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Kalman filter based orbit observability study Shorten, Roy Robert 1979

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KALMAN FILTER BASED ORBIT OBSERVABILITY STUDY by ROY ROBERT SHORTEN B.Eng., Sir George Williams University, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of- E l e c t r i c a l Engineering) We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA June, 1979 (c) Roy Robert Shorten, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be gr a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s underst o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f P-~ JCsx^r^>C^»^c^'( *- <^>^9\-*~~M.&~V The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date 5v̂ -*v» ? 1. i i ABSTRACT An Extended Kalman f i l t e r simulation techni que was used to determine the ob s e r v a b i l i t y of s a t e l l i t e o r b i t s from one or two earth s t a t i o n s . It was found that range and range-rate measurements alone were i n s u f f i c i e n t f or o r b i t determination. Either azimuth angle or elevation angle information were also required before an acceptable o r b i t a l estimate was obtained. However, range and range-rate measurements alone proved to be s u f f i c i e n t to improve the state estimates of approximately known o r b i t s . Also i f simultaneous range and range-rate measurements were a v a i l a b l e from two st a t i o n s , o r b i t determination was po s s i b l e . Various common o r b i t a l types were used throughout the study and only one pass of data was considered. i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS i x I INTRODUCTION TO SATELLITE TRACKING 1 1.1 Observable? 1 1.2 Geometry 2 1.3 Measurement Methods 2 1.4 Basic Problem Studied 4 1.5 Method 4 II THE SYSTEM MODEL 6 II.1 Limitations 6 I I . 2 Units 6 I I . 3 Coordinates 7 11.3.1 Azimuth-Elevation Coordinate System 7 11.3.2 Right Ascension Declination Coordinate System 8 11.3.3 O r b i t a l Plane Coordinate System 9 I I . 4 State Model 11 I I . 5 Measurement Model 12 III KALMAN FILTERING AND SMOOTHING .' 15 I I I . l Linear Kalman F i l t e r 15 I I I . 2 Extended Kalman F i l t e r 16 III.3 Linearized Kalman F i l t e r 17 I I I . 4 Smoothing 19 i v IV SIMULATION PROCEDURES 20 IV. 1 Generating The Measurement Data 20 IV. 1.1 Algorithm Used To Generate Data 20 IV. 1.2 Types of Orbit 22 IV.2 Simulation Algorithm 26 IV. 3 I n i t i a l i z a t i o n 30 V SIMULATION RESULTS 32 V. 1 Range and Range-rate From One Station 32 V.2 Range, Range-Rate and Azimuth from One Station 35 V.3 Range, Range-Rate, Azimuth, and Elevation 53 V.4 Range and Range-Rate from Two Stations 53 V. 4.1 Orbit 2 Results 53 V.4.2 Orbit 1 Results 71 V.4.3 Other O r b i t s ' Results 80 V.5 Known Orbit Improvement 80 VI CONCLUSIONS 99 REFERENCES 100 L I S T OF TABLES I I . 1 S t a t e E q u a t i o n 12 I I . 2 Measurement E q u a t i o n s 14 I I I . l L i n e a r Ka lman F i l t e r E q u a t i o n s 15 I I I . 2 E x t e n d e d K a l m a n F i l t e r E q u a t i o n s 17 I I I . 3 L i n e a r i z e d K a l m a n F i l t e r E q u a t i o n s 18 I V . 1 S a t e l l i t e S t a t e C a l c u l a t i o n 21 I V . 2 C o o r d i n a t e s o f H e c t o r McLeod B l d g 24 I V . 3 O r b i t s Used i n S t u d y 25 I V . 4 S i m u l a t i o n A l g o r i t h m 29 V . l O R B I T . 2 R e s i d u a l s T r y 1 49 V . 2 O R B I T . 2 R e s i d u a l s T r y 2 50 V . 3 O R B I T . 2 R e s i d u a l s T r y 3 51 V . 4 O R B I T . 2 R e s i d u a l s T r y 4 52 v i L I S T OF FIGURES I I . 1 A z i m u t h - E l e v a t i o n C o o r d i n a t e S y s t e m 7 I I . 2 R i g h t A s c e n s i o n D e c l i n a t i o n C o o r d i n a t e S y s t e m 9 I I . 3 O r b i t a l P l a n e C o o r d i n a t e Sys t em O b l i q u e V i e w 10 I I . 4 O r b i t a l P l a n e C o o r d i n a t e S y s t e m P e r p e n d i c u l a r V i e w 11 I V . 1 T y p i c a l Measurements ( O R B I T . 1 ) 23 I V . 2 T y p i c a l S i m u l a t i o n Run R e s u l t 28 V . 1 P o s i t i o n E r r o r w i t h Range and R a n g e - R a t e 33 V . 2 V e l o c i t y E r r o r w i t h Range and R a n g e - R a t e 34 V . 3 O r b i t 1 P o s i t i o n E r r o r w i t h 3 Measuremen t s 36 V . 4 O r b i t 1 V e l o c i t y E r r o r w i t h 3 Measuremen t s 37 V . 5 O r b i t 2 P o s i t i o n E r r o r w i t h 3 Measuremen t s 38 V . 6 O r b i t 2 V e l o c i t y E r r o r w i t h 3 Measuremen t s 39 V . 7 O r b i t 3 P o s i t i o n E r r o r w i t h 3 Measuremen t s 40 V . 8 O r b i t 3 V e l o c i t y E r r o r w i t h 3 Measuremen t s 41 V . 9 O r b i t 4 P o s i t i o n E r r o r w i t h 3 Measurements 42 V . 1 0 O r b i t 4 V e l o c i t y E r r o r w i t h 3 Measuremen t s 43 V . l l O r b i t 5 P o s i t i o n E r r o r w i t h 3 Measurements 44 V . 12 O r b i t 5 V e l o c i t y E r r o r w i t h 3 Measuremen t s 45 V . 1 3 O r b i t 6 P o s i t i o n E r r o r w i t h 3 Measuremen t s 47 V . 14 O r b i t 6 V e l o c i t y E r r o r w i t h 3 Measuremen t s 48 V . 1 5 O r b i t 1 P o s i t i o n E r r o r w i t h 4 Measuremen t s 54 V . 16 O r b i t 1 V e l o c i t y E r r o r w i t h 4 Measuremen t s 55 V . 1 7 O r b i t 5 P o s i t i o n E r r o r w i t h 4 Measuremen t s 56 V . 18 O r b i t 5 V e l o c i t y E r r o r w i t h 4 Measuremen t s 57 V . 1 9 O r b i t 2 P o s i t i o n E r r o r 1 / 2 ° L a t i t u d e S e p a r a t i o n 58 v i i V . 2 0 O r b i t 2 V e l o c i t y E r r o r 1 / 2 ° L a t i t u d e S e p a r a t i o n 59 V . 2 1 O r b i t 2 P o s i t i o n E r r o r 1 ° L a t i t u d e S e p a r a t i o n 60 V . 2 2 O r b i t 2 V e l o c i t y E r r o r 1 ° L a t i t u d e S e p a r a t i o n 61 V . 2 3 O r b i t 2 P o s i t i o n E r r o r 5 ° L a t i t u d e S e p a r a t i o n 63 V . 2 4 O r b i t 2 V e l o c i t y E r r o r 5 ° L a t i t u d e S e p a r a t i o n 64 V . 2 5 O r b i t 2 P o s i t i o n E r r o r 1 / 2 ° L o n g i t u d e S e p a r a t i o n 65 V . 2 6 O r b i t 2 V e l o c i t y E r r o r 1 / 2 ° L o n g i t u d e S e p a r a t i o n 66 V . 2 7 O r b i t 2 P o s i t i o n E r r o r 0 . 1 ° L o n g i t u d e S e p a r a t i o n 67 V . 2 8 O r b i t 2 V e l o c i t y E r r o r 0 . 1 ° L o n g i t u d e S e p a r a t i o n 68 V . 2 9 O r b i t 2 P o s i t i o n E r r o r 3 ° L o n g i t u d e S e p a r a t i o n 69 V . 3 0 O r b i t 2 V e l o c i t y E r r o r 3 ° L o n g i t u d e S e p a r a t i o n 70 V . 3 1 O r b i t 1 P o s i t i o n E r r o r 2 ° L o n g i t u d e S e p a r a t i o n 72 V . 3 2 O r b i t 1 V e l o c i t y E r r o r 2 ° L o n g i t u d e S e p a r a t i o n 73 V . 3 3 O r b i t 1 P o s i t i o n E r r o r 2 ° L a t i t u d e S e p a r a t i o n 74 V . 3 4 O r b i t 1 V e l o c i t y E r r o r 2 ° L a t i t u d e S e p a r a t i o n 75 V . 3 5 O r b i t 1 P o s i t i o n E r r o r 3 ° L o n g i t u d e S e p a r a t i o n 76 V . 3 6 O r b i t 1 V e l o c i t y E r r o r 3 ° L o n g i t u d e S e p a r a t i o n 77 V . 3 7 O r b i t 1 P o s i t i o n E r r o r 1 ° L a t i t u d e S e p a r a t i o n 78 V . 3 8 O r b i t 1 V e l o c i t y E r r o r 1 ° L a t i t u d e S e p a r a t i o n 79 V . 3 9 O r b i t 3 P o s i t i o n E r r o r 2 ° L o n g i t u d e S e p a r a t i o n 81 V . 4 0 O r b i t 3 V e l o c i t y E r r o r 2 ° L o n g i t u d e S e p a r a t i o n 82 V . 4 1 O r b i t 4 P o s i t i o n E r r o r 2 ° L o n g i t u d e S e p a r a t i o n 83 V . 4 2 O r b i t 4 V e l o c i t y E r r o r 2 ° L o n g i t u d e S e p a r a t i o n 84 V . 4 3 O r b i t 5 P o s i t i o n E r r o r 2 0 ° L o n g i t u d e S e p a r a t i o n 85 V . 4 4 O r b i t 5 V e l o c i t y E r r o r 2 0 ° L o n g i t u d e S e p a r a t i o n 86 V . 4 5 O r b i t 6 P o s i t i o n E r r o r 2 ° L a t i t u d e S e p a r a t i o n 87 V . 4 6 O r b i t 6 V e l o c i t y E r r o r 2 ° L a t i t u d e S e p a r a t i o n 88 v i i i V . 4 7 300 Km E r r o r i n " a " 90 V . 4 8 One D e g r e e E r r o r i n " L " 91 V . 4 9 One Degree E r r o r i n " p " 92 V . 5 0 One Degree E r r o r i n " i " O r b i t 3 93 V . 5 1 One D e g r e e E r r o r i n " i " O r b i t 4 94 V . 5 2 One Degree o f E r r o r i n " i " 95 V . 5 3 One Degree E r r o r i n " L " 97 V . 5 4 0 . 0 1 E . R . E r r o r i n " a " 98 i x ACKNOWLEDGEMENTS I w o u l d l i k e t o e x p r e s s my t h a n k s t o my s u p e r v i s o r D r . E . V . Bohn and D r . A . C . Soudack f o r t h e i r a s s i s t a n c e i n p r e p a r i n g and h e l p i n r e v i e w i n g t h i s t h e s i s . I w o u l d a l s o l i k e t o t h a n k D r . V . J . M o d i f o r h i s h e l p and e n c o u r a g e m e n t . 1 CHAPTER I_ INTRODUCTION TO S A T E L L I T E TRACKING J^. O b s e r v a b l e ? A s y s t e m I s o b s e r v a b l e i f f r o m a g i v e n s e t o f measurements y o u c a n d i s t i n g u i s h t h e s t a t e s o f t h e s y s t e m . I f t h e s y s t e m i n q u e s t i o n happens t o be a s a t e l l i t e i n o r b i t a r o u n d t h e e a r t h , t h e s t a t e s c a n be c o n s i d e r e d i t s p o s i t i o n and v e l o c i t y . D e t e r m i n i n g a s a t e l l i t e ' s p o s i t i o n and v e l o c i t y i s c a l l e d o r b i t d e t e r m i n a t i o n . N e w t o n ' s i n v e r s e s q u a r e l a w , w h i c h r e l a t e s t h e d i s t a n c e be tween two masses t o t h e f o r c e of a t t r a c t i o n be tween them, i s t h e p r i m a r y e f f e c t g o v e r n i n g s a t e l l i t e m o t i o n . I t a l s o makes t h e p r o b l e m n o n - l i n e a r . O b s e r v a b i l i t y f o r a l i n e a r s y s t e m i s r e l a t i v e l y s i m p l e t o d e t e r m i n e . I t has been p r o v e n t h a t t h e r a n k o f t h e o b s e r v a b i l i t y m a t r i x d e t e r m i n e s w h e t h e r t h e s y s t e m i s o b s e r v a b l e o r n o t . I n t h e n o n - l i n e a r c a s e h o w e v e r , o b s e r v a b i l i t y i s much more d i f f i c u l t t o d e t e r m i n e [3]- C u r r e n t l y t h e p r i m a r y method o f d e t e r m i n i n g n o n - l i n e a r o b s e r v a b i l i t y i s s i m u l a t i o n . T h i s i s t h e a p p r o a c h t a k e n i n t h i s t h e s i s . A q u a s i - r e a l i s t i c mode l i s emp loyed and s i m u l a t i o n r u n s a r e p e r f o r m e d u s i n g v a r i o u s t y p e s o f measurements i n o r d e r t o d e t e r m i n e i f t h e s t a t e s c a n be d i s t i n g u i s h e d . The mode l i s q u a s i - r e a l i s t i c b e c a u s e as w i t h any mode l a s s u m p t i o n s were made and s e c o n d a r y e f f e c t s n e g l e c t e d . A s i m u l a t i o n i s o n l y as good as i t s a s s u m p t i o n s and m o d e l . A d i s c u s s i o n of t h e s e l i m i t a t i o n s i s p r e s e n t e d i n c h a p t e r I I . 