Effect of sampling rate and record length on the determination of stability and control derivatives

(NASA-TIY-72858) EFFECT O F S A H P L I N G RATE A N D R E C O R D LENGTH ON THE DETERMIYATTON OF STABILITY A N D CONTROL D E R I V A T I V E S ( N A S A ) 1Q0 p RC A07/rF A 0 1 CSCL 0 1 C Unclas G 3 / 0 8 3 8 9 0 1 NASA T e c h n i c a l Memorandum 72858 EFFECT OF SAMPLING RATE AND RECORD LENGTH ON THE DETERMINATION OF STABILITY AND CQNTROL DERIVATIVES

Martin J . B r e n n e r , Kenneth W . Iliff, a n d Robert K . Whitman

D e c e m b e r 1978

NASA T e c h n i c a l Memorandum 72858 EFFECT OF SAMPLING R.4TE AND RECORD LENGTH ON THE DETERMINATION OF STABILITY AND CONTROL DERIVATIVES Martin J . B r e n n e r . K e n n e t h W . Iliff, a n d Robert K . Whitman D r y d e n F l i g h t R e s e a r c h C e n t e r E d w a r d s . California Nat~onal Aero~aclf~cs and Space Adrnin~strat~on

EFFECT O F SAMPLING RATE AND RECORD LENGTH O N THE DETERMINATION O F STABILITY AND CONTROL DERIVATIVES M a r t i n 3 . Brenner, Kenneth W. I l i f f , and Robert K. Whitman NASA Hugh L. Dryden F l i g h t Research Center INTRODUCTION S t a b i 1 i t y and c o n t r o l d e r i v a t i v e s e x t r a c t e d from f l i g h t data have been used f o r many years t o p r c v i d e f i n a l v e r i f i c a t i o n o f t h e p r e d i c t e d f u l l - s c a l e a i r c r a f t aerodynamic c h a r a ~ t e r i s t i c s and f o r t h e v e r i f i c a t i o n o f p r e d i c t i o n techniques. The f l i g h t - d e t e r m i n e d d e r i v a t i v e s can be compared w i t h c a l c u l a t e d d e r i v a t i v e s and wind-tunnel p r e d i c t i o n s , and t h i s comparison can be used t o update p r e d i c t i o n methods f o r t h e improvement o f f u t u r e a i r c r a f t designs.

Many areas need t o be s t u d i e d t o assess t h e r e l i a b i l i t y o f t h e f l i g h t - m e a s u r e d s t a b i l i t y and c o n t r o l d e r i v a t i v e s .

Among these areas i s t h e e f f e c t o f sam- p l i n g r a t e and r e c o r d length.

The d e s i r e t o minimize t h e amount o f data proc- e s s i n g r e q u i r e d f o r m y g i v e n f l i g h t program c r e a t e s a need f o r more economi- c a l data hand1 i n g techniques. A r e d u c t i o n i n sampling r a t e and/or r e c o r d l e n g t h would s i g n i f i c a n t l y economize on computer u t i l i z a t i o n f o r processing f l i g h t data i n such an a n a l y s i s . L i t t l e e f f o r t has been devoted t o t h i s sub- j e c t i n a i r c - a f t parameter e s t i m a t i o n .

The computation t i m e r e q u i r e d t o perform a d e r i v a t i v e e s t i m a t i o n i s d i - r e c t l y p r o p o r t i o n a l t o t h e number o f data p o i n t s . The number o f data p o i n t s i s determined by t h e sampl i n g r a t e and t h e r e c o r d l e n g t h .

The lowest sam- p l i n g r a t e s p o s s i b l e f o r d e r i v a t i v e a n a l y s i s a r e examined i n t h i s r e p o r t , a l - though t h e sampling r a t e r e q u i r e d f o r f i l t e r i n g i n t h e data a c q u i s i t i o n sys- tem i s u s u a l l y h i g h e r t h a n t h a t necessary t o o b t a i n s t a b i l i t y and c o n t r o l de- r i v a t i v e s . This r e p o r t presents t h e r e s u l t s o f d e t e r m i n i n g s t a b i l i t y and con- t r o l d e r i v a t i v e s from f l i g h t d a t a u s i n g t h e maximum 1 i k e l ihood e s t i m a t i o n tech- nique ( r e f . 1). Several l a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l maneuvers were analyzed a t d i f f e r e n t sampling r a t e s and r e c o r d l e n g h t s t o assess t h e e f f e c t o f sampling r a t e and/or r e c o r d l e n g t h . Time s h i f t i n g e f f e c t s ( r e f . 2), which a l s o i n f l u e n c e t h e q u a l i t y o f t h e estimates, a r e n o t i n v e s t i g a t e d i n t h i s r e - p o r t .

normal accel erati on, g lateral acceleration, g nondimensional roll ing-moment coefficient nondimensional pi tchi ng-moment coefficient nondimensional yawing-moment coefficient norrdimensional side-force coefficient nondimensional normal-force coefficient roll rate, deg/sec or rad/sec roll i ng angular acceleration, deg/sec pitch rate, deg/sec or rad/sec 2 2 dynamic pressure, kN/m (Ib/ft ) yaw rate, deg/sec or rad/sec yawing angular acceleration, deg/sec velocity , m/sec (ftjsec) angle of attack, deg or rad angle of sideslip, deg or rad aileron deflection, deg or rad differential tail deflection, deg or rad blended combination o f spoiler and differential tail deflections, deg or rad elevator deflection, deg or rad rudder deflection, deg or rad pitch angle, deg or rad roll angle, deg or rad

Subscripts: p, 9 , r, a, fi p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o t h e ba, 6c1, 6 s u b s c r i p t e d v a r i a b l e s METHOD OF ANALYSIS A maximum 1 i k e l ihood e s t i m a t i o n (MLE) method o f a n a l y s i s , described i n r e f e r e n c e 1, was used t o determine a complete s e t o f l i n e a r s t a b i l i t y and con- t r o l d e r i v a t i v e s fvom t h e maneuvers performed i n f l i g h t . The method i s a d i g i t a l computational technique t h a t determines t h e b e s t s e t o f c o e f f i c i e n t s ( s t a b i l i ty and c o n t r o l d e r i v a t i v e s ) o f t h e 1 i n e a r i r e d equations o f motion. T h i s technique minimizes a weighted i n t e g r a l squared e r r o r between flight-measured and estimdted time h i s t o r i e s .

The r e s u l t i s t h a t t h e estimated t i m e h i s t o r y tends t o match t h e f l i g h t time h i s t o r y .

Examples o f matches o f l a t e r a l - d i r e c t i o n a l d a t a f o r each o f t h e f i v e a i r c r a f t i n v e s t i g a t e d a r e shown i n f i g u r e 1, and t h e matches f o r t h e l o n g i t u d i n a l data a r e shown i n f i g u r e 2. The s o l i d l i n e i s t h e measured data and the dashed l i n e i s t h e MLE estimated data. The elements o f t h e w e i g h t i n g m a t r i c e s f o r each v e h i c l e (discussed i n general i n r e f e r e n c e 3) a r e g i v e n i n reference 2.

The Cramer-Rao bound ( r e f s . 3 and 4) p r o v i d e s an e s t i m a t e o f t h e degree o f confidence t h a t should be placed i n t h e d e r i v a t i v e s e x t r a c t e d from f l i g h t T h i s bound, a l s o c a l l e d t h e u n c e r t a i n t y l e v e l , data. p r o v i d e s an e s t i m a t e o f t h e lower bound o f t h e covariance o f t h e parameters estimated from a g i v e n s e t o f f l i g h t data.

TEST AIRCRAFT A N D DATA SYSTEM F l i g h t t e s t data from several types o f a i r c r a f t were analyzed t o o b t a i n r e s u l t s t h a t were independent of t h e a i r c r a f t c o n f i g u r a t i o n . The c h o i c e o f a i r c r a f t was based l a r g e l y on t h e a v a i l a b i l i t y of t h e proper dynamic response data f o r determining s t a b i l i t y and c o n t r o l d e r i v a t i v e s . The d a t a a r e from f l i g h t t e s t programs conducted a t t h e NASA Dryden F l i g h t Research Center.

These a i r c r a f t represent a wide v a r i e t y o f a i r c r a f t c o n f i g u r a t i o n s .

Data from f i v e a i r c r a f t , r e f e r r e d t o as a i r c r a f t A, B, C, D, and E, were used. A i r c r a f t A was a PA-30 a i r c r a f t , a l i g h t , twin-engine general a v i a t i o n a i r p l a n e described i n reference 5. An unpowered remotely p i l o t e d 3/8-scale model o f t h e F-15 a i r p l a n e ( r e f . 6) was a i r c r a f t B. The J e t S t a r a i r p l a n e , a

A i r c r a f t D was low-winged e x e c u t i v e j e t t r a n s p o r t , was a i r c r a f t C ( r e f . 7 ) .

an F-111A a i r p l a n e ( r e f . a ) , a f i g h t e r w i t h a v a r i a b l e sweep wing. The HL-10 l i f t i n g body research v e h i c l e ( r e f . 9) was a i r c r a f t E.

Table 1 l i s t s t h e f l i g h t c o n d i t i o n s f o r which t h e d a t a from t h e f i v e a i r c r a f t were acqui red.

Each i n d i v i d u a l case represents a s i n g l e maneuver.

A1 1 o f t h e t e s t s were performed a t a nominal l o a d f a c t o r o f l g w i t h s t a b i l i t y augmentation systems o f f . The sense c f t h e rudder d i r e c t i o n f o r a i r c r a f t C 2nd t h e sense o f l a t e r a l c o n t r o l f o r a i r c r a f t D a r e o p p o s i t e t o t h e sense o f these c o n t r o l s f o r t h e o t h e r a i r c r a f t .

