L e t L i b e t h e l i n e segment forming t h e i t h s i d e o f t h e envelope. The minimum d i s t a n c e , R i , from t h e p o i n t , E:, t o t h i s segment i s t h e p e r p e n d i c u l a r d i s t a n c e t o t h e segment i f t h e i n t e r s e c t i o n of t h e p e r p e n d i c u l a r o c c u r s w i t h i n L i . Otherwise, i t is t h e lesser o f t h e d i s t a n c e s t o t h e end p o i n t s of L i .

- -

To c a l c u l a t e R i f i r s t d e f i n e u n i t v e c t o r s , u , n a l o n g and normal t o t h e l i n e segment Li ( s e e f i g . 3 7 ) .

- - 1

- Pi+l - P i T

u = - = ( c o s a , s i n 0 ) T

n = ( s i n o,-cos a)

Then t h e l i n e segment, L i , c a n be g i v e n as - where 01 measures t h e d i s t a n c e a l o n g u from t h e p o i n t , Ff. The perpendicu-

l a r from F A i n t e r s e c t s t h e e x t e n s i o n of L i a t

- -

ai = (5; - p i ) u

and t h e n t h e minimum d i s t a n c e t o L i i s

-

If ai G 0

P i I

F i n a l l y , t h e c o n t r o l margin is t h e s m a l l e s t s u c h d i s t a n c e among the s i d e s of t h e polygon

CM(pc) 2 min{Ri, i = 1, . . ., N)

Admissible F l a p S e t t i n g s The f l a p s c h e d u l e w a s c a l c u l a t e d e m p i r i c a l l y by d e t e r m i n i n g t h e r a n g e of a c c e p t a b l e f l a p s e t t i n g s a t each f l i g h t c o n d i t i o n and s e l e c t i n g t h e maximum such s e t t i n g . The c o m p u t a t i o n a l s t e p s r e q u i r e d are d e s c r i b e d below; t h e y u t i l i z e t h e p r e v i o u s l y d e f i n e d trim s o l u t i o n and c o n t r o l margin a l g o r i t h m s as w e l l as t h e c o n s t r a i n t l i s t from t h e t e x t .

1.

Define g r i d s { V E ~ } ,{Auj) and { 6 f i ) which c o v e r t h e d e s i g n r a n g e s of i n t e r e s t f o r V E , and 6 f g i v e n i n zo,G&o.

2. G e n e r a t e and s t o r e a c c e l e r a t i o n envelopes f o r t h e s e g r i d s The s t a n d a r d parameter v a l u e s , ps, are used.

- 3 . For each v a l u e of x i n t h e set g i v e n by d e t e r m i n e t h o s e 6 f i n (6 } f o r which f!L

CM6f(?,ps) > 0.25 g

and a t l e a s t one t r i m s o l u t i o n e x i s t s s a t i s f y i n g t h e remaining c o n s t r a i n t s .

T h i s i s done by enumerating s o l u t i o n s f o r a g i v e n t h r o t t l e g r i d c o v e r i n g i t s - p e r m i t t e d r a n g e and e v a l u a t i n g t h e c o n s t r a i n t f u n c t i o n s { f i ( x , p,, u ) } l i s t e d i n t h e t e x t ( f i g . 1 5 ) .

These s t e p s produced t h e p r i n c i p a l e m p i r i c a l d a t a on which t h e s c h e d u l e w a s based; t h a t i s , t h e a c c e p t a b l e r a n g e a t each p o i n t i n a g r i d c o v e r i n g a

s u b r e g i o n of t h e d e s i g n f l i g h t r e g i m e , z 0 , go, g i v e n by

-

AN = 1,

p = Fs

The f l a p s c h e d u l e w a s s e l e c t e d a p p r o x i m a t e l y as t h e maximum a c c e p t a b l e f l a p a t each p o i n t and a s c h e d u l e , which is a f u n c t i o n of o n l y two v a r i a b l e s , F n ( V ~ ,AU), is produced. S t e p 3 is r e p e a t e d a t o t h e r AN i n a g r i d ( A N k } c o v e r i n g t h e d e s i g n r a n g e of t o d e t e r m i n e i f t h e a c c e p t a b l e f l i g h t AN envelope f o r t h e s c h e d u l e can b e u s e f u l l y maximized by e i t h e r a d j u s t i n g t h e s c h e d u l e o r i n c l u d i n g AN as an independent v a r i a b l e . S i m i l a r l y , s t e p s 2

-

and 3 are r e p e a t e d o v e r a g r i d (pm} c o v e r i n g Po t o d e t e r m i n e i f p should be a n independent v a r i a b l e i n t h e f l a p s c h e d u l e .

Optimum Nozzle Schedule The optimum n o z z l e s c h e d u l e w a s determined e m p i r i c a l l y by c a l c u l a t i n g t h e r a n g e of a c c e p t a b l e n o z z l e v a l u e s a t e a c h f l i g h t c o n d i t i o n and a t t h e scheduled f l a p s e t t i n g f o r t h a t f l i g h t c o n d i t i o n , and t h e n s e l e c t i n g t h e v a l u e w i t h t h e maximum f i x e d - n o z z l e r e g u l a t o r c o n t r o l margin CM6f,v(2, 5).

An optimized t h r o t t l e s c h e d u l e f o r maximum f i x e d - t h r o t t l e c o n t r o l margin,

-

CM6f,6t(x, p) c a n b e c a l c u l a t e d a n a l o g o u s l y . The r e q u i r e d c o m p u t a t i o n a l s t e p s are d e s c r i b e d below; t h e y u t i l i z e t h e p r e v i o u s l y d e f i n e d t r i m s o l u t i o n and c o n t r o l margin a l g o r i t h m s (appendix B ) , t h e c o n s t r a i n t l i s t from t h e t e x t ( f i g . 1 5 ) , and t h e f l a p s c h e d u l e , Fk (VE, Au) .

1.

D e f i n e g r i d s { V E ~ } , {A,.}, AN^) and {vn} which cover t h e d e s i g n

r a n g e s f o r V E , A,, AN and v J i n Eo, a0.

2. G e n e r a t e and s t o r e f i x e d - n o z z l e a c c e l e r a t i o n e n v e l o p e s f o r t h e s e g r i d s

-

where t h e s t a n d a r d parameter v a l u e s ps are used and i s t a k e n from t h e 6fi, j t h e f l a p s c h e d u l e , 6 f i , j = F"(VE Au.).

i' J - 3. For each v a l u e of x i n t h e g r i d , c a l c u l a t e t h e t r i m s o l u t i o n s corresponding t o {vn}, and t h e c o r r e s p o n d i n g v a l u e s of t h e c o n s t r a i n t para- meters { f i ( % , Ps, E} and c o n t r o l margin, CM6f,v(x, p s ) . These d a t a g i v e t h e range of n o z z l e s e t t i n g s f o r which t r i m s o l u t i o n s e x i s t , t h e s u b s e t of t h i s range f o r which t h e y are a c c e p t a b l e , -and t h e a c c e p t a b l e n o z z l e a n g l e having maximum c o n t r o l margin, v;'.

These s t e p s produced t h e e m p i r i c a l d a t a from which t h e s c h e d u l e , v * ( x ) , g i v e n i n t h e t e x t w a s d e f i n e d . These d a t a a r e a l l t a k e n a t t h e s t a n d a r d parameter - v a l u e s ps, b u t s t e p s 2 and 3 c a n be r e p e a t e d o v e r a g r i d {pm> c o v e r i n g go t o d e t e r m i n e i f t h e a c c e p t a b l e f l i g h t envelope of t h e s c h e d u l e can b e u s e f u l l y

-

maximized by i n c l u d i n g p as a n independent v a r i a b l e i n t h e f l a p s c h e d u l e .

The e f f e c t s of off-nominal parameter v a l u e s on t h e f l i g h t envelope b o u n d a r i e s are s i g n i f i c a n t s i n c e t h e STOL approach a t t e m p t s t o e x p l o i t t h e margins of t h e a c c e p t a b l e f l i g h t envelope, p a r t i c u l a r l y However, a s t u d y of t h e s e e f f e c t s and t h e p o s s i b i l i t i e s f o r maximizing t h e a c c e p t a b l e envelope f o r t h e d e s i g n e d s c h e d u l e by i n c l u d i n g t h e p a r a m e t e r s as independent v a r i a b l e s w a s o u t s i d e t h e scope of t h e p r e s e n t e f f o r t .

7 0

I APPENDIX D ATTITUDE AND ATTITUDE RATE C O M M A N D S AND STABILITY AXIS SPECIFIC FORCE C O M M A N D S The a t t i t u d e and a t t i t u d e rate commands a s s o c i a t e d w i t h t h e e x e c u t i o n of t h e a c c e l e r a t i o n commands and t h e nominal p a t h are c a l c u l a t e d i n t h e trim map a s i n d i c a t e d i n f i g u r e 25. The E u l e r a n g l e a t t i t u d e commands, (a, 0 , Y ) and body axis rate commands ( p , q , r ) c a n b e c a l c u l a t e d from t h e v a l u e s of t h e a n g l e s {a, 6, QV, y , Y , ) and t h e i r r a t e s f o r t h e nominal p a t h and a c c e l e r a t i o n commands, as d e r i v e d i n ' t h i s appendix. I n a d d i t i o n , t h e Trimmap r e q u i r e s t h e t r a n s f o r m a t i o n of t h e i n p u t a c c e l e r a t i o n commands t o s t a b i l i t y axis components of t h e a p p l i e d s p e c i f i c f o r c e . T h i s t r a n s f o r m a t i o n w a s g i v e n i n t h e t e x t ( e q s . ( 5 ) - ( 7 ) ) assuming z e r o s t e a d y - s t a t e s i d e s l i p a n g l e . A more g e n e r a l f o r m u l a t i o n f o r t h e case of nonzero s i d e s l i p is d e r i v e d i n t h i s appendix.