2 7_. Geomet ry S a t e l l i t e s o f t h e e a r t h d e s c r i b e p a t h s w h i c h a r e c o n i c s e c t i o n s . We s h a l l c o n c e r n o u r s e l v e s o n l y w i t h c i r c u l a r and e l l i p t i c a l o r b i t s . A l l o r b i t s d e s c r i b e a p l a n e i n s p a c e c a l l e d t h e o r b i t a l p l a n e w h i c h , i n t h e p r e s e n c e o f a p u r e l y c e n t r a l f o r c e , i s f i x e d i n i n e r t i a l s p a c e . Measurements o f s a t e l l i t e p o s i t i o n a l o n g t h i s o r b i t a r e o b t a i n e d f r o m a s t a t i o n on t h e s u r f a c e o f t h e e a r t h , w h i c h i s i t s e l f r o t a t i n g . T h i s r o t a t i o n o f t h e e a r t h c o m p l i c a t e s t h e measurement p r o c e s s as t h e p o s i t i o n and v e l o c i t y o f t h e m e a s u r e r must c o n t i n u a l l y be u p d a t e d and t a k e n i n t o a c c o u n t . O n l y a s m a l l p a r t o f t h e o r b i t c a n be s e e n f r o m any one e a r t h s t a t i o n . The r e s t o f t h e o r b i t i s h i d d e n b e h i n d t h e e a r t h . T h i s b r i n g s i n t h e c o n c e p t o f ' p a s s ' , a p a s s b e i n g t h e p o r t i o n o f an o r b i t d u r i n g w h i c h t h e s a t e l l i t e i s above t h e h o r i z o n o f t h e m e a s u r i n g s t a t i o n . A p a s s c a n l a s t f r o m z e r o m i n u t e s i n t h e c a s e o f a s a t e l l i t e w h i c h i s f o r e v e r h i d d e n f r o m some s t a t i o n , t o m i l l i o n s o f y e a r s as i n t h e c a s e o f a g e o s y n c h r o n o u s s a t e l l i t e . The l e n g t h o f t i m e t h e s a t e l l i t e i s i n v i e w a f f e c t s i t s o b s e r v a b i l i t y . _3. Measurement Me thods H i s t o r i c a l l y t h e f i r s t method u s e d t o t r a c k s a t e l l i t e s was v i s u a l s i g h t i n g . I t was u s e d by t h e a n c i e n t s t o t r a c k t h e moon, t h e p l a n e t s , and t h e s t a r s , a l l o f w h i c h t h e y b e l i e v e d c i r c l e d t h e e a r t h . V i s u a l t r a c k i n g c a n s u p p l y a n g u l a r i n f o r m a t i o n o n l y . I t i s i m p o s s i b l e t o e s t i m a t e r ange v i s u a l l y as c a n be s e e n f r o m a n c i e n t e s t i m a t e s o f t h e d i s t a n c e t o t h e moon. The a n g u l a r a c c u r a c y o f v i s u a l t r a c k i n g i s s u r p r i s i n g l y g o o d , b u t i t i s d i f f i c u l t d u r i n g d a y l i g h t t o t r a c k a n y t h i n g w i t h l e s s b r i g h t n e s s t h a n t h e moon. The v i s u a l method was l a t e r 3 augmented by the use of optical equipment such as telescopes. These enabled much better accuracy to be obtained and also allowed daylight measurements, but significantly increased the cost. When cameras were added to the optical tracking equipment, extremely high accuracies were obtained by measurements relative to known stellar background positions. The recent introduction of radar to track satellites has drastically changed the type and quantity of measurements that can be performed. While the angular information supplied from radars is not nearly as accurate as from optical methods, radar does supply an accurate range and range-rate measurement. If the sa t e l l i t e we wish to track i s active, that is i f i t carries on board a radio transceiver which w i l l receive and transmit radio signals, other types of tracking equipment can be used. Large phased antenna arrays and/or large steerable dish antennas can be used to obtain angular information, however both of these are extremely expensive. Conversely range and range rate measurements are relatively inexpensive. To measure range one only has to accurately measure a signal's round trip travel time to and from the s a t e l l i t e . The accuracy is determined by the time resolution possible. Accuracies of f i f t y centimeters are possible using sophisticated laser techniques [7]. Inexpensive equipment could resolve to about one kilometer without major problems. Range-rate measurements are accomplished by measuring the Doppler shift caused by sa t e l l i t e motion. Again relatively inexpensive equipment can resolve to about three meters per second. 4 4_. B a s i c P r o b l e m S t u d i e d G i v e n t h a t r ange and r a n g e - r a t e measurements were r e l a t i v e l y i n e x p e n s i v e w h i l e a n g u l a r measurements e i t h e r by o p t i c a l methods o r r a d a r were v e r y e x p e n s i v e , t h e q u e s t i o n a r o s e as t o w h e t h e r r a n g e and r a n g e - r a t e were s u f f i c i e n t f o r o r b i t d e t e r m i n a t i o n . From t h e c o n t r o l s t a n d p o i n t , t h e q u e s t i o n w a s : " I s t h e s y s t e m o b s e r v a b l e ? " . I f t h e s y s t e m were n o t o b s e r v a b l e , what was t h e minimum number o f measurements r e q u i r e d t o make i t o b s e r v a b l e , and how much c o u l d be a c c o m p l i s h e d u s i n g r a n g e and r a n g e - r a t e a l o n e ? Were a n g u l a r measurements o f s a t e l l i t e p o s i t i o n r e q u i r e d f o r s a t e l l i t e t r a c k i n g ? And f i n a l l y , "What t e c h n i q u e c o u l d be d e v e l o p e d t o answer t h e above q u e s t i o n s ? " 5_. M e t h o d The f i r s t p r o b l e m e n c o u n t e r e d when one i s a t t e m p t i n g t o d e t e r m i n e o b s e r v a b i l i t y i s one o f m e t h o d . How does one show o b s e r v a b i l i t y ? As m e n t i o n e d e a r l i e r t h e t h e o r e t i c a l a p p r o a c h i s e x t r e m e l y a r d u o u s f o r n o n - l i n e a r s y s t e m s . S i m u l a t i o n o f f e r s t h e o n l y a l t e r n a t i v e . An a l g o r i t h m must be c h o s e n s u c h t h a t i f i t f a i l s t o o b s e r v e t h e s y s t e m , t h e s y s t e m i s , t o a h i g h p r o b a b i l i t y , n o t o b s e r v a b l e . The a l g o r i t h m must be r e l a t i v e l y s i m p l e so t h a t p r o g r a m c o m p l e x i t y i s n o t a p r o b l e m , and y e t s o p h s t i c a t e d enough t o h a n d l e t h e measurement n o i s e e x p e c t e d and t h e s y s t e m mode l u s e d . E s c o b a l [1] t r e a t s a number o f t h e c l a s s i c a l methods o f o r b i t d e t e r m i n a t i o n , however a l l a r e d e t e r m i n i s t i c and do n o t a l l o w n o i s e . I f we c o n s i d e r t h e p r o b l e m of c o n t a m i n a t e d measurements we have a c h o i c e be tween l e a s t s q u a r e s , Ka lman f i l t e r i n g and some f o r m of p a r a m e t e r e s t i m a t i o n . S i n c e i t c a n be shown t h a t t h e s e q u e n t i a l a l g o r i t h m s o b t a i n e d f r o m l e a s t s q u a r e s a r e e s s e n t i a l l y t h e same as t h o s e 5 o b t a i n e d f r o m K a l m a n f i l t e r i n g [8], and s i n c e t h e n o n - l i n e a r i t y o f t h e p r o b l e m makes p a r a m e t e r e s t i m a t i o n p a r t i c u l a r l y d i f f i c u l t , t h e Ka lman f i l t e r was c h o s e n as t h e b a s i c e x a m i n a t i o n t o o l - The b a s i c K a l m a n f i l t e r i s an a l g o r i t h m whereby measurements a r e w e i g h t e d as t h e y a r r i v e a c c o r d i n g t o t h e e r r o r i n t h e s t a t e e s t i m a t e ( e r r o r c o v a r i a n c e e s t i m a t e ) . As e s t i m a t e s o f t h e s t a t e d e r i v e d f r o m a s y s t e m mode l ge t w o r s e , t h e measurements, a r e w e i g h t e d h i g h e r . As t h e s t a t e e s t i m a t e s ge t b e t t e r , t h e measurements a r e w e i g h t e d l o w e r and t h e r e f o r e h a v e l e s s a f f e c t on f u t u r e s t a t e e s t i m a t e s . C h a p t e r I I I f u r t h e r expands on t h e Ka lman f i l t e r . The method d e c i d e d on t h e r e f o r e c o n s i s t s o f w r i t i n g a number o f Ka lman f i l t e r b a s e d smoo the r s t o t e s t v a r i o u s c o m b i n a t i o n s o f m e a s u r e m e n t s . 6 CHAPTER I I THE SYSTEM MODEL _1. L i m i t a t i o n s S t a t e o f t h e a r t o r b i t d e t e r m i n a t i o n p rograms s u c h as [ 5 ] , t a k e i n t o a c c o u n t many f a c t o r s o f s e c o n d and t h i r d o r d e r . These i n c l u d e e a r t h g r a v i t a t i o n a l a n o m a l i e s , l u n a r and s o l a r g r a v i t y , s o l a r w i n d , a t m o s p h e r i c d r a g , e a r t h s h i n e e t c The mode l u s e d i n t h i s s t u d y i s t h e c l a s s i c a l N e w t o n i a n c e n t r a l f o r c e m o d e l . The i n c l u s i o n o f s e c o n d a r y and t e r t i a r y e f f e c t s w o u l d add l i t t l e t o t h e b a s i c q u e s t i o n o f o b s e r v a b i l i t y , w h i l e c a u s i n g a c o m p l e x i t y w h i c h w o u l d o b s c u r e b a s i c p r o b l e m s and e f f e c t s . A l l t h e measurements were assumed t o have been made f r o m t h e same l o c a t i o n on t h e e a r t h . The l o c a t i o n c h o s e n was t h e H e c t o r McLeod B u i l d i n g o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . T h i s r e p r e s e n t s an i n t e r m e d i a t e l a t i t u d e . o f a p p r o x i m a t e l y 50 d e g r e e s , so i t i s b e l i e v e d t h a t r e s u l t s o b t a i n e d w i t h t h i s s t a t i o n w i l l h o l d t r u e f o r most o t h e r l a t i t u d e s . The s t a t i o n c o o r d i n a t e s a r e assumed t o be known e x a c t l y . N o i s e w i t h a n o r m a l d i s t r i b u t i o n was i m p r e s s e d on a l l measu remen t s . The s t a n d a r d d e v i a t i o n b e i n g t a i l o r e d t o t h e t y p e o f measurement . I t i s assumed t h r o u g h o u t t h a t t h e s a t e l l i t e i s a c t i v e and w i l l r e t u r n a s i g n a l s e n t t o i t w i t h a known d e l a y , and i t was w i t h t h i s measurement t e c h n i q u e i n mind t h a t t h e n o i s e s t a n d a r d d e v i a t i o n s were c h o s e n . 2. U n i t s The g e o c e n t r i c s y s t e m of u n i t s i s u s e d t h r o u g h o u t t h i s t h e s i s . T h i s s y s t e m a v o i d s t h e use o f l a r g e numbers and i s a more " n a t u r a l " 7 s y s t e m . I t u se s u n i t s d e f i n e d by t h e s y s t e m i t s e l f . The g e o c e n t r i c s y s t e m has as i t s u n i t o f d i s t a n c e t h e r a d i u s o f t h e e a r t h ( E . R . ) . One E . R . i s e q u a l t o 6 , 3 7 8 . 15 k m . I t s u n i t o f t i m e i s t h e mean s o l a r d a y , and i t s u n i t o f mass i s t h e mass o f t h e e a r t h . The g r a v i t a t i o n a l c o n s t a n t (k ) i n t h e g e o c e n t r i c s y s t e m i s 107 .0867 E . R . ^ ^ ' / D a y . 