The t e s t a i r c r a f t were instrumented t o measure t h r e e - a x i s l i n e a r accelera- t i o n s and angular r a t e s , E u l e r angles, angle of sides1 i p , angle o f a t t a c k , con- t r o l s u r f a c e d e f l e c t i o n s , v e l o c i t y , and a l t i t u d e . The data were recorded on a n i ne-bi t p u l s e code modul a t i on (PCM) magnetic tape system. The b a s i c sampl i ng r a t e s o f t h e PCM system were 200 samples p e r second p e r channel f o r a i r c r a f t A, B, and C; 20 samples p e r second p e r channel f o r a i r c r a f t 0; and 50 samples p e r second p e r channel f o r a i r c r a f t E. Data from a i r c r a f t A, B , and C were t h i n n e d t o 50 samples p e r second f o r processing. The t a b u l a t e d values i n t a b l e 1 f o r t h e sampling r a t e s a r e r e f e r r e d t o as t h e b a s e l i n e sampling r a t e s .

Before being encoded and recorded by t h e P C M system, t h e data were f i l - t e r e d w i t h a f i r s t - o r d e r , low-pass a n t i - a l i a s i n g f i l t e r . The 200 samples p e r second data f o r a i r c r a f t B were a l s o d i g i t a l l y f i l t e r e d t o remove h i g h f r e - quency s t r u c t u r a l resonance b e f o r e t h i n n i n g .

PESULTS AND DISCUSSION The h i g h q u a l i t y data from a i r c r a f t A were s e l e c t e d f o r more e x t e n s i v e a n a l y s i s than t h e data from t h e o t h e r f o u r a i r c r a f t . Once conclusions had been drawn from t h e a i r c r a f t A data, t h e r e s u l t s from t h e o t h e r a i r c r a f t were used t o generai i ze these conclusions. Consequently, treatment o f a i r c r a f t A w i l l be more thorough than t h a t o f t h e o t h e r a i r c r a f t , and comparisons w i l l be c i t e d where a p p l i c a b l e . A i r c r a f t B, C , D , and E r e p r e s e n t a broad c l a s s o f a i r c r a f t f o r a wide v a r i e t y o f f l i g h t c o n d i t i o n s (Mach number, angle o f a t - t a c k , e t c . ) . The acceptable d a t a from these f o u r a i r c r a f t showed g r e a t e r non- l i n e a r i t i e s ( i n d i c a t e d by poorer matches b e t w e ~ n computed and a c t u a l f l i g h t data) than d i d t h e d a t a from a i r c r a f t A. L a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l maneuvers f o r each a i r c r a f t (except a i r c r a f t E , where a l o n g i t u d i n a l maneuver was n o t used) were analyzed. The t y p e o f i n p u t f o r each maneuver i s l i s t e d i n t a b l e I. The values o f t h e elements o f t h e w e i g h t i n g m a t r i c e s used i n t h e maximum l i k e l i h o o d e s t i m a t o r a r e shown i n t a b l e 2. The values o f t h e d e r i v - a t i v e estimates computed from t h e b a s e l i n e sampling r a t e s ( t a b l e 1) a r e des- i g n a t e d as b a s e l i n e d e r i v a t i v e values. These values a r e presented i n t a b l e 3.

Experience has shown t h a t d a t a obtained a t 50 samples p e r second f o r r e c o r d l e n g t h s a t l e a s t t w i c e t h e c h a r a c t e r i s t i c t i m e o f t h e a i r c r a f t have p r o v i d e d s a t i s f a c t o r y s t a b i l i t y and c o n t r o l d e r i v a t i v e estimates. Therefore, i t was n o t necessary t o study sampling r a t e s h i g h e r t h a n 50 samples p e r second.

The a i r c r a f t A data base c o n t a i n e d 1 a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l maneuvers w i t h c o n t r o l i n p u t s o f two d i f f e r e n t amplitudes. F i v e small ampli- tude maneuvers w i t h n e a r l y i d e n t i c a l c o n t r o l i n p u t s and f o u r 1 arge ampl i tude maneuvers w i t h n e a r l y i d e n t i c a l c o n t r o l i n p u t s were used f o r t h e l a t e r a l - d i r e c t i o n a l a n a l y s i s ; and s i m i l i a r l y , two groups o f f o u r l o n g i t u d i n a l maneu- vers were analyzed. The n e a r l y i d e n t i c a l c o n t r o l i n p u t s were o b t a i n e d b y up- 1 i n k i n g ( r e f . 10) t h e computer-generated c o n t r o l commands. The r a t i o between t h e l a r g e and small amplitude s i m i l a r l y - s h a p e d c o n t r o l i n p u t s i s a f a c t o r o f 1 . 4 f o r t h e l o n g i t u d i n a l maneuvers, and a f a c t o r of 2.0 f o r t h e l a t e r a l - d i r e c t i o n a l maneuvers.

A comparison o f t h e r e s u l t i n g s t a b i 1 i t y and c o n t r o l d e r i v a t i v e s from t h e l a r g e and small ampl i tude maneuvers r e v e a l s i n f o r m a t i o n about t h e consistency o f t h e estimated d e r i v a t i v e t r e n d s as sampl i ng r a t e and/or r e c o r d l e n g t h a r e reduced. No s t u d y was done on t h e of h e r a i r c r a f t t o show t h e e f f e c t o f r e c o r d l e n g t h o r c o n t r o l i n p u t amplitude.

EFFECT OF SAMPLING RATE AND RECORD LENGTH O N TIME HISTORY MATCPES As expected, t h e matches between f l i g h t and estimated t i m e h i s t o r i e s , when d a t a w i t h very low sampling r a t e s were used i n t h e a n a l y s i s , were l e s s s a t i s f a c t o r y t h a n t h e matches obtained w i t h b a s e l i n e data.

T h i s i s due t o 1 i m i t e d d e f i n i t i o n o f t h e measured s i g n a l s .

For example, f i g u r e s l ( a ) and 2(a) show t h e matches t h a t r e s u l t e d w i t h t h e b a s e l i n e d a t a a t 50 samples p e r second f o r a 1 a t e r a l - d i r e c t i o n a l and a l o n g i t u d i n a l maneuver from a i r c r a f t A, and t h e matches f o r t h e same maneuvers a r e represented a t a reduced sampling r a t e o f f i v e samples p e r second i n f i g u r e s 3(a) and 3(b). The degradation o f t h e matches between estimated and a c t u a l data, as a r e s u l t o f t h i n n i n g t h e data, i s e v i d e n t . It was a l s o observed t h a t as t h e sampling r a t e was reduced, t h e number o f i t e r a t i o n s r e q u i r e d f o r convergence increased. The computation t i m e s t i l l decreased a t lower sampling r a t e s , b u t because o f t h e increased number o f i t e r a t i o n s t h e decrease was n o t p r o p o r t i o n a t e t o t h e r e - d u c t i o n i n number o f data p o i n t s .

Divergence occurred when t h e sampling r a t e approached 5 samples p e r second. I n most cases, t h e maximum l i k e l i h o o d estima- t i o n program converged t o a reasonable answer even i f t h e sampling r a t e was severely reduced.

Reducing r e c o r d l e n g t h d i d n o t show any s i g n i f i c a n t e f f e c t on t h e qual- i t y o f t h e t i m e h i s t o r y matches. The l a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l maneuvers o f f i g u r e s l ( a ) and 2(a) a r e shown again a t t h e same sampling r a t e i n f i g u r e s 4(a) and 4(b), b u t t h i s t i m e w i t h h a l f r e c o r d l e n g t h , and i n f i g - ures 5(a) and 5(b) w i t h f o u r t h r e c o r d length. The d e v i a t i o n from t h e o r i g i n a l maneuvers i s v e r y s l i g h t i n a few o f t h e parameters o r n o t n o t i c e a b l e a t a l l .

The weighted e r r o r sums d e r i v e d from t h e maximum l i k e l i h o o d e s t i m a t o r a r e a measure o f t h e q u a l i t y o f t h e match. The e r r o r sums d i d n o t change appreci- a b l y as r e c o r d l e n g t h decreased. The number o f i t e r a t i o n s r e q u i r e d f o r conver- gence was n e a r l y t h e same as i o r t h e f u l l r e c o r d l e n g t h d e s p i t e any change i n r e c o r d l e n g t h .

EFFECT O F SAMPLING RATE A N D R E C O R D LENGTH O N DERIVATIVE ESTIMATES The sampling r a t e s f o r both l a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l maneu- vers f o r each a i r c r a f t were reduced (by t h i n n i n g t h e data) from t h e i r base- l i n e r a t e s t o as low as 2.5 samples p e r second, o r u n t i l t h e maximum l i k e l i - hood estimation computer a l g o r i t h m f a i l e d t o u n i f o r m l y converge t o an answer.

F i r s t , t h e baseline values o f t h e d e r i v a t i v e s were estimated f o r each a i r c r a f t ( t a b l e 3). To assess the e f f e c t o f reduced sampling r a t e , the r a t e was low- ered and a new s e t o f d e r i v a t i v e s was obtained f o r each maneuver. The d i f - ference i n t h e d e r i v a t i v e s was a t t r i b u t e d t o the decreased number o f data p o i n t s analyzed per u n i t o f time. Since the number o f data p o i n t s used i s a f u n c t i o n o f both the sampling r a t e and the record length, the combined e f f e c t o f sampling r a t e and record l e n g t h was a l s o investigated. To evaluate t h i s combined e f f e c t , o n l y the high q u a l i t y data from a i r c r a f t A were used.

