KALMAN FILTER BASED ORBIT OBSERVABILITY STUDY by ROY ROBERT SHORTEN B.Eng., S i r 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 thesis i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced d e g r e e 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 , I a g r e e 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 that permission f o r extensive copying o f t h i s thesis f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e 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 understood that copying 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 n o t be a l l o w e d w i t h o u t my written permission. Department o f P-~ JCs ^r^>C^»^c^'( x 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 V a n c o u v e r , Canada V6T 1W5 Date 5v^-*v» ? 1. *- <^>^9\-*~~M.&~V ii ABSTRACT An Extended Kalman f i l t e r observability s i m u l a t i o n t e c h n i que was used to determine the of s a t e l l i t e orbits from one or two earth was found t h a t range and r a n g e - r a t e measurements alone were for orbit determination. Either azimuth angle or stations. It insufficient elevation angle i n f o r m a t i o n were a l s o r e q u i r e d b e f o r e an a c c e p t a b l e o r b i t a l e s t i m a t e was o b t a i n e d . However, sufficient range t o be to improve the s t a t e e s t i m a t e s of a p p r o x i m a t e l y known o r b i t s . Also i f simultaneous from two orbital and r a n g e - r a t e measurements a l o n e proved stations, range orbit types were used was c o n s i d e r e d . and range-rate determination throughout measurements was the study possible. were available Various and o n l y one pass common of data iii TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS ix I INTRODUCTION II TO SATELLITE TRACKING 1.1 Observable? 1 1.2 Geometry 2 1.3 Measurement Methods 2 1.4 B a s i c Problem S t u d i e d 4 1.5 Method 4 THE SYSTEM MODEL 6 II.1 Limitations 6 II. 2 Units 6 I I . 3 Coordinates 7 11.3.1 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 III 1 System 7 11.3.2 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 System 8 11.3.3 O r b i t a l Plane C o o r d i n a t e System 9 I I . 4 S t a t e Model 11 I I . 5 Measurement Model 12 KALMAN FILTERING AND SMOOTHING .' 15 I I I . l L i n e a r Kalman F i l t e r 15 I I I . 2 Extended Kalman F i l t e r 16 I I I . 3 L i n e a r i z e d Kalman F i l t e r 17 I I I . 4 Smoothing 19 iv IV SIMULATION PROCEDURES IV. 1 G e n e r a t i n g The Measurement Data 20 IV. 1.2 Types of O r b i t 22 Algorithm IV. 3 I n i t i a l i z a t i o n SIMULATION RESULTS 26 30 32 V. 1 Range and Range-rate From One S t a t i o n 32 V.2 Range, Range-Rate and Azimuth from One S t a t i o n 35 V.3 Range, Range-Rate, Azimuth, and E l e v a t i o n 53 V.4 Range and Range-Rate from Two S t a t i o n s 53 V. 4.1 O r b i t 2 Results 53 V.4.2 O r b i t 1 Results 71 V.4.3 Other O r b i t s ' R e s u l t s V.5 Known O r b i t VI 20 IV. 1.1 A l g o r i t h m Used To Generate Data IV.2 S i m u l a t i o n V 20 CONCLUSIONS REFERENCES Improvement 80 80 99 100 L I S T OF TABLES I I . 1 State Equation I I . 2 Measurement 12 Equations I I I . l L i n e a r Kalman F i l t e r 14 Equations I I I . 2 Extended Kalman F i l t e r Equations I I I . 3 L i n e a r i z e d Kalman F i l t e r IV. 1 Satellite State IV. 2 Coordinates Used of Calculation Hector in Equations IV. 3 Orbits IV. 4 Simulation Algorithm McLeod Study 15 17 18 21 Bldg 24 25 29 V. l ORBIT. 2 R e s i d u a l s Try 1 49 V.2 ORBIT. 2 Residuals Try 2 50 V.3 ORBIT. 2 Residuals Try 3 51 V.4 ORBIT. 2 Residuals Try 4 52 vi L I S T OF FIGURES II. 1 Azimuth-Elevation I I . 2 Right Ascension Coordinate Declination II. 3 Orbital Plane Coordinate II. 4 Orbital Plane Coordinate IV. 1 Typical Measurements IV. 2 T y p i c a l System 7 Coordinate System 9 View 10 System O b l i q u e System P e r p e n d i c u l a r View 11 (ORBIT.1) 23 Simulation Run R e s u l t 28 V. 1 P o s i t i o n Error with Range and Range-Rate 33 V.2 Velocity Error with Range and Range-Rate 34 V.3 Orbit 1 Position Error with 3 Measurements 36 V.4 Orbit 1 Velocity Error with 3 Measurements 37 V.5 Orbit 2 Position Error with 3 Measurements 38 V.6 Orbit 2 Velocity Error with 3 Measurements 39 V.7 Orbit 3 Position Error with 3 Measurements 40 V.8 Orbit 3 Velocity Error with 3 Measurements 41 V.9 Orbit 4 Position Error with 3 Measurements 42 V.10 Orbit 4 Velocity Error with 3 Measurements 43 V.ll 5 Position Error with 3 Measurements 44 V . 12 O r b i t 5 Velocity Error with 3 Measurements 45 V.13 Orbit 6 Position Error with 3 Measurements 47 V . 14 O r b i t 6 Velocity Error with 3 Measurements 48 V.15 Orbit 1 Position Error with 4 Measurements 54 V . 16 O r b i t 1 Velocity Error with 4 Measurements 55 V.17 Orbit 5 Position Error with 4 Measurements 56 V . 18 O r b i t 5 Velocity Error with 4 Measurements 57 V.19 Orbit 2 Position Error 1/2° Latitude 58 Orbit Separation vii V.20 Orbit 2 Velocity Error 1/2° Latitude Separation 59 V.21 Orbit 2 Position Error 1° L a t i t u d e Separation 60 V.22 Orbit 2 Velocity Error 1° L a t i t u d e Separation 61 V.23 Orbit 2 Position Error 5° L a t i t u d e Separation 63 V.24 Orbit 2 Velocity Error 5° Latitude Separation 64 V.25 Orbit 2 Position Error 1/2° Longitude Separation 65 V.26 Orbit 2 Velocity Error 1/2° Longitude Separation 66 V.27 Orbit 2 Position Error 0.1° Longitude Separation 67 V.28 Orbit 2 Velocity Error 0.1° Longitude Separation 68 V.29 Orbit 2 Position Error 3° Longitude Separation 69 V.30 Orbit 2 Velocity Error 3° Longitude Separation 70 V.31 Orbit 1 Position Error 2° Longitude Separation 72 V.32 Orbit 1 Velocity Error 2° Longitude Separation 73 V.33 Orbit 1 Position Error 2° L a t i t u d e Separation 74 V.34 Orbit 1 Velocity Error 2° L a t i t u d e Separation 75 V.35 O r b i t 1 Position Error 3° Longitude Separation 76 V.36 Orbit 1 Velocity Error 3° Longitude Separation 77 V.37 Orbit 1 Position Error 1° L a t i t u d e Separation 78 V.38 Orbit 1 Velocity Error 1° L a t i t u d e Separation 79 V.39 Orbit 3 Position Error 2° Longitude Separation 81 V.40 Orbit 3 Velocity Error 2° Longitude Separation 82 V.41 Orbit 4 Position Error 2° Longitude Separation 83 V.42 Orbit 4 Velocity Error 2° Longitude Separation 84 V.43 Orbit 5 Position Error 20° Longitude Separation 85 V.44 Orbit 5 Velocity Error 20° Longitude Separation 86 V.45 Orbit 6 Position Error 2° L a t i t u d e Separation 87 V.46 O r b i t 6 Velocity Error 2° L a t i t u d e Separation 88 viii V.47 300 Km E r r o r in "a" 90 V . 4 8 One D e g r e e Error in "L" 91 V . 4 9 One D e g r e e Error in "p" 92 V . 5 0 One D e g r e e Error in " i " Orbit 3 93 V . 5 1 One D e g r e e Error in " i " Orbit 4 94 V . 5 2 One D e g r e e of V . 5 3 One D e g r e e Error V.54 0.01 E.R. Error Error in in in " i " 95 "L" 97 "a" 98 ix ACKNOWLEDGEMENTS I would Dr. A.C. reviewing his like to express Soudack this h e l p and my t h a n k s for thesis. their I would encouragement. to my s u p e r v i s o r assistance also like in to Dr. preparing thank Dr. E . V . Bohn and help V . J . Modi and in for 1 CHAPTER INTRODUCTION TO S A T E L L I T E J^. A system Is distinguish to be a the states satellite considered position observable its and attraction motion. It velocity also makes determine. It has determines whether however, Currently the simulation. be model of earth, is the is determination. between types is system the is rank of observable employed o f measurements were o n l y as is made good more approach The m o d e l as and its these l i m i t a t i o n s i s presented inverse the force satellite non-linear. or relatively the not. and difficult taken in simulation i n order to In the to determine secondary assumptions i n chapter II. matrix non-linear this runs [3]- thesis. are if because effects and to observability determine quasi-realistic simple observability determining non-linear model to governing of the be satellite's two m a s s e s effect can Newton's p r i m a r y method is a can happens states Determining primary linear system the much assumptions is distance system i n question is