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UBC Theses and Dissertations

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., 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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