The f i r s t step i n measuring the combined e f f e c t o f sampling r a t e and rec- ord l e n g t h on the d e r i v a t i v e estimates was t o shorten each a i r c r a f t A maneuver t o one-half and then one-fourth o f i t s baseline record length. Next t h e sam- p l i n g r a t e was lowered using each new record l e n g t h t o acquire a d i f f e r e n t s e t The p o r t i o n o f each maneuver where the o f d e r i v a t i v e s f o r every maneuver.

c o n t r o l i n p u t occurred was always included i n the data. A t a f i x e d sampling r a t e , a comparison between the estimated d e r i v a t i v e s determined using f u l l record lengths w i t h the estimates found a t each reduced record l e n g t h shows It should be noted, f o r example, the e f f e c t o f record l e n g t h on the estimates.

t h a t the same number o f data p o i n t s are used f o r the analysis o f the e n t i r e record l e n g t h a t 25 samples per second as are used f o r the h a l f record l e n g t h a t 50 samples per second. Based on t h e d e r i v a t i v e s computed a t 50 samples per second and f u l l record length, t h e percentages o f change o f the estimates de- termined a t 25 samples per second were compared t o those found w i t h h a l f rec- o r d l e n g t h a t 50 samples per second. S i m i l a r l y , the d e r i v a t i v e s obtained a t 12.5 samples per second were compared t o those determined w i t h quarter record l e n g t h a t 50 samples per second. The r e s u l t s are shown i n t a b l e s 4(a) ( l a t e r a l - d i r e c t i o n a l maneuvers) and 4(b) ( l o n g i t u d i n a l maneuvers). Compari ng these d i f f e r e n c e s demonstrates the d e s i r a b i l i t y o f lessening sampling r a t e as opposed t o reducing record l e n g t h as a method o f reducing the t o t a l computer time required i n a s t a b i l i t y and c o n t r o l analysis.

The s t a b i l i t y and c o n t r o l d e r i v a t i v e s f o r each s e t o f maneuvers from a i r - c r a f t A are presented as functions o f sampling r a t e i n f i g u r e s 6, 7, 8, and 9 f o r f u l l record l e n g t h (top), h a l f record l e n g t h (middle), and quarter record 1 ength (bottoa). The maneuvers are grouped according t o type (1 a t e r a l - d i r e c t i o n a l and l o n g i t u d i nal ) and amp1 i tude o f c o n t r o l i nput. A1 1 the 1 a t e r a l - d i r e c t i o n a l maneuvers from a i r c r a f t A have o n l y a i 1 eron inputs since a i r c r a f t A i s known t o have a rudder i n p u t response which i s inadequate f o r determining a complete s e t o f s t a b i l i t y and c o n t r o l d e r i v a t i v e s . Since the rudder maneuvers do n o t 1 end themselves t o y i e l d i n g high qua1 i t y s t a b i 1 i t y d e r i v a t i v e s f o r the maximum sampling r a t e , it was f e l t t h a t it and c o n t r o l would n o t be v a l i d t o evaluate the e f f e c t o f sampling r a t e and record l e n g t h on these data.

F i g u r e 6 shows t h e p l o t s o f t h e l a t e r a l - d i r e c t i o n a l s t a b i l i t y and c o n t r o l d e r i v a t i v e s versus sampi i n g r a t e f o r t h e f i v e s m a l l e r (16,1 < 10 degrees) con- A p o i n t i s p l o t t e d t r o l i n p u t l a t e r a l - d i r e c t i o n a l maneuvers from a i r c r a f t A.

on t h e r i g h t t h a t represents t h e r o o t mean square average o f those d e r i v a t i v e s computed a t 50 samples p e r second, and t h e v e r t i c a l b a r a s s o c i a t e d w i t h t h i s syn~bol i s t h e average o f t h e corresponding u n c e r t a i n t y 1 eve1 s. These uncer- t a i n t y l e v e l s show t h e amount o f s c a t t e r t h a t m i g h t be expected a t t h e h i g h e s t sampling r a t e . S i m i l a r p l o t s o f t h e f o u r l a r g e r (16,1< 20 degrees) i n p u t l a t e r a l - d i r e c t i o n a l maneuvers a r e shown i n f i g u r e 7. F i g u r e s 10, 11, 12, and 13 show t h e l a t e r a l - d i r e c t i o n a l d e r i v a t i v e s versus sampl i ng r a t e f o r a i r c r a f t 0 , C, D, and E, r e s p e c t i v e l y . The l o n g i t u d i n a l s t a b i l i t y and c o n t r o l d e r i v a - t i v e s versus sampling r a t e obtained from t h e f o u r s m a l l e r ( 1 6 , l < 1.1 degrees) c o n t r o l i n p u t l o n g i t u d i n a l maneuvers from a i r c r a f t A a r e shown i n f i g u r e 8.

F i g u r e 9 represents s i m i l a r r e s u l t s f o r t h e f o u r l a r g e r (16,1< 1.5 degrees) i n p u t l o n g i t u d i n a l maneuvers . The l o n g i t u d i n a l d e r i v a t i v e s as a f u n c t i o n o f sampling r a t e f o r a i r c r a f t B, C, and D a r e p l o t t e d i n f i g u r e s 14, 15, and 16, r e s p e c t i v e l y .

L a t e r a l - D i r e c t i o n a l D e r i v a t i v e s E f f e c t o f sampling r a t e . - The l a t e r a l - d i r e c t i o n a l r e s u l t s f o r a i r - c r a f t A summarized i n t a b l e 4(a) i n d i c a t e t h a t , except f o r C , and C , t h e change due t o sampling r a t e r e d u c t i o n s from 50 t o 12.5 samples p e r second i s n o t g r e a t e r than two percent. The percentages o f change i n C and C '6 a nb a A study o f t h e p l o t s a r e l a r g e because t h e b a s e l i n e values a r e c l o s e t o zero.

i n f i g u r e s 6 and 7 r e v e a l s t h e f o l l o w i n g s i g n i f i c a n t observations o f t h e e f f e c t o f sampling r a t e on t h e a i r c r a f t A d e r i v a t i v e s .

1. The change i n t h e non-control s t a t i c d e r i v a t i v e s i s i n s i g n i f i c a n t t o as low as f i v e samples p e r second. They show very c o n s i s t e n t t r e n d s w i t h respect t o sampling r a t e r e d u c t i o n .

2. The r o t a r y d e r i v a t i v e s and t h e c o e f f i c i e n t o f t h e p a r t i a l d e r i v a - t i v e s o f s i d e f o r c e and yawing moment w i t h r e s p e c t t o t h e c o n t r o l i n p u t ( C y and Cn ) a r e n o t p r e d i c t e d as w e l l when sampling r a t e i s reduced f o r t h e 'a 6a U s u a l l y , t h e t r e n d s o f these estimates, cases w i t h s m a l l e r amp1 i t u d e i n p u t s .

a r e constant, b u t more s c a t t e r about t h e t r e n d was e v i d e n t i n some cases.

3 . Rates as low as 5 t o 10 samples p e r second a r e acceptable t o compute d e r i v a t i v e estimates f o r a i r c r a f t A.

The l a t e r a l - d i r e c t i o n a l p l o t s i n f i g u r e s 10, 11, 12, and 13 demonstrate gocd agreement between a i r c r a f t A and t h e a i r c r a f t B, C , D, and E f i n d i n g s .

S t a t i c d e r i v a t i v e s show c o n s i s t e n t trends, y e t t h e r o t a r y d e r i v a t i v e s and t h e

c o e f f i c i e n t o f the ~ a r t i a l d e r i v a t i v e o f s i d e f o r c e w i t h r e s ~ e c t t o the con-

t r o l motion ( C ) , occasional l y show increased f l u c -

, c , Cy6 , and Cy

' 6 a ' 6 r 6 1 C2 t u a t i o n s as sampling r a t e i s reduced. Nevertheless, 10 samples p e r second a r e s u f f i c i e n t t o compute dersivative estimates w i t h l i t t l e v a r i a t i o n from t h e base1 i n e values, and 5 samples per second are acceptable f o r most estimates.

E f f e c t o f record length. - Table 4(a) shows t h a t t h e e f f e c t o f reducing

the record lengths o f the a i r c r a f t A maneuver i s much more pronounced, w i t h up t o nearly 100 percent change i n the estimates o f Cp Changes i n C and r • Y6 a Fig- C are again l a r g e p a r t l y because o f t h e i r very small baseline values.

%

a ures 6 and 7 show t h a t the trends o f the estimates w i t h respect t o sampling r a t e and record l e n g t h f o r the smaller and l a r g e r amplitude maneuvers are s i m i l a r . The greater s c a t t e r i n the small amplitude cases i s character- i s t i c o f each record length. For f u l l record lengths (top p l o t s ) , the e s t i - mates do n o t change s i g n i f i c a n t l y u n t i l a sampling r a t e o f about i n general

f i v e samples per second i s used, the exceptions being C and C . A t

'6 '6 a a one-half record length, the trends are s t i l l obviaus and very s i m i l a r t o f u l l record l e n g t h trends, b u t w i t h s l i g h t l y more s c a t t e r present. The quarter rec- ord l e n a t h lots do n o t i n d i c a t e we1 1 defined trends i n the d e r i v a t i v e s . e w e -

c i a l l y i n C' , Cpr, Cng , and Cp f o r the smaller inputs. The c o n t r o l deriva-

r B t i v e s do n o t get worse when record l e n g t h i s decreased as the e n t i r e pulse was r e t a i n e d when the l e n g t h o f the maneuver was shortened, thus increasing t h e r e l a t i v e information o f these d e r i v a t i v e s .

No study was done on the e f f e c t o f record l e n g t h on the maneuvers f o r other a i r c r a f t .

Longitudinal D e r i v a t i v e s E f f e c t o f sampling r a t e . - The l o n g i t u d i n a l r e s u l t s f o r a i r c r a f t A de- p i c t e d i n t a b l e 4(b) i n d i c a t e t h a t , except f o r C and C ,the change due ma zb e t o sampling r a t e reduction from 50 t o 12.5 samples p e r second i s n o t more than three percent. The C and CZ changes are l a r g e because the 50 samples per '"a second values are small. ( C i s small due t o the a f t center o f g r a v i t y loca- ma D e r i v a t i v e estimate trends i n f i g u r e s 8 and 9 are very s i m i l a r t o each t i o n . ) other, y e t greater s c a t t e r i n the smaller hmpl i t u d e c o n t r o l i n p u t maneuvers e x i s t s , as was noted f o r the l a t e r a l - d i r e c t i o n a l maneuvers discussed e a r l i e r .

and C are exceptions t o t h e o t h e r w i s e very c o n s i s t e n t t r e n d s i n t h e m a 6 e d e r i v a t i v e s as sampling r a t e i s reduced f o r f u l l r e c o r d lengths. These trends a r e c o n s i s t e n t when sampling r a t e i s reduced t o 10 samples p e r second.

