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On the dynamics of spacecraft with a slewing appendage Mah, Harry Wayne 1986

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ON  THE  DYNAMICS  OF  SPACECRAFT  WITH  A  SLEWING  APPENDAGE  by HARRY B.ENG.,  A  TECHNICAL  THESIS THE  WAYNE  UNIVERSITY  SUBMITTED  IN  REQUIREMENTS MASTER  MAH  OF  OF  PARTIAL FOR  THE  APPLIED  N.S.,  1983  FULFILMENT DEGREE  OF  SCIENCE  in FACULTY DEPARTMENT  We  accept to  THE  OF  OF  thesis  required  UNIVERSITY  OF  July,  ©  SCIENCE  MECHANICAL  this  the  APPLIED  HARRY  ENGINEERING  as  conforming  standard  BRITISH  COLUMBIA  1986  WAYNE  MAH,  1986  OF  In  presenting  requirements  this for  an  BRITISH  COLUMBIA,  freely  available  that  permission  scholarly  I  agree for  purposes or  understood  that gain  in  advanced  for  Department  f i n a n c i a l  thesis  by  degree  that  the  reference  extensive may his  copying  shall  partial  not  be or  be  OF  and  Date:  J u l y ,  1986  study.  I  this  granted  by  the  allowed  of  make  further  Head  of It  thesis my  OF it  agree  thesis  this  without  the  UNIVERSITY  representatives.  publication  COLUMBIA  of  shall  of  MECHANICAL ENGINEERING  THE U N I V E R S I T Y OF B R I T I S H 2075 Wesbrook Place Vancouver, Canada V6T 1W5  THE  Library  permission.  DEPARTMENT  the  copying  her or  at  fulfilment  for my is for  written  ABSTRACT A  relatively  l i b r a t i o n a l appendage  dynamics free  developed. coupled  The  that  combinations  to  the  governing  the of  design  would  of  of and  a  undergo  are  formulation r i g i d any  inertia, The  become  geometric, fundamental and  construction  involve  arbitrary  complex  of  the the  slewing  components.  i i  studying with  a  slewing  r i g i d motion  nonautonomous  numerically.  can  s a t e l l i t e s  for  s a t e l l i t e  nonlinear,  solved  system  parameters.  experiments, which  to  equations  suggests  history  general  A  orbital  under and  information Orbiter  study  c r i t i c a l  slewing is  time  relevant  based  proposed motion  and  parametric  unstable  is  of  space  station,  structural  TABLE 1.  2.  CONTENTS  INTRODUCTION  1  1.1  Preliminary  1.2  A  1.3  Purpose  Brief  Remarks  Review and  of  THE  2.1  Spacecraft  Model  2.2  Development  Potential  Librational  2.5  S p e c i f i c Study  Investigation  7  General  Equations  of  Motion  energy  Equations Terms  16  of  for  S a t e l l i t e  ..11 11  Motion the  18  Equations  Configuration  of  Motion  Selected  ...19  for 19  central body appendage  Governing parameters configuration  2.5.3  Appendage  rotations  2.5.4  Appendage  time  System  5  energy  C y l i n d r i c a l c y l i n d r i c a l  2.5.2  the  2  7  the  2.2.2  2.4  2.6  of  Kinetic  Governing  of  Literature  PROBLEM  2.2.1  2.3  1  the  Scope  F O R M U L A T I O N OF  2.5.1  3.  OF  with  a 19  for  the  two  cylinder 21 24  history  parameters  Parameters  26 26  2.6.1  Instantaneous  center  of  mass  27  2.6.2  Evaluation  system  parameters  27  RESULTS  AND  of  DISCUSSION  29  3.1  Computational  Considerations  3.2  Effect  of  Appendage  Time  3.3  Effect  of  S a t e l l i t e  Inertia  3.4  Effect  of  Impulsive  Disturbance  3.5  Effect  of  Orbit  History  E c c e n t r i c i t y  i i i  29  and  29 Geometry  48 54 59  4.  CONCLUDING  REMARKS of  61  4.1  Summary  Conclusions  4.2  Recommendations  for  61  Future  Work  62  BIBLIOGRAPHY  65  APPENDIX  A  -  A N G U L A R MOMENTUM V E C T O R  APPENDIX  B  -  SATELLITE  INERTIA  APPENDIX  C  -  EQUATIONS  OF  APPENDIX  D  -  LIBRATIONAL  APPENDIX  E  -  TRANSFORMATION  TERMS  FOR  APPENDIX  F  -  CENTER OF  TERMS  . ...  APPENDIX  G  -  NONDIMENSIONAL  SATELLITE  APPENDIX  H  -  NONDIMENSIONAL  A N G U L A R MOMENTUM V E C T O R  APPENDIX  I  -  NONDIMENSIONAL INERTIA MATRIX  DERIVATIVE  APPENDIX  APPENDIX  J  K  -  -  {H}  MATRIX  [I]  ...78  MOTION  79  TERMS  MASS  88  FOR  SLEW MANEUVERS  96 99  INERTIA  OF  MATRIX  109 111  SATELLITE 112  NONDIMENSIONAL DERIVATIVE MOMENTUM V E C T O R EQUATIONS  74  OF  APPENDAGE TIME  iv  ANGULAR 113 HISTORIES  114  LIST  OF  FIGURES  Figure 2.1  Page Geometry a  2.2  2.3  2.4  2.5  2.6  2.7  3.1  slewing  3.3  orbiting  spacecraft  with  appendage  8  Domains a s s o c i a t e d with c e n t r a l and appendage components of spacecraft and c o r r e s p o n d i n g reference coordinate systems  10  Schematic diagram showing position v e c t o r s to mass e l e m e n t s in different domains  12  Eulerian r o l l (0),  20  rotations showing p i t c h a n d yaw (X) l i b r a t i o n s  Specific s a t e l l i t e configuration c y l i n d r i c a l central body with a c y l i n d r i c a l appendage Variation of the inertia with s a t e l l i t e geometry Appendage slew motion to c e n t r a l body Appendage time f i n i t e i n i t i a l (a) (b) (c) (d) (e) (f)  3.2  of  ratio  (\//),  22  $ 23  r e l a t i v e 25  h i s t o r i e s v e l o c i t y :  with  exponential displacement; v e l o c i t y ; acceleration; s i n u s o i d a l or ramp displacement; v e l o c i t y ; acceleration  Appendage  time  h i s t o r i e s  with  (a) (b) (c)  exponential displacement; v e l o c i t y ; acceleration;  (d) (e) (f)  s i n u s o i d a l , ramp v e l o c i t y ; acceleration  or  cubic  zero  i n i t i a l  30 v e l o c i t y :  displacement; 32  System response showing e f f e c t of appendage time h i s t o r i e s with f i n i t e i n i t i a l v e l o c i t y : (a) (b) (c)  exponential; s i n u s o i d a l ; ramp  35 v  3.4  System time  3.5  (a) (b)  sinusoidal; ramp;  (c)  cubic  3.11  of  i n i t i a l  appendage v e l o c i t y :  orbit; orbit; orbit  pitch pitch  response; rate response  .  motion  ."  effect  39 of  duration; i n i t i a l p o s i t i o n ; final position  40 effect  of  i n i t i a l p o s i t i o n ; final position  42 appendage  pitch response; r o l l response; yaw r e s p o n s e  System response due p i t c h and r o l l slew (a) (b)  pitch response; r o l l response;  (c)  yaw  43 to appendage motions:  response  System r e s p o n s e due p i t c h and r o l l slew (a)  pitch  (b) (c)  r o l l response; yaw r e s p o n s e  duration v e l o c i t y :  37  System response due to slew motion in r o l l : (a) (b) (c)  3.10  0.001 0.100 0.300  System response showing appendage p i t c h : (a) (b)  3.9  effect  36  System response showing appendage p i t c h : (a) (b) (c)  3.8  zero  System response during slew for various time h i s t o r i e s : (a) (b)  3.7  showing with  System response showing the effect of of slewing maneuver with zero i n i t i a l (a) (b) (c)  3.6  response  h i s t o r i e s  45 to appendage motions:  response; 46 vi  3.12  System response showing e f f e c t of of appendage d u r i n g slew motion: (a) (b) (c)  3.13  3.14  of:  central body i n e r t i a ratio; appendage inertia ratio; appendage/central body mass  r a t i o  in ,(c)  the orbital across the  plane; orbital  .  .  .  .  49  pitch  plane  51 r o l l  pitch response; r o l l response  52  pitch response; r o l l response  53  S y s t e m r e s p o n s e f o r an adverse combination of s a t e l l i t e inertia and geometry: (a) (b) (c)  3.18  effect  System r e s p o n s e due to appendage r o l l slew motion as a f f e c t e d by hinge l o c a t i o n a c r o s s the orbital plane: (a) (b)  3.17  showing  System r e s p o n s e due to appendage slew motion as a f f e c t e d by hinge location in the orbital plane: (a) (b)  3.16  response  47  System r e s p o n s e due to appendage slew motion as a f f e c t e d by hinge location: (a) (b)  3.15  pitch response; r o l l response; yaw r e s p o n s e  System (a) (b) (c)  path  pitch response; r o l l response; yaw r e s p o n s e  55  System r e s p o n s e due to appendage p i t c h slew motion as a f f e c t e d by an i m p u l s i v e disturbance: (a) (b)  in the orbital ,(c) across the  plane; orbital  v i i  plane  56  3.19  System r e s p o n s e due to appendage r o l l slew motion as a f f e c t e d by an impulsive disturbance in the o r b i t a l plane: (a) (b)  3.20  57  System r e s p o n s e due to appendage r o l l slew motion as a f f e c t e d by an impulsive d i s t u r b a n c e across the orbital plane: (a) (b)  3.21  pitch response; r o l l response  pitch response; r o l l response  System  response  due  r o l l slew motion as orbit eccentricity: (a) (b)  58 to  appendage  affected  pitch response; r o l l response  by  60  vi i i  ACKNOWLEDGEMENT The  author  appreciation the  preparation  have  been The  by  to  the  Canada,  wishes Dr. of  to  V . J . the  express Modi  for  t h e s i s .  his  sincere  guidance  thanks  given  His  help  and  in  this  thesis  and  throughout  encouragement  invaluable. investigation  Natural Grant  Science No.  reported and  A-2181.  Engineering  Research  was  supported  Council  of  LIST a  OF  position vector Figure 2.3  c,  c  position  Q  for  vectors,  nondimensional 3/(1+ecos0)  SYMBOLS mass  Figure  c  e  c  M  gravitational  d  a  position  vector,  -c  - c  0  d  s  position  vector,  -c  - c  0  dm ,  dm  a  s  element  gravitational  constant,  3n/R  constant, + ,  h  3 c  +  Appendix  1$,  Appendix  position  f3  projection  f  nondimensional eccentricity -2esin0/(1+ecos0)  of  f  Figure onto  and  central  *2.3  body  axes  [M ]f a  function,  position  vector,  -c  h  position  vector,  Figure  i,j,k  unit  vectors  along  x,y,z  ^a'^a'^a  unit  vectors  along  x  a  , y  a  , z  a  axes,  respectively  ^S'Ds'^s  unit  vectors  along  x  s  , y  s  , z  s  axes,  respectively  {/}  direction  m,m ,m a  s  cosines  to  body  for  s a t e l l i t e , appendage respectively  generalized coordinate degree of freedom  r  position  coordinate freedom  a  position_vector  for  mass  +h  +f  +a,  mass  Figure  x  2.3  respectively  vector  central  for  for  Q  axes,  vector  vector  - c  Figure  2.3  and  qi  -c  +f,  along  R  c  with  x,y,z  generalized degrees of  a  0  unit  axes  {q}  r  +h  x , y , z ;  g  respect  - c  B  B  d i f f e r e n t i a l elements for appendage body, respectively, Figure 2.3 vector,  appendage,  2.3  f  e  of  i  f c  for  ^  masses,  librational  librational  element  element 2.3  body  dm dm , a  r  position vector Figure 2.3  s  s  for  mass  element  dm ,  position vector for body, Figure 2.3  mass  element  of  t  0  + s ,  central  time  x , y , z  a ' Y a '  x  z  a  srYs> s  x  - c - c  s  z  C  s a t e l l i t e body Figure 2.2  coordinates  with  origin  at  C,  appendage body Figure 2.2  coordinates  with  origin  at  0  central Figure  body 2.2  coordinates  instantaneous  C  G  D  a  center  center of mass slew motion , D  s  domains body of  of  of  mass  s a t e l l i t e  associated s a t e l l i t e ,  with appendage Figure 2.2  [E]  unit  {H}  a n g u l a r momentum w i t h r e s p e c t due to s l e w i n g appendage  [I]  s a t e l l i t e  inertia  diadic  [l ]  appendage  inertia  diadic  a  r  [l ] [M ]  x  O  a  , O  body  s  a ' Y a '  z  a  a  n  d  used to motion  inertia  transformation  a  S  appendage  and  central  matrix  central  s  0 ,  force  arbitrary inertia the equations of  D  at  ,  s a t e l l i t e  before  center  I  origin  of  E  a  of  with  a  X  Z  a  on  x  defining e  x,y,z  rotation  between  s  P  hinge  central  {Q}  generalized of freedom  force  vector  Qi  generalized freedom  force  for  xi  axes  nondimensionalize  c e n t e r s of mass f o r appendage respectively, Figure 2.3 location  body  diadic  matrix 'Y>  to  i  f c  body, for  ^  and  central  Figure  2.3  l i b r a t i o n a l  l i b r a t i o n a l  body,  degrees  degree  of  R  position vector mass element dm  R  C  ,R  kinetic  U  force  to  contributing motion  X,Y,Z , Y  0  i n e r t i a l , Z  0  the  energy  orbital  coordinate coordinate  system system  appendage slew motion r o l l , respectively  aj,/3i  i n i t i a l appendage p o s i t i o n and r o l l , respectively final  e  appendage  respectively  orbit  e c c e n t r i c i t y  inertia  ratio  for  appendage,  $  s  inertia  r a t i o  for  central  17  mass  6  true  n  universal  gravitational  vector  normalized  £  hinge p  mass  appendage  a  X  of  location density  l i b r a t i o n a l yaw,  I  S  the  E  at  C  0  and  pitch  pitch  and  a , x  I  (  central  Z  / I  S  (  X  body,  m /m a  s  anomaly  on  constant  coordinates  central  ratio  of  slew motion durations r o l l , respectively  T ,rp  I a , z /  body,  to  at  pitch  coordinates,  a  of  origin  to  coordinates,  £  ratio  origin  coordinates,  position  r o l l ,  with with  a,B  af,Bf  a  the  to  effective potential energy c o n t r i b u t i n g l i b r a t i o n a l equations of motion  e  0  of  energy  potential  X  center  effective kinetic energy l i b r a t i o n a l equations of  e  U  the  s c a l a r d i s t a n c e and p o s i t i o n vector, respectively, from the center of force to s a t e l l i t e ' s instantaneous center of mass  C  T T  from  appendage for  coordinates  respectively  xi  i  specifying  body to  appendage  defining  central pitch  p i t c h ,  body and  r o l l  and  co, {co}  s a t e l l i t e  angular  velocity  vector  ^a'f^a^  appendage angular central body  velocity  Aa,A/3  s p e c i f i e d changes in c o o r d i n a t e s due appendage slew motion, pitch and r o l l , r e s p e c t i v e l y  vector  relative  to  to  Dots and primes represent d i f f e r e n t i a t i o n with respect to t a n d 6, respectively, unless otherwise defined; superscript * implies nondimensional parameter unless otherwise indicated.  