distinguished. simulation orbit been proven t h a t the the velocity. problem a the o f measurements y o u observability quasi-realistic can the the If around called them, for This using various system. and is between from a g i v e n set orbit relates Observability case in the TRACKING Observable? if position square law, which of of I_ model. is A performed the as states with neglected. any A A discussion 2 7_. G e o m e t r y Satellites We s h a l l concern All orbits the presence the of on rotation of position and purely surface the velocity i n space called orbit in the during which station. A pass obtained earth, the which forever the is case of satellite is of the a the of zero of the a i n view affects its orbit is hidden pass being above the minutes to satellite. as the portion the any of an measuring of a millions of years length and earth. case The the from the a This updated behind h o r i z o n of i n the from process c a n be seen in space. rotating. c o n t i n u a l l y be some s t a t i o n , geosynchronous itself measurement orbit is is must 'pass', from hidden from which measurer satellite last o r b i t a l plane which, are concept can orbits. orbit rest the elliptical p o s i t i o n along this The brings the sections. inertial one This and conic in Only a s m a l l part station. are fixed the force, taken i n t o account. earth circular complicates of which is of earth paths with central satellite the describe only a plane a of earth ourselves describe Measurements station of satellite of as time in the observability. _3. M e a s u r e m e n t M e t h o d s Historically sighting. and the It the was u s e d stars, all tracking can supply estimate range distance to surprisingly with less first by t h e of which as moon. good, but brightness ancients they angular visually the method u s e d it to track information can be seen angular is difficult the track believed The than to satellites the circled only. from It ancient accuracy of was moon, the the earth. is impossible The visual planets, Visual estimates visual method of to the tracking d u r i n g d a y l i g h t to t r a c k moon. visual is anything was later 3 augmented by the use of o p t i c a l equipment such as telescopes. enabled much better accuracy to be obtained measurements, but s i g n i f i c a n t l y increased These and also allowed daylight the cost. When cameras were added to the o p t i c a l tracking equipment, extremely high accuracies were obtained by measurements r e l a t i v e to known s t e l l a r background positions. The recent drastically introduction of radar changed the type and quantity performed. While the angular information nearly optical as accurate as from to track satellites of measurements that supplied methods, from radars radar does has can be i s not supply an accurate range and range-rate measurement. If the s a t e l l i t e we wish to track i s active, that i s i f i t carries on board a radio signals, other antenna arrays obtain transceiver which w i l l receive and transmit radio types of tracking equipment can be used. Large phased and/or angular large steerable information, however dish both antennas of these can be used to are extremely expensive. Conversely range and range rate measurements are r e l a t i v e l y inexpensive. To measure range one only has to accurately measure a signal's round t r i p t r a v e l time to and from the s a t e l l i t e . The accuracy is determined centimeters by the time are possible resolution using possible. sophisticated Accuracies laser of fifty techniques [7]. Inexpensive equipment could resolve to about one kilometer without major problems. Range-rate Doppler s h i f t measurements are accomplished by measuring the caused by s a t e l l i t e motion. Again r e l a t i v e l y equipment can resolve to about three meters per second. inexpensive 4 4_. B a s i c P r o b l e m S t u d i e d Given that inexpensive radar range while and angular were very expensive, range-rate were sufficient standpoint, the question system were not required range position either the arose required question for orbit was: alone? for the was the as optical to system relatively methods or range and whether From the control observable?". m i n i m u m number of If the measurements a n d how much c o u l d b e a c c o m p l i s h e d u s i n g Were satellite c o u l d be d e v e l o p e d t o a n s w e r by were determination. "Is what observable, range-rate measurements measurements observable, t o make i t and range-rate angular tracking? the above measurements And f i n a l l y , of satellite "What technique questions?" 5_. M e t h o d The f i r s t observability mentioned problem encountered is earlier non-linear must system be and y e t the Simulation a such high sophsticated model of o r b i t If offers that if that Escobal determination, we c o n s i d e r a c h o i c e between parameter estimation. algorithms obtained least from l e a s t not program can only to number squares are alternative. An the The not of system, algorithm a problem, expected the shown that essentially and classical d e t e r m i n i s t i c and do contaminated As for noise Kalman f i l t e r i n g be determine arduous observe t h e measurement problem of it extremely complexity i s a to observability? observable. however a l l are squares, Since is fails [1] t r e a t s the show the it probability, attempting one approach enough t o h a n d l e used. is How d o e s theoretical chosen to method. r e l a t i v e l y s i m p l e so allow noise. have the be is, system methods of systems. a l g o r i t h m must the one when one not m e a s u r e m e n t s we and some the the form of sequential same as those 5 obtained from Kalman problem makes filter was filter is filtering parameter chosen as estimation the basic to estimate). As e s t i m a t e s of worse, measurements, are the the the on Kalman filter. Kalman future method filter measurements. error measurements affect The since the particularly examination state decided based in the the state weighted estimates. derived therefore smoothers to a and III of basic Kalman they system model get estimates get various have expands writing arrive covariance therefore further the Kalman as state of the (error As t h e consists test The weighted from lower Chapter difficult, estimate weighted higher. are on state non-linearity tool- an a l g o r i t h m whereby measurements a r e according better, [8], a n d a on number combinations less the of of 6 CHAPTER II THE SYSTEM MODEL _1. L i m i t a t i o n s State into earth of the account art many factors gravitational atmospheric classical drag, orbit of earthshine effects observability, p r o b l e m s and and third order. lunar and solar gravity, etc add causing the location on the Building of the measurements The m o d e l little a were earth. The University intermediate l a t i t u d e . of that obtained with used to complexity would station assumed to location of British 50 station deviation being assumed t h r o u g h o u t that sent a with coordinates tailored the satellite known i n mind t h a t the delay, This geocentric system avoids system the use of of was the study wind, is the secondary and question of will the so hold basic it same McLeod of was most known other exactly. measurement. with an believed on a l l measurements. type it is for be i s a c t i v e and w i l l and the represents true to from Hector This degrees, assumed noise standard 2. The are to made Columbia. 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 technique include obscure been chosen approximately this have Noise with it take solar of which [5], These this basic The to in the latitudes. standard as effects. All results such f o r c e m o d e l . The i n c l u s i o n would while programs second anomalies, Newtonian c e n t r a l tertiary determination It The is return a signal this d e v i a t i o n s were measurement chosen. Units units large is used numbers throughout and is a this more thesis. "natural" 7 system. It s y s t e m has uses as units its unit defined of by distance E.R. is equal to 6 , 3 7 8 . 