3 . C o o r d i n a t e s 3 .1 A z i m u t h - E l e v a t i o n C o o r d i n a t e S y s t e m Measu remen t s f r o m a s t a t i o n on t h e s u r f a c e o f t h e e a r t h o f a s a t e l l i t e ' s p o s i t i o n a r e r e f e r e n c e d i n i t i a l l y t o t h e a z i m u t h - e l e v a t i o n c o o r d i n a t e s y s t e m . T h i s i s a r o t a t i n g c o o r d i n a t e s y s t e m w h i c h has as i t s o r i g i n t h e o b s e r v i n g s t a t i o n , s ee F i g u r e I I . 1 . North A z i m u t h - E l e v a t i o n C o o r d i n a t e S y s t e m . F i g u r e I I . 1 The f u n d a m e n t a l p l a n e i s d e f i n e d as b e i n g t a n g e n t i a l t o t h e e a r t h a t t h e 8 o b s e r v e r . T h i s means t h a t t h e l o c a l v e r t i c a l i s n o r m a l t o t h e f u n d a m e n t a l p l a n e . The p o s i t i v e X d i r e c t i o n i s d e f i n e d as b e i n g due S o u t h . The p o s i t i v e Y d i r e c t i o n i s t h e r e f o r e due E a s t and p o s i t i v e Z i s s t r a i g h t u p . Two a n g l e s w h i c h d e f i n e t h e d i r e c t i o n o f t h e s a t e l l i t e i n t h i s s y s t e m a r e t h e e l e v a t i o n and a z i m u t h a n g l e s . The e l e v a t i o n i s t h e a n g l e be tween t h e s a t e l l i t e and t h e f u n d a m e n t a l p l a n e measured i n a p l a n e p e r p e n d i c u l a r t o t h e f u n d a m e n t a l p l a n e . The a z i m u t h i s t h e a n g l e measured f r o m N o r t h i n t h e f u n d a m e n t a l p l a n e t o t h e p r o j e c t i o n o f t h e s a t e l l i t e r a d i a l o n t o t h e f u n d a m e n t a l p l a n e . Range i s s i m p l y t h e d i s t a n c e o f t h e s a t e l l i t e f r o m t h e o r i g i n measured a l o n g t h e r a d i a l . 3 . 2 R i g h t A s c e n s i o n D e c l i n a t i o n C o o r d i n a t e Sys t em The r i g h t a s c e n s i o n d e c l i n a t i o n c o o r d i n a t e s y s t e m i s an i n e r t i a l o r f i x e d s y s t e m . I t s c e n t e r i s t a k e n as t h e c e n t e r o f t h e e a r t h , b u t i t does n o t r o t a t e w i t h t h e e a r t h . The f u n d a m e n t a l p l a n e i s t h e p l a n e o f t h e e q u a t o r and t h e p o s i t i v e X d i r e c t i o n p o i n t s t o w a r d t h e v e r n a l e q u i n o x , s ee F i g u r e I I . 2 . The p o s i t i v e Z d i r e c t i o n i s up t h r o u g h t h e N o r t h p o l e and t h e Y d i r e c t i o n c o m p l e t e s t h e r i g h t handed s y s t e m . T h e r e a r e two ways t o d e f i n e a l o c a t i o n i n t h i s s y s t e m . The f i r s t i s t o s i m p l y s p e c i f y t h e X , Y , and Z c o o r d i n a t e s . T h i s i s t h e s y s t e m u s e d i n t h e s y s t e m m o d e l . The s e c o n d i s t o s p e c i f y t h e r i g h t a s c e n s i o n , t h e d e c l i n a t i o n and t h e r a n g e . The r ange i s s i m p l y t h e d i s t a n c e f r o m t h e c e n t e r t o t h e l o c a t i o n t o be s p e c i f i e d . The r i g h t a s c e n s i o n i s t h e a n g l e measured i n t h e f u n d a m e n t a l p l a n e be tween t h e X a x i s and t h e p r o j e c t i o n o f t h e r a d i a l v e c t o r on to t h e f u n d a m e n t a l p l a n e . The d e c l i n a t i o n i s t h e a n g l e be tween t h e l o c a t i o n and t h e e q u a t o r measured i n a p l a n e n o r m a l t o t h e e q u a t o r . 9 Z North Pole R i g h t A s c e n s i o n D e c l i n a t i o n C o o r d i n a t e S y s t e m . F i g u r e I I . 2 3.3 O r b i t a l P l a n e C o o r d i n a t e S y s t e m I t i s o f t e n much e a s i e r t o a n a l y z e an o r b i t a l p r o b l e m f r o m a s y s t e m w h i c h t a k e s as i t s f u n d a m e n t a l p l a n e , t h e o r b i t a l p l a n e o f t h e s a t e l l i t e . Such a s y s t e m i s f i x e d i n i n e r t i a l s p a c e i f t h e r e a r e no p e r t u r b a t i o n s . R e a l i s t i c a l l y , h o w e v e r , t h e r e a r e a l w a y s p e r t u r b a t i o n s and t h e o r b i t a l p l a n e w i l l g r a d u a l l y d r i f t . S i n c e i t i s a n n o y i n g t o h a v e t h e c o o r d i n a t e s y s t e m d r i f t t h e f u n d a m e n t a l p l a n e i s u s u a l l y t a k e n as t h e o r b i t a l p l a n e a t some epoch t i m e . The c e n t e r o f t h i s s y s t e m i s t h e c e n t e r o f t h e e a r t h . The p o s i t i v e X a x i s p o i n t s t o w a r d s t h e p e r i f o c u s . One o f t h e a n g l e s d e f i n e d i n t h i s c o o r d i n a t e s y s t e m i s o f some i m p o r t a n c e . T h i s i s t h e e c c e n t r i c anomaly ( E ) . See F i g u r e I I . 3 10 O r b i t a l P l a n e C o o r d i n a t e S y s t e m O b l i q u e V i e w F i g u r e I I-3 and F i g u r e I I . 4 . I t i s d e f i n e d as t h e a n g l e measured i n t h e f u n d a m e n t a l p l a n e f r o m t h e X a x i s t o t h e p o i n t on a c i r c l e t h a t c i r c u m s c r i b e s t h e a c t u a l e l l i p s e o f m o t i o n measured a t t h e c e n t e r o f t h e c i r c u m s c r i b e d c i r c l e . 11 O r b i t a l P l a n e C o o r d i n a t e Sys t em P e r p e n d i c u l a r V i e w . F i g u r e I I . 4 h_. S t a t e M o d e l The s y s t e m mode l d e c i d e d on has s i x s t a t e s , t h r e e o f p o s i t i o n and t h r e e o f v e l o c i t y . A mode l b a s e d on t h e s e t h r e e s t a t e s i n an i n e r t i a l r e f e r e n c e frame was d e t e r m i n e d t o be t h e s i m p l e s t a v a i l a b l e . A t t i t u d e s t a t e s e t c . we re n e g l e c t e d as i r r e l a v a n t . The mode l i s b a s e d s o l e l y on N e w t o n ' s i n v e r s e s q u a r e l a w . The s a t e l l i t e s t a t e v e c t o r X i s d e f i n e d as (x]_, X2, X 3 , X 4 , X5, X6) T w h e r e : x i i s t h e v e l o c i t y i n t h e X d i r e c t i o n x £ i s t h e v e l o c i t y i n t h e Y d i r e c t i o n X 3 i s t h e v e l o c i t y i n t h e Z d i r e c t i o n X 4 i s t h e d i s p l a c e m e n t i n t h e X d i r e c t i o n 12 X5 i s t h e d i s p l a c e m e n t i n t h e Y d i r e c t i o n , and x g i s t h e d i s p l a c e m e n t i n t h e Z d i r e c t i o n . The s t a t e e q u a t i o n can be s e e n i n T a b l e I I . 1 . — - K • -k X, -k Xr -k X, S t a t e E q u a t i o n . T a b l e I I . 1 5_. Measurement M o d e l T h e r e a r e f o u r t y p e s o f measurements c o n s i d e r e d i n t h i s t h e s i s . They a r e : R a n g e : The d i s t a n c e f r o m t h e o b s e r v i n g s t a t i o n t o t h e s a t e l l i t e , o b t a i n e d by m e a s u r i n g r o u n d t r i p s i g n a l t i m e s . 13 R a n g e - R a t e : The speed w i t h w h i c h t h e s a t e l l i t e i s a p p r o a c h i n g o r r e c e d i n g a l o n g t h e l i n e o f s i g h t , o b t a i n e d by m e a s u r i n g t h e D o p p l e r s h i f t . A z i m u t h : The a z i m u t h a n g l e t o t h e s a t e l l i t e measured i n t h e a z i m u t h e l e v a t i o n c o o r d i n a t e s y s t e m . E l e v a t i o n : The e l e v a t i o n a n g l e t o t h e s a t e l l i t e measured i n t h e a z i m u t h e l e v a t i o n c o o r d i n a t e s y s t e m . The e q u a t i o n s f o r t h e s e measurements i n t e rms o f t h e s y s t e m s t a t e s c a n be f o u n d i n T a b l e I I . 2 . The n o i s e p r e s e n t i n t h e measurements was a l s o m o d e l e d . Random numbers w i t h a G a u s s i a n d i s t r i b u t i o n were added t o t h e measurements b e f o r e t h e y were u s e d . The s t a n d a r d d e v i a t i o n s u s e d were Range 0 .0001 E . R . Range R a t e 0 . 0 4 E . R . / D a y A z i m u t h 0 .01 r a d i a n s E l e v a t i o n 0 .01 r a d i a n s . I t was f e l t t h a t t h e s e v a l u e s r e p r e s e n t e d an e a s i l y a c h i e v a b l e measurement a c c u r a c y . 14 Range |p| [<x 4 - s 4 ) 2 + (X 5 - v 2 + <v v 2 ] 1 / 2 • R a n g e - R a t e |p| ( x 4 - s 4 ) ( x 1 - s 1 ) + ( x 5 - s 5 ) ( x 2 - s 2 ) + ( x 6 - s 6 ) ( x 3 - s 3 ) IPI E l e v a t i o n A n g l e (h) , - 1 s i n p-R |>l A z i m u t h A n g l e (A) - 1 cos /•V p-N m iNiJ s y m b o l d e f i n i t i o n s S= S t a t i o n S t a t e V e c t o r S l X v e l o c i t y S 2 Y v e l o c i t y S 3 Z v e l o c i t y S 4 X d i s p l a c e m e n t S 5 Y d i s p l a c e m e n t S 6 Z d i s p l a c e m e n t R= S t a t i o n P o s i t i o n V e c t o r R l t h r o u g h R„ = S. t h r o u g h S . 3 4 o N= V e c t o r p o i n t i n g due N o r t h i n p l a n e t a n g e n t t o t he e a r t h a t t he s t a t i o n p= S t a t i o n to s a t e l l i t e p o s i t i o n v e c t o r p= P r o j e c t i o n o f p on p l a n e t a n g e n t t o the e a r t h a t s t a t i o n . Measurement E q u a t i o n s . T a b l e I I . 2 15 CHAPTER I I I KALMAN F I L T E R I N G AND SMOOTHING 1_. L i n e a r Ka lman F i l t e r A f i l t e r , i n t h e e s t i m a t i o n s e n s e o f t h e w o r d , i s an a l g o r i t h m w h i c h e s t i m a t e s t h e s t a t e v e c t o r a t t h e c u r r e n t t i m e b a s e d upon p a s t measurements c o r r u p t e d by n o i s e . The l i n e a r Ka lman f i l t e r i s s u c h an a l g o r i t h m . I t m i n i m i z e s t h e e s t i m a t i o n e r r o r i n a w e l l d e f i n e d s t a t i s t i c a l s e n s e . The l i n e a r K a l m a n f i l t e r e q u a t i o n s a r e p r e s e n t e d i n T a b l e I I I . l . S y s t e m M o d e l Measurement M o d e l V W V v k ~ N ( 0 , R ) S t a t e E s t i m a t e E x t r a p o l a t i o n v - ) = vA-i ( + ) E r r o r C o v a r i a n c e E x t r a p o l a t i o n V - ) = 0 k - i p k- i ( + ) 0 k- i T + \-i S t a t e E s t i m a t e Upda te V + ) = \ ( - ) + K k [ v H k V - ) ] E r r o r C o v a r i a n c e Upda te v->- [ i - wpk<-> Kalman G a i n M a t r i x v v-VtHkV-V + v"1 L i n e a r Ka lman F i l t e r E q u a t i o n s . T a b l e I I I . l T h i s f i l t e r i s a p p l i c a b l e o n l y t o l i n e a r s y s t e m s b u t fo rms t h e b a s i s o f t h e e x t e n d e d K a l m a n f i l t e r . 16 2. E x t e n d e d K a l m a n F i l t e r I n o r d e r t o be a b l e t o use t h e power o f t h e Ka lman f i l t e r on non l i n e a r s t a t e e s t i m a t i o n p r o b l e m s , t h e E x t e n d e d K a l m a n f i l t e r must be u s e d . I t i s b a s i c a l l y t h e same as t h e l i n e a r f i l t e r b u t w i t h t h e s t a t e and measurement e q u a t i o n s l i n e a r i z e d abou t t h e c u r r e n t b e s t e s t i m a t e . The e x t e n d e d K a l m a n f i l t e r e q u a t i o n s a r e p r e s e n t e d i n T a b l e I I I . 2 . T h i s f i l t e r forms t h e b a s i s o f our a t t a c k on t h e o r b i t o b s e r v a b i l i t y p r o b l e m . 