x i i i  1.  1.1  Preliminary The  Space  v e r s a t i l i t y l o g i c a l  in  step  operational launch  INTRODUCTION  Remarks Shuttle  is  operational  undertaking  would bases  be  to  for  diverse  use  it  the  maintenance,  manufacture  favourable  microgravity  environment,  technology  committed  i t s e l f  the  nineties  early  Transportation crew.  The  relative and  involve  to  slewing  and  presents  other  spacecraft.  using  Mobile  complex  In complex: systems  dynamics,  its  utmost  slewing  system  as  to  inertia  ferrying a  host  space  antennas, in  Transfer  of  load  Manipulator h i s t o r i e s  has  station  and  members,  case  over  can  phase  the  proposed  space  (MRMS)  create  may  and  control  problems.  generality  the  problem  can  by  a  the  one  or  f l e x i b l e  more  fundamental  d i s t r i b u t i o n ,  1  be  To  dynamics  f l e x i b i l i t y ,  require  quite  t r a n s l a t i n g ,  platform.  station  challenging  s t a b i l i t y  of  can  instruments,  of  the  involving  subassemblies  operational  the  by  construction  p o s s i b i l i t i e s  System  and  U.S.  s c i e n t i f i c  as  oriented  Space  material  state  the  earth  of  based  of  in  the  a  of  s a t e l l i t e  end,  structural  steady of  motion  supported  appreciation the  time  of  Shuttle  the  its  next  products  and  this  systems  Remote  slewing  transient  the  motion  G a l i l e o a  the  for  motion  To  The  construction  of  construction  using  Even  slewing  telescopes  the  System  program  platforms.  programs.  proved  exploration,  and  applied  has  missions.  in  s c i e n t i f i c  and  get as  f l e x i b l e  some  affected o r b i t a l  by  2  parameters,  slewing  it  is  planned  of  complexity.  slewing with  1.2  to  A  dynamics  Brief  and at  thesis.  a  a  Of  of  aspects  extends  flexible  complexity  involving well  as  while  r i g i d  well  as  Dow  with  s p e c i f i c  v e l o c i t i e s . solar  panels  acceleration M i s r a  1  7  et  1  0  a l .  "  are  1  configurations  during with  studied  is  the  inertia,  orbit.  has  an  had  the  papers  his  and  motion a  its  "  7  .  spacecraft  central  to  and  Honeycutt by  1  of  led  the  body.  analyses  point  f l e x i b l e  1  as  8  masses,  f l e x i b l e  a s s o c i a t e s  uniform  analytical  l i b r a t i o n a l  from  cover  thoroughly  where  is  present  that  quite  appendage  investigated  attempt  of  Lang  fixed  any  r i g i d  scope  often  and  and  of  l i t e r a t u r e  treated  and  dynamics  relative  outwardly  deployment  the  the  6  general  dynamics  considered  1  arbitrary  the  of  several  Cherchas  5  J a n k o v i c  the  assumptions.  .  2  considers  order  body  dynamics  problem  authors 1  a  vast  beyond  represented  9  in  of  forces,  increasing  thesis  enormous  this  deployment  the  other  b o d i e s  of  and  simplifying  several  indeed  appendages  of  C l o u t i e r  is  interest  involving  an  l i b r a t i o n a l  s a t e l l i t e  considerable  in  appendage,  to  there  of  generalized  Literature  pertaining  review  this  platform  the  spacecraft  and  problem  step,  r i g i d  unwarranted  appendages  the  r i g i d  Fortunately,  specific  as  of  comprehensive  considered  The  f i r s t  Review  flexible a  a  to  Literature  h i s t o r i e s  approach  As  reference  time  3  members "  1  "  as  appendages  deployment  dynamics  of  correlated  p r e d i c t i o n . dynamics  of  the  CTS  measured Ibrahim  a  body  tip  and  deploying  3  two  plate  plane.  type  The  properties general beam  of  on  l i b r a t i o n a l  Interaction  between was  combinations  of  9  takes  .  into  a  of  central in  the  The  appendages  indeed  control  and  Bainum  angle  dictates  a  Barba of  the  by  means  2  Several  need and  problem of  Ibrahim  formulation  of  mass  and  intriguing  with  the  an  is  the  for  controlled  the  c o n t r o l l i n g devices  of  a  control  of  of  slewing  rapid  large  accuracy.  mathematical  gave  by  appendages.  slewing  motions  The  proffered  optimal  pointing  slewing and  problem.  require  that  slewing  dynamics  s a t e l l i t e  presented  factors  telescoping  missions  degree  are  analysis  s t o i c h a s t i c  movable  and  the  deploying  high  gyroscopic  by  with  met  the  extensible  s a t e l l i t e  a  of  representing  a  an  .  i n s t a b i l i t y .  with  and  1  to  spacecraft a  8  and  as  which  1  certain  presented  center  M o d i  of  interest  2  that  f l e x i b l e ,  been  and  f l e x i b i l i t y ,  rise  interest  proposed  Aubrun  Lips  i n e r t i a ,  with  relevant  by  A  deploying  body  on  0  spacecraft  maneuvers  has  represents  Sellappan  spacecraft.  deploying  of  well  flexible  noted gave  plate  investigated.  formulation  shifting  class  been  of  general  the  with  orbiting  and  dynamics,  was  particular  has  Also  it  the  was  presented  parameters  with  challenge  spinning  and  r i g i d  appendage.  a  been  appendages  is  response  l i b r a t i o n  rather  account  occur  the  to  v e l o c i t i e s  s a t e l l i t e s  has  system  type  This  changing also  of  spacecraft  beam/plate 1  for  studied  recently,  class  M o d i  the  normal  deployment  appendages  deployment  members  effect  formulation  type  More  flexible  of  This  maneuvers. description spacecraft  geometrical  4  interpretations the of  "momentum  2  7  ,  have  gyroscopic minimize and  angle,  2  have  r e l i e d  on  solar and  optimal  large  Juang  et  to  developed  f  l  Carrington  several  of  ,  2  and  5  5  step  a  between  of  form  and  J u n k i n s  3  "  7  3  ,  9  spacecraft 1  "  5  5  ,  M e i r o v i t c h  with 5  5  of  Junkins  et  M o d i  3  3  3  0  been  a l .  as 3  0  and  1  introduced while  for  F a d a l i  feedback  3  6  tracking.  The  for  Control  by  slewing  analyzed  contributions considerable  process  suitable  presented  Dwyer* .  use  maneuvers.  spacecraft,  and  the  thrusters  electromagnets  and  were  have  the  hand,  strategies  planning  and  on  reconstruction  Dwyer  feedback  but  for  solution  maneuver  different  opted  J u n k i n s  r i g i d  a  Chen  Exploiting  and  and  that  torques,  slewing  and  number  via  laws  from  Lunscher  optimization  nonlinear  9  other  closed  trajectory  flexible  and  a  used  i n v e s t i g a t o r s '  Breakwell "  gyrostat.  Vadali  maneuvers  3  control  2  A  consumption.  