A t t i t u d e Commands The n o t a t i o n TAB w i l l d e n o t e t h e t r a n s f o r m a t i o n of v e c t o r s r e f e r r e d t o axes A i n t o v e c t o r s r e f e r r e d t o a x e s B . The a x e s of i n t e r e s t h e r e are i n e r - t i a l o r runway ( r ) , p a t h ( p ) , s t a b i l i t y ( s ) , and body ( b ) .

The a t t i t u d e of t h e a i r c r a f t body a x e s w i t h r e s p e c t t o i n e r t i a l a x e s i s g i v e n by t h e s t a n d a r d E u l e r a n g l e s , @, 0 , Y , measured by g y r o s on t h e a i r c r a f t .

The c o r r e s p o n d i n g t r a n s f o r m a t i o n i s ( r e f . 10) cos 0 c o s Y cos 0 s i n Y - s i n 0 0 s i n 0 cos Y - c o s 0 s i n Y s i n 0 s i n 0 s i n Y + c o s 0 c o s Y s i n @ c o s 0

0 s i n 0 cos Y + s i n 0 s i n Y cos 0 s i n 0 s i n Y - s i n 0 cos 0 cos @ c o s 0

I

T h i s can a l s o b e e x p r e s s e d i n t h e form of a sequence of r o t a t i o n s a b o u t a s i n g l e axis ( r e f . 1 0 ) where t h e s u b s c r i p t i n d i c a t e s t h e axis a b o u t which t h e r o t a t i o n o c c u r s

I

7 1

. ... . . . ,. . . I-.---..-

L (x) z 0 c o s x s i n x

r o 0 - s i n x cos O I x

cos x 0 -sin x L2(x) 1

[si: x 0 co: .]

[ co; x si, x H ]

L3(x) E - s i n x cos x E x p r e s s i o n s f o r t h e E u l e r a n g l e s can b e g i v e n i n terms of t h e a n g l e s { a , B , yV, y} by e q u a t i n g e q u a l t r a n s f o r m a t i o n s between i n e r t i a l and body axes Tbr = T T T (D3) b s s p p r

-

The t r a n s f o r m a t i o n t o p a t h a x e s is d e f i n e d from t h e a i r v e l o c i t y v e c t o r VA u s i n g e q u a t i o n ( 6 ) of t h e t e x t T = L2(Y)L3(YV) ( D 4 ) P r The t r a n s f o r m a t i o n from s t a b i l i t y a x e s t o body axes i s simply The remaining t r a n s f o r m a t i o n from s t a b i l i t y t o p a t h a x e s i s d e f i n e d w i t h t h e a i d of s k e t c h ( e ) . The p a t h axes { u , - E, E} are a l o n g t h e air v e l o c i t y v e c t o r (u) and normal t o i t w i t h iii and i n t h e h o r i z o n t a l and v e r t i c a l p l a n e s , r e s p e c t i v e l y . The - s t a b i l i t y axes {TS, Ts, ES} a r e , r e s p e c t i v e l y , a l o n g t h e p r o j e c t i o n of VA i n t h e body p l a n e of symmetry (zb, j b ) , a l o n g t h e body wing S k e t c h (e).- S t a b i l i t y and p a t h a x e s .

- -

a x i s j b , and i n t h e p l a n e of symmetry normal t o is. The s t a b i l i t y a x e s are o b t a i n e d from p a t h a x e s by f i r s t r o t a t i n g about t h e a i r v e l o c i t y v e c t o r d i r e c - - t i o n u , through t h e a n g l e Qv, and then a b o u t t h e ks a x i s through t h e s i d e - s l i p a n g l e @, i n a l e f t - h a n d e d d i r e c t i o n .

The a t t i t u d e commands can t h e r e f o r e b e c a l c u l a t e d from t h e remaining f i v e a n g l e s . I n t h e Trimmap, cxc i s o b t a i n e d from t h e trim s o l u t i o n a l g o r i t h m , Bc is u s u a l l y taken as zero b u t c a n be nonzero i n t h e case of decrabbed f l i g h t s o is r e t a i n e d h e r e f o r g e n e r a l i t y , Qvc i s one of t h e c y l i n d r i c a l c o o r d i n a t e s of t h e t o t a l a c c e l e r a t i o n - command, and ( y , Yv) are g i v e n by t h e commanded a i r v e l o c i t y v e c t o r , VAG.

To d e r i v e f o r m u l a s f o r t h e E u l e r a n g l e s , r e p r e s e n t t h e e l e m e n t s of t h e m a t r i x on t h e r i g h t - h a n d s i d e of e q u a t i o n (D7) as [ a i - ] and t h e n , u s i n g J e q u a t i o n (Dl) o b t a i n Tr TI - < @ < - 2 2 o r , f o r c o m p u t a t i o n a l convenience -1 a 2 3 Q = s i n ____ c o s 0 The r e q u i r e d t e r m s f o r t h e c a l c u l a t i o n of t h e E u l e r a n g l e s are -

cos COS B s i n y + s i n B s i n @v cos y ] - s i n a c o s @v cos y

a 1 3 - -

a Z 3 = - s i n B s i n y + c o s 6 s i n @v cos y

a 3 3 = - s i n COS B s i n y + s i n 6 s i n @v cos y) + c o s cx c o s 9, c o s y

a

= s i n \Y [ c o s cx s i n B c o s @v - s i n a s i n

11 V

+ c o s Yv[cos COS B c o s y - s i n 6 s i n @ s i n y) - s i n cx c o s @ s i n y ]

V V

a = s i n Yv[cos COS cos y - s i n 6 s i n @ s i n y) - s i n cx c o s @ s i n y]

1 2 V V

+ c o s \Yv[-cos cx s i n B c o s @v + s i n cx s i n

The h e a d i n g a n g l e e x p r e s s i o n c a n b e s i m p l i f i e d by d e n o t i n g t h e c o e f f i c i e n t s

of s i n Yv, cos Y v i n a12 as A, B and t h e n w e have

a = A s i n Yv + B cos 'r, = B cos Yv - A s i n Y , a l 1 -1 B -

Y = Yv + t a n

J

A For p a s s e n g e r o p e r a t i o n s i n STOL a i r c r a f t , t h e a n g l e s a, 6, y , 0 c a n b e t a k e n as s m a l l a n g l e s and second and h i g h e r - o r d e r terms (H.O.T.'s) i n t h e s e a n g l e s c a n be n e g l e c t e d t o o b t a i n t h e a p p r o x i m a t i o n

0 = y + a cos CP, + 6 s i n Ov + H . O . T . ' s

0 = CPV + H . O . T . ' s

(D11)

Y = YV + ci s i n CP - 6 c o s CPv + H . O . T . ' s

V A t t i t u d e Rate Commands The body a x i s a t t i t u d e commands are c a l c u l a t e d from t h e f i v e independent a n g l e s , { a , 6, CPV, y , Yv}, and t h e r e f o r e t h e body r a t e commands can b e con; s t r u c t e d from t h e a n g u l a r rates of t h e s e a n g l e s and o r i e n t e d a l o n g t h e i r a p p r o p r i a t e a x e s a b o u t which t h e r o t a t i o n o c c u r s ; t h a t i s , from t h e v e c t o r i d e n t i t y A l l v e c t o r s are u n i t v e c t o r s from one o r a n o t h e r of t h e a x i s frames p r e v i o u s l y d e f i n e d , and a l l can b e transformed t o body a x e s u s i n g e q u a t i o n s (D4)-(D6); t h a t is, t h e body a x i s r a t e commands are o r s i n a s i n 9 , - c o s a s i n f3 cos 9 ,

cos a s i n a cos f3 -cos a s i n 9 , - s i n c1 s i n 6 cos

-cos a [ s i n y c o s 6 + s i n 6 s i n 9 , c o s y ] - s i n a c o s 9 , cos y

- s i n y s i n 6 + cos f3 s i n 9 , cos y

- s i n a [ s i n y cos f3 + s i n f3 s i n 9 , cos y ] + c o s a cos 9 , cos y

7 4 The s m a l l a n g l e a p p r o x i m a t f o n - i s :bt?ined by n e g l e c t i n g t e r m s which are second

o r h i g h e r i n y, a, B , i, B , @ , , y, Yv

(w), = ( : ) & + ( ) b + [ ) 6 v +(cot @ , ); +(si: ..>. + H.O.T.'s (D14)

- s i n aV cos @ , . .