17 S y s t e m M o d e l X ( t ) = f ( x ( t ) , t ) + w ( t ) : w ( t ) ~ N(0 , Q ( t ) ) Measurement M o d e l V V x ( t k ) ) + \ vk~N(0'V S t a t e E s t i m a t e P r o p a g a t i o n X ( t ) = f ( X ( t ) , t ) E r r o r C o v a r i a n c e P r o p a g a t i o n P ( t ) = F ( X ( t ) , t ) P ( t ) + P ( t ) F T ( X ( t ) , t ) + Q ( t ) S t a t e E s t i m a t e Upda te V + ) = V _ ) + W V V - ) ) ] E r r o r C o v a r i a n c e U p d a t e P k ( + ) = [ I - K k H k ( X k ( - ) ) ] P k ( - ) G a i n M a t r i x v v->HkT (v-»t vv-» Pk (->\\ (-»+ V"' where F ( X ( t ) , t ) = df(X(t),t) , and S x ( t ) H k ( X k ( - ) ) = 5 h k ( X ( t k ) ) d x ( t k ) E x t e n d e d Ka lman F i l t e r E q u a t i o n s . T a b l e I I I . 2 _3. L i n e a r i z e d Ka lman F i l t e r I f i n s t e a d o f l i n e a r i z i n g abou t t h e b e s t e s t i m a t e as i n t h e e x t e n d e d K a l m a n f i l t e r , t h e s t a t e i s l i n e a r i z e d abou t some known 18 t r a j e c t o r y t h e f i l t e r becomes what i s known as t h e l i n e a r i z e d K a l m a n f i l t e r . T h i s f i l t e r w o r k s w e l l i n r e d u c i n g e r r o r due t o s m a l l p e r t u r b a t i o n i n t h e s t a t e v e c t o r . The l i n e a r i z e d K a l m a n f i l t e r e q u a t i o n s a r e p r e s e n t e d i n T a b l e I I I . 3 . S y s t e m M o d e l X ( t ) = f ( X ( t ) , )+ w ( t ) ; w ( t ) ~ N ( 0 , Q ( t ) ) Measurement M o d e l zk= h k(X(t k))+ v k ; V N ( o , y S t a t e E s t i m a t e P r o p a g a t i o n X ( t ) = f ( X ( t ) , t ) + F ( X ( t ) , t ) [ X ( t ) - X ( t ) ] S t a t e E s t i m a t e U p d a t e v + ) = v- ) + V W ^ k ^ - H k ( X ( t k ) ) [ X k ( - ) - X ( t k ) ] ] E r r o r C o v a r i a n c e P r o p o g a t i o n P ( t ) = F ( X ( t ) , t ) P ( t ) + P ( t ) F T ( X ( t ) , t ) + Q ( t ) E r r o r C o v a r i a n c e Upda te G a i n M a t r i x H k T ( X ( t k ) ) + where F ( X ( t ) , t ) = d f ( X ( t ) , t ) , and 3 x ( t ) H k ( X ( t k ) ) = d h k ( X ( t k ) ) f ) x ( t k ) L i n e a r i z e d Ka lman F i l t e r E q u a t i o n s . T a b l e I I I . 3 19 _4. S m o o t h i n g S m o o t h i n g d i f f e r s f r o m f i l t e r i n g i n t h a t i t i t u s e s a l l t h e measurements f r o m an e n t i r e pas s t o e s t i m a t e t h e s t a t e a t some t i m e " t " d u r i n g t h e p a s s . I n t h e l i n e a r c a s e i t c a n be shown [2] t h a t t h e o p t i m a l smoothe r i s a l i n e a r c o m b i n a t i o n o f two K a l m a n f i l t e r s , one f i l t e r i n g t h e d a t a i n t h e f o r w a r d d i r e c t i o n t o t i m e " t " , and t h e o t h e r f i l t e r i n g b a c k w a r d s t o " t " . The e q u a t i o n s f o r t h i s smoothe r a r e x ( t | T ) = P ( t | T ) [ P - I ( t ) x ( t ) + P b - I ( t ) x b ( t ) ] p - 1 ( t | T ) = p - 1 ( t ) + P b - 1 ( t ) where t h e s u b s c r i p t b i n d i c a t e s t h e b a c k w a r d s e s t i m a t e . I f we a r e o n l y i n t e r e s t e d i n t h e i n i t i a l s t a t e , i e . t=0 , and we assume t h a t our i n i t i a l e r r o r c o v a r i a n c e i s v e r y l a r g e w i t h r e s p e c t t o t h e e r r o r c o v a r i a n c e a f t e r t h e d a t a has been p r o c e s s e d , t h e n t h e i n i t i a l s t a t e i s a f u n c t i o n o n l y o f t h e b a c k w a r d s f i l t e r . I f we assume t h a t t h e n o n l i n e a r smoothe r i s a l s o a l i n e a r c o m b i n a t i o n o f t h e f o r w a r d and b a c k w a r d f i l t e r s , t h e n t h e same e q u a t i o n s h o l d t r u e i n t h e n o n - l i n e a r c a s e - 20 CHAPTER I V SIMULATION PROCEDURES 1_. G e n e r a t i n g The Measurement D a t a 1.1 A l g o r i t h m U s e d To G e n e r a t e D a t a A c t u a l t r a c k i n g d a t a was i m p o s s i b l e t o o b t a i n due t o t h e many d i f f e r e n t t y p e s o f measurement and o r b i t r e q u i r e d . S i m u l a t e d measurement d a t a was t h e r e f o r e u s e d . T h i s d a t a was o b t a i n e d by a s s u m i n g an o r b i t and u s i n g t h e a l g o r i t h m w h i c h f o l l o w s t o c a l c u l a t e t h e s a t e l l i t e p o s i t i o n and v e l o c i t y a t a p p r o p r i a t e measurement i n t e r v a l s . F rom t h i s known p o s i t i o n and v e l o c i t y , measurements as t h e y w o u l d have a p p e a r e d t o an e a r t h t r a c k i n g s t a t i o n were c a l c u l a t e d . The e q u a t i o n s f o r t h i s a l g o r i t h m a r e l i s t e d i n T a b l e I V . 1 and T a b l e I I . 2 . S t ep 1: The d e s i r e d o r b i t a l p a r a m e t e r s a r e r e a d i n t o t h e c o m p u t e r . They a r e : O r b i t a l I n c l i n a t i o n i L o n g i t u d e o f t h e A s c e n d i n g Node . . . L Argument o f t h e P e r i g e e P E c c e n t r i c i t y e S e m i - m a j o r a x i s a Time o f P e r i f o c a l p a s s a g e S tep 2 : C a l c u l a t e t h e d i r e c t i o n c o s i n e s f o r c o n v e r t i n g f r o m t h e o r b i t a l p l a n e c o o r d i n a t e s y s t e m t o t h e a s c e n s i o n d e c l i n a t i o n D i r e c t i o n C o s i n e s P^= cos ( p ) c o s ( L ) - s i n ( p ) s i n ( L ) c o s ( i ) Py= c o s ( p ) s i n ( L ) + s i n ( p ) c o s ( L ) c o s ( i ) P = s i n ( p ) s i n ( i ) Q^= - s i n ( p ) c o s ( L ) - c o s ( p ) s i n ( L ) c o s ( i ) Q = - s i n ( p ) s i n ( L ) + cos ( p ) c o s ( L ) c o s ( i ) Q = c o s ( p ) s i n ( i ) z Mean M o t i o n . 1 / 2 . 3 /2 n= k u / a Mean Anomaly M= n ( t - T ) P E c c e n t r i c Anomaly 0 0 -, E= M+ iS" - J (me)s in(mM) —: m m m=l S t a t e i n O r b i t a l p l a n e c o o r d i n a t e s E= 1 nr a X u = a ( c o s ( E ) - e) Y^= a ( l - e 2 ) 1 / 2 s i n ( E ) X_= - a E s i n ( E ) Y u ) = a E ( l - e 2 ) 1 / 2 c o s ( E ) S t a t e i n R i g h t A s c . D e c . C o r d s . p = X • P + Y • Q [* J [Px Q x] W * P y Q y N Z P Q L J _ z z j S a t e l l i t e S t a t e C a l c u l a t i o n T a b l e I V . 1 22 s y s t e m . S tep 3: Compute t i m e s i n c e p e r i f o c a l p a s s a g e . S t ep 4 : Compute mean a n o m a l y . S t ep 5: Compute E c c e n t r i c a n o m a l y . S t ep 6: C a l c u l a t e c o o r d i n a t e s o f t h e s a t e l l i t e i n t h e o r b i t a l p l a n e c o o r d i n a t e s y s t e m . S tep 7: C a l c u l a t e c o o r d i n a t e s i n r i g h t a s c e n s i o n d e c l i n a t i o n s y s t e m . S tep 8: C a l c u l a t e v e l o c i t i e s i n o r b i t a l p l a n e c o o r d i n a t e s y s t e m . S tep 9: C a l c u l a t e v e l o c i t i e s i n t h e r i g h t a s c e n s i o n d e c l i n a t i o n s y s t e m . S tep 10: C a l c u l a t e r a n g e , r a n g e - r a t e , a z i m u t h , and e l e v a t i o n f r o m g round s t a t i o n t o s a t e l l i t e . S t ep 1 1 : I f e l e v a t i o n a n g l e i s g r e a t e r t h a n one d e g r e e ( i e . i f t h e s a t e l l i t e i s n o t h i d d e n b e h i n d e a r t h ) s t o r e t h e measurements c a l c u l a t e d and t h e s a t e l l i t e s t a t e . S tep 12: I nc r emen t c u r r e n t t i m e . S tep 13: Go t o s t e p 3 . The e a r t h s t a t i o n c o o r d i n a t e s u s e d a r e t h o s e o f t h e H e c t o r M a c L e o d b u i l d i n g a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a and a r e l i s t e d i n T a b l e I V . 2 . T y p i c a l measurement c u r v e s c a n be s e e n i n F i g u r e I V . 1 . 1.2 Types o f O r b i t I n o r d e r t o a c c o u n t f o r p o s s i b l e o r b i t t y p e dependancy by t h e s m o o t h i n g a l g o r i t h m s , a s e t o f s t a n d a r d o r b i t s was d e v e l o p e d w h i c h c o v e r t h e many d i f f e r e n t t y p e s o f o r b i t c u r r e n t l y i n u s e . T a b l e I V . 3 l i s t s H O 0 CO C (D 0 n> 3 o 1-1 n CD LU<=" UJ CC.,. CD CD • CD CD CD CD CD CD CD © CD CD 1 1 1 1 0.0 40.0 80.0 120.0 ,160.0 TIME (SEC.) (X10 1 ) 200.0 *1 (W c CD < cp a CC(n DEC • cc-v CD CD CD CD CD CD CD CD CD CD CD CD CD o CD o.o 40.0 —1 80.0 TIME 120.0 ,160.0 (SEC.) (X10 1 ) 200.0 —.r- >- CC O ceo LU<R. LD 1 cn 1 240.0 CD CD O O CD CD CD CD CD CD CD tt) CD CD CD CD 1 1 1 1 1 1 0.0 40.0 80.0 120.0 ,160.0 200.0 240.0 TIME (SEC.) (X10 1 ) o cc ce-d- cc _Jo" .CM O f T ) • CD CD CD CD CD CD CD CD CD CD CD CD CD CD n 1 1 1 1 1 1 2 4 0 0 0.0 40.0 80.0 120.0 ,160.0 200.0 240.0 TIME (SEC.) (X10 1 ) E a s t L o n g i t u d e 2 3 6 . 7 5 ° 24 L a t i t u d e 4 9 ° 1 5 ' 45" A l t i t u d e 310 f e e t C o o r d i n a t e s o f H e c t o r McLeod B l d g . T a b l e I V . 2 t h e o r b i t s u s e d and a s s i g n s t o each a r e f e r e n c e number . T h i s r e f e r e n c e number w i l l a p p e a r on many f i g u r e s t o f o l l o w . F o r i n s t a n c e " O R B I T . 3 " w i l l i n d i c a t e t h a t o r b i t a l d a t a f o r t h e o r b i t w i t h r e f e r e n c e number 3 was u s e d . Among t h e o r b i t s i n T a b l e I V . 3 a r e c i r c u l a r o r b i t s , e c c e n t r i c o r b i t s , p o l a r o r b i t s , c l o s e o r b i t s , and d i s t a n t o r b i t s . These r e p r e s e n t a good c r o s s - s e c t i o n o f t h e i n f i n i t e number o f o r b i t a l p o s s i b i l i t i e s . No. Inclination Ascending node Argument of perigee Semi-major axis Eccentricity Time of Peri- focal passage 1 45° 45° 45° 1.5 0.1 2345345.329167 2 90° 0° 0° 1.05 0. 2444055.5 3 90° 0° 0° 1.5 0. 2444055.5 4 90° 0° 0° 1.5 0.2 2444055.5 * 5 0° 90° 0° 6.6227 0. 2444055.5 6 45° 0° 0° 1.5 0. 2444055.5 * geosynchronous orbit. 26 2. S i m u l a t i o n A l g o r i t h m I n t h i s s t u d y we w i s h t o a r r i v e a t a good s t a t e e s t i m a t e a t some epoch t i m e . T h i s e s t i m a t e c o u l d t h e n be u s e d f o r f u t u r e o r b i t p r e d i c t i o n - F o r o u r p u r p o s e s , t h e epoch t i m e was c h o s e n t o be t h e t i m e o f t h e f i r s t measurement ( t = 0 ) . Due t o t h e n o n - l i n e a r n a t u r e o f t h e p r o b l e m , an o p t i m a l e s t i m a t e canno t be computed d i r e c t l y . A r e p e t i t i v e a l g o r i t h m c a n , h o w e v e r , be u s e d . I t w i l l be shown i n t h e s i m u l a t i o n r e s u l t s t h a t t h e e s t i m a t e a t t=0 can be made t o a p p r o a c h t h e c o r r e c t e s t i m a t e by r e p e t i t i v e f i l t e r i n g . The p r i n c i p a l r e s u l t o b t a i n e d f r o m t h e s i m u l a t i o n r u n s d e s c r i b e d i n t h e n e x t c h a p t e r a r e t h e compute r p r o d u c e d p l o t s o f t h e e r r o r i n t h e s t a t e e s t i m a t e c a l c u l a t e d by t h e s m o o t h e r . These show t h e m a g n i t u d e s o f t h e e r r o r i n t h e p o s i t i o n e s t i m a t e s and t h e v e l o c i t y e s t i m a t e s p l o t t e d a g a i n s t t i m e . These e r r o r s c o u l d be c a l c u l a t e d b e c a u s e when t h e s i m u l a t e d measurement d a t a was g e n e r a t e d and s a v e d , t h e s a t e l l i t e s t a t e v e c t o r was a l s o s a v e d . These s t a t e s were t h e n r e a d by t h e s i m u l a t i o n p r o g r a m t o g e t h e r w i t h t h e measurement d a t a and compared w i t h t h e s m o o t h e r ' s e s t i m a t e s . Due t o t h e s t a t e i n i t i a l i z a t i o n p r o c e d u r e d e s c r i b e d i n t h e n e x t s e c t i o n , t h e t i m e o f c l o s e s t s a t e l l i t e a p p r o a c h t o t h e e a r t h s t a t i o n i s u s e d f o r t h e i n i t i a l g u e s s . The s m o o t h e r , t h e r e f o r e , n o r m a l l y s t a r t s somewhere i n t h e m i d d l e o f t h e measurement d a t a . From t h i s p o i n t t h e f i l t e r sweeps f o r e w a r d i n t i m e u s i n g t h e e x t e n d e d K a l m a n f i l t e r a l g o r i t h m . When t h e f i n a l measurement i s r e a c h e d , t h e f i n a l s t a t e e s t i m a t e becomes t h e i n i t i a l s t a t e e s t i m a t e , t h e d i r e c t i o n o f sweep i s r e v e r s e d , and t h e e r r o r c o v a r i a n c e i s r e - i n i t i a l i z e d . The f i l t e r t h e n sweeps b a c k w a r d s i n t i m e t o t h e f i r s t measurement u s i n g t h e same d a t a . 