problem  Batten  through  Skaar  control  energy  on  the  performance,  slew  or  execute  and  and  control  find  while  to  obtained  single  involving  dynamics  3  angle  slew  simultaneous  f i e l d  pressure  Frauenholz  improve  laws  3  On  ,  to  interaction  T u r n e r  a l .  control.  wheels.  6  of  a  and  2  problem  not of  transfer  e l l i p s o i d " .  K r a n t o n  slewing  Dwyer  the  "energy  time  the  momentum  them  the  momentum  magnetic  Junkins  among  attention  reaction  E a r t h ' s  ,  and  the  attempting  c o n t r o l ,  as  u t i l i z e d  5  and  maneuver  angular  well  the  2  approached  8  active  "  2  means,  focusing  internal  2  energy  addressed  either  Kane  the  sphere"  i n v e s t i g a t o r s  K r a i g e  of  of  by  by  s i g n i f i c a n c e .  5  The quite is  area  relevant  the  study  control  of  auxiliary  to  Other  by  et  a l .  6  robust  spacecraft Bell and  and  control  motion a  et  modal  Swigert  a  6  6  of  Purpose From  apparent  The  for  a  that has  review the  preliminary  analysis  a  Voyageur  high  a l .  Also,  5  9  systems  S t i e b e r  a  6  for  a  flexible  dual-spin  Jahanshahi  6  while  2  requirements  Observation  scheme  large  examined  1  of  pointing  for  System.  controlling  arm.  Laurenson  flexible  structures  of  an  Telescope  of  Earth  attitude was  were:  control  and  the  array  the  is  interest  Space  and  of  effect  torque  6  the  presented  5  while  shapes  on  the  structure.  of of  the the  dynamics  of  Yuan  flexible  considerable  purpose  et  ;  7  pointing  study an  5  the  attitude  rotating  Scope  and  developed  flexible  and  ;  which  solar  J o s h i  Hayati  the  in  6  performance  lightweight  the  appendages  and  a  of  8  and  presented  have  by  antenna  by  "  5  5  systems  particular  analyzed  Dougherty  evaluated  investigated  6  systems  by  approaches  a l .  pointing  Of  rotating  station,  pointing  3  et  a  pointing  studied  analysis  response  1•3  6  a l .  of  with  s a t e l l i t e s ,  was  L i n  Alberts  Chretien  unit,  The  appendage  investigation.  beam-pointing  spacecraft.  or  J a h a n s h a h i  0  communications the  by  instrument  control  Broquet  this  s a t e l l i t e  antenna,  pointing  instrument  done  a  considered.  gain  of  this of  Investigation relevant  of  literature,  s a t e l l i t e s  ground  to  investigation the  problem  in  with  it  is  slewing  cover. is the  to  attempt  hope  that  a it  w i l l  6  provide with  insight  slewing  system  operating The  with in  of  consisting geometry  of  is  s a t e l l i t e  as  throughout of  investigation with  and  system  for  some  directions  useful for  by  by  body  with  general  about  of is  more  the  studies.  one  body  exhibits  with  of a  and  and  orbit  nonautonomous  challenging  task.  physical  may  models. and  c y l i n d r i c a l  situation,  nonlinear, a  s p e c i f i c  motion,  simple  which  a  conditions  presents  remarks  the  arbitrary  slew  i n i t i a l  sophisticated  of  model  of  body  relatively  behaviour  appendage  analysis  appendage  highly  the  other.  methodology  concluding  future  where the  of  forces.  bodies  central  equations on  r i g i d  formulation  response  the  s a t e l l i t e s  simple:  spacecraft  r i g i d  the  this  kept  and  the  of  appreciation  purposely  geometry,  of  system  of  better  environmental  c y l i n d r i c a l  Even  appreciation  a  rotation  integration  coupled  is  behaviour  d i s t r i b u t i o n ,  affected  e c c e n t r i c i t y .  gain  connected,  inertia  numerical  for  followed a  any  i n i t i a t e d  two  configuration,  To  central  of  inertia  is  appendage,  dynamical  model  r i g i d  relative  This  Emphasis  a  motion  and  specified  the  absence  study  equations  the  appendages.  behaviour,  spacecraft  and  into  help The  thoughts  thesis on  ends  possible  2. The general bodies one  study  is  i n i t i a t e d  spacecraft of  model  arbitrary  body  other.  FORMULATION OF  executes  The  with  s p e c i f i e d of  the  consisting  geometry  equations  THE  and  consideration of  two  inertia  relative  motion  PROBLEM  are  of  a  connected,  r i g i d  d i s t r i b u t i o n ,  rotation  developed  about for  where  the  the  general  model. Next, central  a  particular  body  s a t e l l i t e  and  parameters  configuration. are  expressed  2.1  Spacecraft  arbitrary  the  is  central  mass.  The  a  and  capable body  s a t e l l i t e  define  of  the  coordinate of  force.  appendage frame  X  o  .Y o  0  inertia  of  the  is  a  .Z,, o  for  the  of  of  this motion  parameters.  The  X,Y,Z  motion. with  to  its  with  r i g i d  in  the  of  location  C  orthogonal,  origin  at  3  7  an R  to  C^ o  in  c  and  at  E,  about  center  of  true  the  absence  so  The  center  of  i n e r t i a l  orbiting is  of  arbitrary,  the  origin  of  rotation  instantaneous  respect its  2 . 1 . ) .  s a t e l l i t e  vector  the  bodies  connected  (Figure  negotiate  having  An  two  relative  position  location  of  joint  shift  free  represents  slew  set  slewing  d i s t r i b u t i o n ,  executing  spacecraft  system C  A  variations  appendage  comprised  causing  anomaly C  geometry  considered.  history  ( f r i c t i o n l e s s )  trajectory.  mass  the  time  s a t e l l i t e  s p e c i f i e d 6  is  possible  in  of  c y l i n d r i c a l  Model  ideal  appendage  specify  terms  geometry  an  of  appendage  Variations in  Consider  through  the  case  of  center any  reference  oriented  that  Z^ o  F i g u r e 2.1  Geometry of o r b i t i n g s p a c e c r a f t with a slewing appendage.  9 and  X  are  Q  along  respectively, s a t e l l i t e k frame  '  X 0  i t s  any  '  Y 0  appendage at  while  body  has  origin ^  Z 0  by  t  a  set  domain,  body  are  referred  is  ,  thus  O  .  ,z  an  given  The (central central  is  to,  with  horizontal, the  x,y,z  of  any of  orbit  with  coincides  normal.  unit  vectors  the  orbital  with  librations  the  o r b i t a l  and  s a t e l l i t e  frame  Eulerian  X  . Y , . Z o o o  rotations  i l l u s t r a t e d  and  mass  center  of  for  the  The  as  for m ,  appendage  are  t o t a l  g  mass  1  in  , q  2  , q 3 .  Figure central  0 .  g  m of  x,y,z is  q  the  D ,  The  relative  axes  schematically  respectively,  at  with  O  =  g  m„ s  i s  unit  a  +  g  specified the  as,  s a t e l l i t e  nta .  central  vectors  i  body ,]  ,k  coordinate .  coordinate  frame  Similarly  S S S  body  located  at  0 3  system  x  ,y_,z with a a. OL =  unit  i #5 »^ a  a  a  s a t e l l i t e body body  and  body  frame  x  orientation  frame  x , y , z  which  is  is  frame  appendage) ,y  5 The  and  by  appendage  vectors  to  parameters  S S S  is  and  respectively.  