F i v e samples per second i s t o l e r a b l e f o r most estimates.

The i m p o r t a n t observations o f t h e e f f e c t o f sampling r a t e on t h e e s t i - mated l o n g i t u d i n a l d e r i v a t i v e s , based o n l y on t h e data analyzed, a r e sum- mari zed below: 1. shows the l e a s t c o n s i s t e n t t r e n d of t h e estimates as sam- C Z 6e p l i n g r a t e i s reduced. The o t h e r l o n g i t u d i n a l d e r i v a t i v e s a r e v e r y c o n s i s t - ent.

2. Ten samples p e r second a r e more than adequate t o e s t i m a t e d e r i v - a t i v e s .

15, and 16 show good comparison between r e s u l t s f o r a i r c r a f t Figures 14, 0 , C, and D, and t h e a i r c r a f t A r e s u l t s . C f l u c t u a t e s i n most cases as z6 e sampling r a t e i s reduced, w h i l e t h e o t h e r d e r i v a t i v e s a r e steady.

Ten sam- p l e s per second i s s a t i s f a c t o r y f o r d e r i v a t i v e e x t r a c t i o n f o r a l l a i r c r a f t .

E f f e c t o f r e c o r d length. - Table 4(b) shows t h a t t h e e f f e c t o f re-

ducing r e c o r d l e n g t h i s much g r e a t e r than l e s s e n i n g sampling r a t e . F u r t h e r - more, f i g u r e s 8 and 9 show t h a t t h e d e r i v a t i v e values obtained a t q u a r t e r r e c o r d l e n g t h a r e very s i m i l a r t o those d e r i v e d a t h a l f r e c o r d length.

and C changes a r e l a r g e f o r t h e same reason mentioned i n t h e p r e v i - ' m (Y z6 e ous s e c t i o n .

Observations o f t h e e f f e c t o f r e c o r d l e n g t h on t h e l o n g i t u d i n a l d e r i v a t i v e s correspond very we1 1 w i t h t h e e f f e c t s on t h e l a t e r a l - d i r e c t i o n a l estimates discussed e a r l i e r .

EFFECT O F CONTROL INPUT SHAPE The e f f e c t s o f t i m e d u r a t i o n and r a p i d i t y o f t h e c o n t r o l inpu'; on t h e d e r i v a t i v e e s t i m a t e t r e n d s w i t h reduced sampling r a t e and r e c o r d l e n g t h were s t u d i e d f o r l a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l maneuvers from a i r c r a f t A.

The maneuvers were performed d t h s l c w l y - v a r y i n g c o n t r o l i n p u t s and had t h e same i n p u t energy as t h e standard pulses p r e v i o u s l y discussed. The t i m e d u r a t i o n and r a p i d i t y o f t h e c o n t r o l i n p u t s had no s u b s t a n t i a l e f f e c t on the trends o f t h e d e r i v a t i v e s . The more s l o w l y - v a r y i n g c o n t r o l i n p u t s , however, demonct.ratsd much more s c a t t e r w i t h reduced r e c o r d l e n g t h and/or sampling r a t e , as wouid be expected s i n c e t h e e n t i r e p u l s e may n o t be represented i n a shortened maneuver. As a r e s u l t , t h e e f f e c t s o f sampling r a t e and r e c o r d l e n g t h r e d u c t i o n s were n o t as obvious w i t h a l o n g e r , slower- v a r y i n g c o n t r o l i n p u t . Hence, t h e maneuvers w i t h sharp, l a r g e r pulses were

presented i n t h i s r e p o r t t o demonstrate t h e e f f e c t s o f sampling r a t e and/or r e c o r d l e n g t h on t h e d e r i v a t i v e s .

I n summary, i t has been determined from t h e d a t a f o r t h e l a t e r a l - d i r e c t i o n a l and l o n g i t u d i n a l d e r i v a t i v e s t h a t i f it i s d e s i r a b l e t o o b t a i n accurate estimates of t h e c o n t r o l d e r i v a t i v e s ( e s p e c i a l l y C

, Cn , and

a a ) and t o a l e s s e r degree t h e r o t a r y d e r i v a t i v e s (Cn , Cp . Cn , and Cm ), r P Q e sampling r a t e s o f 10 samples p e r second o r h i g h e r should be used i n t h e anal- y s i s . The d e r i v a t i v e s t h a t have t h e g r e a t e s t e f f e c t on t h e a i r c r a f t response, l i k e Cn , were n o t a f f e c t e d u n t i l very low r a t e s and s h o r t r e c o r d l e n g t h s B were reached. An i m p o r t a n t aspect o f a l l t h e d e r i v a t i v e t r e n d s from b o t h types o f a i r c r a f t A maneuvers i s t h e increased s c a t t e r i n t h e estimates as t h e amp1 i t u d e o f t h e c o n t r o l i n p u t s i s decreased.

The lowest t o l e r a b l e sampl i n g r a t e f o r e x t r a c t i n g reasonably accurate s t a b i l i t y and c o n t r o l d e r i v a t i v e s i s n e a r l y i d e n t i c a l ( 5 t o 10 samples p e r second) f o r a l l these a i r c r a f t . Any v e h i c l e being t e s t e d should be s t u d i e d t o determine t h e lowest acceptable sampling r a t e w i t h t h e requirements o f i n d i v i d u a l t e s t i n g and computation f a c i l i t i e s . Since t h e sampling r a t e r e q u i r e d f o r d i g i t a l data f i l t e r i n g i s usual l y h i g h e r than t h a t necessary t o o b t a i n s t a b i l i t y and c o n t r o l d e r i v a t i v e s , t h e data must be f i l t e r e d b e f o r e it i s thinned. The e f f e c t o f sampling r a t e needs t o be checked r e g u l a r l y t o v e r i f y t h a t t h e e s t i m a t i o n process y i e l d s s u f f i c i e n t l y good estimates.

EFFECT OF SAMPLING RATE AND RECORD LENGTH O N UNCERTAINTY LEVELS T I I ~ v e r t i c a l l i n e s through t h e data p o i n t s a t t h e b a s e l i n e sampling r a t e s i n f i g u r e s 6 through 16 i n d i c a t e u n c e r t a i n t y l e v e l s . The u n c e r t a i n t y l e v e l s a t t h e h,ighest sampling r a t e represent t h e i n h e r e n t e r r o r i n t h e base1 i ne estimates. These l e v e l s a r e bounds t h a t a p a r t i c u l a r d e r i v a t i v e f o r a p a r t i c u l a r maneuver should remain w i t h i n t o be considered s a t i s f a c t o r y as sampling r a t e i s reduced. For t h e a i r c r a f t A maneuvers ( f i g u r e s 6, 7, 8, and 9 ) , t h e trends o f t h e u n c e r t a i n t y l e v e l s a r e c o n s i s t e n t w i t h t h e theo- r e t i c a l d e f i n i t i o n o f u n c e r t a i n t y l e v e l . The u n c e r t a i n t y l e v e l s a r e t h e smal- l e s t f o r maximum r e c o r d l e n g t h , and as t h e e r r o r due t o reduced r e c o r d l e n g t h ( l e s s i n f o r m a t i o n o f t h e a i r c r a f t response) increases, t h e s i z e of t h e uncer- t a i n t y l e v e l increases. The u n c e r t a i n t y l e v e l s a r e a l s o l a r g e r f o r t h e s m a l l e r amplitude c o n t r o l i n p u t cases ( f i g u r e s 6 and 8) as compared t o t h e l ~ r g e r amplitude maneuvers ( f i g u r e s 7 and 9). A major d e t e r m i n i n g f a c t o r i n any s t a b i l i t y and c o n t r o l a n a l y s i s i s t h e accurate d e f i n i t i o n o f c o n t r o l motion.

The r e s o l u t i o n s o f t h e s i g n a l s used f o r processing w i l l a f f e c t t h e estimated a i r c r a f t response t o t h e c o n t r o l i n p u t t o a g r e a t e r degree f o r a small i n p u t

than f o r a l a r g e r one. Consequently, l e s s i n f o r m a t i o n i s p r e s e n t i n t h e d a t a and more e r r o r i s i n t r o d u c e d i n t o t h e estimated d e r i v a t i v e s .

As an example o f t h e i n f l u e n c e on t h e u n c e r t a i n t y l e v e l s due t o reduc- t i o n s i n t h e sampling r a t e , f i g u r e 17 shows d e r i v a t i v e p l o t s , w i t h confidence l e v e l s , o f t h e l a t e r a l - d i r e c t i o n a l a i r c r a f t A maneuver o f f i g u r e l ( a ) . The l e v e l s a r e f a i r l y constant w i t h reduced ,-ampling r a t e . The disagreement a t 2.5 samples p e r second was found t o be a r e s u l t o f m i s s i n g t h e i n i t i a t i o n o f c o n t r o l motion.

Therefore, t h e v e h i c l e appeared t o s t s r t responding a t a d i f - f e r e n t i n i t i a l t i m e p o i n t than t h e c o n t r o l motion. Consequently, t h e c o n t r o l t i m e h i s t o r y r e s u l t e d i n unacceptable p r e d i c t i o n s o f motion, degrading t h e estimated d e r i v a t i v e s .

The u n c e r t a i n t y l e v e l s g e t l a r g e r wb .n g r e a t e r s c a t t e r e x i s t s i n t h e d e r i v a t i v e s a t t h e lower sampling r a t e s ;,d s h o r t e r r e c o r d lengths. The confidence l e v e l trends a r e as expected when l e s s i n f o r m a t i o n i s g i v e n i n t h e data. Trends i n t h e confidence l e v e l s from t h e maneuvers o f t h e o t h e r a i r c r a f t and a i r c r a f t A were i n good agreement.