can b e g e n e r a t e d from t h e nominal accelera- The a n g u l a r rates f o r a , , y , Y , t i o n commands, u s i n g e q u a t i o n s (5) and ( 7 ) of t h e text d

i ; - - (E - m) + H . O . T . ' s

vc g c o s y - ac n d t

Of t h e s e a n g u l a r rates Yv h a s a s t e a d y v a l u e d u r i n g t u r n s (of t h e o r d e r of 3 t o 5 d e g / s e c ) , varies t r a n s i e n t l y a t t u r n e n t r y and e x i t a n d - i s z e r o o t h e r - w i s e , and a p p e a r s i n b o t h q , r. The f l i g h t - p a t h a n g l e r a t e y o c c u r s t r a n s i e n t l y i n r e l a t i v e l y s h o r t d u r a t i o n f l a r e s and p i t c h o v e r s and a f f e c t s q p r i n c i p a l l y . The r o l l rate, 6 , , o c c u r s o n l y t r a n s i e n t l y a t t u r n e n t r y and e x i t . It is c a l c u l a t e d from t h e l a t e r a l j e r k , which i s a v a i l a b l e from t h e command g e n e r a t o r as a r e s u l t of i t s t r a n s i t i o n maneuver fommand g e n e r a t i o n c a l c u l a t i o n s . Of t h e remaining rates i n e q u a t i o n (D14), Bc(t) can b e assumed can b e c o n s t r u c t e d as t h e d e r i v a t i v e of t h e

z e r o a l m o s t everywhere, and G C

o u t p u t of t h e trim s o l u t i o n a l g o r i t h m . However, f o r t h e nominal p a t h , c1 and y t r a n s i e n t s are a s s o c i a t e d b u t t h e c1 t r a n s i e n t u s u a l l y h a s a h i g h e r frequency and is less s i g n i f i c a n t as a rate command f o r f o l l o w i n g t h e nominal maneuver.

S t a b i l i t y Axis S p e c i f i c Force Commands The i n p u t s t o t h e b a s i c Trimmap are t h e s t a b i l i t y a x i s components of t h e a p p l i e d s p e c i f i c f o r c e commands. T h i s a p p l i e d s p e c i f i c f o r c e command, r e f e r -

-

enced t o p a t h axes, i s r e a d i l y computed from t h e commanded a c c e l e r a t i o n ac and a i r v e l o c i t y v e c t o r V X , u s i n g e q u a t i o n s ( 3 ) and ( 6 ) from t h e t e x t ac c o s y T h i s i s transformed t o s t a b i l i t y axis components u s i n g (D6) - s i n B c o s Qv - s i n B s i n Qv

6 ) = E3(-B)E1(QV) E) = 1:: ," c o s B c o s Qv c o s B s i n

(D17) - s i n Qv c o s QV These t h r e e e q u a t i o n s c o n t a i n f i v e unknowns: Normally, B , Qv, %, +, A,.

it i s assumed t h a t B y Ay are z e r o a f t e r which 4, A,, Qv c a n b e s o l v e d

w i t h t h e s a m e r e s u l t p r e v i o u s l y g i v e n by e q u a t i o n ( 7 ) . More g e n e r a l l y , t h e s t e a d y - s t a t e v a l u e of Bc need n o t b e z e r o , as i n decrabbed f l i g h t i n t h e p r e s e n c e of lateral winds, and c a n be assumed g i v e n i n d e p e n d e n t l y of t h e t r a j e c t o r y command. The v a l u e of A i n t h i s c a s e is n o t a r b i t r a r y b u t f o l l o w s from Bc and t h e d i r e c t i o n a r e q u i l i b r i u m of t h e a i r c r a f t ; i n s t e a d y s t a t e , B g e n e r a t e s b o t h l a t e r a l a c c e l e r a t i o n ( p r o p o r t i o n a l t o Cy6B) and yawing moment ( C n 6) which is balanced by a d d i t i o n a l l a t e r a l f o r c e from t h e rudder. The r e s u B t i n g l a t e r a l a c c e l e r a t i o n i n s t e a d y s t a t e i s then where b y R , are, r e s p e c t i v e l y , t h e wingspan and t h e moment a r m of t h e r u d d e r c e n t e r of p r e s s u r e a b o u t t h e a i r c r a f t c e n t e r of g r a v i t y . Consequently, e q u a t i o n ( D 1 7 ) c a n now be solved f o r Ax, A,, Qv assuming t h a t A,, p4n, An and Bc are g i v e n and t h a t Ay is computed from 6 , . The s o l u t i o n i s A - s i n B U CAU Ax = c o s Bc -

A, = - + An2 -

G2

where

A - s i n BcAu

* Y A = Y c o s Bc A"

c = sin-1

i 4 n 2 + An2

E q u a t i o n s (D16) , ( D 1 8 ) , and ( D 1 9 ) comprise t h e map of t h e a p p l i e d s p e c i f i c f o r c e commands t o s t a b i l i t y a x e s used i n t h e Trimmap of f i g u r e 25.

REFERENCES Advanced 1. Meyer, George; and C i c o l a n i , L u i g i S.: A Formal S t r u c t u r e f o r Automatic F l i g h t C o n t r o l Systems. NASA TN D-7940, 1975.

2. C i c o l a n i , L u i g i S.; and Weissenberger, S t e i n : A N o n l i n e a r T r a e c t o r y Command Generator f o r a D i g i t a l F l i g h t C o n t r o l System. NASA TP-1221, 1978.

3 . Benner, Margaret S . ; McLaughlin, M i l t o n ; Sawyer, R i c h a r d H . ; Van Gunst, Roser; and Ryan, John J.: A F l i g h t I n v e s t i g a t i o n w i t h a STOL A i r p l a n e NASA TN D-7669, F l y i n g Curved Descending I n s t r u m e n t Approach P a t h s .

1974.

4. C l e v e l a n d , W i l l i a m B . ; Vomaske, Richard F.; and S i n c l a i r , S. R. M . : Augmentor Wing Jet STOL Research A i r c r a f t D i g i t a l S i m u l a t i o n Model.

NASA TN X-62,149, 1972.

5. W h i t t l e y , D. C . ; and Cook, J. L . : Comparison of Model and F l i g h t T e s t Data f o r a n Augmentor-Wing STOL Research A i r c r a f t . AGARD Conf. P r o c .

N o . 1 8 7 , F l i g h t Ground T e s t i n g F a c i l i t i e s C o r r e l a t i o n , J u n e 1 9 7 5 , p . 1 7 - 1 .

6. Grossmith, S. W . : Augmentor Wing Jet STOL Research A i r c r a f t Update and Powered-Lift V e h i c l e C e r t i f i c a t i o n S t a n d a r d s . Canadian Aeronaut.

Space J., v o l . 21, no. 7 , S e p t . 1975, pp. 254-261.

S c o t t , B. C . ; M a r t i n , P. W . ; Hynes, C . S . ; and Bryder, R. B . : P r o g r e s s 7 .

Toward Development of C i v i l A i r w o r t h i n e s s Criteria f o r Powered-Lift A i r c r a f t . NASA TM X-73,124, 1976.

8. H e f f l e y , Robert K . ; S t a p e l f o r d , Robert L . ; and Rumold, Robert C . : A i r - w o r t h i n e s s Criteria Development f o r Powered-Lift A i r c r a f t . NASA CR-2791, FAA-RD-76-195, 1977.

9. C i c o l a n i , L . S . ; and Meyer, G . : D i g i t a l S i m u l a t i o n of V/STOL A i r c r a f t f o r A u t o p i l o t Research. Large-Scale Dynamic Systems, NASA SP-371, 1975.

10. E t k i n , Bernard: Dynamics of Atmospheric F l i g h t . John Wiley & Sons, I n c . , New York, 1972.

11. P e c s v a r a d i , Thomas: Four-Dimensional Guidance Algorithms f o r A i r c r a f t i n a n A i r T r a f f i c Environment. NASA TN D-7829, 1975.

12. L e e , Homer Q . ; Neuman, Frank; and Hardy, Gordon H . : 4-D A r e a N a v i g a t i o n System D e s c r i p t i o n and F l i g h t T e s t R e s u l t s . NASA TN D-7874, 1975.

13. Murrey, W . , e d . : Numerical Methods f o r Unconstrained O p t i m i z a t i o n .

Academic P r e s s , London and New York, 1972.

14. Powell, M. D. D.: A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations. AERE-R,5947, AERE, Harwell, Berks, UK, Nov. 1968.

15. Apostol, Tom M.: Mathematical Analysis. Addison-Wesley, Reading, Mass., 1957.

TABLE 1.- TABULATED AWJSRA ENGINE MODEL - ONE ENGINE, CORRECTED HOT THRUST, COLD THRUST, INLET MASS F'LOW

An LOW ANI THROTTLE - POWER RELATION

FUEL ~ Tabulated 103.515 92.5 95.0 98.0 0 53.6 84.0 89.5 parameter . -~ h o t s

21 , 506 26 , 098 31,198

0 0 380 4552 LO , 688 16 , 521

TH

21,692 26 , 256 31 , 325

60 0 509 4858 LO,928 16 740 - % N

24 , 309 28 , 518 33,125

250 0 2473 9350 14 , 368 19 , 911

~~ 15,712

, 246 14,684

0 0 363 3444 7385 11,096 13 Tc

14 , 632 15,809

60 0 378 3486 7414 11 074 13,142 - % N

16 , 380

250 0 596 4068 8039 11,870 13,836 15 , 315

. .- .