27 T h i s p r o c e d u r e r e p e a t s i t s e l f f o r two more sweeps o f t h e d a t a . The a r r o w s p r e s e n t on most o f t h e f o l l o w i n g g raphs i n d i c a t e t h e d i r e c t i o n o f sweep when t h e c u r v e n e a r e s t t o t h e a r r o w was c a l c u l a t e d . D u r i n g a l l t h e s e sweeps t h e e s t i m a t e d s t a t e and t h e t r u e s t a t e a r e compared and t h e i r d i f f e r e n c e compu ted . Two e r r o r m a g n i t u d e s a r e compu ted , t h e p o s i t i o n e r r o r w h i c h i s t h e s q u a r e r o o t o f t h e sum of t h e s q u a r e s o f t h e t h r e e p o s i t i o n s t a t e e r r o r s ; and t h e speed e r r o r w h i c h i s t h e s q u a r e r o o t o f t h e sum o f t h e s q u a r e s o f t h e t h r e e v e l o c i t y s t a t e e r r o r s . These e r r o r q u a n t i t i e s a r e t h e n p l o t t e d on two s e p a r a t e g r a p h s , and a p e r f o r m a n c e measure o f t h e s m o o t h i n g a l g o r i t h m r e s u l t s . I t s h o u l d be n o t e d t h a t t h e a b s c i s s a i s a l w a y s t h e t i m e s i n c e t h e f i r s t measurement o f t h e pas s - The a l g o r i t h m u s e d i n t h e s i m u l a t i o n r u n s i s d e s c r i b e d i n somewhat more d e t a i l i n T a b l e I V . 4 . The s t e p numbers c o r r e s p o n d t o t h e c i r c l e d numbers on t h e t y p i c a l r e s u l t shown i n F i g u r e I V . 2 . 28 1 I n i t i a l i z e K a l m a n F i l t e r a t Minimum Range Time (T^) u s i n g computed i n i t i a l gue s s ( X ^ n ^ t ) and I n i t i a l E r r o r C o v a r i a n c e ( P . . ) . i n i t X = X . o i n i t P = P . . „ o i n i t 2 Use E x t e n d e d K a l m a n F i l t e r on d a t a b e t w e e n T ' and F i n a l Measurement T ime (T,.) . m r 3 R e - i n i t i a l i z e K a l m a n F i l t e r a t T^ u s i n g f i n a l s t a t e e s t i m a t e ( X ( T f ) ) and I n i t i a l E r r o r C o - v a r i a n c e . X Q = X ( T f ) P = P . . o i n i t A Use E x t e n d e d K a l m a n F i l t e r b a c k w a r d s i n t i m e on d a t a b e t w e e n T^ and F i r s t Measurement Time ( T s ) . 5 R e - i n i t i a l i z e K a l m a n F i l t e r a t T g u s i n g f i n a l s t a t e e s t i m a t e o f b a c k w a r d sweep ( X ( T g ) ) and I n i t i a l E r r o r C o v a r i a n c e . X = X ( T ) o s P = P . . „ o i n i t 6 Use E x t e n d e d K a l m a n F i l t e r f o r w a r d i n t i m e on d a t a be tween T and T, . . s f 7 R e - i n i t i a l i z e K a l m a n F i l t e r a t T f u s i n g l a t e s t s t a t e e s t i m a t e a t T^ and I n i t i a l E r r o r C o v a r i a n c e . V X ( V p = p, .„ o i n x t S i m u l a t i o n A l g o r i t h m ( C o n t i n u e d n e x t page) 29 8 Use E x t e n d e d K a l m a n F i l t e r b a c k w a r d s i n t i m e on d a t a be tween T,. and T . f s 9 F i n a l E s t i m a t e E r r o r a t T . s S t a t e E s t i m a t e a t T c a n be u sed f o r f u t u r e s o r b i t p r e d i c t i o n . S i m u l a t i o n A l g o r i t h m ( C o n t i n u e d f rom p r e v i o u s page) T a b l e I V . 2 o a o.o ONE STN 3 NNTS — Simulation Type 0RB]T.6 Orbit Type TRY A Initialization Used 10.0 20.0 30.0 ,40.0 T I M E ( S E C . ) ( X 1 0 1 ) 50.0 60.0 T y p i c a l S i m u l a t i o n Run R e s u l t F i g u r e I V . 2 30 _3. I n i t i a l i z a t i o n The i n i t i a l i z a t i o n o f t h e s m o o t h i n g a l g o r i t h m s t u r n e d ou t t o be an i m p o r t a n t c o n s i d e r a t i o n . Nea r t h e b e g i n n i n g o f t h e s t u d y t h e i n i t i a l s t a t e e s t i m a t e was made q u i t e a r b i t r a r i l y , however i t s o o n became a p p a r e n t t h a t t h e t h i s guess was i n some c a s e s c r i t i c a l due t o t h e f a c t t h a t t h e e x t e n d e d K a l m a n f i l t e r i n i t i a l l y u se s a p p r o x i m a t i o n s o f t h e s t a t e and measurement f u n c t i o n s l i n e a r i z e d abou t t h i s f i r s t g u e s s . The v a l i d i t y o f t h e s e a p p r o x i m a t i o n s t h e r e f o r e depended on t h e a c c u r a c y o f t h i s g u e s s . The f o l l o w i n g a l g o r i t h m was d e v e l o p e d t o c a l c u l a t e f o u r d i f f e r e n t i n i t i a l gues se s one o f w h i c h was c l o s e enough t o t h e c o r r e c t i n i t i a l s t a t e t o a l l o w t h e smoothe r t o c o n v e r g e . S t ep 1: Range d a t a was s c a n n e d and t h e minimum range f o u n d . S t ep 2 : T h i s minimum range was u s e d as an a l t i t u d e t o p o s i t i o n t h e s a t e l l i t e d i r e c t l y above t h e m e a s u r i n g e a r t h s t a t i o n . The t i m e a t w h i c h t h e minimum range measurement was made was u s e d as t h e i n i t i a l t i m e . S t ep 3: V e l o c i t i e s were computed w h i c h w o u l d p u t t h e s a t e l l i t e i n t o a c i r c u l a r o r b i t i n one o f f o u r d i r e c t i o n s . I n f i g u r e s t o f o l l o w t h i s i n i t i a l d i r e c t i o n w i l l be i n d i c a t e d by t h e " T R Y " number . TRY 1 i s due N o r t h TRY 2 i s due S o u t h TRY 3 i s due E a s t TRY 4 i s due West T h i s a l g o r i t h m was u s e d t o i n i t i a l i z e a l l t h e s m o o t h i n g a l g o r i t h m s e x c e p t t h e l i n e a r i z e d s m o o t h e r . The i n i t i a l v a l u e o f t h e e r r o r c o v a r i a n c e was a r r i v e d a t by t r i a l 31 and e r r o r , b u t once a r e a s o n a b l e v a l u e h a d been f o u n d i t was u s e d t h r o u g h o u t a l l t h e s i m u l a t i o n r u n s . I t was a l s o f o u n d n e c e s s a r y t o r e i n i t i a l i z e t h e e r r o r c o v a r i a n c e a f t e r e a c h sweep w i t h i n a r u n b e c a u s e t h e e r r o r c o v a r i a n c e became so s m a l l a f t e r one o r two sweeps t h a t c o n v e r g e n c e became v e r y s l o w . The i n i t i a l e r r o r c o v a r i a n c e was s e t t o 0 . 1 f o r t h e v a r i a n c e o f t h e p o s i t i o n s t a t e s and t o 1 0 0 . 0 f o r t h e v e l o c i t y s t a t e v a r i a n c e s . A l l o t h e r c o v a r i a n c e s were s e t t o z e r o . 32 CHAPTER V SIMULATION RESULTS The p l o t s t o f o l l o w r e p r e s e n t t h e r e s u l t s o f t h i s s i m u l a t i o n s t u d y . The r e a d e r i s r e f e r r e d t o t h e p r e v i o u s c h a p t e r f o r an e x p l a n a t i o n o f f o r m a t and t i t l e s . J_. Range and R a n g e - r a t e F rom One S t a t i o n An E x t e n d e d Ka lman f i l t e r b a s e d smoothe r was w r i t t e n w h i c h was t o c a l c u l a t e a p r e l i m i n a r y o r b i t e s t i m a t e u s i n g o n l y r a n g e and r a n g e r a t e d a t a f r o m one s t a t i o n . As p r e v i o u s l y s t a t e d , i f f e a s i b l e , s u c h a p r o g r a m w o u l d g r e a t l y r e d u c e t h e c o s t o f t r a c k i n g e q u i p m e n t . Even a s i m p l e d i p o l e a n t e n n a c o u l d be u s e d i f s i g n a l s t r e n g t h s were s u f f i c i e n t . T h i s w o u l d do away w i t h t h e need f o r l a r g e s t e e r a b l e d i s h a n t e n n a s o r l a r g e p h a s e d a r r a y s , a t l e a s t a t l o w e r f r e q u e n c i e s . A l s o no o p t i c a l equ ipment w o u l d be n e e d e d . The r e q u i r e d equ ipment w o u l d c o n s i s t o f an a c c u r a t e t i m e r f o r m e a s u r i n g r o u n d t r i p t r a v e l t i m e o f a s i g n a l t o and f r o m t h e s a t e l l i t e , and an a c c u r a t e f r e q u e n c y c o u n t e r f o r m e a s u r i n g d o p p l e r s h i f t . As c a n be s e e n f r o m F i g u r e V . 1 and F i g u r e V . 2 t h e smoothe r d i d n o t c o n v e r g e f o r any s e t o f i n i t i a l c o n d i t i o n s . T y p i c a l l y t h e p o s i t i o n e r r o r w o u l d i n c r e a s e l i n e a r i l y w i t h t i m e f r o m t h e p o i n t o f n e a r e s t a p p r o a c h . Such b e h a v i o r c a n be u n d e r s t o o d by a s s u m i n g t h e f i l t e r t o have l i t t l e o r no e f f e c t on t h e e r r o r i n t h e i n i t i a l g u e s s . I n s u c h a c a s e t h e d i s t a n c e be tween t h e assumed p o s i t i o n o f t h e s a t e l l i t e and t h e t r u e p o s i t i o n w o u l d be l e a s t when t h e y were n e a r e s t t o t h e s t a t i o n and w o u l d i n c r e a s e a p p r o x i m a t e l y l i n e a r l y as t h e y d i v e r g e d . Such b e h a v i o r X*A a^riSx^ a^B-g - aSuB^ put? aSuB-g joaag uoT5T s od cc o CC cc o 0 . 0 4 0 . 0 &) 3 OQ (D i r 8 0 . 0 1 2 0 . 0 , 1 6 0 . 0 TIME (SEC.) (XlO 1 ) ONE. STN O R B I T . 1 TRY 2 2 HNT5 2 0 0 . 0 2 4 0 . 0 0 . 0 I 4 0 . 0 0Q C CC CC o CC CC UJ o o UJ > 0 . 0 4 0 . 0 8 0 . 0 TIME T r 1 2 0 . 0 , 1 6 0 . 0 (SEC.) (X10 1 ) ONE. STN O R B I T . 1 TRY BO.O TIME 2 HNTS 1 1 1 2 0 . 0 , 1 6 0 . 0 ISEC.) IX101 ) I 2 0 0 . 0 - 1 2 4 0 . 0 2 0 0 . 2 4 0 . 0 0 . 0 4 0 . 0 —I 1 1 1 1 8 0 . 0 1 2 0 . 0 , 1 6 0 . 0 2 0 0 . 0 2 4 0 . 