Located ,y  C  modified  mass,  m  x  system  absence  of  The  m  aligned  relative  2.2.  ,  at  v e r t i c a l  Orientation  spacecraft  Corresponding  is  0  the  n  The  D  Y  rotation.  described  local  coordinate  instant  J  the  ,z  5 of  determined  by  is  for  taken  the  to  be  entire  system  p a r a l l e l  to  the  .  o the  specified  x , y , z  by  the  axes  a  x  a  .y J  a  ,z„ a  relative  transformation  order  and  number  matrix of  to  the  [ M l ,  rotations  Figure  2.2  Domains a s s o c i a t e d with c e n t r a l and appendage components of spacecraft and corresponding reference coordinate systems.  11  used  to  arrive  at  frame  x  a  ,y  a  ,z  from  a  s a t e l l i t e  body  axes  x , y , z . Let mass  s  and  elements  (Figure  and  instantaneous  x  a  =  x  c  determines  with  appendage  relative  2.2.1  The  of  to  of  Kinetic  i  a  +  a  domains  y  +  y  a  J  point  the  be  5  s  mass  respect  w i l l  Development  s  hinge  0  2.2  i  =  center  diadic  in  of D_ s  a  positions  inertia  dm  positions  s  locates  Q  the  s  vector  h  define  dm  2.3).  The  vector  a  to  J  +  i  relative  P  with Due  central  General  of  to  respectively  to  the  point  respect  to  c  and  C  O  rotation  body, of  , a  position  C  P.  D  k a  a  the  and  d i f f e r e n t i a l  k  s  + z  a  functions  the  z  the  Q  .  The  , of  and  f  the  the  s a t e l l i t e  time.  Equations  of  Motion  energy  kinetic  energy  T  U  =  of  P'P  the  dm  system  can  be  written  ,  as  (2.1)  m  where the  m  is  the  position  respect  to  E.  total  of If  a  mass  of  d i f f e r e n t i a l r  is  the  the  spacecraft  element  position  of  vector  and  mass of  R dm  the  defines with element  12  Figure  2.3  Schematic to  mass  diagram  elements  showing in  position  different  vectors  domains.  13  mass  with  respect  to  R  where  r  for  r  =  r  =  r  a  R c  different  =  g  =  the  =  - c  -  c  - c -  c  o  s  +  =  r  h  CJ i s  the  s a t e l l i t e ; velocity of  R  f  of  change  the  of  r  the  Substituting  T  =  R  1/ 2  **in  +  2R  It  may b e  a  c  angular  2R  ot  noted  then  a  domain? domain,  time  (2.2)  that  ot  •(wxr)  c  velocity  (tangential  and -|f  dt  +  to  g  (wxr)  + | f - | f+  c  D  respect  reference  .R,  defined  ,  equation(2.2)  ~  c  +  orbital  the  x , y , z ,  ot  s a t e l l i t e ; in  as  | f+  +  librational is  c  i s  D  +  C  where  system  ,  R with  R  coordinate  , '  domains  Differentiating  R  +  +  Q  body  of  and  represents  coordinate  into  radial) the  time  system  equation(2.1)  (wxr)-(wxr)  +  the  2|f»(wxr)  dt  dm  rate  x , y , z .  14  "g+r = =  where  u>  velocity  - c  -^r  =  - c  a  ot  with of  reference  respect  the  the  coordinates equations  or  of  + w  the  kinetic  motion  of  +  to  the  the  the  D  in  the  D  central  and c  mass  and  domain,  a  velocity  body of  domain  g  C  of i s  in  the the  the  x , y , z .  do  the  terms not  without  contribute  l i b r a t i d n a l  with the  in  angular  center  energy  for  of  a)  relative  v e l o c i t i e s  component  formulation  x(f  system  differentiation  effective  a  instantaneous  coordinate  Note,  to  =  represents  a  appendage  due  -gr;  q  and q  kinetic  equations  of  generalized to  Lagrange's  degrees  involved. energy  motion,  of  freedom  Hence,  the  contributing  T  ,  c a n be  in  written  as,  T  =  jJ  Since  (wxr).(wxr)  R  f  But  /  m  r  dm =  C  is  R ^  m  c  0  ,  a  •  constant  (c3xr)  so  +  the  dm =  2R  . (5xr)  over  R  the  • Z J X /  c  kinetic  + 2 - | | - (5xr)  m  r  dm  spacecraft,  dm  energy  .  expression  form,  T  e  =  m  (wxr).(wxr)  dm +  J  m  .  -^.(wxr) ot  dm  .  takes  the  1 5  Clearly,  the f i r s t  IS  can  be w r i t t e n  part  (wxr)  m  of  the kinetic  • (c3xr)  energy,  dm ,  as  i{co} tl]{o,} ; T  and  the second  term  of  the kinetic  'm If* " (  can  be e x p r e s s e d  as  xF> d m  (Appendix  energy,  ' A)  {o>} {H} . T  So a  the kinetic  compact  T  form  energy  of  c a n be e x p r e s s e d  as  = i { " } [ l H w } + {a>} {H} ,  (2.3)  T  T  e  the system  where:  {to}  angular  [I]  time  v e l o c i t y  dependent  vector;  s a t e l l i t e  inertia  matrix;  in  16  {H}  time  dependent  respect  to  rotation  T  the  energy  body,  inertia  change.  the  lesser  the  s a t e l l i t e ' s  the  time  through products  2.2.2  use of  of  written  due  change  to  to to  the  l i b r a t i o n a l  coupling the  between  s a t e l l i t e  relative  with  the  (Appendix  rotation  and  primarily  appendage degree  relative  products due  respect  relocation  center inertia  p a r a l l e l - a x i s  inertia  pure  of  moments  is  s a t e l l i t e  the  relative  appendage  undergoes  instantaneous  dependent  Potential The  the  due  with  and  to  the  body.  s a t e l l i t e ' s  This  frame  vector  s a t e l l i t e ;  the  body  a  the  energy  of  body  to  rotation  of  to  due  rotation  reorientation and  of  momentum  appendage  l i b r a t i o n a l  appendage  central  the  kinetic  central  As  of  body;  kinetic  {u} {H}  s a t e l l i t e  central  rotation  angular  of  to  mass.  theorems  the  of  to  or  matrix  about  [I]  for  the  central  s h i f t i n g  of  Computation is  of  achieved  moments  and  system  can  B).  energy  gravitational  potential  energy  of  the  as  f  TT =  dm  Til  (2.4)  be  17  After the  binomial  potential  U  expansion  energy  =  can  be  and  truncation  written  -u (m +m ) / R  xy  / / x  y  -M(m +m )/R s  +  A  3M/(2R  3 c  C  s e r i e s ,  as  3  C +  2  + 6 ( 1  the  -u/(2R )*  5 a C I (1-3/ ) xx x  [  of  I  +1 -  ){/}  T  M  (1-3/  yy  yz  / / y  / ( 2 R  [I]{/}  +  zx  tr  )  3  )  +1  z  C  2  y  I  / / z  zz  (1-3/  x  '  2  z  )  )] J  [ I ]  ,  where:  u  universal  R„ c  distance  {/  }  gravitational from  the  center  s a t e l l i t e  center  vector  d i r e c t i o n  along  of R  C  with  constant;  of  of  force  to  the  mass; cosines  respect  to  respectively,  {/ ,/  , /  components  s a t e l l i t e  x  z  }  for  body T  unit  axes  vector x , y , z ,  ;  ^ x x ' ^ v v ' ^ z z ^ 1  I  x y '  T  )>  1  the  s a t e l l i t e  expression rotation.  is The  d i f f e r e n t i a t i o n degrees the  inertia  matrix [ I ] .  y z ' * z x  Here the  of  of  f i r s t  treated the  of  as  represents a  point  contribution  f i r s t  freedom  effective  term  two the in  terms  w i l l  Lagrange's  potential  energy  mass  due  potential  so  potential  to  and  the  its  f i n i t e  disappear  energy  for  equations far  energy  as  rest  to  of  the  size  and  upon the  of  the  due  l i b r a t i o n a l  motion.  