- JT 86.6 94.7 m - % kg/se 19.8 36.7 58.4 72.4 80.4 E 6 ~ m

3782.9 E % kg/hr

2206.9 2452.7 2965.0 0 551.6 864.7 1401.0 6G ~~ ~ Note: Throttle-power relation: 1 3 ~ = max(0.0618 NH, 1.3866 NH - 107.17) TABLE 2.- AWJSRA DRAG AND L I F T C O E F F I C I E N T S -

5.5 9.5 1 3 . 5 1 7 . 5 1 1 9 . 5 27.5

F l a p 5.6 - 0 0.073 0.095 0.136 0.188 0 . 2 6 1 0.375 0.443 0 . 8 1 1

.2 -. 1 2 4 -. 105

-. 062 -. 005 .070 .200 .269 ,648

.4 -. 324 -. 299 -. 256 -. 1 9 8 .092

-.117 .018 .484

.6 -. 520 -. 495 -. 446

-. 378 -. 297 -. 1 6 1 -. 087 .321

-. 718 -. 688

.8 -. 638 -. 570 -. 478 -. 337 -. 257 .146

-1.116

1 . 2 -1.078 -1.023 -. 949 -. 846 -. 690 -. 610 -. 1 8 4

2 . 0 - 1 . 9 0 1 -1.850 -1.780 -1.562

-1.674 -1.294 -1.304 -. 847

.~ . . -. . - F l a p = 3 0 . 0 ~~ - - - _ 0 0 . 1 6 0 0.220 0.270 0 . 3 3 0 0.370 0.410 0.430 0.500

.2 - .070 .020 .110 .190 .270 .340 .3%0 .510

. 4 -. 270 -. 1 6 0 -. 060 .040 .140 .240 .290 .480

.6 -.470 -. 350

-. 230 -.110 .ooo .120 .170 - 3 9 0

.8 -. 660 -. 530

- .400 -. 270 -. 1 4 0 -. 010 .050 .300

1 . 2 -1.025 -. 870 - . 7 2 0

-. 560 - .405 -. 250 -.175 .140

2.0 -1.540 -1.370

-1.180 -. 990 - .800 -. 620 -. 520 - .140

- . .

F l a p 5 0 . 0 ~~ - .

0 0.300 0.330 0.300 0.315 0.370 0 . 4 0 0 0 . 4 4 0 0 . 4 7 0 0.485 0 . 5 5 0 .2 .080 .130 .180 .240 .305 .375 .455 .540 .580 .750

.4 - . l o o -. 020 .065 ,160

.260 .375 .500 .620 .680 .930

.6 -. 255 -. 1 5 5 - .050 ,080

,220 .375 .520 .670 .740 1 . 0 5 0 .8

-. 380 -. 280 -. 1 6 0 -. 015 .150 .325 .515 .710 .820

1 . 2 6 0

1 . 2 -. 665 -. 550 -. 415 -. 250 -. 050 .178 .700 .840

.435 1 . 4 0 2 2 . 0 -1.250 -1.110

-. 925 -. 720 -. 480 -. 200 . l o o .440 .610 1 . 3 2 0

~- _ _ - . F l a p = 6 5 . 0 ._ .- ~ ~. .. .

0 0 . 3 0 0 0 . 3 0 0 0 . 3 2 0 0 . 3 4 0 0 . 3 7 0 0 . 4 1 0 0.450 0.500 0.520 0.620 .2 .200 .240 .270 .320 .620 .660 .860 .380 .460 .540 . 4 .130 .170 - 2 3 0 ,320 .430 ,540 .660 .800 .860 1 . 1 2 0 .6 .040 .110 .220 .330 .480 .640 1 . 0 5 0 1 . 3 8 0 .810 .970 .8

-. 050 .050 .180 .340 ,510 .700 .890 1 . 1 0 0 1 . 2 0 0 1 . 6 6 0

1 . 2 -. 220 -. 1 1 0 .040 .215 ,425 .655 .890 1 . 1 5 0 1 . 2 8 5 1 . 9 2 5

2 . 0 -. 640 -. 490 -. 290 -. 060

,200 .470 .760 1 . 0 7 0 1 . 2 4 0 2.140 _ _ _ - -- . . -- ._ - _ _ F l a p = 72.0 __= - .- 0.290 0 . 3 0 0 0 . 3 8 0 0 . 4 2 0 0 0 . 3 2 0 0.335 0 . 3 5 0 0.455 0 . 4 8 0 0.575 ,240 .265 .2 .300 .340 .470 .545 .400 .630 .670 .845 .195 .245 .4 .305 .380 .590 .715 .475 .825 .875 1.100 .160 .230 .6 .330 .447 1 . 0 6 0 .725 .895 .575 1 . 1 4 0 1 . 4 9 0 .140 .235 .820 1 . 0 0 6 .8 .365 .495 1 . 2 1 5 1 . 3 2 0 .650 1 . 7 7 0 -.OOO .125 .835 1 . 0 6 2 1.2 .282 .437 ,625 1.317 1 . 4 5 0 2.035 -.320 -.140 .795 1 . 0 8 0 2.0 .065 .290 .530 1 . 4 0 0 1 . 5 7 5 2.315 I I .~ - . - ~ -- T A B L E 2.- CONCLUDED.

~ _ _

1 0- 1.004 1 . 3 5 3 1 . 6 7 6 1 . 7 4 8 1 . 6 2 4

0 . 6 i 2 -0.. 9 2 8 -0.-577 -0.207 0 . 2 1 5

.660 1 . 0 7 3 -. 1 7 2 .252 1 . 4 4 4 1 . 7 8 2 1 . 9 5 0 2.040

I .2 -. 9 3 5 -. 5 6 3

I

.720 1 . 1 4 4 -. 9 4 8 -. 5 5 6 -. 1 6 3 .308 1 . 5 2 5 1 . 8 7 4 2.032 2.254

I - 4 1 . 5 9 4 1 . 9 4 3 2 . 1 2 0 2 . 3 5 2

.756 1 .6 1 . 2 1 2 -. 9 6 5 -.572 -. 1 4 9 .319

i 1.269 1 . 6 5 1 2 . 0 5 0 2 . 2 1 8 2.462

. 8 2 0 -. 9 7 1 -. 577 -. 1 4 4 . 3 3 4

I - 8

.888 1 . 3 6 4 - 1 . 0 5 2 -. 5 9 1 1 . 8 0 7 2 . 2 2 2 2.405 2 . 6 5 1

1 1 . 2 -. 127 . 3 9 1

2.078 2.589 2-. 8 1 8

1 . 0 2 8 1 2 . 0 1 . 5 6 5 - 1 . 1 4 9 -. 6 1 4 -. 0 7 9 . 4 8 1 3 . 0 5 5

. ~ ~~~ ! - - . ~ _ _ _ - - f ( j 1 . 4 4 6 1 . 6 7 3 1 . 7 6 1 0 . 8 3 1 1 . 1 5 3 -0.717 - 0 . 2 9 2 0.101 0 . 4 7 4 1 . 5 9 8 1 . 2 1 . 5 2 2 1 . 8 8 0 -. 1 7 8 . 2 8 9 .703 1 . 1 1 6 2 . 2 2 3 2 . 5 1 2 2 . 6 3 1 2 . 7 2 3 2 . 7 6 7 3.077 I .4 2.017 2 . 3 9 5 .186 .675 1 . 1 3 9 1 . 5 7 3 3.216 3 . 1 7 4

I .6 2.237 2.644 . 3 4 3 .842 1.327 1 . 7 9 3 3.027 3 . 3 4 6 3 . 4 7 3 3 . 2 3 6

! .8 2.367 2.825 . 4 3 0 . 9 5 0 1 . 4 3 5 1 . 9 1 0 3 . 2 3 8 3 . 5 7 5 3.713 3 . 3 2 0