0 TIME (SEC.) (X10 1 ) ; 35 by t h e f i l t e r can be assumed t o i n d i c a t e t h a t t h e s y s t e m i s u n o b s e r v a b l e . O t h e r t y p e s o f o r b i t were t r i e d w i t h t h e same r e s u l t . _2. R a n g e , R a n g e - R a t e and A z i m u t h f r o m One S t a t i o n The d i m e n s i o n o f t h e measurement v e c t o r must be i n c r e a s e d f r o m two i f t h e s y s t e m i s t o be made o b s e r v a b l e . T h e r e a r e two ways o f i n c r e a s i n g t h e measurement v e c t o r d i m e n s i o n . The f i r s t i s t o add an a d d i t i o n a l measurement o f a d i f f e r e n t t y p e f r o m t h e same s t a t i o n . The s e c o n d i s t o u se t h e same t y p e o f measurements b u t add a s e c o n d s t a t i o n . T h i s s e c t i o n d e a l s w i t h t h e f i r s t o f t h e s e p o s s i b i l i t i e s , t h e o t h e r i s d e a l t w i t h i n s e c t i o n 4 . The s m o o t h i n g a l g o r i t h m was m o d i f i e d t o a c c e p t a t h i r d measurement . A z i m u t h a n g l e was c h o s e n b u t e l e v a t i o n a n g l e was t r i e d and w o u l d have w o r k e d e q u a l l y w e l l . S i m u l a t i o n r u n s were made u s i n g a l l t h e o r b i t s o f T a b l e I V . 3 . F i g u r e V . 3 and F i g u r e V . 4 show t h e r e s u l t s o f t h e O r b i t 1 s i m u l a t i o n . E v e r y i n i t i a l i z a t i o n c o n v e r g e d i n t h i s c a s e , a r a r e o c c u r r e n c e d u r i n g t h i s s t u d y d u e , as w i l l be d i s c u s s e d l a t e r , t o t h e a l g o r i t h m ' s s e n s i t i v i t y t o i n i t i a l i z a t i o n . F i g u r e V . 5 and F i g u r e V . 6 show t h e r e s u l t s f o r O r b i t 2 . O n l y i n i t i a l i z a t i o n T r y 2 c o n v e r g e d i n t h i s c a s e . F i g u r e V . 7 and F i g u r e V . 8 show t h e r e s u l t s f o r O r b i t 3 , where T r y 4 was t h e o n l y i n i t i a l i z a t i o n t o c o n v e r g e . W i t h O r b i t 4 two i n i t i a l i z a t i o n s w o r k e d , T r y 2 and T r y 4 as shown i n F i g u r e V . 9 and F i g u r e V . 10 . O r b i t 5 , t h e g e o s y n c h r o n o u s o r b i t p r o v e d t o be u n o b s e r v a b l e . (See F i g u r e V . 11 and F i g u r e V . 12 ) I t w i l l be shown, i n t h e n e x t two s e c t i o n s , t h a t t h e a d d i t i o n o f an e l e v a t i o n a n g l e measurement o r a s e c o n d s t a t i o n was r e q u i r e d t o o b t a i n c o n v e r g e n c e w i t h O r b i t 1 P o s i t i o n E r r o r w i t h 3 Measurements F i g u r e V . 3  POSITION ERROR (E.R.) 0.0 0.04 0.08 0.12 0.16 o  40 o o T I 1 1 1 H a g-0 9'0 fr'D 2'0 O'O c a ' 3 ) yoda3 NQIJLISQCI o a I 1 1 I To 8 -0 9'D t>' 0 Z'D O'O r a ' 3 ) yoaa3 NoniGOd O r b i t 3 P o s i t i o n E r r o r m w i t h 9"I Z'l 8'0 t o O'O c a ' 3 ) aoaa3 NQinsod 3 Measuremen t s F i g u r e V . 7  42 8'C P'O NOIi lSOd UJ CO 8'0 —1 1 1 g-D P'Q 2'D c a ' 3 ) yoaa3 NQiiisod 0" 0 r ? / 7 8'D fr'O C d ' 3 1 clQeJcG NOIi lSOd O r b i t . 4 P o s i t i o n E r r o r w i t h 3 Measuremen t s F i g u r e V . 9 O'OST O'OOT D'OS O'O ( A b a / ' a '3) aoaa3 UI3Q13A o'oee T 1 ~ r O'Ot̂ e 0 '097 0 ' 0 8 O'O ( j L d a / ' a ' 3 i aoaa3 untrraA O r b i t 4 V e l o c i t y ' E r r o r w i t h 3 Measurements F i g u r e V - 1 0  45 IT 09 ( A d a / trot- D'oe tro '3) yOdcJ3 XI13013A 0'08 T~ r D - 0 9 D"Ot> O'Oc 0"Q (Aua/ 'd'3) yoyy3 UIDQIBA a'091 ( j L t i a / 0 ' 0 8 D'Dt' O'O y-3) yoyy3 UIOCTGA z in >— ' <o in i - UJ 5 >- Z OS Q£ o o t- CD ID(_) U J CO LU • I 1 1 r Q-027 0'08 0'0t> O'O u u a v y 3 ) aoya3 ui3cn3A O r b i t 5 V e l o c i t y E r r o r w i t h 3 Measurements F i g u r e V . 1 2 46 t h i s o r b i t . F i n a l l y , t h e r e s u l t s w i t h O r b i t 6 a r e shown i n F i g u r e V . 13 and F i g u r e V . 1 4 . As c a n be s e e n f r o m a l l t h e s e f i g u r e s , t h e smoothe r r e q u i r e d a number o f sweeps t o c o n v e r g e and t h e r a t e o f c o n v e r g e n c e v a r i e d w i t h t h e o r b i t and t h e i n i t i a l i z a t i o n u s e d . The number o f sweeps was f i x e d a t 4 i n t h e b e l i e f i f c o n v e r g e n c e had n o t o c c u r r e d by t h e n , t h e o b s e r v a b i l i t y was n o t s a t i s f a c t o r y . F o r n o n - l i n e a r sy s t ems o b s e r v a b i l i t y seems t o be a m a t t e r o f d e g r e e r e a t h e r t h a n a ye s o r no c h o i c e as w i t h l i n e a r s y s t e m s . The r a t e o f c o n v e r g e n c e o f t h e s e f i g u r e s g i v e s a c r u d e measure o f t h e d e g r e e o f o b s e r v a b i l i t y . I n p r a c t i c e i f one were u s i n g a s m o o t h i n g a l g o r i t h m of t h i s s o r t t o t r a c k a s a t e l l i t e , one w o u l d need some method o f s e l e c t i n g t h e i n i t i a l i z a t i o n t o u s e . I n t h e p r e s e n c e o f no p r i o r i n f o r m a t i o n a t a l l , t h e c h o i c e can o n l y be made a f t e r a l l h ave been t r i e d . The d i f f e r e n c e s be tween t h e e s t i m a t e d measurements and t h e a c t u a l ones ( t h e r e s i d u a l s ) , p r o v i d e a b a s i s f o r an i n t e l l i g e n t c h o i c e . T a b l e V . 1 , T a b l e V . 2 , T a b l e V . 3 , and T a b l e V . 4 l i s t t h e r e s i d u a l s f o r t h e f o u r i n i t i a l i z a t i o n t r i e s o f O r b i t 2 . The r e s i d u a l s o f T r y 2 a r e somewhat s m a l l e r t h a n t h o s e o f t h e o t h e r i n i t i a l i z a t i o n s , i n d i c a t i n g t h a t T r y 2 ' s s t a t e e s t i m a t e s h o u l d be u s e d . 47 8 0 ' 0 t>0'C •y - 3) aoaa3 Nouisod ££'0 fc77-Q 9 T 0 8 0 ' 0 O'O {•a'!) yoaa^ Noiiisod t>'D E - 0 a'3) Z ' D V O aoaa3 Nouisod O'O 9 T 0 T T ~ 2 7 ' 0 8 0 ' 0 t>0'0 r a ' 3 ) aoaai Nouisod O'O O r b i t 6 P o s i t i o n E r r o r w i t h 3 Measuremen t s F i g u r e V . 1 3 ^ T ' A a j n S x j s a u a m a j n s B a f l £ H ^ T M j o j j g i C a x o o x a A 9 ^ T ^ a n VELOCITY ERROR ( E . R . / D A T J VELOCITY ERROR ( E . R . / D R Y ) ( X l O 1 ) 0.0 20.0 10.0 60.0 80.0 l I I I m o —I Q Q =C 50 2! - t co m ^ —' u> w • —\ VELOCITY ERROR (E R . / D R T ) 0.0 80.0 160.0 240.0 320.0 J VELOCITY ERROR ( E . R . / D R Y ) 0.0 40.0 80.0 120.0 160.0 H Q a » n z - c cu n 8^ Range - . 8 3 8 1 E + 0 0 - . 1 3 0 7 E + 0 0 - . 2 5 8 6 E + 0 0 - -2416E+00 - . 4 7 1 5 E - 0 1 - . 3 8 0 1 E - 0 1 - . 2 9 9 6 E - 0 1 - - 2 3 9 5 E - 0 1 - - 2 8 6 5 E - 0 1 - . 4 2 3 0 E - 0 1 0 . 1 5 4 9 E - 0 2 0 . 1 0 8 7 E - 0 2 0 . 3 0 7 7 E - 0 3 0 . 4 2 6 4 E - 0 3 0 . 1 0 8 5 E - 0 3 0 . 1 4 1 7 E - 0 4 - . 5 1 1 5 E - 0 4 - . 1 0 2 0 E - 0 2 - - 4 4 2 1 E - 0 3 - . 8 8 9 9 E - 0 3 - . 3 4 5 7 E - 0 3 0 . 5 6 6 4 E - 0 4 ORBIT R a n g e - R a t e - . 3 3 9 2 E + 0 3 - . 9 2 0 3 E + 0 2 0.1826E+03 0 .1772E+03 0.2386E+02 0 .3641E+02 0 .4186E+02 0-1821E+02 - . 6040E+02 - . 5 6 2 0 E + 0 2 0.4509E+01 0.4037E+01 0.5349E+01 0 .4489E+01 0 .2085E+01 - . 1507E+01 - . 2 5 6 8 E + 0 0 0-1424E+01 0.7822E+00 0 .6302E+00 0 .6076E+00 0 .2649E+00 2 R e s i d u a l s TRY T a b l e V . 1 A z i m u t h - . 4 4 3 3 E + 0 0 - . 2 4 6 5 E + 0 0 0.1101E+01 - . 2 9 7 0 E + 0 0 0 .3150E+01 0 .1301E+01 0.3966E+00 0 . 8 9 3 8 E - 0 1 0 .3589E+00 0 . 9 0 2 0 E - 0 1 - . 1848E-01 - . 1 1 0 8 E + 0 0 - . 1 3 2 4 E + 0 0 - . 7 3 8 3 E - 0 1 - . 2 6 4 0 E - 0 1 0 . 1 7 8 9 E - 0 1 0 . 4 1 2 3 E - 0 1 0 . 2 6 1 6 E - 0 2 - . 8 0 9 6 E - 0 1 - - 7 3 9 9 E - 0 2 0 . 1 3 5 3 E - 0 1 - . 1 4 4 2 E - 0 1 Range - . 1 1 4 0 E - 0 1 - . 2 1 9 6 E - 0 2 - . 1321E-02 - . 1 2 9 1 E - 0 2 - . 2 6 5 1 E - 0 2 - - 3 5 8 4 E - 0 2 - . 7135E-03 - . 5 5 6 0 E - 0 3 - . 1650E-03 - . 6 2 8 1 E - 0 3 - . 1056E-02 - . 1 8 0 0 E - 0 3 - . 1100E-02 - . 1 0 6 9 E - 0 2 - . 1540E-03 - . 9 3 3 8 E - 0 4 - . 3 5 7 4 E - 0 3 0 . 2 2 8 2 E - 0 3 - . 1222E-04 0 . 1 9 7 5 E - 0 3 - . 1 5 9 3 E - 0 3 0 . 1 3 3 8 E - 0 4 ORBIT R a n g e - R a t e - -3845E+01 - . 1052E+01 0.4325E+01 0.2932E+01 0.2207E+00 0.3131E+01 0.1499E+01 0.1103E+01 - . 1209E+01 - . 9 9 9 7 E + 0 0 - . 3 4 4 0 E + 0 0 0.1137E+01 - . 2 7 6 0 E - 0 1 - . 1314E+01 - . 5 4 0 2 E + 0 0 0 .1679E+00 0 .1502E+00 - . 1 0 3 3 E + 0 0 - . 2 3 9 2 E + 0 0 - . 8 0 4 1 E - 0 1 0 . 2 5 1 4 E - 0 1 0 .2932E+00 2 R e s i d u a l s TRY T a b l e V . 2 A z i m u t h - . 3 8 4 9 E - 0 1 0 . 3 1 2 5 E - 0 1 - . 2 5 1 5 E - 0 1 - . 2 8 8 7 E + 0 0 - . 1 5 2 9 E + 0 0 0 . 3 0 0 9 E - 0 1 - . 9 8 4 2 E - 0 2 0 . 7 7 4 0 E - 0 2 0 . 1 9 4 0 E - 0 1 - . 1 8 3 9 E - 0 1 - . 9 7 1 0 E - 0 2 - . 7 9 8 0 E - 0 1 0 . 6 0 0 4 E - 0 1 0 . 3 8 7 1 E - 0 2 - . 1 8 4 7 E - 0 1 - . 2 3 5 9 E - 0 2 - - 3 1 3 6 E - 0 2 0 . 9 0 5 4 E - 0 2 0 . 1 4 0 8 E - 0 2 - . 7 7 1 4 E - 0 2 - . 2 4 8 1 E - 0 1 - . 1 4 1 5 E - 0 1 Range - -6224E+00 - . 1 0 4 4 E + 0 0 - . 6 7 2 9 E + 0 0 - . 6 4 7 1 E - 0 1 - . 6 3 9 5 E - 0 2 0 . 1 9 9 4 E - 0 2 0 . 1 1 1 8 E - 0 2 0 . 6 0 1 3 E - 0 3 - . 3 7 2 9 E - 0 3 - . 2 6 1 6 E - 0 3 - . 4 8 4 3 E - 0 2 - . 3 7 6 0 E - 0 3 - . 1 1 4 1 E - 0 2 - . 1 0 5 3 E - 0 2 0 . 8 4 0 0 E - 0 4 - . 7 3 3 0 E - 0 4 - . 1 7 1 4 E - 0 3 - . 2 3 1 2 E - 0 2 -.9902E-03 - . 3 9 2 8 E - 0 3 - . 3 8 3 1 E - 0 3 - . 8 3 1 1 E - 0 4 ORBIT 2 R a n g e - R a t e - . 2 4 2 3 E + 0 3 - . 1 1 1 4 E + 0 3 0 .1362E+03 0 .1434E+02 - . 3907E+01 - . 5798E+01 - . 4745E+01 - . 2817E+01 0.1578E+01 - . 8 3 6 4 E + 0 0 - . 8827E+01 0.1217E+01 - . 7 9 3 7 E + 0 0 - . 3 4 5 9 E + 0 0 - . 1 4 4 8 E + 0 0 0 . 7 6 5 4 E - 0 1 - . 2 1 2 1 E + 0 0 0.3295E+01 0 .9765E+00 0.1101E+01 0.8845E+00 0 .4567E+00 R e s i d u a l s TRY T a b l e V . 3 A z i m u t h - - 2 9 5 3 E - 0 1 0 . 6 8 4 1 E - 0 1 0 . 3 7 6 9 E - 0 1 0 .1522E+00 0 . 8 6 2 8 E - 0 1 0 .1038E+00 0 . 4 7 2 5 E - 0 1 0 . 4 7 9 0 E - 0 1 - . 2 2 3 1 E - 0 1 - . 9 7 4 0 E - 0 1 - - 1 5 2 9 E - 0 1 0 .1386E+00 0 . 8 0 8 2 E - 0 1 - . 2 2 9 5 E - 0 1 0 . 4 7 8 6 E - 0 2 0 . 1 5 7 0 E - 0 1 - . 1 8 9 7 E - 0 1 - . 2 9 6 0 E - 0 1 - . 2 7 6 0 E - 0 2 - . 2 2 8 9 E - 0 1 - . 5 7 2 6 E - 0 2 0 . 7 8 9 3 E - 0 2 Range R a n g e - R a t e A z i m u t h - . 8 8 8 9 E - 0 1 - . 2 7 3 5 E - 0 1 - . 3 6 3 8 E - 0 1 - . 2 0 5 1 E - 0 1 - . 5 6 1 7 E - 0 3 3280E+00 - . 3662E+00 - . 4 2 5 7 E - 0 1 - . 2 0 7 6 E + 0 0 - .1486E+01 - . 1354E+01 - -4716E+00 - . 8 4 2 1 E - 0 1 - . 1 9 4 5 E + 0 0 - . 6 8 6 7 E - 0 1 - . 3 1 9 7 E - 0 1 - . 3 2 3 8 E - 0 1 1511E-01 - . 7 2 0 5 E - 0 2 - . 2 0 7 5 E - 0 3 - . 3 0 8 8 E - 0 3 0 . 6 5 8 2 E - 0 3 ORBIT - . 5 3 2 7 E + 0 2 - . 2061E+02 0.4473E+02 0.8351E+01 - . 2 0 6 8 E + 0 0 0 .1188E+03 - . 6 7 8 6 E + 0 2 0.7297E+01 - . 1 2 5 2 E + 0 3 - . 8053E+02 - . 5 3 1 3 E + 0 3 - . 4 2 7 2 E + 0 3 - . 1 6 8 7 E + 0 2 0.4245E+02 - . 1 6 7 2 E + 0 2 0.2982E+02 0.2354E+02 0.1269E+02 - . 6797E+01 - .7731E+01 - . 3620E+01 - . 1663E+01 2 R e s i d u a l s TRY T a b l e V . 4 - . 1 2 9 7 E + 0 0 0 . 7 3 9 5 E - 0 1 0 .6343E+00 0.2925E+00 0.5272E+01 0.4274E+01 0.3332E+01 0.3313E+01 0.2214E+01 0 .7935E+00 - . 4 7 4 0 E + 0 0 0.3740E+01 0 .1944E+01 0 .6598E+00 0-4191E+00 0 . 4 0 0 7 E - 0 1 - . 7 9 5 1 E - 0 1 0-3247E+00 0 .2604E+00 - . 2 2 3 4 E - 0 1 - . 1230E-01 - . 2 8 2 8 E - 0 1 53 3 .