15 k m . I t s and its unit of mass constant (k) i n the is the geocentric the system the radius unit mass of of 3. of time the system i s itself. the is The earth the earth. (E.R.). mean The geocentric One solar day, gravitational 107.0867 E . R . ^ ^ ' / D a y . Coordinates 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 System Measurements satellite's position coordinate system. its the origin from are This a station on referenced is a the initially rotating observing station, see surface to of the coordinate Figure the earth of a azimuth-elevation system which has as at the II.1. North A z i m u t h - E l e v a t i o n Coordinate System. Figure I I . 1 The f u n d a m e n t a l plane is d e f i n e d as b e i n g t a n g e n t i a l to the earth 8 observer. This fundamental means plane. that The the local positive X direction 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 straight this up. Two a n g l e s system are the angle between plane perpendicular measured define elevation the satellite therefore and the the plane. from North i n the fundamental plane of the the satellite fundamental from the The r i g h t fixed does not the equator Its rotate equinox, two simply the ways declination angle to specify system center the to the measured projection of declination is taken earth. The positive II.2. The as X define the a The second range. location the the the i n a plane normal to to is The be to the radial is plane vector angle between the equator. the onto is the in the a angle of the simply the radial. simply The is the up the the right the the it plane of the vernal through the There first is system used ascension, distance from to in the the ascension is the X axis and the fundamental l o c a t i o n and but system. The right between the is handed is the earth, toward system. This an i n e r t i a l o r the plane right this specified. fundamental of points specify range is in is projection system i s Z direction X , Y , and Z c o o r d i n a t e s . the in in satellite a l o n g the center direction location due measured Range fundamental positive being System the Y d i r e c t i o n completes model. and is the the Figure N o r t h p o l e and are with and see center the o r i g i n measured ascension d e c l i n a t i o n coordinate system. the The a z i m u t h to the and p o s i t i v e Z i s plane plane. 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 as to The e l e v a t i o n fundamental fundamental onto defined angles. the radial normal d i r e c t i o n of azimuth and is is due E a s t to satellite distance which vertical plane. equator The measured 9 Z North Pole Right Ascension Declination Coordinate Figure 3.3 O r b i t a l It which is Plane Coordinate o f t e n much e a s i e r takes satellite. as Such perturbations. and the have as the center is the of importance. will at This the epoch The angles the the orbital inertial however, some is in gradually earth. the plane, fixed system d r i f t plane of One 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 Realistically, plane II.2 System fundamental system coordinate orbital perifocus. some a orbital the the its System. there drift. space are time. positive defined in eccentric The X this if it plane center axis is no annoying this points (E). are usually coordinate anomaly the perturbations is of of there always Since fundamental plane system See taken system towards system Figure to is the is of II. 3 10 Orbital Plane Coordinate Figure and F i g u r e I I . 4 . plane from actual circle. the ellipse It is X axis of defined to motion the as measured II-3 the point System O b l i q u e View angle on a at the measured circle center that of i n the fundamental circumscribes the the circumscribed 11 Orbital Plane Coordinate System P e r p e n d i c u l a r Figure View. II.4 h_. S t a t e M o d e l The s y s t e m m o d e l three of reference states frame were inverse was determined neglected square is the as to on be six these the irrelavant. l a w . The s a t e l l i t e (x]_, X2, X 3 , X 4 , X5, X 6 ) xi on h a s v e l o c i t y . A model based etc. Newton's decided T three X direction x£ i s the v e l o c i t y i n the Y direction X3 i s the v e l o c i t y i n the Z direction X4 i s the i n the three states simplest The m o d e l X direction of position inertial available. Attitude is in and an state vector where: v e l o c i t y i n the displacement states, based X is solely on defined as 12 X5 i s the displacement i n the Y d i r e c t i o n , and xg i s the displacement i n the Z direction. The s t a t e e q u a t i o n c a n be s e e n i n T a b l e II.1. - — -k X, -k X r -k X, K • State Equation. Table II.1 5_. M e a s u r e m e n t There They are four types of Model measurements considered in this thesis. are: Range: The distance from the observing o b t a i n e d by m e a s u r i n g round t r i p s i g n a l station times. to the satellite, 13 Range-Rate: receding The along speed the with line of which the satellite sight, obtained by to satellite is approaching measuring or the Doppler the azimuth i n the azimuth shift. Azimuth: The elevation coordinate Elevation: elevation azimuth angle The e q u a t i o n s these be found i n T a b l e II.2. numbers before noise with a in Gaussian the satellite the i n terms measurements distribution The s t a n d a r d Range It to measurements present they were u s e d . Rate were added to 0.01 radians Elevation 0.01 radians. measurement accuracy. these values used states modeled. the can Random measurements were E.R. Azimuth that system also E.R./Day felt the was 0.04 was measured of deviations 0.0001 Range in system. for The measured system. The e l e v a t i o n a n g l e coordinate the represented an easily achievable 14 Range [ < x - s ) + (X 2 |p| 4 4 5 • Range-Rate - v 2 + <v v 2 ] 1 / 2 (x -s )(x -s )+(x -s )(x -s )+(x -s )(x -s ) 4 |p| 4 1 1 5 5 2 2 6 6 IPI E l e v a t i o n Angle (h) , -1 sin p-R |>l Azimuth Angle (A) cos -1 /•V p-N i iJ m symbol definitions N S= S t a t i o n S t a t e V e c t o r R= S l S 2 S 3 S 4 S 5 S 6 X velocity Y velocity Z velocity X displacement Y displacement Z displacement Station Position Vector R l through R„ = S. t h r o u g h 3 4 S. o N= V e c t o r p o i n t i n g due N o r t h i n tangent to the p= Station to s a t e l l i t e p= P r o j e c t i o n of to the Measurement Table earth earth p at Equations. II.2 at the plane station position on p l a n e station. vector tangent 3 3 15 CHAPTER III KALMAN F I L T E R I N G AND SMOOTHING 1_. L i n e a r A which filter, in estimates measurements the It statistical sense. estimation state corrupted algorithm. Table the at noise. The by the linear Filter sense vector minimizes The Kalman of the the word, current linear estimation time Kalman error Kalman f i l t e r is an based filter in a equations algorithm are upon is well past such an defined presented in III.l. System Model Measurement Model V State Estimate v- Error Extrapolation State Estimate Covariance Kalman G a i n V Update filter is (+) extended Kalman 0 + ) = -i k-i p k \ ( [i (+)0 k-i T+ k v kV- ) + K [ H \-i ) ] - wk<-> p v v-Vt kV-V v" H Matrix applicable ) = v->- Update Kalman F i l t e r Table the k vA-i ) = V- Linear This v ~N(0,R) Covariance Extrapolation Error V W only filter. to + 1 Equations. III.l linear systems but forms the basis of 16 2. E x t e n d e d K a l m a n In linear used. and order state It to be able use the estimation problems, is basically measurement the equations The e x t e n d e d K a l m a n f i l t e r filter to same as power of the the linearized equations Filter the Extended linear about are Kalman filter the filter but with current best presented f o r m s t h e b a s i s o f o u r a t t a c k on t h e Kalman f i l t e r on n o n must the be state estimate. i n Table III.2. This orbit o b s e r v a b i l i t y problem. 