1 1 . 2 2 . 5 9 4 3 . 0 6 2 . 5 6 0 1 . 0 9 6 1 . 5 9 8 2 . 1 0 2 3 . 5 2 5 3 . 9 1 9 4 . 0 7 4 3 . 5 6 2 4 . 0 4 4 4 . 5 1 9 1 2 . 0 2 . 9 9 8 ~ 3 . 5 2 0 . 7 7 1 - 1.357- 1 . 9 1 5 2 . 4 7 3 4 . 7 1 9 4 . 1 5 0 . - I - = I ~- . -~~ -~ 0 1 . 2 6 8 1 . 5 4 2 - 0 . 1 1 8 0 . 2 5 6 1 . 7 9 5 1 . 9 8 1 2 . 0 5 9 2 . 1 1 2 0 . 5 9 6 0 . 9 4 6 .2 2 . 5 5 9 1 . 9 1 4 2 . 2 3 4 . 4 6 4 . 8 9 3 1 . 2 1 8 1 . 5 9 0 2 . 8 3 0 2 . 9 8 9 3.216 .4' 2 . 3 6 7 2 . 7 5 9 .712 1 . 1 4 3 1 . 5 7 0 1 . 9 9 4 3.067 3 . 3 8 9 3 . 5 0 0 3 . 4 7 5 .6 2 . 7 3 3 3.124 .816 1 . 3 2 1 1 . 8 3 2 2 . 3 1 1 3 . 4 8 3 3 . 7 8 8 3.917 3 . 6 9 4 2 . 9 0 2 3 . 3 2 5 1 . 4 1 7 1 . 9 6 8 2 . 4 2 7 3.702 4 . 0 4 7 4 . 1 8 9 .8 . 8 9 9 3 . 8 5 0 1 . 2 3 , 1 3 1 3 . 5 7 2 1 . 0 2 1 1 . 5 8 7 2 . 1 2 0 2 . 6 2 7 4 . 0 5 9 4 . 4 1 8 4 . 5 6 8 4 . 0 6 4 2 . 0 3 . 6 5 8 4._156 1 . 4 3 0 2 . 0 2 4 - - 2 . 5 5 9 3.095 4.632- 5 . 0 5 6 5 . 2 1 5 4 . 6 3 1 I_ ~ . . . . ___- ~~ ~~ . , - ~~ 1 . 6 1 5 0 . 3 3 0 i . 8 6 0 2 . 0 3 4 2 . 0 9 2 2 . 1 0 8 0 1 . 3 4 4 -0.007 0 . 6 8 9 1 . 0 4 2 . 2 2 . 2 4 8 2.588 1 . 1 6 3 1 . 9 0 4 2 . 9 1 8 3 . 2 0 0 3 . 3 2 8 3 . 3 6 3 . 7 5 3 1 . 5 5 9 .4 2 . 8 4 7 3.217 1 . 1 9 4 1.709 2 . 1 0 3 2.492 3.545 3 . 8 5 8 3 . 9 8 3 3.874 .6 3 . 3 1 3 3 . 7 1 1 1 . 5 1 0 2 . 0 0 1 2 . 4 5 1 2 . 8 9 4 4 . 0 4 3 4 . 3 1 3 4 . 4 2 6 4 . 0 8 0 4 . 3 1 3 4 . 6 1 1 4 . 7 1 4 .8 3 . 5 3 2 3 . 9 5 3 1 . 5 9 4 2.137 2.630 3 . 0 8 7 4.167 3 . 6 6 2 4.116 2 . 1 9 8 4.514 4 . 8 7 3 4.966 4 . 3 7 6 1.2 1 . 6 4 7 2 . 7 1 5 3 . 1 9 4 3 . 9 9 9 4 . 4 7 0 .- 1 . 8 1 5 2 . 4 4 8 2 . 9 8 9 3 . 5 0 9 4.897 5 .25-6 5 . 3 8 9 4 . 7 1 9 - i__ 2.9. . - - _~ 0 0 . 1 0 9 0 . 4 3 6 0 . 7 8 8 1 :039 1 . 3 3 7 1 . 5 4 2 1 . 7 7 3 1 . 9 0 1 1 . 9 9 3 1 . 5 2 1 .2 . 8 7 1 1 . 1 5 2 1 . 4 9 8 1 . 8 0 2 2 . 1 1 2 2.429 2 . 7 4 3 2 . 9 8 1 3.136 2 . 9 5 4 3.519 .4 1 . 3 5 4 1.687 2.056 2 . 4 2 3 2 . 7 9 5 3.154 3 . 7 9 8 3 . 9 1 3 3 . 3 6 5 i. 7 2 1 2.498 2.874 3 . 2 6 1 3 . 6 4 0 3 . 9 7 5 4 . 3 3 7 4 . 4 9 9 .6 2.096 3 . 4 9 8 . 8 2 . 0 6 0 2 . 4 5 0 2.832 3 . 2 2 9 3 . 6 2 6 3 . 9 8 8 4 . 3 8 6 4 . 7 3 8 4 . 8 4 0 3 . 9 1 3 1 . 2 2 . 4 5 2 2 . 8 2 3 3.244 3 . 6 4 4 4 . 0 3 7 4 . 4 5 5 4 . 8 4 9 5.272 5 . 3 0 3 4 . 3 5 9 5 . 5 6 2 2-. 0 3 . 0 3 1 3 . 4 3 5 3.876 4 . 2 8 7 4 . 7 3 8 5 . 1 2 1 5.996 6 . 0 3 7 5 . 0 8 9 - .~ I I I I 11111I I I

I

TABLE 3.- AWJSRA ELEVATOR TRIM SETTINGS - DEG

Flap = 5.6 . -

-

-15.0 -7.9 1 - 3,5 0. -5.9

-12.7 -8.2 ~ 2.7 .2 -6.4 -11.9 -8.6 1 . 9 . 4 -6.8 -12.4 -9.1 . 7 . 6 -7.4 -9.6 -8.0

-12.6 -. 2 .8

-10.8 -13.2 -2.1 1 . 2 -9.5 -15.4 -14.2 -13.5 -7.2 2.0

--

- F l a p = 30.0 - ~. . -- - - I - 11.2 7.6 3.8 0.2 -2.7 -7.2 0. -11.7 -23.3 -5.0 -10.1 9.8 6 . 7 3.3 .1 -2.5 -6.4 . 2 -9.7 -18.7 -4.8 -8.5 -5.2 -7.9 -17.7 -3.0 -6.9 9.4 6.6 3.5 . 6 -1.8 . 4 8.6 5.9 3.0 .5 -1.9 -5.0 -7.8 -19.1 -3.8 -6.6 . 6 7.6 5.0 2.2 -.3 -2.4 -4.9 .8 -7.5 -20.3 -3.9 -6.5 4.9 2.6 -6.3 -8.0 -21.1 -5.7 -7.3 1 . 2 -2.3 - - - - . O -6.0 -10.6 2.0 -10.8 . - . -23.3 . . -10.5 -10.5 ~ _ _ _ I_ F l a p = 50.0 - O S -38. 7 -2.6 -4.8 11.1 -6.6 - 2 -16.4 -5.2 5 . 9 -.9 -3.0 . 4 -7.3 -8.3 -.3 -2.1 -4.0 -5.8 -18.9 1 . 4 . 6 -20.7 -9.3 -9.7 -2.8 -3.9 -5.0 -6.2 -7.8 .8 -22.1 -11.3 -11.9 -6.4 -7.1 -8.0 -9.1 -10.3 1 . 2 -14.7 -14.7 -12.1 -12.9 -13.5 -14.7 -26.5 -11.8 -20.1 -20.0 - - -19.7 -20.2 -20.3 -20.0 2.0 -31.6 -20.0