̂ R a n g e , R a n g e - R a t e , A z i m u t h , and E l e v a t i o n The s m o o t h i n g a l g o r i t h m was m o d i f i e d t o expand t h e measurement v e c t o r once a g a i n . T h i s t i m e e l e v a t i o n a n g l e was a d d e d . T h i s was done t o see i f any s i g n i f i c a n t improvement o c c u r r e d and a l s o t o o b s e r v e i f O r b i t 5 c o n v e r g e d w i t h f o u r measu remen t s . F i g u r e V . 15 and F i g u r e V . 16 show t h e r e s u l t s o b t a i n e d w i t h O r b i t 1. These can be compared w i t h F i g u r e V . 3 and F i g u r e V . 4 w h i c h showed t h e r e s u l t s w i t h t h r e e measu remen t s . Some improvement i n c o n v e r g e n c e speed c a n be s e e n b u t n o t enough t o j u s t i f y t h e e x t r a e x p e n s e . F i g u r e V . 17 and F i g u r e V . 18 show t h a t w i t h 4 t y p e s o f measurement O r b i t 5 c a n be made o b s e r v a b l e . 4 .̂ Range and R a n g e - R a t e f r o m Two S t a t i o n s The s a t e l l i t e - o b s e r v e r s y s t e m has b e e n shown t o be u n o b s e r v a b l e u s i n g r ange and r a n g e - r a t e f r o m one s t a t i o n . The a d d i t i o n o f a s e c o n d s t a t i o n some d i s t a n c e f r o m t h e f i r s t , a l s o s u p p l y i n g r a n g e and r a n g e - r a t e i n f o r m a t i o n , m i g h t make t h i s s y s t e m o b s e r v a b l e . To c h e c k t h i s s i t u a t i o n , t h e smoothe r u s e d i n p r e v i o u s s e c t i o n s was m o d i f i e d , and s i m u l a t i o n r u n s made u s i n g two s t a t i o n s . The p r i m a r y s t a t i o n r e m a i n e d t h e H e c t o r McLeod B u i l d i n g , w h i l e t h e s e c o n d was l o c a t e d f r o m 0 . 1 t o 20 d e g r e e s due E a s t o r N o r t h . R e s u l t s i n d i c a t e d t h a t o r b i t d e t e r m i n a t i o n c o u l d be a c c o m p l i s h e d i f t h e s t a t i o n s were f a r enough a p a r t . 4 . 1 O r b i t 2 R e s u l t s I n i t i a l l y t h e s e c o n d s t a t i o n was p l a c e d 1/2 deg ree l a t i t u d e away f r o m t h e p r i m a r y s t a t i o n , and O r b i t 2 , t h e c l o s e p o l a r o r b i t , was u s e d as t h e o r b i t u n d e r o b s e r v a t i o n . F i g u r e V . 19 and F i g u r e V . 20 show e n c o u r a g i n g b u t u n s a t i s f a c t o r y b e h a v i o r f o r T r y 1 i n i t i a l i z a t i o n . D o u b l i n g t h e s e p a r a t i o n t o 1 d e g r e e , F i g u r e V . 2 1 and F i g u r e V . 22 , <7C ONE STN 4 HHTS O R B I T . 1 TRY 1 0 . 0 4 0 . 0 1 r 8 0 . 0 1 2 0 . 0 , 1 6 0 . 0 TIME (SEC.) (XlO 1 ) l 1 2 0 0 . 0 2 4 0 . 0 ONE STN 4 HNTS O R B I T . 1 TRY 3 8 0 . 0 TIME 1 2 0 . 0 , 1 6 0 . 0 (SEC.) (XlO 1 ) 2 0 0 . 0 2 4 0 . 0 ONE STN 4 HNTS O R B I T . 1 TRY 2 >-9. a O CC U J o 0 . 0 4 0 . 0 8 0 . 0 TIME 1 2 0 . 0 , 1 6 0 . 0 (SEC.) (X101 ) 2 0 0 . 0 - 1 2 4 0 . 0 ONE STN O R B I T . 1 TRY 4 4 HNTS CC a ^8- CC O ceo U j 3 ~ 0 . 0 4 0 . 0 8 0 . 0 TIME T r 120 .0 , 1 6 0 . 0 (SEC.) (X10 1 ) 2 0 0 . 0 2 4 0 . 0 Ul  O r b i t 5 V e l o c i t y E r r o r w i t h 4 Measuremen t s , F i g u r e V . 1 8 58 O r b i t 2 P o s i t i o n E r r o r 1 / 2 ° L a t i t u d e S e p a r a t i o n F i g u r e V . 1 9 59   62 a p p r o x i m a t e l y h a l v e d t h e f i n a l s t a t e e s t i m a t e e r r o r b u t was s t i l l u n s a t i s f a c t o r y . I t was n o t u n t i l t h e s e p a r a t i o n was i n c r e a s e d t o 5 deg rees l a t i t u d e , F i g u r e V . 2 3 and F i g u r e V . 2 4 , t h a t s a t i s f a c t o r y c o n v e r g e n c e was o b t a i n e d . I n o r d e r t o d e t e r m i n e i f t h e d i r e c t i o n o f s t a t i o n s e p a r a t i o n had any e f f e c t , t h e s t a t i o n s were r e p o s i t i o n e d 1/2 deg ree l o n g i t u d e ' a p a r t . The r e s u l t s shown i n F i g u r e V . 25 and F i g u r e V . 26 d e m o n s t r a t e a marked improvement o v e r t h e 1/2 deg ree l a t i t u d e s e p a r a t i o n , an improvement e q u i v a l e n t t o t h e 5 deg ree l a t i t u d e c a s e . I n t u i t i v e l y one c o u l d e x p e c t t h a t a s a t e l l i t e w i t h a g round t r a c k p e r p e n d i c u l a r t o a l i n e drawn be tween t h e two e a r t h s t a t i o n s c o u l d be b e t t e r o b s e r v e d t h a n a s a t e l l i t e w i t h a p a r a l l e l g round t r a c k . T h i s t u r n e d ou t t o be t h e c a s e . A l s o , t h e b e s t i n i t i a l i z a t i o n i s no l o n g e r T r y 1 as w i t h 5 deg ree l a t i t u d e , b u t T r y 2 and T r y 3 . T h i s phenomena o f t h e b e s t i n i t i a l i z a t i o n c h a n g i n g w i t h t h e s e p a r a t i o n was o b s e r v e d f r e q u e n t l y . L o n g i t u d i n a l s e p a r a t i o n had t o be d e c r e a s e d t o 0 . 1 d e g r e e , F i g u r e V . 27 and F i g u r e V . 28 , b e f o r e c o n v e r g e n c e c e a s e d , and when i n c r e a s e d t o 3 d e g r e e s , F i g u r e V . 2 9 and F i g u r e V . 3 0 , T r y 3 i n i t i a l i z a t i o n no l o n g e r c o n v e r g e d w h i l e T r y 4 i m p r o v e d m a r k e d l y . C o n v e r g e n c e seems t o g r a d u a l l y s l o w w i t h d e c r e a s i n g s t a t i o n s e p a r a t i o n u n t i l , i n t h e l i m i t , t h e s y s t e m becomes t o t a l l y u n o b s e r v a b l e as shown i n s e c t i o n 2 . The s t a t i o n s e p a r a t i o n r e q u i r e d f o r o b s e r v a b i l i t y i s a f u n c t i o n o f t h e d i r e c t i o n o f s e p a r a t i o n a n d , as w i l l be shown n e x t , t h e t y p e o f o r b i t .  2 STNS 5 DEE flPRRT ORBIT.2 TRY J 0.0 80.0 1 T 160.0 240.0 TIME (SEC) 320.0 480.0 2 STNS 5 DEG fiPRRT DREUT.2 TRY 3 0.0 80.0 160.0 240.0 TIME (SEC) 320.0 400.0 480.0 65 O r b i t 2 P o s i t i o n E r r o r 1 / 2 ° L o n g i t u d e S e p a r a t i o n F i g u r e V . 2 5   68 69 O r b i t 2 P o s i t i o n E r r o r 3 L o n g i t u d e S e p a r a t i o n F i g u r e V . 2 9 O C A o VELOCITY ERROR (E.R./DRY) 320.• 0.0 80.0 160.0 240.0 J ° _ l 1 I I VELOCITY ERROR (E.R./DRY) 0.0 80.0 160.0 240.0 1 I 1 —i o I U TO 3D -c cu cn / 3 m m o o 320.0 _J 71 4 . 2 O r b i t 1 R e s u l t s When O r b i t 1, t h e i n c l i n e d e c c e n t r i c o r b i t , was u s e d as t h e o r b i t u n d e r o b s e r v a t i o n , a s e p a r a t i o n o f e v e n 2 d e g r e e s l o n g i t u d e was i n s u f f i c i e n t as shown i n F i g u r e V . 31 and F i g u r e V . 32 . Howeve r , a s e p a r a t i o n o f 2 d e g r e e s l a t i t u d e showed good c o n v e r g e n c e . (See F i g u r e V . 33 and F i g u r e V . 34 ) L o n g i t u d i n a l s e p a r a t i o n h a d t o be i n c r e a s e d t o 3 d e g r e e s , F i g u r e V . 35 and F i g u r e V . 3 6 , b e f o r e c o n v e r g e n c e became s a t i s f a c t o r y . T h i s c a n be e x p l a i n e d by t h e f a c t t h a t a t 50 d e g r e e s l a t i t u d e , s t a t i o n s one d e g r e e a p a r t i n l o n g i t u d e a r e abou t t h r e e f i f t h s as d i s t a n t as s t a t i o n s one d e g r e e a p a r t i n l a t i t u d e due t o t h e c o n v e r g e n c e o f t h e l i n e s o f l o n g i t u d e t owards t h e p o l e s . R e d u c i n g t h e l a t i t u d i n a l s e p a r a t i o n t o 1 deg ree c a u s e d no d i s c e r n a b l e d e c r e a s e i n p e r f o r m a n c e . I n f a c t , as c a n be s e e n f r o m F i g u r e V . 37 and F i g u r e V . 38 , t h e r e d u c t i o n c a u s e d t h e T r y 1 i n i t i a l i z a t i o n t o c o n v e r g e , w h i c h i t had n o t done f o r t h e 2 d e g r e e s e p a r a t i o n . The i d e a l s e p a r a t i o n d i r e c t i o n changes w i t h t h e t y p e o f o r b i t and i n some c a s e s f r o m p a s s t o p a s s o f t h e same o r b i t . The e r r a t i c b e h a v i o r o f t h e d i f f e r e n t i n i t i a l i z a t i o n t r i e s u n d e r s c o r e s t h i s a l g o r i t h m ' s s e n s i t i v i t y t o i n i t i a l i z a t i o n . I f we e n v i s a g e a s i x d i m e n s i o n a l i n i t i a l i z a t i o n s p a c e , d i s c o n t i n u o u s r e g i o n s e x i s t f r o m w h i c h c o n v e r g e n c e i s p o s s i b l e . As t h e s t a t i o n s a r e s e p a r a t e d by i n c r e a s i n g d i s t a n c e s , t h e measurement e q u a t i o n s c h a n g e , s h i f t i n g t h e s e r e g i o n s o f c o n v e r g e n c e . A g i v e n i n i t i a l i z a t i o n c a n t h e r e f o r e pas s i n and ou t o f c o n v e r g e n c e . A t h e o r e t i c a l e x p l a n a t i o n o f why and how t h e c h a n g i n g measurement e q u a t i o n s s h i f t t h e s e r e g i o n s i s beyond t h e s c o p e o f t h i s t h e s i s . I t c a n be s a i d , h o w e v e r , t h a t i n g e n e r a l t h e l a r g e r t h e s e p a r a t i o n t h e b e t t e r t h e o b s e r v a b i l i t y . I n c r e a s i n g s t a t i o n s e p a r a t i o n must be t r a d e d     76 r 9" ! r-S! 7/1 8"0 *"D i'H'3) hQUhl NOIilSOd V 2 c a 3) aoaa3 Nonisod O'O r 9'1 in m >• or Q£ n o H T Z'l 8'0 t'O O'O c a - 3 ) aoaa3 N o u i s o d 8'0 g-n *'0 Z'Q °' D c a - 3 ) aoaa3 NQiiisod O r b i t 1 P o s i t i o n E r r o r 3 L o n g i t u d e S e p a r a t i o n F i g u r e V . 3 5   8 £ " A u o i i B j B d a s a p n rjT 3 B i T o VELOCITY ERROR (E.R./DRY) 0.0 40.0 80.0 120.0 160.0 _| , 1 1 1 1 H O W 50 so VELOCITY ERROR (E.R./DRY) 0.0 80.0 160.0 240.0 320.0 VELOCITY ERROR (E.R./DRY) 0.0 40.0 80.0 120.0 160.0 o O b 80 o f f a g a i n s t a r e d u c e d measurement t i m e d u r i n g w h i c h t h e s a t e l l i t e i s v i s i b l e t o b o t h s t a t i o n s . 4 . 3 O t h e r O r b i t s ' R e s u l t s S i m u l a t i o n r u n s were made u s i n g t h e o t h e r o r b i t s o f T a b l e I V . 3 . A l l were f o u n d t o c o n v e r g e i f s t a t i o n s e p a r a t i o n was s u f f i c i e n t . F i g u r e V . 39 and F i g u r e V . 40 show t h e r e s u l t s o b t a i n e d w i t h O r b i t 3 and a 2 d e g r e e l o n g i t u d e s t a t i o n s e p a r a t i o n . F i g u r e V . 4 1 and F i g u r e V . 42 show O r b i t 4 ' s r e s u l t s a l s o w i t h a 2 deg ree l o n g i t u d e s e p a r a t i o n . O r b i t 5 , t h e g e o s y n c h r o n o u s o r b i t , r e q u i r e d a 20 d e g r e e s e p a r a t i o n i n l o n g i t u d e t o a c h i e v e t h e r e s u l t s shown i n F i g u r e V . 4 3 and F i g u r e V . 4 4 . F i n a l l y , t h e r e s u l t s u s i n g O r b i t 6 and a 2 d e g r e e l a t i t u d e s e p a r a t i o n a r e shown i n F i g u r e V - 4 5 and F i g u r e V . 4 6 . 