17 System Model Measurement State X(t)= f ( x ( t ) , t ) + w(t): V V k x(t Model \ ))+ w(t)~N(0,Q(t)) v k~ 'V N(0 Estimate X(t)= f(X(t),t) Propagation Error Covariance P(t)= F(X(t),t)P(t) + P(t)F (X(t),t)+ T Q(t) Propagation State Estimate V Update Error V + ) = W _ ) + VV- ) ) ] Covariance P (+)= [I- K H (X (-))]P (-) k k k k k Update v v-> k v-»t H Gain M a t r i x P where vv-» T( k ->\\ -» V"' ( ( + df(X(t),t) F(X(t),t)= ,and Sx(t) 5h (X(t )) H (X (-))= k k k k dx(t ) k Extended Kalman F i l t e r Table Equations. III.2 _3. L i n e a r i z e d K a l m a n F i l t e r If extended instead Kalman of linearizing filter, the about state is the best estimate linearized about as some in the known 18 trajectory filter. the filter This filter the becomes what works well state vector. perturbation in equations are presented i n Table X(t)= System Model Measurement State Model is known in reducing The the linearized error due linearized Kalman to Kalman III.3. f ( X ( t ) ,)+ w ( t ) z = h (X(t ))+ v k as k k k ; w(t)~N(0,Q(t)) ; V N(o,y Estimate X(t)= f(X(t),t)+ F(X(t),t)[X(t)- X(t)] Propagation State Estimate v Update v- +)= VW^k^ )+ -H (X(t ))[X (-)k Error k k X(t )]] k Covariance P(t)= F(X(t),t)P(t)+ P(t)F (X(t),t)+ T Propogation Error Covariance Update Gain Matrix H where F(X(t),t)= df(X(t),t) T k (X(t )) k , 3x(t) H (X(t ))= k k dh (X(t )) k k f)x(t ) k L i n e a r i z e d Kalman F i l t e r Table III.3 Equations. and + Q(t) small filter 19 _4. S m o o t h i n g Smoothing measurements during the optimal differs from from an e n t i r e pass. In smoother is data pass the a filtering the filtering backwards to filtering to linear linear i n the that it the state estimate case it can combination of forward "t". in direction The e q u a t i o n s to for uses at be shown two Kalman time this x(t|T)=P(t|T) [P- (t)x(t)+P - (t) I it "t", all the some t i m e "t" [2] that the filters, one and smoother the other are x (t)] I b b p- (t|T)=p- (t)+P - (t) 1 1 1 b where the subscript interested initial in error covariance after a function If the true initial covariance the the state, is data has very we assume that forward i n the n o n - l i n e a r the backwards estimate. ie. and t=0, large with been p r o c e s s e d , o n l y of the backwards combination of the hold b indicates we If assume respect then the we a r e to only that the initial our error state is filter. nonlinear smoother and b a c k w a r d f i l t e r s , case- is also t h e n t h e same a linear equations 20 CHAPTER IV SIMULATION PROCEDURES 1_. G e n e r a t i n g The M e a s u r e m e n t 1.1 A l g o r i t h m U s e d To G e n e r a t e Actual tracking different types measurement d a t a was an orbit and data of the position From t h i s known p o s i t i o n and appeared to for a l g o r i t h m are Step this 1: The They was therefore satellite an earth and listed orbital at to and obtain orbit which velocity, station to as calculated. i n T a b l e I V . 1 and T a b l e parameters are the read many Simulated by assuming calculate measurement measurements were to obtained follows appropriate due required. T h i s d a t a was algorithm velocity they The the intervals. would have equations II.2. into the computer. are: i Inclination Longitude of Argument of the the A s c e n d i n g Node . . . L Perigee P Eccentricity e Semi-major a axis Time of P e r i f o c a l 2: impossible used. tracking desired Orbital Step Data measurement using Data Calculate orbital the plane passage direction coordinate cosines system to for the converting ascension from the declination Direction Cosines P^= c o s ( p ) c o s ( L ) - sin(p)sin(L)cos(i) Py= c o s ( p ) s i n ( L ) + sin(p)cos(L)cos(i) P = sin(p)sin(i) Q^= - s i n ( p ) c o s ( L ) - cos(p)sin(L)cos(i) Q = -sin(p)sin(L)+ cos (p)cos (L)cos ( i ) Q = z cos(p)sin(i) . 1 / 2 /. a Mean M o t i o n n= k u Mean A n o m a l y M= n ( t - T 0 P 3/2 ) -, 0 E c c e n t r i c Anomaly E= M+ iS" - J (me)sin(mM) —: m m m=l State in Orbital E= plane coordinates 1 nr a X = a(cos(E)- e) u Y^= a ( l X_= e ) 2 e ) 2 u) Asc. Dec. Cords. sin(E) -aEsin(E) Y = aE(lState i n Right 1 / 2 1 / 2 cos(E) p = X•P+ Y•Q [* J [ x P * L Z P J y Q Q x] y W N P Q _ z zj Satellite State Calculation Table IV.1 22 system. Step 3: Compute time since Step 4: Compute mean Step 5: Compute Eccentric Step 6: Calculate perifocal anomaly. anomaly. coordinates coordinate 7: Calculate Step 8: Calculate velocities 9: of the satellite in the orbital right ascension declination plane system. Step Step passage. coordinates Calculate in in orbital velocities in plane the coordinate right system. system. ascension declination system. Step 10: Calculate range, ground s t a t i o n Step 11: If elevation satellite is calculated Step 12: Increment Step 13: Go t o building I V . 2. at Typical 1.2 T y p e s o f In order many and elevation than degree from angle not is greater hidden behind satellite one earth) store the (ie. if the measurements state. time. 3. coordinates U n i v e r s i t y of measurement used British curves are those Columbia c a n be seen of and the are H e c t o r MacLeod listed in Table i n Figure IV.1 . Orbit to account smoothing a l g o r i t h m s , the azimuth, satellite. current station the to and t h e step The e a r t h range-rate, different for a set types of possible of s t a n d a r d orbit orbit orbits currently type dependancy by was d e v e l o p e d w h i c h in use. Table IV.3 the cover lists H O 0 —.r>CC O CD CD ceo CD CO C • (D 0 n> 3 CD © CD o 1 40.0 0.0 n CD LU<R. CD CD 1-1 CD CD CD LD CD 1 cn CD tt) CD 1 80.0 1 120.0 TIME (SEC.) CD CD CD CD CD CC.,. O CD CD LU<=" UJ O CD 1 ,160.0 200.0 1 240.0 CD CD CD 1 40.0 0.0 1 80.0 (X10 ) 1 1 120.0 1 ,160.0 1 200.0 1 240.0 1 200.0 1 240.0 TIME (SEC.) (X10 ) 1 *1 (W c CD CD cp < CD CD • CD CD CD a CD o cc CD CD ce-d- CD CC(n DEC • CD CD CD CD CD cc-v .CM CD 40.0 —1 80.0 CD cc CD o.o CD CD _Jo" CD CD CD o CD CD CD CD OfT) 120.0 TIME (SEC.) (X10 ,160.0 1 ) 200.0 n 2 4 0 0 0.0 1 40.0 1 80.0 1 120.0 1 ,160.0 TIME (SEC.) (X10 1 ) CD 24 East Longitude 236.75° Latitude 49° 15' Altitude 310 Coordinates 45" feet of H e c t o r McLeod Bldg. Table I V . 2 the orbits number will was used will and appear indicate that assigns to on figures many orbital each data for a reference to follow. the orbit number. For with This reference instance "ORBIT.3" reference number 3 used. Among orbits, the polar orbits orbits, a good c r o s s - s e c t i o n in close of the Table IV.3 orbits, are circular and d i s t a n t i n f i n i t e number orbits, orbits. These eccentric represent of o r b i t a l p o s s i b i l i t i e s . No. I n c l i n a t i o n Ascending Argument of Semi-major E c c e n t r i c i t y Time of P e r i node perigee focal passage axis 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 0° 90° 0° 6.6227 0. 2444055.5 45° 0° 0° 1.5 0. 2444055.5 * 5 6 * geosynchronous o r b i t . 26 2. In epoch this study time. This predictionof the results can, the estimate by r e p e t i t i v e next chapter estimate error against are i n the was Due section, filter time of the reversed, sweeps be the will can estimate for chosen to non-linear be be shown made to at some future be orbit the nature computed d i r e c t l y . o b t a i n e d from the computer in A the time of the repetitive simulation approach the correct the s i m u l a t i o n runs d e s c r i b e d i n produced smoother. and could These in state initial the sweeps estimate t=0 used t i m e was to It state of velocity calculated and s a v e d , were measurement the error T h e s e show t h e the be states plots then data estimates the read when the of plotted satellite by the magnitudes because and in the state simulation compared with the estimates. the algorithm. be d a t a was g e n e r a t e d with the somewhere used. errors saved. to for then Due p o s i t i o n estimates together a good epoch cannot at the These also smoother's the c a l c u l a t e d by the time. program used could at filtering. result s i m u l a t e d measurement vector be Algorithm arrive (t=0). however, The p r i n c i p a l the estimate measurement that estimate state to an o p t i m a l e s t i m a t e algorithm the we w i s h For our purposes, first problem, Simulation initialization closest guess. middle of foreward in When the final becomes the initial and