CONCLUDING REMARKS

-

The e f f e c t s o f sampling r a t e and r e c o r d l e n g t h on f l i g h t determined s t a b i 1 i t y and c o n t r o l d e r i v a t i v e s were determined by reducing t h e sampl i n g r a t e and r e c o r d l e n g t h from t h e i r base1 i n e values. The d e r i v a t i v e s f o r t h e data were e x t r a c t s d by u s i n g t h e maximum l i k e l i h o o d e s t i m a t ' m method and analyzed as a f u n c t i o n o f sampling r a t e . The combined e f f e c t o f sampling r a t e and r e c o r d l e n g t h was a l s o i n v e s t i g a t e d .

Several types o f a i r c r a f t were s t u d i e d t o determine t h e e f f e c t s o f Sam- p l i n g r a t e and r e c o r d l e n g t h on t h e d e r i v a t i v e s . A wide v a r i e t y o f a i r c r a f t c o n f i g u r a t i o n s and f l i g h t cond:tions were used t o d i s t i n q u i s h between t h e r e s u l t s dependent upon t h e c l a s s o f a i r c r a f t and t h e qua1 i t y o f t h e data, and those e f f e c t s t h a t were independent o f these v a r i a b l e s . The e f f e c t s o f Sam- p l i n g r a t e d e r i v e d from t h e h i g h q u a l i t y PA-30 ( a i r c r a f t A ) data were v e r i f i e d w i t h some lower q u a l i t y ( b u t s t i l l acceptable) data from various o t h e r a i r - c r a f t i~ d i f f e r e n t f l i g h t c o n d i t i o n s . C o n f i r m a t i o n o f t h e r e s u l t s by these o t h e r a i , t r a f t was convincing. G e n e r a l i z a t i o n s f o r a l l a i r c r a f t should n o t be made because t h e data base was small, b u t t h e f o l l o w i n g conclusi3ns about t h e e f f e c t s o f r e d u c t i o n s i n sampling r a t e and r e c o r d l e n g t h o f t h e maneuvers used i n t h i s study may be drawn.

1. Excluding d i g i t a l f i 1 t e r i n g and data t i m e - s h i f t i n g c o n s i d e r a t i o n s , a sampling r a t e o f 5 t o 10 sample: per. second has been found t o be adequate t o o b t a i n reasonably accurate s t a b i ! i c y and c o n t r o l d e r i v a t i v e s by u s i n g t h e maximum 1 i k e l ihood e s t i m a t i o n method.

2 . Reducing t h e sampling r a t e i s more d e s i r a b l e than r e d u c i n g t h e r e c o r d l e n g t h as a method o,f lessening t h e t o t a l computation t i m e r e q u i r e d

for estimation without greatly degrading the quality of the estimates. I f the record length and sampling rate are reduced simultaneously, the magni- tude of the tolerable reductions is smaller.

3. Reducing sampling rate and record length degraded the accuracy of the derivative estimates. The determining factor is the accurate definition of control motion. The small ampl itude inputs demonstrate greater degradation than the larger ampl i tude control inputs. The less significant control deriv- atives and the lateral-directional rotary derivatives were affected more than the other derivatives by the reduction of sampling rate and/or record length.

Dryden F l i ~ h t Research Center National Aeronautics and Space Administration Edwards, Cal ifornia, Crtober 31, 1978

REFERENCES

1. I 1 i f f , Kenneth W. ; and Taylor, Lawrence W. , J r . : Determination

o f Stabi 1 i t y D e r i v a t i v e s From F l i g h t Data Using a Newton-Raphson M i n i m i z a t i o n Technique. NASA T N 0-6579, 1972.

2. Steers, Sandra Thornberry; and I l i f f , Kenneth W.: E f f e c t s o f Time- S h i f t e d Data on F l ight-Determi ned Stabi 1 i t y and Control Derivatives.

NASA TN 0-7830, 1975.

3. Maine, Richard E. ; and I 1 i f f , Kenneth W. : A FORTRAN Program f o r Determining A i r c r a f t Stabi 1 i t y and Control D e r i v a t i v e s From F l i g h t Data. NASA TN 0-7831, 1975.

4. I 1 i f f , Kenneth W. ; and Maine, Richard E. : Further Observations on Maximum L i k e l i hood Estimates o f Stabi 1 i t y and Control Character- i s t i c s Obtained From F l i g h t Data. A I A A 77-1133, Aug. 1977.

5. Fink, Marvin P. ; and Freeman, Delma C . , Jr.: Full-Scale Wind- Tunnel I n v e s t i g a t i o n o f S t a t i c Longitudinal and L a t e r a l Character- i s t i c s o f a L i g h t Twin-Engine Airplane. NASA T N D-4983, 1969.

6. Holleman, E u c l i d C.: Summary o f F l i g h t Tests To Determine the Spin and C o n t r o l l a b i l i t y C h a r a c t e r i s t i c s of a Remotely P i l o t e d , Large- Scale (3/8) F i g h t e r Airplaric: Model. NASA TN D-8052, 1976.

7. Clark, Daniel C . ; and K r o l l , John: General Purpose Airborne Simu- lator-Conceptual Design Report. NASA CR-544. 1966.

8. Sisk, Thomas R . ; Matheny, N e i l W.; K i c r , David A.; and Manke, John A. : A p r e l i m i nary Flying-Qua1 i t i e s Eva1 u a t i o n o f a Variable- Sweep Fighter-Type A i r c r a f t . NASA T M X-1583, 1968.

9. Pyle, Jon S . : L i f t and Drag C h a r a c t e r i s t i c s o f the HL-10 L i f t i n g Body During Subsonic G l i d i n g F l i g h t . NASA TN D-6263, 1971.

10. Edwards, John W . ; and Deets, Dwain A . : Development o f a Remote D i g i t a l Augmentation System and A p p l i c a t i o n t o a Remotely P i l o t e d Research Vehicle. NASA TN 0-7941, 1975.

TABLE 1. TEST CONDITIONS

(A) AIRCRAFT A

-

P 8 BASELINE v, CASE KWH' SAMPLING RATE, Q, O.IANEUVER) DEG ( L B / F T ~ ) INPUT SAHPLES/SECOND

)ABLE 1, - TEST CONDITIONS (CONTINUED) (B) AIRCRAFT B, C , D, AND E LATERAL-DIRECTIONAL MANEUVERS

-

* B , , BASELINE v, KFVM M/ EC : ~ C H QJ SAMPLIN RATE, CASE (FTjsEc) NUMBER DEG (LB/FT'I 1 NPUT sAMPLE~FSECOND ~ I R C R A F T B 50 0,44 2,00(41,1) 5,13 129,7(425,4) 1: 2 &r 0,60 4,40(92,0) 4,74 175,5(575,8) 1: 3 50 6c1 50 0.32 1,55(32,4) 13,911 95,7(314,1) 1 : l O ~

dC1

i

50 0,24 1,28(26,7) 17,25 74,2(243,3)

1 1:13

6~

t

1 : 1 4 ~ 50 0,24 1,32(27,6> 17,52 74,1(243,2) AIRCRAFT C 182 :29 3,60(75,2) 0,40 50 9,70 126,4(414,8) bar 6r AIRCRAFT D 11,30(236,0) 7.00 0,80 239,9(784,0) 20 7:5

B*

1la87(?48,O) 6,50 0 , 8 1 245.1(804,9) 20 7:6 6 , 1On25(214,O) 11,50 264,3(867,0) 20 7:9 6r

20,49(428,0) 367,9(1207,0) 20 7: 17 O g 0 1,24 1 5,50

6r 14,65(306,0) 248,1(814,0> 20 8:4 0,78 5,OO

k 2

AIRCRAFT E *

19 :2 1 1.22 1 16.60 1 361.8(1187.0) 1 7,09(148.0) 1 4. 6 . 1 50

TABLE 1, - TEST CONDITl0llS (CONTINUED) r- I I I 1 I

4 ! BASELINE I

" , 'IACH ~ S E C I ~ll/n*, I S A M P L I N G RATE, j CASE ; NUMBER of6 (FTISEC) (LB/FT 1 I N P U T ; SAMPLES~SECOND 1

T A B L E 2 ELEMENTS OF W E I G l l T I N G MATRICES USED I N THE ERROR M I N I M I Z A T I O N F O R A I R C R A F T A, B j C, D j AND E i A I R C R A F T WEIGHTING

E D c B

MATRICES S I GNAL A

-

i 0 0 0 ' 0 0 I; 0 0 0 0 0 31,430 66,500 100 , 000 35,000 19,100 A~ Lateral- 33,05Cl 31,500 86,100 13,000 55,000 Directional r 122,600 383,000 480,000 2OOJ300 lJ41OJO00 P 2,610 1 1 , 590 35,000 66,300 4,293

577,300 74,300 250,909 100 , 000 450,000

P

-------

10,00!l 3,500 4,890 6,300 an

-------

8 1,100,000 41,599 2,009,000 400,000 Longitudinal

-------

q 115,000 364,000 70,080 130,000

-------

(Y 225,000 571,000 100,000 1,000,000 4 b

M ( V O h M 3 C V h W 3 = r l A z r = r L n h W h

8 8 8 8 8 g S B S

' *

O C O O O O O O 'J

o o o o o q c 3 ? g . . - -

N L n h M - L n N M h d 0 0 h L n W m C V M O h

Y 3 2 s 2 2 5 5 5 2

U 0 0 0 0 0 0 0 0 0 O q ~ O o o O O o - - - . - . .