-

_. --- . ~- 10.8 ~ 6.7 3.c 0.4 -1.1 -2.2 -3.6 -.5 -1.4 -2.1 -2.7 7.8 4.6 1 . 4 -2.2 -2.9 -3.6 -3.6 4.9 1.8 -.6 -3.8 -4.4 -4.5 -4.8 1.0 -1.4 -3.0 -6.2 -6.3 -6.4 -6.6 -2.1 -3.9 -5.2 .8 -.l -10.9 -11.0 -11.0 -10.8 -7.9 -9.3 -10.2 1 . 2 -6.2 -18.2 -18.0 -18.0 -17.6 -16.4 -17.3 -17.9 2.0 -15.6 I _ .

- ~~~

.. - = - -

-1.6 -1.2 -0.8 -1.3 0 . 1 9.9 .1 . 2 .8 . 3 .8

fl; 8.2

1 . 0 -3.0 1 . 7 . 5 . 4 5.7 1 . 2 .6

-6.8 11.0 1 . 4 -. 7 -. 9 3.3 -. 5

-.4 .1 -1.9 -2.8 -6.8 7.7 .8 -2.6 -9.1 2.9, -.7 -2.5 -4.7 -5.9 -3.3 1.2 -7.3 -12.6 2.0 ._ -6.9- -9.7 -11.5 -10.0

-5.0 - 3 . 0 -

~ - _ _ - AUTOMATIC CONTROL LOGIC

- -

-

ESTIMATION + TRAJECTORY CONTROL A A A INPUT j l f : Rv vr TRAJECTORY

1 PATH ATTITUDE

COMMANDS AIC CONTROL z ERROR RC,. V C ? REGULATOR FORCE ENGINE COMMAND ACTUATORS TRIMMAP GENERATOR Aar 4ERODYNAMICS I acr act (6f. 6t. Y, 6eTIc Figure 1.- Automatic control logic - AWJSRA.

W

I BASIC TRIMMAP

(Au, AN. V E I q AUTOMATIC 6 f c w CON FlGU RATION SCHEDULE

1 , 6F vc I

I I 1 -

ELEVATOR (AUc, ANc) TRIM ffc w SOLUTION MAP (aapp)r ALGORITHM ffC FUNCTION

*

act, TO STAB1LlTY AXES & COMPUTE VELOCITY ffc

- EULER

ANGULAR RATES $vc, T C , I t , ,

* ATTITUDE &

($, 8. *Ic

ANGULAR -

MAP TO SPHERICAL COORDINATES F i g u r e 2.- AWJSRA Force Trimmap.

PATH AXES I RUNWAY AXES (a) Transformation angles. (b) Path and runway axes.

Figure 3.- Axis systems and transformation angles.

MAXIMUM - 2.0 REGION OF NOMINAL ACCELERATION COMMANDS FOR AWJSRA

/

m LOCUS OF STEADY - 51.5 r - - - TURNS ($" = 20') a I W ' 0 LOCUS OF STATIC E

,f / EQUILIBRIUM

- 1.0

-

Y

Y = -30 ' 30

n

L _ _ _ ,--J

v) A a E -

g .5

z

I I I I

-.5 0 .5

-1 .o

LONGITUDINAL SPECIFIC FORCE, A,, g Figure 4.- Applied specific force commands - steady flight conditions.

1.2 FLARE

v\

STEADY LEVEL 1.1 FLIGHT STEADY FLIGHT AT Y = -7.5"

A,, /

1 .o

P I T C H O V E R 4 .9 .8 -.3 -.2 -.I 0 .I A,, 9 F i g u r e 5.- Applied s p e c i f i c f o r c e commands - t r a n s i e n t commands f o r f l i g h t - p a t h a n g l e changes.

1. STRAIGHT/LEVEL FLIGHT 2. DESCENDING TURN 3. DECELERATING LEVEL FLIGHT 4-5. LEVEL TURN WITH STEADY WIND 6. DECELERATING GLIDE SLOPE LONGITUDl N A L APPLl ED SPECIFIC FORCE, A,, deg F i g u r e 6 . - STOL t e s t approach p a t h : hodograph of a p p l i e d s p e c i f i c f o r c e command.

DUCT AND NOZZLE WING

REAR SPAR \

FLAP HINGE POINT\^

FLmx%,.

CHOKE AUGMENTOR FLAP TH TU ENGINE EXHAUST NOZZLES F i g u r e 7.- A u g m e n t o r w i n g j e t STOL research a i r c r a f t ( A W J S R A ) .

v = 75 6f. deg \ 5.6 ALPHA, deg V, = 120 knots A,, = O -5 AN = 1 w =w, r = I 6 = I I I I I I I -10 10 15 20 25 30 35 40 THROTTLE, deg (a) Level f l i g h t .

F i g u r e 8.- Trim s o l u t i o n s f o r AWJSRA.

- v . d e s 6 30 60 70 A n 85 ALPHA, deg V, =65knots A, = -.1305 A, = .9914 w = w, -5 7 = 1 6 = 1

I I I u

I -10 30 35 40 10 15 20 25 THROTTLE, deg (b) G l i d e s l o p e .

F i g u r e 8.- Concluded.

1( v = 6 6 f . deg L t 5.6 \

/ ?

MINIMUM

I -

NOZZLE /I 15 ALPHA, deg (] ACCEPTABLE TRIM v, = 120 knots 0 MAXIMUM A, = NOMINAL -5 A, = 1 THROTTLE w = ws ' - \ - 45 .- 1 A r = 6 = L U I U I K U L MARGIN I --I -10 I . I I I 15 20 25 30 35 40 THROTTLE, deg ( a ) Level f l i g h t .

F i g u r e 9.- A c c e p t a b l e trim s e t t i n g s f o r AWJSRA.

MAXIMUM NOZZLE MAXIMUM NOMINAL THROTTLE MINIMUM CONTROL MARGIN ALPHA, 5 - deg MINIMUM LIFT ACCEPTABLE 0 - TRIM SETTINGS VE = 65 knots A, = 0.1305 -5 - A, = 0.9914 ..

w = w, 7 = 1

I

s = 1

-10 1

I 1 ~~ I I I 15 20 25 30 35 40 THROTTLE, deg (b) G l i d e s l o p e .

F i g u r e 9. - Concluded.

9 1 P FROM

COMMAND -

AUTOMATIC GENERATOR ' 0 CONFIGURATION SCHEDU LE I I TRIM SOLUTION ALGORITHM (REGULATOR CONTROLS)

t

( a ) S e p a r a t i o n i n t o scheduled and f e e d b a c k c o n t r o l s .

FROM COMMAND

- 1 CONFIGURATION AUTOMATIC {

F*(X), u*(X) TO ACTUATORS

(2) VE C

I - c Vc

c 6 t c

TO ATTITUDE 1

cat- COMMAND

I I

LOGIC (b) Basic Trimmap - AWJSRA.

F i g u r e 10.- Basic Trimmap s t r u c t u r e - AWJSRA.

9 2 1.4 1.2 m 5 1.c a w- a U u .a - n .

U J J a B a

p .6

.4 1 I I 1 .2 -.4 -.2 0 .2 .4 LONGITUDINAL SPECIFIC FORCE, A,, g F i g u r e 11.- R e g u l a t o r c o n t r o l regime - f i x e d n o z z l e mode; vE = 65 k n o t s , 6f = 65", v = 5 4 . 2 " , p = ps.

1.6 6, = 65" v, = 65 knots 1.4 - - P =P, 1.2 o l w- a

g 1.0

U

-

Y

L v)

2 .8

E

z .6 .4 I I I I I .2 ~ -.6 -.4 -.2 0 .2 .4 LONGITUDINAL SPECIFIC FORCE, g F i g u r e 12.- C o n t r o l regimes f o r t h e r e g u l a t o r modes - g l i d e s l o p e ; fif = 65", VE = 65 k n o t s , 5 = ps.

9 4 1.4 1.2 CONTROL REGIME ~ 1 .o w- a

P

L L .8 n v) J a

i?

P

.6 .4 .2 -.4 F i g u r e 13.- R e g u l a t o r c o n t r o l regime f o r t h r e e c o n t r o l s and mode s w i t c h i n g diagram - g l i d e s l o p e .

6 t = 6 4 - 1

+l

FIXED FIXED NO SOLUTION THROTTLE ALPHA

I MODE

t l l

NOTE: ( I(-) INDICATES VALUES RETAINED FROM THE PREVIOUSCONTROL CYCLE F i g u r e 14.- Flow diagram - Trimmap r e g u l a t o r c o n t r o l l o g i c .

Parameter Steady flight limits Regulator usage limits - Flap 6f E [5.6", 6 (VJl fmax Throttle

6 , E hmin (6f). 6tmaxl

Nozzle v E [6,1041 Lift margin LM(Z,iT,i) 2 LMmin(6f) Pitch 9 E [-IOo, 15OI 9 > 2" a t touchdown Angle of attack 01 E [-10.5', 0 1 ~ ~ ~ , ~ ~ ~ ( V ~ ) l Control margin CMg (Z,B) > 0.25 f Elevator S (x,u) E [-17O, 7'1 6 , E [-25O, 15OI eT ( a ) C o n s t r a i n t list.

FLAP, deg 0 50 100 150 200 EQUIVALENT AIRSPEED, knots (b) F l a p extremes.

F i g u r e 15.- O p e r a t i o n a l c o n s t r a i n t s on c o n t r o l s e t t i n g s - AWJSRA.

THROTTLE SERVO STOP u l a l U M MAX, NOM i 20 26 = STAGNATION I ', / I ' I - ' I

221 1 I 'I

- _I

50 100

.9 1 .o 1.1

FLAP, deg STAGNATION TEMPERATURE RAT1 0, Tof288.16 K ( c ) T h r o t t l e extremes.

(d) Maximum t h r o t t l e f o r r e g u l a t o r usage.

I-- %TALL LM = 0.15

------- ---

o(MAX, REG - MINIMUM LIFT MARGIN BOUNDARY OL. deg 6 t = 25.6' 6 t = 6 t min 0 - 6f = 6fMA)( (v,) ii = R

I I I -

-10 ( e ) Upper l i m i t s on a n g l e of a t t a c k f o r l i f t margin c o n s t r a i n t and r e g u l a t o r usage.

F i g u r e 15.- Concluded.

, , ... .. .. . .. .

ADM ISS I B LE FLAP RANGE n ( f ) A , = -0.15 J U

r --\

6o r x

( c ) % = 0.12

( g ) AU = -0.18 50 100 150 200 50 100 150 200 VE, knots F i g u r e 16.- Admissible f l a p r a n g e - s t a n d a r d case (5 = G s , AN = 1).

BOUNDARY OF ACCEPTABLE FLIGHT CONDITIONS (p=p,,A,=I)

/

.2 1; -.

.

\\ . I k 6 f = 5 . 6 d

- 6

I I A,, 9

-. 1

I c , '.

I , .

.

2 -.2 L , .

6 f =65

- 3 - 1 ' I-

6 f = 65 50 L 20 . 5 . 6

I - . 3 __ 50 100 150 200 VE, knots F i g u r e 1 7 . - Nominal f l a p s c h e d u l e : c o n t o u r p l o t .

(a) Flap command rates for speed aF* changes Q , - <%, V E ) .

a V E 2r = - 332 deglg

.4 r

-.2 L