the error backwards i n time satellite The the using measurement state the first data. the is estimate, is the re-initialized. measurement The u s i n g the the filter final d i r e c t i o n of is starts point Kalman the next station this extended the the normally From reached, in earth therefore, measurement time described approach to smoother, covariance to procedure state sweep is filter then same data. 27 This procedure The arrows direction During repeats i t s e l f of all present on sweep when t h e these sweeps and computed, the p o s i t i o n e r r o r the of the square errors. root of measurement that estimated to is the state computed. which the are then p l o t t e d the the abscissa root and the quantities of the the state always the are magnitudes of the are sum o f speed e r r o r which three velocity on two s e p a r a t e smoothing a l g o r i t h m r e s u l t s . is the calculated. true error square of indicate a r r o w was Two squares data. graphs and the measure of the following sum o f time since the is state graphs, It should the first of the pass - The a l g o r i t h m u s e d more d e t a i l the curve nearest difference the These e r r o r noted of three p o s i t i o n state errors; and a p e r f o r m a n c e be two more sweeps most the compared squares their for i n Table numbers o n t h e typical i n the s i m u l a t i o n runs I V . 4 . The result step numbers is described i n correspond shown i n F i g u r e I V . 2 . to the somewhat circled 28 Initialize Time 1 and (T^) Kalman F i l t e r using Initial at Minimum Range computed i n i t i a l guess Error Covariance Use E x t e n d e d K a l m a n F i l t e r T ' and m Kalman F i l t e r state estimate 3 on d a t a F i n a l Measurement Time Re-initialize (X(T )) f and n t (P. . ) . init X = X. o init P = P. .„ o init 2 (X^ ^ ) between (T,.) . r at T^ u s i n g Initial final Error Co- variance. X = X(T ) Q f P = P. . o init Use E x t e n d e d Kalman F i l t e r A o n d a t a b e t w e e n T^ a n d backwards First in Measurement time Time (T ). s Re-initialize Kalman F i l t e r at T using g 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 5 Initial Error final and Covariance. X = X(T ) o s P = P . .„ o init 6 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 Re-initialize 7 X ( p = p, o Simulation and in T^ and at T Initial f using Error V .„ inxt Algorithm time T,.. f Kalman F i l t e r s t a t e e s t i m a t e at V s forward (Continued next page) latest Covariance. 29 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 8 on d a t a between Final 9 Estimate State Estimate orbit in time T,. a n d T . f s E r r o r at at T prediction. s T . s c a n be u s e d for future S i m u l a t i o n A l g o r i t h m (Continued from p r e v i o u s page) Table IV.2 o a ONE STN 3 NNTS — Simulation Orbit Type Initialization Used 0RB]T.6 TRY A o.o Type 10.0 20.0 TIME Typical 30.0 (SEC.) ,40.0 (X10 1 S i m u l a t i o n Run R e s u l t Figure IV.2 ) 50.0 60.0 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 important state that consideration. estimate apparent that the state was the this these guess. different quite g u e s s was filter initial guesses s t a t e to a l l o w the Step 1: Range Step 2: This data was satellite time at Step 3: which the initial Velocities circular follow critical due uses therefore to and was above initial soon became to the developed close to the guess. The accuracy of calculate enough fact of first on t h e an to the four correct converge. the used this be the approximations depended was to minimum r a n g e as the an found. altitude measuring to position the station. The was made was used earth minimum r a n g e measurement time. were computed orbit this i n some c a s e s l i n e a r i z e d about scanned study it o f w h i c h was directly the out however smoother minimum r a n g e of turned arbitrarily, algorithm one initial beginning initially functions following the the approximations The as smoothing algorithms Near Kalman and measurement of the made this extended validity of in one initial which of would four put the directions. direction will be satellite In indicated into figures by a to the "TRY" number. TRY 1 i s due N o r t h TRY 2 i s due South TRY 3 i s due East TRY 4 i s due West This algorithm except the was used linearized The i n i t i a l value to initialize all the smoothing algorithms smoother. of the error covariance was arrived at by trial 31 and error, throughout but once all the reinitialize the error for velocity error the became reasonable simulation covariance convergence 0.1 the a runs. It covariance after became very variance value slow. of state variances. the so small The A l l other been found was also found each one error states or it a run two covariance and to c o v a r i a n c e s were s e t was necessary sweep w i t h i n after initial position had 100.0 to to because sweeps was used set for zero. that to the 32 CHAPTER V SIMULATION RESULTS The p l o t s The reader format to f o l l o w represent is and referred to the the results previous chapter Extended calculate data Kalman f i l t e r one station. would greatly large would accurate from the doppler arrays, would timer be for error have and explanation of using w r i t t e n w h i c h was to The an of tracking large lower range such a equipment. Even a trip accurate steerable equipment travel dish frequency sufficient. antennas Also no consist of signal a counter for or optical would time rate feasible, frequencies. required round if and i f s i g n a l s t r e n g t h s were for at o n l y range stated, cost need was for any f r o m F i g u r e V . 1 and F i g u r e V . 2 t h e set increase Such or no distance increase of initial linearily behavior effect between t r u e p o s i t i o n w o u l d be would an of to an and measuring shift. little case the the least needed. satellite, would approach. the at smoother previously measuring As c a n be s e e n converge As c o u l d be u s e d do away w i t h phased equipment for study. F r o m One S t a t i o n estimate reduce s i m p l e d i p o l e antenna This based a preliminary orbit from program simulation titles. J_. R a n g e a n d R a n g e - r a t e An of t h i s can be on t h e the least approximately conditions. with time error assumed when t h e y linearly in Typically from understood by the the initial as the nearest they the point assuming p o s i t i o n of were smoother the diverged. nearest filter In satellite the not position of guess. to did to such a and the station and Such behavior X*A a^riSx^ a^B-g - aSuB^ put? aSuB-g joaag u o T 5 T d s o ONE. STN 2 HNT5 ORBIT. 1 TRY 2 cc o CC cc o 0.0 40.0 80.0 i 120.0 r ,160.0 TIME (SEC.) (XlO &) 200.0 0.0 240.0 I 40.0 ) 1 BO.O 1 120.0 1 I 200.0 ,160.0 TIME ISEC.) IX10 ) -1 240.0 1 3 OQ (D ONE. STN 2 HNTS ORBIT. 1 TRY 0Q C CC CC o CC CC UJ o o UJ > 0.0 40.0 80.0 T 120.0 —I r ,160.0 TIME (SEC.) (X10 1 ) 200. 240.0 0.0 40.0 80.0 1 120.0 1 ,160.0 TIME (SEC.) (X10 1 ) 1 200.0 1 240.0 ; 35 by the filter can be assumed to indicate that the system is unobservable. Other types o f o r b i t were t r i e d w i t h the same result. _2. R a n g e , R a n g e - R a t e a n d 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 of if the system is the to increasing the additional measurement second i s This dealt with be made measurement to use section measurement the deals of with i n section the type be There dimension. different same t y p e must observable. vector a vector The from of these the angle was Table I V . 3. occurrence Every during algorithm's show the this case. where Figure initializations Figure V . 10 next measurement study two or the angle were of is same to add an station. as will 2. Only Figure The station. other is 5, Figure sections, a second 2 the and in be V.8 Try show to 4 the that the station was Figure results addition of to With in It an obtain Orbit rare the Figure V.6 converged in Orbit Orbit 3, 4 two V.9 and to be proved be 1 to Figure will of a for orbit ) the later, 2 have orbits case, Try shown V . 12 required of V . 