( V = r w m C n m O N W - - T C O - 4 4 M M O r ( L n h W U 3 M M N N M q n m o ~ h d m h ~ L I C m d m z r m d m rg 2-

C , O O O O O O " O d' C V m m m U 3 4 n w

( U M m w L n N L n m 'J O O O C , O O O O O E L " m m m - = i f . f a

v

c ? o o o y q c 3 q - - -

9 C ? C 1 9 ? C ? C !

I I I I I I I I I I I I .

--- # I

3 2 z 2 i 2 S ~ E s r

= O h O C V C O L n a J

*

M ~ > ~ W ~ I D O W W ~ n z r z r r n ~ m w m $ ( V M c o h m h N 2 7 'J C 9 d 0 3 d - l O d O C ) rg ~ o d o w h o m I

. - . - - N

1 I I I " Z E ! S S ~ Z 2 2

C . 9 9 9 ? C 1 0 = ?

I I I I M C O W C D O L n C D W W

a

R % Z J N % z S S 8 ,

g s q G > g s 2 g

d . - ! d - l 4 4 d y d

- - - - . l % k G ! 3 2 2 E i E

I I I I I I I I I'

U a v i J q L n < r . ? ~ q 3

J N W w - 7 c U s M 4 d f l 4 4 4 4 4 I I I 1 I I l

z s z ? , % s ~ s 2

m C ~ O c n C , O , O m m C Q 1 3 O 4 0 r 1 0 4 0 0 O O C J O O O O O ~

o o o o o o ~ q 0.0-

- - - - - M ; T 4 Z M W D

i

h O h m M m N w W N ( V w m c 9 m - l

,,,,,, ,,, 2 s z a s s a s z

h C O - i m h U l W W O ~1

0 0 ~ 0 0 0 - ? 4 . - - -

M N h L n W C O h O 3 I I I I I I I I O M L n W - l C V ~ J L n 'J W L n s U 3 m L n W h L n 4 4 d l - l 4 4 r ( d

- . - - - . - . -

4 W h W N O C u C V 0 0 a O V \ h D h C n L n 0 - ) 0 0 h d - i W h C " " W U L n W o 0

u

a d = r = r - l r l W h h W

m

g s g s z ~ g > ~

% x s e q h q a g - .

~ - 4 ~ z r ~ ~ 4 9 a 4

I I I I I I I I I I I I I I I I I h - h - h h h

u

s z E ? s E z E ? s

g 5 : 2 = , g s g g g

\ \ h 0 0 W h h W h h W

$

~ ~ ~ ~ Q S E l ~

s

8 8 8 8 8 8 8 8 8

= ? 9 = ? 0 9 9 = ? 9 9

I I I I I I I I I N L n * L " L n N O M W

TABLE 3 . - BASELINE DERIVATIVES

(CONTINUED) (B) AIRCRAFT B , C, D , AND E LATERAL-DIRECTI ONAL HANEWERS

TABLE 3, - BASELINE DERIVATIVES (CONTINUED) C cm cz cm A1 RCRAFT CASE La 6 , za ' 3 be B F-15 RPRV 2 : 1 -, 065409 - ,006057 -6,2785 -, 002019 - ,010208 2:2 -, 048862 -,004474 -6,5334 -,006970 -, 009925 2 : 4 -, 042239 - ,008746 -3,3774 -, 0082% -. 003850 5 : lA -,079106 - ,005367 -6,53377 -, 008547 - ,010398 C JETSTAR 184A:27 -, 094349 -. 014897 -15.177 -, (313728 -, 016588

D F-111A 5 : 4 -. 095696 - ,027798 -43.5252 -. 002445 - ,04019 1

6 : 12 -, 045465 - ,047253 -23,3456 -. ,005089 - ,02921:s 7 : 18 -, 049059 - ,050011 -25,986 - ,001827 -. 02690 8 : 3 -, 103698 - ,9?0826 -36,72383 ,006117 -. 032379

TABLE 4, - PERCENT DIFFERENCE I N

DERIVATIVES FOR AIRCRAFT A (ALL NUMBERS IN PERCENT) FULL RECORD LENGTH ( 1 2 . 5 SPS) FULL RECORD LENGTH ( 2 5 SPS) 50 SPS

I l/4 RECORD LENGTH ( 5 0 SPS) I

2 5 SPS 1 / 2 RECORD LENGTH ( 5 0 SPS) ASTER I SK DESIGNATES AVERAGE COMPUTED W 1 TH EXTREME VALUES a AVERAGES TAKEN EXCLUDING THESE VALUES ARE I N PARENTHESESa

TABLE 4, - PERCENT DIFFERENCE I N

DERIVATIVES FOR AIRCRAFT A (ALL NUMBERS IN PERCENT)

1.1 Aircraft 4 1 6. pula*.

Figure 1. Typical match between estimated and measured fl i g h t time h i s t o r i e s f o r a l a t e r a l - d i rectional maneuver a t the base1 i n e sampling r a t e and f u l l record length.

- .- ..- .- MEASURED . . - - - - .

- C S T I M T I D

" ?I

drg -

(b) A i r c r a f t 8 1 pulme.

(C) Aircreft CI 6 , pulse.

rlqure 1. Continuad

fdl A i r c r a f t D; & p u l s e .

Pigurt . Continuad.

. - - - - . - -

ESTIMATED ( a ) Aircraft A .

Figure 2. Typical match between estimated and measured time histories for a longitu- dinal maneuver a t the baseline sampling rate and full record length.

MEASURED

- - - - - - ESTlMATEO

(b) Aircraft B .

Figure 2 . Continued

MEASURED

-----

- - - - -

- - - ESTIWATED

- deg t~ ( c ) A i r c r a f t C.

Figure 2 . Continued.

MEASURED

---

ESTIC(ATE0 - ?

deg o (d) Aircraft D.

Figure 2 . Concluded.

Figure 3. Matches between estimated and measured flight time histories for Aircraft A maneuver w l t h sampl ing rate reduced t o five samples per second.

RASURtO

-- ---- -

- - - - - - - ESTIWTCD

0 : w ~ ~ ?.OO 4'.00 6:00 T I M L / S E L (b) Longitudinal.

Fiqurm 3 . Concluded.

(a) Lateral-directional . ( b ) Longitudinal .

Figure 4. Matches between estimated and measured flight time histories for Aircraft A maneuver with record length reduced to one-half.

M t CSURLO -* ~ C$TIMTtD - - - ( a ) Latera: -directional (b) Longitudinal Figure 5. Matches between estimated and measured flight time histories for Aircraft A maneuver with record length reduced one-quarter.

u flRNEUVER 1

A flRNEUVER 2 0 nRNEUVER 3 O tlRNEUVER 4 o HRNEUVER 5 X RVERROE VRLUE AT 50 SRflPLES/SEC FULL RECORD LENOTH

I

HRLF RECORD LENOTH OURRTER RECORD LENOTW

l v

-0 03 I-

0 10 2 0 30 4 0 5 0 SAHPLINO RATE I N SAtlPLES/SEC .

(a) cvo.

.

Estimated lateral-directional derivatives a8 a function Figure 6.

of aampling rate for Aircraft A maneuvers with mmall amplitude control inputr.

FULL RECORD LENOTH HALF RECORD LENOTH v

e

Q l l H R T F R RECDRO LENGTH

T

S A H P L I N O RATE I N SAMPLESISEC Figure 6 . Continued

C1 RRhEUVER 1 A IlRNEUVER 2 r+J NRNEUVER 3 6 IRNEUVER 4 ol NRNEUVER S X RVtRROE VALUE RT 5 0 SRHPLES/SEC dALF R F C O R O L C I b T W 0 * 0 ° WARTER RFCORO LEH6TM

0 * 0 ° i

I I 1 I I -0.75 0 10 2 0 3 0 40 60 SRHPLINO RRTE I N SRttPLES/SEC Figure 6. Continued.

FULL R E C O R D LENGTH , per t a d A I A A -0 .so 10 2 0 3 0 40 50 SAtiPLING RRTE I N SRtlPLES/SEC (dl C .

4, Figure 6 . Continued.

FULL RECORD LENGTH

.I I

per dcq 0 -05

t

0.15 I HALF RECORD LENGTH

9 U I R T E R R E C O R D LENGTH I 1 1 I 0 .oo 0 10 2 0 30 4 0 50 SAtlPLING RATE I N SAHPLES/SEC Figure 6 . Continued.

I

R HRMEUVLR & A HRNEUVLR 2 O IlRNEUVER 3 e tlRNEUVER 4 w HRNEUVER 5 X RVEKROL VALUE fir 50 SRHPLES/SEr FULL RECORD LENGTH H l L F RECOUD LENGTH

O.OO i

-0 a30 0 10 20 30 40 50 SAHPLINO RATE I N SAHPLES/SEC Figure 6 . Continued.

I

9 flRNEUVtS I A IIRNtUYER 2 O IlRNElJVER 3 O IIRNEUVER 4 o HRNEUVER 5 X RVERflGE VRLUE AT 53 SAnPLES/SEC FULL P I C O R 0 LEN6TH WALF RECORD LEhl6TH

,*lo 1

I 1 1 I 1 - 1

-0.20 1

0 10 2 0 30 40 5 0

SAHPLINO RATE I N SAHPLES/SEC ( 9 ) C .

" L T Figure 6. Continued.

E l NANFUVER 1 A HANEUVER 2 0 RANEUVER 3 e HANLULLH 4 V HANEUVkR 5 X AVERROE VALUE AT SO SAHPLES/QEC Y

0, • 2.50r

FULL RECORD LENGTH 9, PUkRTER RECORD LENGTH -5.00 ~----A-J I I 1 0 10 ' 7 3 0 40 50 SAMPLING i<HTE IN SRMPLES/SEC Figure 6. i ' o n t i n u c t ' .

R MANEUVER 1 A tlANEUVER 2 O nRNEUVER 3 Q MANEUVER 4 v HANEUVER 5 X AVERAGE VALUE A T SO SAMPL.ES/SEC FULL RECORD LENGTH HALF RECORD LENGTH QUARTER RECORD LEN61 H I 1 I 2

0 .oo

0 10 20 3 0 40 50 S A M P L I N G R A T E I N SflNPLES/SEC Figure 6. C o n t i n u e d .