Hence,  derivation  of  18  the  equations  written  of  motion  where  c  =  u  Governing The  using  c a n be  finally  as  U  2.3  are concerned,  = c  e  M  / 2 {/}[I]{/}  (2.5)  3n/R, . c 3  Equations  governing  of  Motion  equations  the Lagrange  of  motion  c a n now b e  obtained  p r i n c i p l e ,  (2.6)  where  q^ and  coordinate freedom. in  motion,  +  and force  in in  for  Appendix  the  {H*})  +[M]  [M]  [I*]  T  f  +{H*}  of  C .  e  [q]  +[([M] ) ]  +[I*]  {q}  {q}  v  T  f  {q}  should  parameters  t  ]  g  ]  of  that  [I*],  t  ]  [I*])  [L  e  c  v  ]  [I*]  degree  of  motion  i s  equations  given of  follows  -[M [M]  +{H*})  generalized  [I*]  g  c  +  as  -[M ])  {N}  be emphasized (matrices  v  +[q]([M  - [ N  e  It  l i b r a t i o n a l  1  appear  ]([I*] )  the  the equations  form,  [I*]  -[N  {N}  i*"*  +[q]([M  [M ]  represent  The nondimensional  vector-matrix  [I*]  T  +[M]  respectively,  The development  detail  [M]  ,  [M] ])  {q}  ([I*]  +[M]  T  {N}  [I*]  ([N ] y  +[M] ([I*]  {N}  T  {/}  =  {Q*}  .  q  the nondimensional  [ i * ] ,  and vectors  system  {H*},  {H*})  are  19  functions slew  of  motion.  obtained described  v a l i d  chosen  2.4  to  Hence,  orbital  by  the  general  it  for  should  any  coordinate  (Figure  a  set  be  noted  for of  of  of  that  the  motion  X^,Y o o  modified  at  be  can  easily be  section  equations  for  the  of  x , y , z  .Z^ o  of  appendage  Eulerian  Equations axes  can  that  model  modified  the  body  frame  motion  the  of  motion  rotations s a t e l l i t e .  Motion  relative  any  2.1.  instant  to t  the can  be  J  Eulerian  rotations  as  follows  2.4):  \p ( p i t c h ) <t> ( r o l l ) X  The  (yaw)  about about  about  l i b r a t i o n a l  modified  Eulerian  S p e c i f i c  2.5.1  spacecraft  l i b r a t i o n a l  Terms  of  and  configuration  sequence  describe  by  configuration  equations  s a t e l l i t e  orientation  described  2.5  the  any  Librational The  s a t e l l i t e  for  Furthermore, are  the  Consider aligned  with  appendage  be  z"  the its a  giving  Q  x'  giving giving  terms  for  rotations  S a t e l l i t e  C y l i n d r i c a l  Y  x",y",z" x , y , z  this  central of  cylinder  body body  with  with  axis  in  cylinder  a  for  c y l i n d r i c a l  S i m i l a r l y , z  sequence  Appendix  Selected  a  a  symmetry.  ;  particular  l i s t e d  as  ;  .  Configuration  central  axis  are  x ' , y ' , z '  aligned  with let with  of  D.  Study  appendage axis  z  the its  axis  20  Figure  2.4  Eulerian rotations showing a n d yaw (X) l i b r a t i o n s .  pitch  ,  r o l l  (tf>),  21  of  symmetry As  a  moments  (Figure  result  of  inertia  let  the  this for  character  configuration,  each  body  of  c y l i n d e r ,  a  are  the  equal and  transverse due  the  to  the  products  of  vanish.  Let to  of  inertia  axisymmetric  2.5).  the  hinge  center  the  of  location  the  placement  axial  of  the  on  the  appendage  be  end  of  the  appendage  hinge  on  the  central  restricted cylinder  cylinder  and  be  arbitrary. Let  2.5.' 2  the  density  Governing  by  the  ratio  close  approaching The  $ <  5 o  the  to  of  a  represents  describes  a  parameters  a  squat, are  inertia  ratio  of  appendage  of  appendage  TJ  parameters i n e r t i a l  c y l i n d e r ,  combinations  the  consider  location a  of  normalized  I  two  5fZ I  a fz  body,  to  hinge  coordinate  on  2.6).  „  5fX  the  system  ;  „ a fx m_/m_ o 5  describe  cylinder  A  that  follows:  suffice  the  while  as  central  a  i n e r t i a .  (Figure  c y l i n d e r ,  for  of  represented  dish  c y l i n d e r ,  to  be  rod,  selected  central  ratio  f l a t  can  moment  slender  of  specify  cylinder  transverse  ratio  possible  two  cylinder  inertia  The  To  axial  zero  inertia  mass  T?  its  two  for  properties  of to  known.  ion  inertia  ratio  be  parameters  conf igurat The  ratio  a l l  s a t e l l i t e . central  x  y  z  22  Figure  2.5  Specific s a t e l l i t e configuration c e n t r a l body w i t h a c y l i n d r i c a l  c y l i n d r i c a l appendage.  Figure  2.6  Variation geometry.  of  the  inertia  ratio  $  with  s a t e l l i t e  ro 00  24  located  at  cylinder the  0  with  s  radius  central  £  and  x  ,  £  1-  ,  y  normalized  with  cylinder.  The  reference  geometric  1 = <« ,« ,* > x  specifies  the  center  mass  of  Thus,  hinge for  y  s a t e l l i t e  to  P  with  central  c y l i n d r i c a l  2.5.3  two  relative  specified  by  Consider appendage  spacecraft the axes  x , y , z  two  is  to  respect  to  0 ,  the  g  cylinder. $  .  17,  £  specify  any  a  c y l i n d r i c a l  a  the  line  s a t e l l i t e  a  (appendage  pitch)  B  (appendage  roll)  of  at  by  to  about about  the  with  a  2.7).  =  x"  for  center to  to  of  mass, the  ,z  the  of body  rotations:  x " , y " , z " ; a a a ,y  of  orientation  r e l a t i v e  x  be  body.  p a r a l l e l  set  by  the  therefore  The  giving giving  determined  central  following  y' a  be  appendage  x _ , y _ , z  the  can  x ' , y ' , z ' a a a  (Figure  axes  can  direction body  the  system  x,y,z  coordinate  space  central  located  specified  in  relative  coordinate  axes  line  The  angles  then,  body  appendage  of  angles.  appendage  the  of  rotations  orientation  specifying  a  length  appendage.  Appendage The  of  the  T  parameters  orientation  half  central  parameter  s possible  the  z  position the  using  .  Figure  2.7  Appendage  slew  motion  relative  to  central  body.  26  The  resulting  relative  angular  derivatives  2.5.4  Appendage The  one  path  position  curve. specify r o l l  are  B.  Two  the  final  are  given.  defined  time  c3  by  specified  by  a  The  slew  the  parameters  are  motion  duration  for  is  cubic, once  the  moves time  required  composed  function  specified  and  E.  be  the  appendage  respective  it  s i n u s o i d a l ,  p o s i t i o n ,  their  as  functions  ramp,  and  appendage  history  of  ]  3  the  time  shape  and  a  [M  parameters  can  appendage's  matrix  Appendix  history  completely  as  in  traversed  the  is  l i s t e d  vector  another  exponential, history  velocity  to  the If  transformation  the  time  of  completely a  and  ( i . e . ,  then  the  i n i t i a l the  history  pitch  known  etc.)  of  to  from  p o s i t i o n ,  slew  history  time  maneuver  curves  are  follows:  i n i t i a l  position  in  pitch  and  r o l l ,  respectively; final  position  in  pitch  and  r o l l ,  respectively; duration  of  slew  motion  in  pitch  and  r o l l ,  respectively.  