5 and converge. as would the discussed geosynchronous V . 11 a n d all this Figure measurement. and results initialization only i n i t i a l i z a t i o n Try tried using converged due, a third was made initialization. V . 7 and Orbit (See to Orbit worked, . unobservable. the for ways two p o s s i b i l i t i e s , the F i g u r e V . 4 show t h e initialization sensitivity 4 was elevation S i m u l a t i o n runs this results Try but F i g u r e V . 3 and simulation. are 4. chosen worked e q u a l l y w e l l . two add a second 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 Azimuth from first of measurements but first increased shown, elevation convergence in angle with Orbit 1 Position Error with 3 Measurements Figure V.3 0.0 o POSITION ERROR (E.R.) 0.04 0.08 0.12 0.16 40 o o T I g-0 1 1 fr'D 9'0 ca'3) 1 Ha O'O 2'0 yoda3 NQIJLISQCI o a I 8 0 - 1 1 9'D t>' 0 ra'3) Orbit 3 I Z'D yoaa3 NoniGOd Position Errorm To O'O 9"I Z'l ca'3) w i t h 3 Measurements 8'0 t o aoaa3 N Q i n s o d Figure V.7 O'O 42 P'O 8'C NOIilSOd UJ CO —1 8'0 g-D 1 P'Q 1 2'D c a ' 3 ) yoaa3 NQiiisod Orbit.4 Position Error with r 0" 0 3 Measurements 8'D ?/7 Cd'31 fr'O clQeJcG N O I i l S O d Figure V.9 O'OST O'OOT (Aba/'a'3) Orbit D'OS O'O aoaa3 UI3Q13A 4 Velocity' Error with o'oee T O'Ot^e 1 0'097 ( j L d a / ' a ' 3 i aoaa3 3 Measurements Figure V-10 ~ r 0'08 O'O untrraA 45 IT 09 (Ada/ trot- D'oe '3) yOdcJ3 XI13013A tro r T~ 0'08 D 09 - D"Ot> O'Oc ( A u a / ' d ' 3 ) yoyy3 UIDQIBA 0"Q CD ID(_) UJ CO • z in >— ' <o in i UJ 5 >- Z OS Q£ o o t- a'091 0'08 (jLtia/ Orbit 5 D'Dt' O'O y-3) yoyy3 UIOCTGA Velocity Error with 1 Q-027 1 uuavy3) 3 Measurements r 0'08 Figure 0'0t> aoya3 ui3cn3A V.12 O'O LU I 46 this orbit. Finally, the results with Orbit 6 are shown i n F i g u r e V . 13 and F i g u r e V . 1 4 . As can be seen number o f sweeps orbit in the b e l i e f was a and t h e not the these figures, initialization used. degree The number The r a t e degree of than smoother systems a convergence of of sweeps o c c u r r e d by t h e n , For n o n - l i n e a r reather the required r a t e of convergence v a r i e d w i t h i f convergence had not of systems. all t o c o n v e r g e and t h e satisfactory. matter of from yes these or was fixed as a gives a crude satellite, one to use. In would the c h o i c e c a n o n l y b e made between the provide a e s t i m a t e d measurements basis for an of those of estimate Orbit the 2. The other s h o u l d be need residuals no p r i o r of linear measure tried. and t h e actual ones choice. Table residuals for of Try 2 the are indicating sort selecting i n f o r m a t i o n at been intelligent the of method a l l have initializations, used. some presence after T a b l e V . 3 , and T a b l e V . 4 l i s t tries be of o b s e r v a b i l i t y . initialization the to with I n p r a c t i c e i f one w e r e u s i n g a s m o o t h i n g a l g o r i t h m o f t h i s track 4 the o b s e r v a b i l i t y choice figures the at o b s e r v a b i l i t y seems no a the all, The differences (the residuals), V.1 , four Table V.2 , initialization somewhat that to smaller Try 2's than state 47 80'0 t>0'C ££'0 9T0 fc77-Q {•a'!) • y 3 ) aoaa3 Nouisod - T t>'D E 0 - Orbit a'3) 6 Z'D VO O'O 9T0 27'0 ra'3) T ~ 80'0 aoaai aoaa3 Nouisod Position Error with 3 Measurements 80'0 O'O yoaa^ Noiiisod Figure V.13 t>0'0 Nouisod O'O ^T'A VELOCITY ajnSxj ERROR s a u a m a j n s B a f l £ H^TM j o j j g VELOCITY (E.R./DATJ 0.0 iCaxooxaA ERROR 20.0 l 9 ^T^an (E.R./DRY) 10.0 I (XlO ) 1 60.0 I —I Q =C 50 80.0 I Q 2! - t co m ^ w —' u> —\ • m o VELOCITY 0.0 80.0 ERROR (E R . / D R T ) 160.0 240.0 320.0 J VELOCITY 0.0 40.0 ERROR (E.R./DRY) 80.0 120.0 160.0 H Q a » n z - c cu n 8^ Range Range-Rate Azimuth -.8381E+00 -.3392E+03 -.4433E+00 -.1307E+00 -.9203E+02 -.2465E+00 -.2586E+00 0.1826E+03 0.1101E+01 --2416E+00 0.1772E+03 -.2970E+00 -.4715E-01 0.2386E+02 0.3150E+01 -.3801E-01 0.3641E+02 0.1301E+01 -.2996E-01 0.4186E+02 0.3966E+00 --2395E-01 0-1821E+02 0.8938E-01 --2865E-01 -.6040E+02 0.3589E+00 -.4230E-01 -.5620E+02 0.9020E-01 0.1549E-02 0.4509E+01 - . 1848E-01 0.1087E-02 0.4037E+01 -.1108E+00 0.3077E-03 0.5349E+01 -.1324E+00 0.4264E-03 0.4489E+01 -.7383E-01 0.1085E-03 0.2085E+01 -.2640E-01 0.1417E-04 -.1507E+01 0.1789E-01 -.5115E-04 -.2568E+00 0.4123E-01 -.1020E-02 0-1424E+01 0.2616E-02 --4421E-03 0.7822E+00 -.8096E-01 -.8899E-03 0.6302E+00 --7399E-02 -.3457E-03 0.6076E+00 0.1353E-01 0.5664E-04 0.2649E+00 -.1442E-01 ORBIT 2 R e s i d u a l s TRY Table V.1 Range Range-Rate Azimuth -.1140E-01 --3845E+01 -.3849E-01 -.2196E-02 - . 1052E+01 0.3125E-01 - . 1321E-02 0.4325E+01 -.2515E-01 -.1291E-02 0.2932E+01 -.2887E+00 -.2651E-02 0.2207E+00 -.1529E+00 --3584E-02 0.3131E+01 0.3009E-01 - . 7135E-03 0.1499E+01 -.9842E-02 -.5560E-03 0.1103E+01 0.7740E-02 - . 1650E-03 -.1209E+01 0.1940E-01 -.6281E-03 -.9997E+00 -.1839E-01 - . 1056E-02 -.3440E+00 -.9710E-02 -.1800E-03 0.1137E+01 -.7980E-01 - . 1100E-02 -.2760E-01 0.6004E-01 -.1069E-02 -.1314E+01 0.3871E-02 - . 1540E-03 -.5402E+00 -.1847E-01 -.9338E-04 0.1679E+00 -.2359E-02 -.3574E-03 0.1502E+00 --3136E-02 0.2282E-03 -.1033E+00 0.9054E-02 - . 1222E-04 -.2392E+00 0.1408E-02 0.1975E-03 -.8041E-01 -.7714E-02 -.1593E-03 0.2514E-01 -.2481E-01 0.1338E-04 0.2932E+00 -.1415E-01 ORBIT 2 R e s i d u a l s TRY Table V.2 Range Range-Rate Azimuth --6224E+00 -.2423E+03 --2953E-01 -.1044E+00 -.1114E+03 0.6841E-01 -.6729E+00 0.1362E+03 0.3769E-01 -.6471E-01 0.1434E+02 0.1522E+00 -.6395E-02 -.3907E+01 0.8628E-01 0.1994E-02 -.5798E+01 0.1038E+00 0.1118E-02 -.4745E+01 0.4725E-01 0.6013E-03 -.2817E+01 0.4790E-01 -.3729E-03 0.1578E+01 -.2231E-01 -.2616E-03 -.8364E+00 -.9740E-01 -.4843E-02 -.8827E+01 --1529E-01 -.3760E-03 0.1217E+01 0.1386E+00 -.1141E-02 -.7937E+00 0.8082E-01 -.1053E-02 -.3459E+00 -.2295E-01 0.8400E-04 -.1448E+00 0.4786E-02 -.7330E-04 0.7654E-01 0.1570E-01 -.1714E-03 -.2121E+00 -.1897E-01 -.2312E-02 0.3295E+01 -.2960E-01 -.9902E-03 0.9765E+00 -.2760E-02 -.3928E-03 0.1101E+01 -.2289E-01 -.3831E-03 0.8845E+00 -.5726E-02 -.8311E-04 0.4567E+00 0.7893E-02 ORBIT 2 R e s i d u a l s TRY Table V.3 Range Range-Rate Azimuth -.8889E-01 -.5327E+02 -.1297E+00 -.2735E-01 -.2061E+02 0.7395E-01 -.3638E-01 0.4473E+02 0.6343E+00 -.2051E-01 0.8351E+01 0.2925E+00 -.5617E-03 -.2068E+00 0.5272E+01 3280E+00 0.1188E+03 0.4274E+01 -.3662E+00 -.6786E+02 0.3332E+01 -.4257E-01 0.7297E+01 0.3313E+01 -.2076E+00 -.1252E+03 0.2214E+01 -.1486E+01 -.8053E+02 0.7935E+00 -.1354E+01 -.5313E+03 -.4740E+00 --4716E+00 -.4272E+03 0.3740E+01 -.8421E-01 -.1687E+02 0.1944E+01 -.1945E+00 0.4245E+02 0.6598E+00 -.6867E-01 -.1672E+02 0-4191E+00 -.3197E-01 0.2982E+02 0.4007E-01 -.3238E-01 0.2354E+02 -.7951E-01 1511E-01 0.1269E+02 0-3247E+00 -.7205E-02 -.6797E+01 0.2604E+00 -.2075E-03 -.7731E+01 -.2234E-01 -.3088E-03 -.3620E+01 - . 1230E-01 0.6582E-03 -.1663E+01 -.2828E-01 ORBIT 2 R e s i d u a l s TRY Table V.4 53 3^. R a n g e , R a n g e - R a t e , A z i m u t h , The smoothing vector once to if see again. any 5 converged show the results V.3 enough to with elevation four with V.4 4 types extra to angle Orbit which was 1. showed expense. and These measurement T h i s was also to V . 15 a n d can the Figure the added. Figure i n convergence o f measurement expand occurred measurements. Some i m p r o v e m e n t the modified improvement Figure justify that with was time obtained and measurements. This significant Orbit Figure algorithm and E l e v a t i o n be O r b i t 5 c a n be made if V . 16 compared with with three c a n be V . 17 a n d observe Figure results speed done seen Figure but V . 18 not show observable. 4^. R a n g e a n d R a n g e - R a t e f r o m Two S t a t i o n s The using satellite-observer range station some range-rate this range-rate distance information, situation, simulation the and system the runs from the might make using due East or 4.1 O r b i t as the the the second primary s t a t i o n , orbit encouraging Doubling this stations. The were f a r of To station located and check and remained from 0.1 orbit enough second was m o d i f i e d , primary that a range observable. sections indicated unobservable supplying system The be addition also s e c o n d was stations to to 20 determination apart. 