C ) HANEUVER 1 A IlANEUVER 2 (I; HRNEUVER 3 6 IlRNEUVER 4 9, HANEUVER 8 X AVERAOE VALUE A T SO sAnPLEs/sEc T FULL RECORD LEWCTH Y A L F RECORD LENGTH P

I

I I 1 1 I -0.10

0 10 20 3 0 4 0 50

SAHPLINO RATE I N SAHPLES/SEC Figure 6. Concluded.

FULL RECIRO LLNOTH CJ ttRMUVQ t

0*0° r A . flRNEUVEl 2

HRLF RECORD LENOTt!

O.OO r

OURRTER K C O R O LENDTH

0 ' 0 ° r

-0.03 0 10 2 0 30 40 5 0 S A U f L I N G RATE I N SflflPLES/SEC Estimated lateral-directional derivative8 as a function Figure 7.

of sanpling rate for Aircraft A mansuvera witti large iuq-litude control inputm.

FULL RECORD LENOTH HRLF RECORO LENOTH

O a R S I

OURRTER RECORD LENOTH -0.10 - 1 - 0 1 0 2 0 30 4 0 50 SAHPLING RATE 1.N SfiRPLES/SEC Ib) Figure 7 . continued.

FULL RECUR0 LCNOTH a NRNCUVER 1 NAYUVER 4 X AVER OE VAL E RT 6% L I M P L ~ ~ / S C C HRLF RECORD LCNOTH

O.OO i

OUARlER RECORO LENRTH c , pcr rmd d) -0 50 -0.75 0 10 20 3 0 4 0 50 SflHPLINO RRTE I N SRHPLES/SEC

c ) c i

P Figure 7. Continued

D MRNE'JVER 1 FULL RECORO LENOTH A MANEUVER 2 O MANEUVER 3

+ MANEUVER 4

X AVERAOE VALUE AT 6 0 8AMPLE8/8EC 1 AVSRn!ETA~NTr 1.EYEL HRLF RECORD LENOTH

c4 r

per rad

0 .oo

1 .OC f OUARTER RECORD LENOTH I 1 1 1 1

-0 .so 1

0 - 1 0 - 20 30 40 5 0 SAMPLINO RATE I N SAMFLES/SEC Figure 7 . Continued.

O MANEUVER 1 FULL RECORD LENOTH MRNEUVER 2

0*15 r

Q MANEUVER 3 MANEUVER 4 X RVtR OE VRLW A 5% IR~PLES/B€C

glx

I "6#$TCINTY

HRLF RECORD LENOTW OUARTER RECORD LENOTW

0*15 r

1 1 J

0.00 -

0 I0 20 3 0 4 0 50

SAMPLING R A T E J N SAMPLES/SEC

Figure 7 . Continued.

FULL RECORO LENOTH CJ RAWEUVER 1 A RINEUVER P pcr r i d -0.20

i

HRLF RECORO LENOTH i I OUIRTER RECORO LENOTH

O.OO I

-0.30 1 I 1 0 I 0 10 20 30 40 5 0 SAHPLING R H T E 1 N SAtlPLES/SEC f c , ?

Figure 7. Continued.

DUGmAL PAGE

D B Q U -

a . NAWUVER 1 FULL RECORD LENOTH A NANEUVLR O Q . nRntUVER 3 X RVERAK VAL

60 SRRPL%A€C

I R M R R q UWCL TRlNtY LEVEL HALF RECORD LEWOTH 1 1 I 1 1 -0.20 1 0 10 2 0 30 4 0 5 0 SAHPLINO R A T E I N SAHPLES/SEC ( 1 c "r Figure 7. Continued.

O . nllNEUkF8 I FULL RECORD LENOiH A . nnmuvtt n O . nANEUVEB ¶ a . n n ~ ~ u v ~ i 4 X . RVERRM VRLUE -RT 60 SRtlPLES/S€C 1 p v 5 C B S T ~ I y ~ ~ LEYEL HALF RECORD LENOTH

-s .ooo 0-

10 20 30 4 0 50 SRflPLING RATE I N SAMPLES/SEC

( h ) C

y s + &

Figure 7 . Continued.

o nAntuwR I FULL RECORO LENOTH A tlAWEUW& 2 a . tlnNIUvE1 9 X . AVERAOL VALUE-AT 5 0 I A W L E I M E C

w x

- 0-03 r HRLF RECORD LEWH

OURRTER RECORD LENOTH 1 1 I 1 1 0.00 0 10 20 30 4 0 50 SEHPLINO RRTE I N SRflPLES/SEC Figure 7 . Continued.

D MRNEUWR 1 FULL RECORD LEWTH r m u m o e nAmuvtR 3 ~ r n u v E n 4 HALF RECORO LENOTH I I -9.10 0 10 20 3 0 40 5 0 SAMPLING RATE I N SflHPLES/SEC Figure 7. Concluded

-0 15 0 10 20 30 40 50 SAHPLINL) RATE IN SAHPLES/SEC Estimated longitudinal derivatives as a function of sampling ~ i g u r e 8 .

rate for Aircraft A maneuvers with small amplitude control idputm .

1 1 L 1 J - 0 . 1 0 0 1 0 2 0 3 0 4 0 SO SflHPLINO RATE I N 3AHPLES/SEC

-

ORIGINAL PAGE I S Figure 8. Continued.

og POOR Q U W

-

-10.00 C

"9

p e ~ cad-

- Q @ Q e

20.00

e

HALF RE= LllMllM

0*0° I

-30 00 0 10 2 0 30 40 60 SAHPLINO RATE I N SAHPLES/SEC Figure 8 . Continued.

1 I 1 I I -0.20 1 0 10 20 30 40 SO SAHPLINO ROTE I N SRHPLES/SEC Figure 8. Continued

Figure 8. Concluded

FULL RtcORO LEWOTH

I L . I I

-0 15 0 10 20 3 0 4 0 SO SAHPLENO RATE I N SAHPLfS/SEC

(a) &

Estimated longitudinal derivative8 a8 a function of sampling Figure 9 .

rate for Aircraft A maneuvers with large amplitude control inputs.

FULL RCCORO LCNOlW 0 IIWLUWR 1 r m i u m t 8 I ( M I U I m ¶ * l M W N V U 4 HALF RECORO LENOTH Figure 9 . Continued.

1 , 1 -30.00 0 10 20 30 40 50 SAPlPLrND RATE I N SAPlPLES/SEC ( c ) Cmg .

Figure 9 . Continued

FULL R C C ~ LEIY)M o nnmuvu I r w u v u L Q RnYIUVu 9 + RRNEUVU 4 gr x ; ; ; ; ro%nntL!%/%c vni, @ & a a O a o o r & 8 6 & el 1 & I C ~ ~ ~ ~ N W cz L V be

0*1° i

per de3 -0.10

t

I 1 1 1 I

-0.15 I

0 10 2 0 30 43 S O SAflPLrNO RRTE I N 8RHPLE8/8EC (el Cm6 C Pigure 9. Concluded

SAMPLINC R A T E I N SAHPLES/SEC ( 1 C~ ' ' n f o r case 1 :2.

C 1 L 3 6 Figure P O . Estlmsted l a t e r a l - d i r e c t i o n a l d e r i v a t i v e s f o r f i v e ca;es as I f u n c t i o n s of sampling r a t e . A i r c r a f t 0.

SAMPLINO RATE I N SRHPLES/SEC (b) c Y e , c , Cn for case 1:3.

Figure 1 0 , Continued.

ORIGINAE PAGE IS O F . POOR QUALlm

I I I I I 0.00 0 10 2 0 30 4 0 SO SRHPLINO RATE I N EAHPLES/SEC for case 1:lOB

, Ck , Cn

= ' Yp p p

per deq.

O I ' I

I -I 0.00 l---- 0 10 2 0 9 0 4 0 60 SfillPLIN? RRTE I N SAHPLES/SEC (a) c r , c4, cnj for case 1: 13.

Figure 10. Continued.

I I I I 1 0.00 1 0 10 2 0 % 30 40 50 SAHPLINO RATE IN SAHPLES/SEC

(el Cp . C4. c for case 1:14R

"P Figure 10. Continued.

p e r rod -0 a 2 0 S A M P L I N G R f l T E I N SAMPLES/SEC

(f) C1 , Cn for case 1:2

P P Figure 10. Continued.

SAMPLING RATE I N SAHPLES/SEC (9) C1 , C for case 1:3.

n P P Figure 10. Continued.

0.03 r

I

-

0.02

C

" P

per rab

-

0.01 0 0

a 0

I I I 1 v 0.00 ' 0 10 20 30 40 50 SAMPLING R A T E I N SRMPLES/SEC for case 1: 103.

C1 , c n P P Figure 10. Continued.

OlUClNAL PAGE . o E m R R W

-0 030 0 10 20 30 40 50 SAMPLING RATE I N SAHPLES/SEC (i) C , Cn f o r case 1:13.

IP P Figure 10. Continued.

-

-0.05 cn?

PCv Cad - 0 . 1 0

- OC) 00 0

I I I I 1 -0.15 0 10 2 0 30 40 50 SAMPLING RATE I N SflflPLES/SEC (1) cl , C n P P for case 1: 14B.

?igure 10. Continued,

C

QCY ~ d d (N c , C n for case 1 :2 l r r Figure 10, Continued

SRRPLING RATE I N SARPLES/SEC

(1) C : , C

lr " r for case 1:3.

Figure 10. Continued.

ORIGINAL; PAGE a

OE POOR Q~~

SAMPLING RATE I N SAHPLES/SEC f o r case 1:lOB.

Figure 10. Continued.

car per cad C1 , Cn for case 1:13.

(n) r r Figure 10. C~ntinued.

- 1 moo 0 10 20 30 40 SO SAMPLING R A T E I N SAHPLES/SEC , ) c , c f o r case 1: 148.

r "r Figure 1 0 . Continued.