I I I -.4 50 100 150 200 V,, knots (b) Flap command rates for flight-

a F*

path angle changes % - (b, VE).

a b

Figure 18.- Flap command rates.

1.6 v, = 6 5 knots 6f = 6 5 " if = & 1.4 1.2 m

f

Lu- 0 1 !x

u-

w n v)

< .a

E PI z .6 .4 I I I 1 .2 .4 LONGITUDINAL SPECIFIC FORCE, A,, g e; VE = 65 k n o t s , F i g u r e 19.- C o n t r o l regimes f o r f i x e d - n o z z l e mod 6f = 65", = i j S .

1 0 2 .

OPTIMUM NOZZLE ANGLE uMAX

I I \ \ 2

/ I5 A"

,5

0 -0.1 -0.2 A, = 0.1 / I m 'El d -I CJ

2 20

Lu -I I- - 0 . 1 -0.15 -0.2

+

K I

+

0 20 40 60 80 100 NOZZLE ANGLE, deg (a) VE = 65 knots. (b) VE = 160 knots.

Figure 20.- E f f e c t of nozzle a n g l e on c o n t r o l margin and t h r o t t l e s e t t i n g I--'

0 = ps, 6f = F * ( h , V,>Y AN = 1)-

W .3 .2 Ul

d

w- .1

u

LL w $ 0 J a

z

k u -.l

s

-.2 - . 3 EQUIVALENT AIRSPEED, VE, knots ( a ) Nozzle a n g l e f o r maximum c o n t r o l margin - c o n t o u r p l o t .

F i g u r e 21.- Nozzle a n g l e o p t i m i z a t i o n ; AN = 1, 5 = is, 6 f = F * ( h , V E L . 3 .2

' = 'MAX

m u-

2 .I

LL - :: $ 0 J n -- LM = LM,,, I- UNCONSTRAINED MAXIMUM CM W 2 -.I

s

\

-.2 - . 3 I I 1 I I 1 I 60 80 100 120 140 160 180 200 220 EQUIVALENT AIRSPEED, knots (b) Regions of c o n s t r a i n t s a t u r a t i o n .

F i g u r e 21.- Concluded.

u*(x) = m i n ( l 0 4 , max[Po(x)Pl (x),q(x),61} A, - A 2-0.318 6 - 645.33(Au - A) [ I + 1.57(A, - A ) ] q ( x ) =

A, - A < -0.318

( 1 04 A = m i n ( A , , A,) A , = 0.25(1.12 - AN) - max 0, min 0.15(1.14- A N ) , 0.06]}* m a x b , min [(VE - 100)/20, j}

{ [

A, = 0.0025 [50(4.4 - A N ) - VE] P,(x) = a , - 180 A, a, = min(65, 125(1.52 - AN), 4(vE, - vE)) VEo = max(77,65(2.185 - AN)) P , (x) = a , + mint0, 600(A, - 0.065)} a , = min{80,75(1.867 - AN), 4 [ 2 1 . 8 7 5 ( A ~ + 2.97) - V E I l V,, knots

; * and domains of each of t h e

( a ) F u n c t i o n g e n e r a t o r f o r AN = 1.

g e n e r a t i n g f u n c t i o n s a t F i g u r e 22.- Nozzle s c h e d u l e f o r maximum c o n t r o l margin.

60 50 D . 3 .2 .I A". 9

\ $*, deg

-. 1

-.2

- . 3 + -

50 70 90 110 130 150 170 190 210 230 250 V,, knots (b) Nozzle s c h e d u l e - c o n t o u r p l o t (AN = 1).

F i g u r e 22.- Continued.

E LIFT MARGIN ABUSE REGION

# PITCH ATTITUDE ABUSE REGION

-

.3 1 1 1 I I THROTTLE ABUSE REGION CONTROL MARGIN ABUSE REGION

.2 . I A,. 9

-. 1

-.2 -.3 50 70 90 110 130 150 170 190 210 230 EQUIVALENT AIRSPEED, knots ( c ) Nozzle s c h e d u l e : a c c e p t a b l e f l i g h t e n v e l o p e and envelope a b u s e b u f f e r r e g i o n (AN = 1, p = ps, 6f = F*).

F i g u r e 22.- Concluded.

- .2

u- V 0 . I Y u W n v) - . 3 50 70 90 110 130 150 170 190 210 230 250 EQUIVALENT AIRSPEED, knots (a) Throttle setting, deg.

Figure 2 3 . - Trim solutions for automatic configuration schedule

- -

(AN = 1, P = P S I -

I

I 50 70 90 110 130 150 170 190 210 230 250 V,, knots (b) Angle of a t t a c k , deg.

F i g u r e 2 3 . - Continued.

EQUIVALENT AIRSPEED, knots (c) T r i m e l e v a t o r s e t t i n g , deg.

F i g u r e 2 3 . - Continued.

SOLUTION BOUNDARY STOL APPROACH TRAJECTORY N BOUNDARY I 50 70 90 110 130 150 170 190 210 230 250 V,, knots (d) L i f t margin, g .

I 50 70 90 110 130 150 170 190 210 230 250 VE, knots ( e ) C e n t r a l mode c o n t r o l margin, C M G f y V , g.

F i g u r e 2 3 . - Concluded.

1 1 2

- - LIFTMARGIN

-

= ABUSE REGION

2.0

(IIIII THROTTLE

ABUSE REGION PITCH ATTITUDE 1.8 TRIM SOLUTION EXISTENCE BOUNDAI RY 1.6 m W ' MAX THRUST 1.4

u-

W

a 1.2

n

w

-1 n -1

2 1.0

a z .8 ENGINE OFF .6

I 1 I u

.4

-.3 -.2 -. 1 0 .1 .2 .3

A, LONGITUDINAL APPLIED SPECIFIC FORCE, g I. I I _ . I I d -15 -10 -5 0 5 10 15 deg SIN-' (A,) EQUIVALENT FLIGHT-PATH ANGLE, ( a ) Acceptable f l i g h t envelope and envelope abuse b u f f e r r e g i o n .

F i g u r e 24.- Trim s o l u t i o n s f o r t h e c o n f i g u r a t i o n s c h e d u l e - VE = 80 k n o t s (p = p,).

1 .z 1 . E 1.4 A , , 9 1.2 'EADY

1 .o

= 20° .8 .6 .4

-.3 -.2 -. 1 0 .1 .2 .3

A,.

(b) Nozzle schedule contour plot (VE = 80 knots).

Figure 24.- Continued.

2.0 1.8 1.6 o l w ' 1.4 a

E

LL

-

Y

n

; 1.2

w

-I n n a -I a

z

0 1.0 z .8 .6 17.6 1 I I I I I .4

-.2 -. 1 0 . I .2

- . 3 LONGITUDINAL APPLIED SPECIFIC FORCE, g ( c ) T r i m t h r o t t l e s e t t i n g c o n t o u r p l o t (VE = 80 knots).

F i g u r e 24.- Continued.

2.0 1.8 1 . 6 1.4 A N . g 1 . 2 .a . 6 1 1 . I 1 1 1 .4 - . 2 - . I 0 .I .2 .3 - . 3 A,. 9 (d) T r i m a n g l e of a t t a c k c o n t o u r p l o t (VE = 80 k n o t s ) .

F i g u r e 2 4 . - Continued.

1 1 6 2 . ( 1 .I 1.6 1.4 AN, 9 1.2 1 .o .8 .6 .4 Au. g ( e ) T r i m e l e v a t o r s e t t i n g c o n t o u r p l o t (VE = 80 k n o t s ) .

F i g u r e 24.- Concluded.

11 7 BASIC TRIMMAP CONFIGURATION MAP TO STABILITY SCHEDULE:

AXIS COMPONENTS -

F' (9 EQ. (39 v * (2) FIG.22(al

I I I

6, 6 f . VEr vEc

-

V C ' St, TRIM SOLUTION ALGORITHM u + (6f, v , x. P I

- & r L ' A u f MAP TO STAB1LlTY AN)c FIG. 14 ELEVATOR TRIM 6e_T

APPENDIX B FUNCTION Tpr(Y. *"I AXIS COMPONENTS 6eT:EQS.(A10),(Al 1)

EO. (D4) actp EQS. (D18),(D19) 7 'vc

t '+ I

?Ac# %Ac EULER ATTITUDE MAP TO SPHERICAL COORDINATES EQ. ( 4 ) F i g u r e 25.- AWJSRA Force Trimmap.

h X -2000 -4000 -6000 m 2000 0 TRAJECTORY INPUT PARAM ETE R S Initial position Initial velocity A c c e l e r a t i q Initial - .~ time Leg Po. V. 3,.

7.

XO zO' @V* RC.

( s e d m m m knots m deg deg 9 m 1 0 629.3 -800.6 -1161.6 140 27 0 0 0 2 12 629.3 -914.6 -1161.6 140 27 0 -3 0 -1 524 m 3 45 -894.8 -2439.0 -1036.0 140 180 0 -.02

4 96 -4298.8 -2439.0 -1036.0 120 180 0 0 -1219 1 0

5 127 -5518.3 0 -.035 -1219.5 -1036.0 120 90 -1219 W 6 161 -4298.8 0 -1036.0 96.3 0 -5.9 -.021 7 194 - 2807.9 0 -884.1 83.7 0 -7.5 -.01 -61 0 m 8 294 -2807.9 0 -381.1 65 0 -7.5 0 W 9 379.9 40.0 0 -6.1 65 0 -1.6 0 0 approach ? a t h .

F i g u r e ! 6 . - Four-dimensional s i m u l a t i o n tes 11 9

r

F Iu .-.

NORMAL I L -.-.

ACCELERATION, g LONGITUDINAL MIN ACCEPTABLE ACCELERATION, g NOMINAL A, -.2 - - - 100 SPEED, knots MIN ACCEPTABLE DESCND'G DECEL'G DECEL'G

TURN I LEVEL LEVEL TURN GLIDE I DECELERATING HELIX , GLIDE SLOPE FL?RE

I I I I I I I I I H I - - - - STEADY WIND WIND SHEAR' MODEL ERROR STEADY WIND (a) Acceleration and speed commands.

Figure 27.- Simulation test path.

MAXIMUM ACCEPTABLE STEADY THROTTLE

30 r

S/L DECEL'G DECE L'G DECEL'G GLIDE SLOPE FLARE , GLIDE I HELIX ~ I , I , LEVEL , I I , H +----------1 I I MODEL ERROR STEADY WIND STEADY WIND WIND SHEAR (b) C o n t r o l commands t i m e h i s t o r i e s .

F i g u r e 27.- Continued.

150 - knots - F LIGHT-

--

PATH ANGLE, deg - -10 HEADING ANGLE, 0 deg -200

25 r

I n

PATH ERRORS, m

- /

c +-:e-- - .-.---

I v I I I I I

-3 -