2 Results Initially from first, Results c o u l d be a c c o m p l i s h e d i f t h e shown station. w h i l e the North. been used i n p r e v i o u s two H e c t o r McLeod B u i l d i n g , degrees one from smoother made has the under but station and Orbit observation. unsatisfactory separation to was 1 2, placed the Figure behavior degree, 1/2 close V . 19 for Figure degree polar and Try V.21 latitude orbit, Figure 1 and away was used V . 20 show initialization. Figure V . 22 , <7 C ONE S T N 4 HHTS ONE S T N 4 ORBIT. 1 ORBIT.1 TRY TRY 1 HNTS 2 >-9. a O CC UJ o 0.0 r 1 40.0 80.0 120.0 ,160.0 l 200.0 1 240.0 0.0 40.0 80.0 ,160.0 TIME (SEC.) (X10 ) TIME (SEC.) (XlO ) 200.0 -1 240.0 1 1 ONE S T N 4 120.0 HNTS ONE STN 4 ORBIT. 1 ORBIT. 1 TRY TRY 3 HNTS 4 CC a ^8CC O ceo Uj3~ 80.0 120.0 ,160.0 TIME (SEC.) (XlO ) 1 200.0 240.0 0.0 40.0 80.0 T 120.0 r ,160.0 TIME (SEC.) (X10 1 ) 200.0 240.0 Ul Orbit 5 Velocity Error with 4 Measurements , F i g u r e V.18 58 Orbit 2 Position Error 1/2° Latitude Separation Figure V.19 59 62 approximately halved unsatisfactory. degrees It latitude, c o n v e r g e n c e was In any order effect, was to stations improvement over a with a the and initialization but T r y 2 and T r y 3. Figure but was V.24 degree a Longitudinal direction V . 25 and ground , was still increased that track to 5 satisfactory T h i s phenomena V . 26 an out to with best marked could to be apart. improvement a the expect line observed than a 1 as the a had drawn satellite case. 5 degree Also, latitude, initialization changing frequently. had V . 27 and F i g u r e V . 28 , b e f o r e longitude' demonstrate perpendicular Try of degree separation I n t u i t i v e l y one turned longer station separation, case. This no separation Figure c o u l d be b e t t e r track. is of 1/2 latitude latitude s e p a r a t i o n was o b s e r v e d 3 degrees, error separation repositioned stations ground best the estimate the the were 1/2 with two e a r t h parallel if 5 degree the with V.23 Figure the satellite between the until determine shown i n that not state obtained. the to final Figure The r e s u l t s equivalent the to be decreased convergence to ceased, 0.1 degree, Figure a n d when i n c r e a s e d 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 to longer converged w h i l e Try 4 improved markedly. Convergence separation as shown until, in observability b e shown n e x t , seems to i n the limit, section is gradually 2. a f u n c t i o n of the type of the The the orbit. slow with s y s t e m becomes station decreasing totally separation d i r e c t i o n of separation station unobservable required and, as for will 2 STNS 5 ORBIT.2 TRY J 0.0 80.0 2 STNS 5 DREUT.2 TRY 3 0.0 80.0 DEE flPRRT 1 160.0 T 240.0 480.0 320.0 TIME (SEC) D E G fiPRRT 160.0 240.0 TIME (SEC) 320.0 400.0 480.0 65 Orbit 2 Position Error 1/2° Longitude Separation Figure V.25 68 69 Orbit 2 Position Error Figure 3 Longitude V.29 Separation OCA o 0.0 VELOCITY ERROR (E.R./DRY) 80.0 1 160.0 I 320.• 240.0 1 0.0 J —i TO -c o 3D °_l IU cu cn 3 m / mo o VELOCITY ERROR (E.R./DRY) 80.0 1 160.0 I 240.0 I 320.0 _J 71 4.2 Orbit 1 Results When O r b i t under 1, the observation, insufficient separation as of inclined a eccentric separation shown in 2 degrees Figure latitude orbit, of even V . 31 and showed 2 good degrees, V.36 satisfactory. latitude, as This stations distant as can latitudinal separation reduction not done f o r The be stations of the the caused the the separation sensitivity to initialization is given can be better said, the pass of the these however, to 3 fifths to the Reducing the decrease in converge, orbit. which of why that in is and beyond Increasing exist , it had of orbit and behavior this algorithm's six dimensional a from which convergence by i n c r e a s i n g d i s t a n c e s , these pass general type The e r r a t i c envisage separated therefore the underscores we shifting regions observability. due poles. to degrees three discernable with same tries If are change, can Figure became 50 about a f r o m F i g u r e V . 37 and F i g u r e V . 38 d i r e c t i o n changes to explanation shift no at latitude the was separation. stations initialization equations towards (See increased that in orbit However, convergence fact apart discontinuous regions equations theoretical the caused seen initialization. p o s s i b l e . As t h e measurement 1 degree . t o be i n l o n g i t u d e are longitude initialization space, had the longitude V . 32 before Try 1 i n i t i a l i z a t i o n 2 degree different , by degree c a n be i n some c a s e s f r o m p a s s of apart of to as ideal explained one lines In f a c t , the Figure one d e g r e e convergence performance. and as convergence. separation V . 35 used degrees Figure V . 33 a n d F i g u r e V . 34 ) L o n g i t u d i n a l Figure was regions in and how the the the station out of convergence. A of convergence. A changing scope larger the of the separation this measurement thesis. separation must be It the traded 76 r-S! r 8"0 7/1 9"! i'H'3) hQUhl *"D NOIilSOd V2 O'O c a 3) aoaa3 N o n i s o d m >• or Q£ n o H in T r 9'1 Z'l 8'0 ca 3) - Orbit t'O O'O 8'0 Error 3 *'0 Z'Q °' c a 3 ) aoaa3 NQiiisod - aoaa3 N o u i s o d 1 Position g-n Longitude Separation Figure V.35 D 8£ "A u o i i B j B d a s a p n rjT 3 B i T o 0.0 _| VELOCITY ERROR (E.R./DRY) , 40.0 1 80.0 1 120.0 1 160.0 1 0.0 H O W 50 so 0.0 VELOCITY ERROR (E.R./DRY) 80.0 160.0 240.0 320.0 o O b VELOCITY ERROR (E.R./DRY) 40.0 80.0 120.0 160.0 80 off against visible a reduced to both Simulation were found V . 39 a n d degree Orbit the to the in to V . 40 also geosynchronous results made if show station results the were converge longitude achieve the station the orbit orbit, results a Figure a 6 and a orbits was satellite with longitude Table is Figure 3 and Figure V . 42 separation. a in 2 show Orbit 5, longitude Figure V.44 . latitude I V . 3. Orbit separation V . 4 3 and 2 degree of sufficient. V . 4 1 and 20 d e g r e e shown i n F i g u r e using Orbit other obtained 2 degree required p r e v i o u s l y shown, determination station. extracted instance, fairly suited to well corrected Finally, separation are shown following calculated equal to possible and The q u e s t i o n then arose as from these correct to two for "a small upon? the were runs important the made already If so, to assumed how large it to answer these described i n the it orbit an error filter The i n i t i a l described in A fifth questions. state section trace which could was a s s u m e s a known s t a t e previous could possible, an Kalman from information Was from as orbit. measurements how much linearized differences. algorithm initially accomplish preliminary range-rate perturbations The these questions runs to s i m p l e measurements. priori"? improved using the not range to answer were s i m i l a r is only known or it Improvement using Simulation the the F i g u r e V - 4 5 and F i g u r e V . 4 6 . As be which separation separation. with results using 5_. Known O r b i t one during Results runs Figure 4's time stations. 4.3 Other O r b i t s ' All measurement for was be idealy trajectory. These runs sections with estimate was not I V . 3 but was set to the was added T8 . 