2.6  System In  were  Parameters  section  formulated  described  in  2.3, based  section  the on 2.1.  nondimensional the The  general system  equations  spacecraft parameters  of  motion  model [I*],  27  [I  ],  {H  },  {H  ]  representative The  parameters  the  two  the  pitch  body  c y l i n d r i c a l r o l l  s h i f t i n g  Determination the of and  the  their  mass  v e l o c i t y ,  appendage  2.6.2  rotation  functions  of  ],  parameters  the  s a t e l l i t e  relative  v e l o c i t y , is  and  necessary  angular [I  the  with  for  }.  {H  the  presented  {H  the  in  },  The  s h i f t i n g  with  of  calculation  ]  of  mass.  acceleration  vector  and  central  of  momentum  s a t e l l i t e are  to  center  center  p i t c h ,  Appendix  r o l l F.  parameters [I*],  {H*},  [i*],  configuration  {H*}  and  desired  i n e r t i a l  parameters  $ s  C):  evaluated  the  are appendage  h i s t o r y . The  by  are  sequence.  acceleration  system  system  configuration  s a t e l l i t e ' s  derivatives  sequence  the  mass  mass  cylinder  of  system  [I  and  two  Evaluation The  time  matrix  the  s a t e l l i t e ,  p o s i t i o n ,  respective  for  of  evaluating  parameters  slew  the  of  are  s a t e l l i t e  by  appendage  of  center  inertia  p o s i t i o n , of  of  s a t e l l i t e ' s its  the  motion  s a t e l l i t e .  element  center  of  any  system  appendage  rotation a  for  obtained  the  of  model.  particular  section,  and  causes  motion  e a s i l y  that  equations  general  of  Instantaneous The  the  the  are  for  this  for  2.6.1  of  equations  configuration  In  of  setting  I  = a  r  D  I s  x  *  Then  from  section  =  are  obtained  a 2.3  (Appendix  28  The  elements  [I*]  " "  {H*}  =--  [i*]  •  {H*}  =  of  [I*],  s a t e l l i t e  configuration  appendage  p i t c h ,  in  Appendices  G,  r o l l H,  I,  [  I  J .  /  I  s , x  {H}/(I  {H*},  under  slew  ]  * J)  ;  S ,X  [i*],  {H*}  consideration  sequence  are  for with  the  s p e c i f i c  the  respectively  l i s t e d  3.  3.1  Computational The  using  an  routine method  of  (DGEAR with  The  l i b r a t i o n a l  for  coded  in  of  data  freedom. are  where  succeed,  while  FORTRAN  c o n t r o l  "  7  The  the 6  a l l  coupled integration  integration  implicit  represented  by  three  corresponding  Using  the  to  three  conventional into  degrees  of  value  as  an  i n i t i a l  it  is  necessary  six  freedom  to  derivatives.  double-precision  Adams'  .  8  transferred  updating  using  is  6  and  numerical  on  equations  simultaneously to  based  system  equations  equations  procedure  available  the  for  computer.  is  error  of  nonautonomous  d i g i t a l  d i f f e r e n t i a l  the  DISCUSSION  programmed  I.M.S.L.)  degrees  f i r s t - o r d e r  the  -  b u i l t - i n  second-order  solved  are  470-V8  dynamics  procedure,  nonlinear,  motion  AMDAHL  AND  Considerations  governing  equations  RESULTS  are  problem. use  The  For  the  latest  program  is  variables  throughout.  3.2  Effect  of  Several considered angular  to  Appendage appendage  selected  appendage  and  The  3.2.  l i s t e d  in  time  evaluate  displacement,  Appendix  History history  their  effect  velocity  time  equations  Time  and  h i s t o r i e s for  the  K.  29  functions  were  on  response.  system  acceleration are  various  shown time  in  for  the  Figures  h i s t o r i e s  The  3.1 are  Figure  3.1  Appendage time h i s t o r y with f i n i t e i n i t i a l velocity: (a) exponential displacement; (b) v e l o c i t y ; (c) acceleration.  Appendage Time History Parameters CC;  =  -90°  a  f  =  0°  T  a  =  0.  =  .001  ff  f  =  T  orbit normal  orbit p  =  0  (d)  /  /  / /  /  sinusoidal ramp  T  (e)  T  (f)  _ 0.0  1.0  orbits 3.1  (cont.)  2.0  *10"  3  Appendage time h i s t o r i e s with f i n i t e i n i t i a l v e l o c i t y : (d) s i n u s o i d a l or ramp displacement; (e) v e l o c i t y ; (f) a c c e l e r a t i o n .  Figure  3.2  Appendage time h i s t o r y with zero i n i t i a l v e l o c i t y : (a) exponential displacement; (b) v e l o c i t y ; (c) acceleration.  33  Appendage Time History Parameters  orbit normal  = -90" = 0°  orbit  = .001 orbit Pi  = r  p  =  vertical  0  (d) sinusoidal y/ /'S  ramp cubic  1  (f)  0.0 Figure  3.2  (cont.)  1.0  2.0  orbits  10  -3  Appendage time h i s t o r i e s with zero i n i t i a l velocity: (d) sinusoidal, ramp, or cubic displacement; (e) velocity; (f) acceleration.  34  Figure h i s t o r i e s motions. i n i t i a l  of It  limited  as  not  even  it  to  can be  on  the  concluded  are  motions  slew  on  the  the  that to  become is  duration  of  response. to  the  no  preferable  f i n i t e  such  slew  v i r t u a l l y  a to  response  has  slew  s a t e l l i t e  i l l u s t r a t e s  motion  have  of  time  of  with  acceptable  system  and  durations  imposed  ensure  changing  velocity  3.5(a)  h i s t o r i e s of  motion  i s  of  0.001  responses  slew  slew  seems  Thus  3.2  be  a  the  the  time  limiting  amplitude  effect of  period  by  two  effect  on  the  orders  time  h i s t o r i e s  slew  motions  with that  start".  duration  It  Figure  Note,  Figure  the  shows  cause  must  to  motions  usefulness  the  i n i t i a l  time  slew  3.4  magnitude  "jump  that  readily  the  response  various  to  starting  l i b r a t i o n .  zero  can  system  with  maneuvers  of  response.  3.1  r e s t r i c t i o n s  slewing  smooth  the  seen  C l e a r l y ,  h i s t o r i e s  of  Figure c a n be  Figure  a  shows  velocity  unstable.  the  3.3  to  affect  the  and  begin  for  smooth  motions  variations  Figure  3.2  for  orbit  while  same  to  to  of  to  note  the  of  deviate  time 0.1  s a t e l l i t e  response  slew  system  durations  interest  for  the  the  of  system  shows  0.3  the  smooth  T  slew  motions  of  time  have  negligible  becomes short effect  the  slew  but  o r b i t s ,  smooth  duration, on  system  extended  respectively. h i s t o r i e s Note,  remain  (approximately relatively  present  with  time  h i s t o r i e s  smooth  motion  substantially.  duration  f l  to  3.5(b),(c)  h i s t o r i e s  response  short as  appendage  Figures  and  that  response  large.  time  the  the  0.1  do  same  orbit) Thus  history  response.  