40 80 120 160 200 240 280 320 360 400 0 DESCND'G DECEL'G DECEL'G DECEL'G T U R N , LEVEL LEVELTURN GLIDE , HELIX GLIDE SLOPE FLfRE I I - 1 I I H STEADY WIND WIND SHEAR MODEL ERROR ( c ) T r a j e c t o r y e r r o r s .

F i g u r e 27.- Concluded.

1 2 2 ACCEPTABLE FLIGHT ENVELOPE FOR F = & 2 0 6f = F* v = v* .IO I LEVEL TURN A", 9 NO WIND PATH -.lo -.20 DESCENT -.30

I I I -L

70 90 110 130 150 VE, knots ( a ) T r a j e c t o r y a c c e l e r a t i o n and speed l o c u s .

F i g u r e 28.- S i m u l a t i o n test p a t h .

1 2 3 I T.D. T.D.

t

FLAP, deg START 0 20 40 60 ao NOZZLE, deg (b) Locus of c o n f i g u r a t i o n commands.

F i g u r e 28.- Concluded.

1 2 4 V = AIRCRAFT VELOCITY VECTOR is, k, = STABILITY AXES = ANGLES OF V, T VECTORS FROM BODY LONGITUDINAL AXIS a, v L, D = WING-BODY AERODYNAMIC FORCE COMPONENTS LT = T A I L LIFT = ENGINE RAM DRAG AND THRUST FORCES D R ~ T H F i g u r e 29.- Engine and aerodynamic f o r c e s - AWJSRA.

6,= max { 0.0618N. 1.3866NH-107.17 ] THROTTLE 3o LEVER ANGLE CORRECTED 6 , , deg - MASS FLOW 50 - 0 5 0 100 I POWER SETTING, rpm/121.35,NH I I .

4a CORRECTED FUEL FLOW 2000 I 1 I CORRECTED HOT THRUST

20 V, knots 1

5 , k N CORRECTED COLD THRUST 10 I 0 - 50 70 90 110 CORRECTED POWER SETTING, N H / ~ F i g u r e 30.- AWJSRA e n g i n e model - one e n g i n e , h o t t h r u s t , c o l d t h r u s t , mass flow, f u e l flow and t h r o t t l e - p o w e r r e l a t i o n .

1 2 6 50 1.5 z Y SEA LEVEL

d

V A = 60 knots o z 40 40

1 .o

I I - I - CJ

P

30 .5 STANDARD SEA (ARDC ATMOSPHERE USED) LEVEL TEMP 1 I I I I

20 L

0 1 2 3 4 5 0 100 200 PRESSURE ALTITUDE, km TEMPERATURE, "C AIRSPEED, knots ( a ) E f f e c t of a l t i t u d e (b) E f f e c t of tempera- ( c ) E f f e c t of air- and a i r s p e e d on t u r e on maximum speed on CJ a t maximum c o n t i n u o u s c o n t i n u o u s h o t maximum c o n t i n u o u s h o t t h r u s t .

t h r u s t .

power.

F i g u r e 31.- Engine o u t p u t a t maximum c o n t i n u o u s power ( N = 95.8): two H e n g i n e s 1 2 7 '.5 CL

f

C Y 6 f = 5.6" 6f = 30" -'J I

-2 A - I

- CL -2 0 ' 2 -1 ' 1 3 CD CD C Y CONTOUR VALUES: -10.5'. -6.5". -2.5". 1.5'. 5.5'. 9.5', 13.5', 17.5', 19.5", 27.5' CONTOUR VALUES: 0, .2, A, .6, .8, 1.2,2 C j F i g u r e 3 2 . - L i f t - d r a g p o l a r s - AWJSRA.

1 2 8

-32 ( 1 1 1 1 1 1 1 1 1 1 1

6f = 65" 6f = 50' ELEVATOR, deg -32 , -20 -10 0 10 20 30 -20 -10 0 10 20 30 ANGLE OF ATTACK, deg ANGLE OF ATTACK, deg F i g u r e 3 3 . - AWJSRA e l e v a t o r t r i m s e t t i n g - 6eA(6f, a, CJ).

t

OPE RAT1NG POINT,

\ \

AN F i g u r e 3 4 . - C o n t r o l margin d e f i n i t i o n .

1 2 9 1 .c 1.4 1.2 NORMAL ACCELERATION, AN, .8 .6 .4 .2 -.4 -.2 0 .2 .4 LOGITUDINAL ACCELERATION, A,, g Figure 35.- Acceleration envelope, dd (VE, p); 6f = 65", VE = 65 knots, f P = Ps.

F i g u r e 3 6 . - T e s t f o r e x i s t e n c e of c o n t r o l margin.

AN 5 A" F i g u r e 37.- Minimum d i s t a n c e from t o a l i n e segment, L.

5 ; 1 3 1 2. Government Accession No. 3. Recipient's Catalog No.

1. Report No.

NASA TP-1222 I 4. Title and Subtitle 5. ReDort Date March 1979 CONFIGURATION MANAGEMENT AND AUTOMATIC CONTROL OF AN AUGMENTOR WING AIRCRAFT WITH VECTORED' THRUST 6. Performing Organization Code No.

8. Performing Organization Report 7. AuthorW A-7099 L u i g i S. C i c o l a n i , B. S r i d h a r , * and George Meyer 10. Work Unit No.

505-07-11 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 o r n i a 94035 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address T e c h n i c a l Paper I 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 15. Supplementary Notes * M C P o s t d o c t o r a l Research A s s o c i a t e 16. Abstract An advanced s t r u c t u r e f o r a u t o m a t i c f l i g h t c o n t r o l l o g i c f o r p o w e r e d - l i f t a i r c r a f t o p e r a t i n g i t e r m i n a l areas is under i n v e s t i g a t i o n a t Ames Research C e n t e r . T h i s s t r u c t u r e i s based on a c c e l e r z t i o n c o n t r o l ; a c c e l e r a t i o n commands are c o n s t r u c t e d as t h e sum of a c c e l e r a t i o n on t h e r e f e r e n c e t r a j e c t o r y and a c o r r e c t i v e feedback a c c e l e r a t i o n t o r e g u l a t e p a t h t r a c k i n g e r r o r s . The c e n t r a l element of t h e s t r u c t u r e , termed a Trimmap, u s e s a model of t h e a i r c r a f t aerodynamic and e n g i n e f o r c e s t o c a l c u l a t e t h e c o n t r o l s e t t i n g s r e q u i r e d t o g e n e r a t e t h e a c c e l e r a t i o n commands.

T h i s r e p o r t d e s c r i b e s t h e d e s i g n c r i t e r i a f o r t h e Trimmap and d e r i v e s a T r i m m a p f o r Ames exper i m e n t a l augmentor wing j e t STOL r e s e a r c h a i r c r a f t . The p r i n c i p a l problems a r e a s s o c i a t e d w i t h con- t r o l redundancy ( t h e r e are two more c o n t r o l s t h a n n e c e s s a r y t o g e n e r a t e any g i v e n a c c e l e r a t i o n command) and model n o n l i n e a r i t y . C o n t r o l redundancy is r e s o l v e d u s i n g a s t o r e d c o n f i g u r a t i o n sched u l e which s e l e c t s two of t h e c o n t r o l s ( f l a p and e n g i n e e x h a u s t n o z z l e ) a s a f u n c t i o n of t h e r e f e r e n f l i g h t c o n d i t i o n w h i l e t h e remaining two c o n t r o l s ( e l e v a t o r and t h r o t t l e ) a r e computed on t h e b a s i s of t h e t o t a l a c c e l e r a t i o n command and t h e a i r c r a f t f o r c e model.

The c o n f i g u r a t i o n s c h e d u l e is d e r i v e d f o r maximum c o n t r o l margins. The a l g e b r a i c n o n l i n e a r i t y of t h e f o r c e model i s t r e a t e d u s i n p i e c e w i s e l i n e a r d e s c r i p t i o n s of t h e model o v e r i t s bounded domain.

The a u t o m a t i c c o n t r o l system, i n c l u d i n g t h e Trimmap d e s c r i b e d i n t h i s p a p e r , w a s s u b j e c t e d t o s i m u l a t i o n tests u s i n g a r i g o r o u s STOL approach t r a j e c t o r y . tests d e m o n s t r a t e t h e system These r e s p o n s e t o maneuver commands, Trimmap model e r r o r s , and s t e a d y winds.

The proposed Trimmap m a i n t a i n s a n o p t i m a l c o n f i g u r a t i o n and c o o r d i n a t e s a l l c o n t r o l s d u r i n g any a d m i s s i b l e maneuvers and i n r e s p o n s e t o s t e a d y winds; it compensates a u t o m a t i c a l l y f o r model e r r o r s .

18. Distribution Statement 17. Key Words (Suggested by Authods)) Automatic V/STOL f l i g h t c o n t r o l Unlimited Automatic c o n f i g u r a t i o n s c h e d u l e Trimmap P o w e r e d - l i f t a i r c r a f t STAR Category - 08 22. Price' 21. NO. of Pages 20. Security Classif. (of this page) 19. Security Classif. (of this report)

1 1 4 1 I $6.00

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