82 a i- o i X i o ; m •So • 1 UJ l oo' UJ CL a in to >(M o'oee r 0'091 OS O f O t- CM Q (Ada/ o'ogi i 0*021 o os - cJ'3) AII3013A (Aua/'a'3) Orbit o'ot> o'o aoaas AU3Q13A 3 Velocity 0'021 O'O r 1 O'oe t— Q'08 (Ada/-a-3) r 0'091 -r 1 0'08 (Ada/ a 3) - Error Figure (12 V.40 - Longitude O'O r 1 0'0£1 0'0t> aoaa3 AII3013A D'Ofr aoaa3 A1I3Q13A Separation O'O T <7 - A uoprjjBdas apn^TSuo'i q £ aanST-a Jto.ua uojqiso" ^ POSnrON ERROR (E.R.) 0.0 es 0.5 1.0 1.5 3jqao POSITION ERROR (E.R.) 2.0 0.0 0.1 0.2 0.3 0.4 2< 7 •A 0.0 VELOCITY ERROR (E.R./DRY) 40.0 _i 80.0 1 120.0 —L 160.0 j 3 " 3 T I 0.0 VELOCITY ERROR (E.R./DRY) 20.0 J 40.0 I 60.0 L 80.0 J —1 • u J O TO -c ro cn 0.0 VELOCITY ERROR (E.R./DRY) 40.0 80.0 120.0 160.0 J 0.0 VELOCITY ERROR (E.R./DRY) 80.0 160.0 240.0 o.o VELOCITY ERROR (E.R./DRY) 20.0 40.0 60.0 J_ J J_ VELOCITY ERROR (E.R./DRY) ( X 1 0 ) 1 80.0 0.0 _J 40.0 1 80.0 1 120.0 I fe H O W =0 TO - 4 CO CO 160.0 I O M TO - C CO CO —t TO M B •D 33 CO m o r cn x o o to fe" VELOCITY ERROR (E.R./DRY) (XlO VELOCITY ERROR (E.R./DRY) (X10 3 0.0 40.0 80.0 120.0 160.0 _l 0.0 fe fe 20.0 40.0 1 80.0 60.0 -H TO ca TO INJ -e. CO co —t , , —i a: in " 1 3 m 33 98 m cn m x x o o o r : J 2 STNS 2 ORBIT.6 TRY 2 DEE APART 89 plots produced. without kinks, orbit. This trace, represents which the error F o u r sweeps w e r e made o f o c c u r s was u p d a t e d The obtained measurements some geometry. error of 0.05 other parameters improvement after the are i n the error accomplished sweeps E.R. is from velocity assumed error magnitude of four the the 0.4 from about sweeps of necessary, the ascending node. parts of the occurs. E.R. to improvement improvement degree. F i g u r e V . 50 a n d errors in orbits As an 3 and example p e r i o d was t o o b r i e f shows the a of or 2, the error first velocity of range-rate at of least in dependance of orbit seen E.R.to a 1- A l l definite some is Similarly 8 E.R./day. having results sweep. The p o s i t i o n e r r o r In estimate This cases been the all is four calculated of a one d e g r e e error is actually greater in than sweep a n d t h e p o s i t i o n e r r o r However, argument the third and position. the perigee sweep Figure is in brings V . 49 error 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 for by a shows one similar 4. the the simulation result assumed i n o r b i t the sweep. i n both when data. The v e l o c i t y be 0.2 24 E . R . / d a y which o b t a i n e d when a n axis can error best for half second semi-major the and degree results As each estimates, a high free. one s w e e p . F i g u r e V . 4 8 shows t h e definite the in assumed decreases not orbital about after range line uncorrected trajectory the F i g u r e V . 4 7 shows t h e the uncorrected estimate for that straight assumed estimate demonstrate approximately over were assumed and t h e to the best indicated They d i d , h o w e v e r , orbital decreased data relatively initially c o u l d be used t o i m p r o v e the cases. on t h e the i n the the linearization results is poor results number when a close polar obtained when the observation too few, Figure V-52 of measurements one degree orbit. error in inclination was CO LINEARIZED 2 MNTS QRBJT.l 300 Km Error Figure in V • 47 "A" 91 o CM" LINEARIZED 2 MNTS ORBIT.1 80.0 TIME One 120.0 (SEC) Degree Figure , 160.0 (XlO1 ) Error V.48 i n " L " 200.0 240.0 LINEARIZED 2 HNTS ORBIT.1 LINERRIZED 2 MNTS ORBIT.1 0.0 40.0 80.0 120.0 , 1B0.0 Error i n TIME (SEC) One Degree Figure (XlO 3 ) V.49 200.0 240 93 One Degree Error figure V.50 in Orbit 3 94 One Degree Error Figure in " i " V.51 Orbit 4 0.0 VELOCITY ERROR (E.R./DRY) 1.0 2.0 3.0 POSITION ERROR ( E . R . ) 4.0 0.01 0.02 0.03 o ro HTO C H (0 o :ro cm i-i ro fD w r( l-( O rt r1 3 feo feD VO Ul 96 Finally one degree satellite. the F i g u r e V . 5 3 shows error smoother axis i n the cannot done average, here, enable the correctable it as node c a n be c a n be s e e n position, improve The r e s u l t s the ascending Some i m p r o v e m e n t semi-major totally in little the o r no i m p r o v e m e n t is assumed obtained however i n F i g u r e V . 54 velocity error for already a if . o b t a i n e d when a geosynchronous the is in The i m p r o v e m e n t is being error so small the it. i n d i c a t e t h a t w h i l e some i m p r o v e m e n t c a n b e e x p e c t e d i s wise to s i m u l a t e the before applying engineer type. to it to determine any if c o r r e c t i o n procedure practical the orbit as has problem. This geometry is on been would of the 97 55 LINEARIZED 2 MNIS ORBIT.5 r\i 1 cc UJ 83 CC CC M O CO o o 0J3 I I 1B0.0 80.0 1 1 240.0 TIME (SEC) - 320 0 (X10 J 400.0 480.0 )' LINEARIZED 2 MNTS ORBIT.5 i TIME One r 1 160J) 240JD (SECJ Degree Figure 320.0 (X10 J Error V.53 1 i n"L" 400.0 480 J CO o LINERRIZED 2 HNTS ORBIT.5 0.0 80.0 160.0 1 240.0 TIME (SEC) Figure V.54 (XlO T , 320.0 3 ) 400.0 480.0 99 CHAPTER VI CONCLUSIONS A t e c h n i q u e has been d e v e l o p e d whereby the system can example been be of determined this technique, determined simulation azimuth showed angle observable the orbit bring of the the scope a for of range-rate the only as a this the and that to two the orbit thesis to the observable. also in of four the correct would change explore these this has case, alone were addition the of runs system sensitivity two ways, to first initializations estimate, equations good Similar made itself measurement given In with As a satellite Kalman f i l t e r ' s displayed or sets. station extended one an e a r t h measurements became second smoother s i m u l a t i o n runs simulation runs. measurement system sensitivity initialization beyond and the addition This of determination s i m u l a t i o n run would best range multiple o b s e r v a b i l i t y of number and d e m o n s t r a t e d one between the measurements initializaton- tried a that for that within for showed insufficient through o b s e r v a b i l i t y of a non-linear were as and second changed well. the It is instabilities in p o s s i b l e i n some c a s e s to detail. Finally, s i m u l a t i o n has i m p r o v e on o r b i t a l and range estimates r a t e measurements shown t h a t a r r i v e d at f r o m one it is by o t h e r means, station. using only range 100 REFERENCES [1] P.R. E s c o b a l , Methods & Sons, [2] A. [3] R. Orbit Determination- New Y o r k : John Wiley 1965. Gelb, Press, of Applied Optimal Estimation- Cambridge, Mass.: M.I.T. 1974. Herman and A.J. Observability", Krener, "Nonlinear IEEE T r a n s a c t i o n s Controllability on A u t o m a t i c C o n t r o l O c t and 1977 v o l AC-22 No- 5 p728 [4] A . H . Jazwinski, Stochastic Academic P r e s s , [5] T.D. Orbit [6] J.J. Moyer., Processes Determination " Program, Orbital Formulation Jet of the Double Parameter I n s t i t u t e o f T e c h n o l o g y Dec B . E . Schutz, Reduction et of al., Laser P. Swerling, Determination "A Comparison of Observations of the Method "Modern S t a t e of Least Precision by Weighted Thesis Air Least Force 1972. AAS/AIAA Astrodynamics Conference, [8] York: P r o p u l s i o n L a b May 1 9 7 1 . S q u a r e 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 [7] New 1970. "Mathematical Pollard, & F i l t e r i n g Theory. Vail Estimation a Near-Earth IEEE C o n t r o l Dec 1971 v o l A C - 1 6 N o . 6 p 7 0 7 . from the Transactions for the Satellite", Colorado / J u l y E s t i m a t i o n Methods Squares", Methods 16-18,1973Viewpoint of on Automatic
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Kalman filter based orbit observability study Shorten, Roy Robert 1979
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Title | Kalman filter based orbit observability study |
Creator |
Shorten, Roy Robert |
Date | 1979 |
Date Issued | 2010-03-05 |
Description | An Extended Kalman filter simulation technique was used to determine the observability of satellite orbits from one or two earth stations. It was found that range and range-rate measurements alone were insufficient for orbit determination. Either azimuth angle or elevation angle information were also required before an acceptable orbital estimate was obtained. However, range and range-rate measurements alone proved to be sufficient to improve the state estimates of approximately known orbits. Also if simultaneous range and range-rate measurements were available from two stations, orbit determination was possible. Various common orbital types were used throughout the study and only one pass of data was considered. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-03-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0094655 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Unknown |
URI | http://hdl.handle.net/2429/21583 |
Aggregated Source Repository | DSpace |
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