SAHPLIND R A l E I N SAHPLES/SEC for case 1 : Z .

c-r,, cfib, F i g u r e 1 0 . x n t i n u e d .

1 I 1 1 I 0.00 0 10 20 3 0 4 0 5 0 SRNPLIND RRTE I N SANPLES/SEC Figure 10. Continued.

SRHPLIND RRTE IN SRHPLES/SEC C for case 1:3 ''I 'y6 r p C%rt n&r Figure 10. continued.

1 (3 (3 per deg 0 - 1 -00 I 1 I 1 1 0.00 L 0 10 20 3 0 40 SO SAflPLING RRTE I N SRHPLES/SEC Figure 10. Concluded

I 1 , I I 1

0 .oo

0 10 20 30 4 0 5 0 SRHPLINO RATE I N SAHPLES/SEC Figure 11. Estimated later€;-directional derivatives a s a function of sampling rate.Aircraft C .

S A M P L I N G RATE I N SAMPLES/SEC

-0 m60 0 10 20 30 40 50 SAMPLING R A T E I N SAflPLES/SEC

SRMPLINC RRTE I N SRHPLciVSEC

-0 - 4 0 0 10 ' 20 3 0 4 0 50 SRflPLlNG RRTE I N SRflPLESlSEC 0 CySr' C b r 1 Ch&, Ff GURE 11. CONCLUDED.

I 1 0 .OO 0 5 10 15 2 0 S A H P L I N O RATE I N SAtlPLES/SEC F i g u r e 12. E s t i m a t e d lateral-directional d e r i v a t i v e s o b t a i n e d from f i v e cases a s a f u n c t i o n o f sampling r a t e . A l r c r a f t D.

I 1 1 4 0.00 1 0 5 10 15 2 0 SAHPLINO RATE I N SAHPLES/SEC (a)% C 1 C , ~n COSE 7:6.

B' b'

Figure 12. Continued

SAHPLINO RATE I N SAHPLES/SEC Figure 12. .Continued.

SRPlPLINO RATE I N SAPlPLES/SEC ( 1 CJ, , C n P M I C ~ S E 7r17 'Y,' p b Figure 12. Continued.

Figure 12. Continued.

c n ~ 0 per rad -0.01 SAMPLING RATE I N SAMPLES/SEC

(+) cap , Cnp FOR ChSE 7;s

Figure 12. Continued.

SRMPLINO RRTE I N SRMPLES/SEC Figure 12. Continued.

I per r a d SAHPLING RATE I N SAHPLES/SEC (h) C - , Cn FOR C Q 5 E 7 : 9 .

P Figure 12. Continucd .

--- -.-- -.

it,

per rad -0.04

t

I I I I

-0.06 '

0 5 10 15 20 SAMPLING RATE I N SAHPLES/SEC (i) C i , Cnp FOR C45E 7~17 P Figure 12. Continued.

I I I J -0.15 0 5 10 15 20 SAMPLING RATE I N SAMPLES/SEC Figure 12. Continued.

per rad

l o

pr r a d

I

SAMPLING RRTE I N SRMPLES/SEC

( ~ 1 ca , Cn

FOR CASE 7:5.

I- r Figure 1 2 . Continued.

t - per r a d

per rad -0.50 1

- 1 050 0 5 10 15 20 SAMPLINO RATE I N SAMPLES/SEC (0.1 C , C , FOR cnsr 1-6.

4 +

Figure 12. Continued.

-

0.00 * a % &

C

00 0 0 *t

-

-0.50 per r a d (9

-

-1.00 I I I I -1 a50 0 5 10 1s 20 SRNPLING R R T E I N SAMPLES/SEC (m> ci , FOR CSSE 7:2 T Figure 12. Continued.

SFIMPLING RATE I N SAHPLES/SEC Figure 12. Continued.

p e r r a d SAMPLING RATE I N SAHPLES/SEC a c1 , C, ~ o a CASE 8 ~ 4 .

(01 r Figure 12. Continued.

SAHPLINO RATE I N SAtlPLES/SEC Figure 12. Continued.

1 1 I I -0.60 0 6 10 15 20 SAHPLIND RATE I N SAt!PLES/SEC Figure 12. Continued.

I 1 1 -0 *09

0.1) 6 10 16 20 8

SPHPLINO RATE I N SAHPLES/GEC

C . 3 Cy6; CLh , CyP ma uu M

r F i g u r e 12. Continued.

'+%

SRMPLIND R A T E I N SRMPLES/SEC .

6) Cyd , Ciar, q6 r a e r n ZI7.

v r Figure 12. Continued.

PC' d=9 I 0 per deg -0 -20 SAHPLINO RATE I N SRHPLES/SEC Figure 12. Concluded

-0 .oz

Per deg 0 ' C$, CnB.

Estimated l a t e r a l - d i r e c t i o n a l derivatives as a Figure 13.

function o f sampling rate.Aircraft E .

I I 1 I I

0.00 1

0 10 20 30 40 50 S A H P L I N G RRTE I N SqpPLES/SEC Figure 13. Continued

0 m o o 0 10 20 30 40 50 C , 0 0 0 0 "r pe+ rad -5 e00 I I

-10.00 1 I I I

0 10 20 30 40 50 SAMPLING RATE I N SAMPLES/SEC Figure 13. Continued.

Per deg

I

I 1 I I 1 0.00 1 0 10 20 30 4 0 6 0 SRHPLINO RRTE I N SRHPLES/SEC (d) Cy Cp . C .

'a 6a "8.

Figure 13. Continued.

1 1 I I 1 -0.03 0 10 2 0 3 0 40 50 SAHPLINO RATE I N SAnPLES/SEC Figure 13. Concluded

per deq -0 -05 . 0 0 0 0 0 per dcg -1.00

I

-5 -00

C" 7 0

9, 00O0 per r a d -10.00

/

1 I I I i -15.00 0 10 20 3 0 40 SO SAMPLING RATE I N SAHPLES/SEC (a) C . C . C for case 2: 1.

=a '"a ma Figure 14. Estimated longitudinal derivative; for four maneuvers as a function of sampling rate. Aircraft B .

per rad -20 a00 I 1 1 1 -30.00 1 0 10 20 3 0 4 0 50 SARPLINO RATE IN SRHPLES/SEC F i g u r e 1 4 . Continued

0 0 0 a 0 Cml'-5.00 0

per rad t o

-16.00 0 10 20 30 40 60 SAHPLINO RRTE I N SRtlPLES/SEC Figure 1 4 . Continued.

1 I I I I -15.00 0 10 20 3 0 40 50 SRtlPLINO RATE I N SAHPLES/SEC (d) Cz4, C m d , C m m * c n s ~ s:lA.

e

Figure 1 4 . Continued.

'zg ' ' m s for ca.e 2: 1.

e e Figure 14. Continued.

I I I I I -0.03 0 10 20 30 40 SO SQMPLINO RATE I N SAMPLES/SEC Figure 1 4 . Continued.

PC' deg -0 a04 I 1 I I J

-1 .so

0 10 20 30 4 0 5 0 SARPLINO RRTE I N SRRPLES/SEC (g) C , C for case 2: 4 .

Z8 m8 e e Figure 14. continued.

a

C z -0.50

a 6e pc' de9 0 0 - 1 moo (I)

t

I I I I 1

- 1 a 5 0 '

0 10 20 30 40 50 SAMPLING RATE I N SAHPLES/SEC (h) C , C for case 5: l A .

z6 m6 e e Figure 1 4 . Concluded.

per rad

I

SflHPLINI) RATE I N SAHPLES/SEC Figure 15. Estimated longitudinal derivatives as a function o f sarnpling r a t e , A i r c r a f t C.

SAMPLING RRTE I N SAMPLES/SEC Figure 15, Concluded

cm )

PI

-40 .OO per rad 0 ('0

I

1 I 8 -60.00 0 5 10 1s 20 SRHPLING RRTE I N SAHPLES/SEC Ffgure 16. Estimated iongitudinal derivatives f o r four maneuvers as a function of sampling r a t e . A i r c r a f t D.

per rad -30.00 0 5 10 15 20 SQHPLINO RATE I N SAHPLES/SEC C ~ J cz c. cok, c , ~ ~ ~ ~ E ~ : n .

Figure 16. Continued

SRHPLINO RATE I N SRHPLES/SEC

(el Ck. Cms, cm* , c , , . 7:ts

Figure 1 6 . Continued.

' dq -0.02 1 "a.

per red -40 a00

/ O Q Q

SAHPLINO RATE I N SARPLES/SEC Cd) C%, C*, C , , rr ems€ F3.

S Figure 16. Continued.

-0 -06 0 5 10 15 20 SRMPLING RRTE I N SfiMPLES/SEC

C

(e) C, FOR ChSE 5:4.

6 e mbe ~ i g u r e 16. Continued.

SAMPLING RATE I N SAHPLES/SEC ($1 c~ G F O R C A S E 8.'3 be ' m&e Figure 16. Continued.

PC' d 9

SAMPLING RRTE I N SRHPLES/SEC Figure 16. Continued.

per deg -0 004 F i g u r e 16. Concluded.

C

ycr decj 1 I 1 I I

0.00 '

0 10 20 30 40 50 8AHPLINO RATE r A bAHPLES/SEC Figure 17. Estimated l a t e r a l - d t r e c t t o n a l derlvattves and uncertatnty l e v e l s f o r A l r c r r f t A mareuver as functlon o f sampltng r a t e .

per r ~ d -0 Cnp, per r a d SAPPLIND RATE I N SAHPLES/SEC Figure 17. Continued

I I I I 1

-0.20 '

0 10 20 30 40 50 SAMPLINCI RATE I @ SAMPLES/SEC Figure 17. Continued.

I 1 I I -0.10 1 0 10 2 0 30 40 5 0 SARPLING RATE I N SRflPLES/SEC Figure 17. Concluded