5_. Known O r b i t Improvement As p r e v i o u s l y shown, i t i s n o t p o s s i b l e t o a c c o m p l i s h p r e l i m i n a r y o r b i t d e t e r m i n a t i o n u s i n g o n l y r a n g e and r a n g e - r a t e measurements f r o m one s t a t i o n . The q u e s t i o n t h e n a r o s e as t o how much i n f o r m a t i o n c o u l d be e x t r a c t e d f r o m t h e s e two s i m p l e measu remen t s . Was i t p o s s i b l e , f o r i n s t a n c e , t o c o r r e c t f o r s m a l l p e r t u r b a t i o n s f r o m an o r b i t w h i c h was f a i r l y w e l l known " a p r i o r i " ? I f s o , how l a r g e a n e r r o r c o u l d be c o r r e c t e d o r i m p r o v e d upon? The l i n e a r i z e d K a l m a n f i l t e r was i d e a l y s u i t e d t o answer t h e s e q u e s t i o n s as i t assumes a known s t a t e t r a j e c t o r y . S i m u l a t i o n r u n s were made t o answer t h e s e q u e s t i o n s . These r u n s were s i m i l a r t o t h e r u n s a l r e a d y d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s w i t h t h e f o l l o w i n g i m p o r t a n t d i f f e r e n c e s . The i n i t i a l s t a t e e s t i m a t e was n o t c a l c u l a t e d u s i n g t h e a l g o r i t h m d e s c r i b e d i n s e c t i o n I V . 3 bu t was s e t e q u a l t o t h e i n i t i a l l y assumed o r b i t . A f i f t h t r a c e was added t o t h e T8 . 82 CL a in to >- OS O f ( M O t - o'oee a i- o i X i ; o m • •So 1 UJ l o o ' UJ (Ada/ o'ogi o - os O'O cJ'3) AII3013A CM Q t— 0'021 Q'08 0'0t> O'O ( A d a / - a - 3 ) aoaa3 AII3013A r 0'091 i 1 r 0*021 O'oe o'ot> o 'o ( A u a / ' a '3) aoaas AU3Q13A r 0'091 - r 1 1 r 0'0£1 0'08 D'Ofr O'O ( A d a / - a - 3 ) aoaa3 A1I3Q13A O r b i t 3 V e l o c i t y E r r o r (12 L o n g i t u d e S e p a r a t i o n F i g u r e V . 4 0 T <7 - A aanST-a u o p r j j B d a s apn^TSuo'i q £ J t o . u a u o j q i s o " ^ 3 j q a o POSnrON ERROR (E.R.) POSITION ERROR (E.R.) 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 es 2 <7 • A 3 " 3 T I VELOCITY ERROR (E.R./DRY) 0.0 40.0 80.0 120.0 _ i 1 — L 160.0 j VELOCITY ERROR (E.R./DRY) 0.0 20.0 40.0 60.0 J I L —1 • u J O TO -c ro cn 80.0 J VELOCITY ERROR (E.R./DRY) 0.0 40.0 80.0 120.0 160.0 J VELOCITY ERROR (E.R./DRY) 0.0 80.0 160.0 240.0  VELOCITY ERROR (E.R./DRY) o.o cn x o 20.0 J_ 40.0 J_ 60.0 J 80.0 _J H O W =0 TO - 4 CO CO M B •D 33 VELOCITY ERROR (E.R./DRY) (X10 1 ) 0.0 40.0 80.0 120.0 160.0 1 1 I I fe CO m o r o to fe" —t O M TO TO - C CO CO VELOCITY ERROR 0.0 40.0 (E.R./DRY) 80.0 (XlO3 120.0 fe m x o 160.0 _ l VELOCITY ERROR 0.0 20.0 (E.R./DRY) 40.0 (X10 1 60.0 fe 3 m c n m o r : J x o 80.0 -H ca INJ TO TO -e. CO co —t , , — i a : in 1" 33 98  2 STNS 2 DEE APART ORBIT.6 TRY 2 89 p l o t s p r o d u c e d . T h i s t r a c e , w h i c h i s t h e r e l a t i v e l y s t r a i g h t l i n e w i t h o u t k i n k s , r e p r e s e n t s t h e e r r o r i n t h e i n i t i a l l y assumed u n c o r r e c t e d o r b i t . F o u r sweeps were made o f t h e d a t a and t h e t r a j e c t o r y abou t w h i c h l i n e a r i z a t i o n o c c u r s was u p d a t e d t o t h e b e s t e s t i m a t e a f t e r e a c h sweep . The r e s u l t s o b t a i n e d i n d i c a t e d t h a t t h e r ange and r a n g e - r a t e measurements c o u l d be u sed t o i m p r o v e t h e o r b i t a l e s t i m a t e s , a t l e a s t i n some c a s e s . They d i d , h o w e v e r , d e m o n s t r a t e a h i g h d e g r e e o f dependance on t h e o r b i t a l g e o m e t r y . F i g u r e V . 4 7 shows t h e r e s u l t s o b t a i n e d when an e r r o r o f 0 . 0 5 E . R . i s assumed i n t h e s e m i - m a j o r a x i s o f o r b i t 1- A l l o t h e r p a r a m e t e r s a r e assumed e r r o r f r e e . As c a n be s e e n a d e f i n i t e improvement i n t h e m a g n i t u d e o f t h e e r r o r o c c u r s . The p o s i t i o n e r r o r i s d e c r e a s e d f r o m a p p r o x i m a t e l y 0 . 4 E . R . t o 0 . 2 E . R . - S i m i l a r l y t h e v e l o c i t y e r r o r d e c r e a s e s f r o m abou t 24 E . R . / d a y t o 8 E . R . / d a y . T h i s i s a c c o m p l i s h e d o v e r f o u r sweeps o f t h e d a t a . I n some c a s e s a l l f o u r sweeps were no t n e c e s s a r y , t h e b e s t e s t i m a t e h a v i n g been c a l c u l a t e d a f t e r one sweep . F i g u r e V . 4 8 shows t h e r e s u l t s o f a one d e g r e e e r r o r i n t h e assumed a s c e n d i n g n o d e . The v e l o c i t y e r r o r i s a c t u a l l y g r e a t e r t h a n t h e u n c o r r e c t e d e s t i m a t e f o r h a l f t h e f i r s t sweep and t h e p o s i t i o n e r r o r f o r p a r t s o f t h e s e c o n d sweep . Howeve r , t h e t h i r d sweep b r i n g s a d e f i n i t e improvement i n b o t h v e l o c i t y and p o s i t i o n . F i g u r e V . 49 shows t h e improvement when t h e argument o f t h e p e r i g e e i s i n e r r o r by one d e g r e e . F i g u r e V . 50 and F i g u r e V . 5 1 show i m p r o v e d r e s u l t s f o r s i m i l a r e r r o r s i n o r b i t s 3 and 4 . As an example o f t h e p o o r r e s u l t s o b t a i n e d when t h e o b s e r v a t i o n p e r i o d was t o o b r i e f o r t h e number o f measurements t o o f e w , F i g u r e V - 5 2 shows a s i m u l a t i o n r e s u l t when a one deg ree e r r o r i n i n c l i n a t i o n was assumed i n o r b i t 2 , t h e c l o s e p o l a r o r b i t . CO LINEARIZED 2 MNTS QRBJT.l 3 0 0 Km E r r o r i n F i g u r e V • 47 " A " 91 o CM" LINEARIZED 2 MNTS ORBIT.1 80.0 120.0 , 160.0 TIME (SEC) ( X l O 1 ) 200.0 240.0 O n e D e g r e e E r r o r i n " L " F i g u r e V . 4 8 LINEARIZED 2 HNTS ORBIT.1 LINERRIZED 2 MNTS ORBIT.1 0.0 40.0 80.0 120.0 , 1B0.0 TIME (SEC) (XlO3 ) 200.0 240 O n e D e g r e e E r r o r i n F i g u r e V . 4 9 93 O n e D e g r e e E r r o r i n f i g u r e V . 5 0 O r b i t 3 94 One D e g r e e E r r o r i n " i " F i g u r e V . 5 1 O r b i t 4 VELOCITY 0.0 1.0 ERROR (E.R./DRY) 2.0 3.0 H- TO C H (0 o ro o :ro cm i-i ro fD w r( l-( O rt r1- 3 4.0 POSITION 0.01 ERROR ( E . R . ) 0.02 0.03 fe- o fe- D VO Ul 96 F i n a l l y F i g u r e V . 5 3 shows l i t t l e o r no improvement o b t a i n e d when a one d e g r e e e r r o r i n a s c e n d i n g node i s assumed f o r a g e o s y n c h r o n o u s s a t e l l i t e . Some improvement c a n be o b t a i n e d however i f t h e e r r o r i s i n t h e s e m i - m a j o r a x i s as c a n be s e e n i n F i g u r e V . 54 . The improvement i s t o t a l l y i n t h e p o s i t i o n , t h e v e l o c i t y e r r o r a l r e a d y b e i n g so s m a l l t h e smoothe r c anno t i m p r o v e i t . The r e s u l t s i n d i c a t e t h a t w h i l e some improvement c a n be e x p e c t e d on t h e a v e r a g e , i t i s w i s e t o s i m u l a t e t h e c o r r e c t i o n p r o c e d u r e as has been done h e r e , b e f o r e a p p l y i n g i t t o any p r a c t i c a l p r o b l e m . T h i s w o u l d e n a b l e t h e e n g i n e e r t o d e t e r m i n e i f t h e o r b i t geomet ry i s o f t h e c o r r e c t a b l e t y p e . 97 55 r\i 1 cc U J 83 C C C C M O C O o o LINEARIZED 2 MNIS ORBIT.5 I I 1 1 - 0J3 80.0 1B0.0 240.0 320 0 TIME (SEC) (X10J ) ' 400.0 480.0 LINEARIZED 2 MNTS ORBIT.5 i 1 r 1 6 0 J ) 240JD 3 2 0 . 0 TIME (SECJ (X101 J One Degree E r r o r i n " L " F i g u r e V.53 400.0 480 J CO o LINERRIZED 2 HNTS ORBIT.5 1 T 0.0 80.0 160.0 240.0 , 320.0 TIME (SEC) (XlO3 ) 400.0 480.0 F i g u r e V . 5 4 99 CHAPTER V I CONCLUSIONS A t e c h n i q u e has been d e v e l o p e d whereby t h e o b s e r v a b i l i t y o f a n o n - l i n e a r s y s t e m c a n be d e t e r m i n e d t h r o u g h m u l t i p l e s i m u l a t i o n r u n s . As a good example o f t h i s t e c h n i q u e , t h e o b s e r v a b i l i t y o f an e a r t h s a t e l l i t e has b e e n d e t e r m i n e d f o r a number o f measurement s e t s . I n t h i s c a s e , s i m u l a t i o n showed t h a t r ange and r a n g e - r a t e measurements a l o n e were i n s u f f i c i e n t f o r o r b i t d e t e r m i n a t i o n and t h a t w i t h t h e a d d i t i o n o f a z i m u t h a n g l e measurements t h e s y s t e m became o b s e r v a b l e . S i m i l a r r u n s showed t h a t t h e a d d i t i o n o f a s e c o n d s t a t i o n a l s o made t h e s y s t e m o b s e r v a b l e and d e m o n s t r a t e d t h e e x t e n d e d Ka lman f i l t e r ' s s e n s i t i v i t y t o i n i t i a l i z a t o n - T h i s s e n s i t i v i t y d i s p l a y e d i t s e l f i n two w a y s , f i r s t w i t h i n one s i m u l a t i o n r u n o n l y one o r two o f t h e f o u r i n i t i a l i z a t i o n s t r i e d w o u l d b r i n g t h e smoothe r t o t h e c o r r e c t e s t i m a t e , and s e c o n d be tween s i m u l a t i o n r u n s as t h e measurement e q u a t i o n s were changed t h e b e s t i n i t i a l i z a t i o n f o r a g i v e n o r b i t w o u l d change as w e l l . I t i s beyond t h e s c o p e of t h i s t h e s i s t o e x p l o r e t h e s e i n s t a b i l i t i e s i n d e t a i l . F i n a l l y , s i m u l a t i o n has shown t h a t i t i s p o s s i b l e i n some c a s e s t o i m p r o v e on o r b i t a l e s t i m a t e s a r r i v e d a t by o t h e r means, u s i n g o n l y r a n g e and r ange r a t e measurements f r o m one s t a t i o n . 100 REFERENCES [1] P . R . E s c o b a l , Me thods o f O r b i t D e t e r m i n a t i o n - New Y o r k : J o h n W i l e y & S o n s , 1965. [2] A . G e l b , A p p l i e d O p t i m a l E s t i m a t i o n - C a m b r i d g e , M a s s . : M . I . T . P r e s s , 1974. [3] R . Herman and A . J . K r e n e r , " N o n l i n e a r C o n t r o l l a b i l i t y and O b s e r v a b i l i t y " , I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l Oc t 1977 v o l A C - 2 2 N o - 5 p728 [4] A . H . J a z w i n s k i , S t o c h a s t i c P r o c e s s e s & F i l t e r i n g T h e o r y . New Y o r k : A c a d e m i c P r e s s , 1970 . [5] T . D . M o y e r . , " M a t h e m a t i c a l F o r m u l a t i o n o f t h e D o u b l e P r e c i s i o n O r b i t D e t e r m i n a t i o n P r o g r a m , J e t P r o p u l s i o n Lab May 1971 . [6] J . J . P o l l a r d , " O r b i t a l P a r a m e t e r D e t e r m i n a t i o n by W e i g h t e d L e a s t Squa re E r r o r and K a l m a n F i l t e r i n g M e t h o d s " , M A S c T h e s i s A i r F o r c e I n s t i t u t e o f T e c h n o l o g y Dec 1972 . [7] B . E . S c h u t z , e t a l . , " A C o m p a r i s o n o f E s t i m a t i o n Methods f o r t h e R e d u c t i o n o f L a s e r O b s e r v a t i o n s o f a N e a r - E a r t h S a t e l l i t e " , A A S / A I A A A s t r o d y n a m i c s C o n f e r e n c e , V a i l C o l o r a d o / J u l y 1 6 - 1 8 , 1 9 7 3 - [8] P . S w e r l i n g , "Modern S t a t e E s t i m a t i o n Methods f r o m t h e V i e w p o i n t o f t h e Me thod o f L e a s t S q u a r e s " , IEEE T r a n s a c t i o n s on A u t o m a t i c C o n t r o l Dec 1971 v o l A C - 1 6 N o . 6 p 7 0 7 .

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