The  35  Satellite Parameters = 0.06 = 0.06 = 0.06 = ( 0 0 1)  V r  Appendage Time History Parameters = -90« = 0'  Initial Conditions V(0) = V'(0) = 0 0(0) = crV(O) = 0 A(0) = A'(0) = 0  orbit  a  Eccentricity Tp =  orbit normal  0  e = 0  a  vertical  exponential T „ = .072 orbil T „ = .073 orbit  (b)  sinusoidal T „ = .005 orbif T „ = .006 orbit  (c)  ramp T „ = .003 orbit T _ = .004 orbil  orbits Figure  3.3  System response showing e f f e c t of appendage time h i s t o r i e s with f i n i t e i n i t i a l velocity: (a) exponential; (b) sinusoidal; (c) ramp.  36 Satellite Parameters  V r  = 0.06 = 0.06 = 0.06 = ( 0 0 1)  Appendage Time History Parameters =  -90'  =  0'  Pi = Pi =  fB  = 0  Initial Conditions f(0) = V'(0) = 0 0(0) = 0'(O) = 0 A(0) = A'(0) = 0 Eccentricity e = 0  orbit normal orbit  a vertical  sinusoidal T„ = 1Q- orbit 3  T„ = KT orbit 5  ramp T„ = 1Q-orbit T„ = 10 orbit 3  _5  cubic T„ = 1Q-orbit T„ = 10 orbit 3  _5  orbits Figure  3.4  System response showing e f f e c t of appendage time h i s t o r i e s with zero i n i t i a l velocity: (a) s i n u s o i d a l ; (b) ramp; (c) cubic.  37  Satellite Parameters  <r  =  Co  = 0-06 = 0.06 =(0 0  s  7]  F  Appendage Time History Parameters = -90'  0.06  a, 1)  =  Initial Conditions V<0) = f (0) = 0 0(0) = 0'(O) = 0 X(0) = X'(0) = 0  0' Eccentricity  0. = /S = - = f  T  0  £ = 0  orbit normal  orbit  a a ry r vertical  exponential sinusoidal romp  cubic  orbits Figure  3.5  System response showing the effect of duration of slewing maneuver with zero i n i t i a l velocity: (a) 0.001 orbit; (b) 0.1 orbit; (c) 0.3 orbit.  38  system  response  during  slew  i s  motion.  gravitational  hence  to  time  A  close  motion It  duration, the  same  h i s t o r i e s  that  conditions  slewing  maneuvers,  "time"  of  despite  the  of  responses  during  given  history  variation  i t s  system  motions  at  the  durations.  i s  slew  time  in  torque  duration,  exercise  response  3.5(a)  system  longer  lengthy  libration  the  disturbance  deviation  smooth the  a  to  system  Figure  of  by  motions  with  the  a l l  regardless  Thus  the in  in  Figure  of  chosen, end of  after  3.6.  short result  the  librations  response  slew  in  slew  during  the  slew  motion  i d e n t i c a l . As  state the  of  in  motion.  is  slew  increased  up view  seen  governed  h a s more  the  i l l u s t r a t e d  c a n be  For  f i e l d  influence, various  mainly  the  smooth  response  attention Figure  p i t c h ,  with  except is  3.7 a  time  h i s t o r i e s  for  focused shows  long the  of  time  Effect  and  appendage  orientations  expected,  effect  the  motion  more  i s  Furthermore, traverse  a  it  of  slew  s p e c i f i c  period,  are  longer  concluded  path,  )  steady now o n  as  slew  on well  moment r  as  the  longer  A s c a n be  the  resulting  slew  3.7a). duration  to  amplitude  of  l i b r a t i o n . Figure of  the  slew  3.7(b)  i l l u s t r a t e s  motion  on  system  the  effect  response.  It  of  in  i n i t i a l  during  (Figure  a  motion  system  considered.  that  smaller  from  appendage  ( T  the  function.  gravitational for  affect  history,  duration  pronounced c a n be  slow  effect  cubic  not  cubic  response. final  of  on  the  s p e c i f i c  a  do  the  shows  magnitude that  the  39  Satellite Parameters  r  = 0.06 = 0.06 = 0.06 = ( 0 0 1)  Appendage Time History Parameters a  .  a  f  r  a  = -90" =  u  orbit normal  ^~~\  orbit  0°  = .001 o r b i t  0i =  Initial Conditions V(0) = V'(0) = 0 0(0) = 0'(O) = 0 A(0) = A'(0) -= 0  =  r  p  =  0  Eccentricity e = 0  Q  ^/vertical  exponential sinusoidal ramp cubic  -3600  Figure  3.6  System response during slew motion for time h i s t o r i e s : (a) pitch response; (b) pitch rate response.  various  40  Satellite Parameters  C Ca s  77  T  = 0-06 = 0.06 =0.06 = ( 0 0 1)  Appendage Cubic Time History Parameters  IS, =  fi,  = T,  =  0  Initial Conditions V(0) = V'(0) = 0 0(0) = 0'(O) = 0 A(0) = A'(0) = 0 Eccentricity € = 0  orbits Figure  3.7  System response showing e f f e c t of appendage p i t c h : (a) duration; (b) i n i t i a l p o s i t i o n ; (c) final p o s i t i o n .  41  longer  the  duration  path  of  that  time,  must  be  traversed  the  greater  the  the  appendage's  within  resulting  a  s p e c i f i c  amplitude  of  l i b r a t i o n . Influence system  of  response  orientation determines  of  of  the  to  for  s a t e l l i t e  axis  of  tends the  the  to  align  center  of  relative  to  state  the  of  Figure pitch  of  It  the  the is  change  the  slew  the  3.9  motion.  in  symmetric  of -  the  hence  moment  by  The the  of  a  of  the  inertia  orientation  with state  of  the  gravitational  inertia  position  determines  axis  of  the  The  body  equilibrium  the  shows a l l  the  coupling  yaw yaw  the  is  p i t c h to  is  a  no  slew  note  moment  towards  the  appendage  equilibrium  v e l o c i t y motion  that  do  phase  the  not  state  the  shown  in  symmetric  Note as  on  is  result  s h i f t .  equilibrium  in also  a  affected  by  o r i e n t a t i o n . response  due  l i b r a t i o n a l e x i s t s  apparently  gives  orbital  appendage  there  relative  Now  Strong  s a t e l l i t e  l i b r a t i o n  and  s a t e l l i t e ' s  s a t e l l i t e ' s  appendage's  excited.  d i a d i c ,  moment  interesting  motions  Figure  central  inertia  body  3.7(c).  the  axes.  Therefore,  Figure  on  s a t e l l i t e .  response  the  of  central  symmetric in  body  in  orientation  to  minimum  minimum  force.  during  slew  of  determined  moment  the  3.8.  is  the  Influence response  axis  relative  inertia  s a t e l l i t e  minimum  i l l u s t r a t e d  appendage  s a t e l l i t e  respect the  v i v i d l y  the  the  orientation  is  relative  to  degrees  between  r o l l  unlimited.  indication  appendage  as  to  of  r o l l  freedom  and  yaw,  are  and  Although the  s t a b i l i t y  of  42 Satellite Parameters  C Ca s  77 r  Appendage Cubic Time History Parameters  Initial Conditions V(0) = f'(0) = 0 0(0) = 0'(O) = 0 X(0) = X'(0) = 0  0= 0.06  T  Eccentricity  = ( 0 0 1)  0. =  = 0.06  =  0 6  Q  = .001 orbi t fii = = 0 T  orbit normal  er = 0 orbit  orbit normal  orbits  Figure  3.8  System p i t c h :  response showing e f f e c t of appendage (a) i n i t i a l p o s i t i o n ; (b) final p o s i t i o n .  43  Satellite Parameters <r  = cos  (  = 0.06 0.06 = ( 0 0 1)  s  Appendage Cubic Time History Parameters  Co 7] T  = .001 orbit a  i  = «f =  T  a =  0  Initial Conditions = ^'(0) = 0 0(0) = 0'(O) = 0 X(0) = X'(0) = 0  orbit normal  f(0)  Eccentricity € =0  '"^ orbit  vertical  /3i=0,/3,=90  j3i=-90,/g,=0 <3,=90,^,=0  orbits Figure  3.9  System motion  response in r o l l :  response;  (c)  due to appendage slew (a) p i t c h r e s p o n s e ; (b)  yaw r e s p o n s e .  r o l l  44  a  s a t e l l i t e ,  when  both  does is  Of  course,  general,  r o l l  and  in  pitch  the  in  yaw  p a r t i c u l a r l y Figure  3.12  cases  response  of  and  slew  orbit.  This  a  executing  r o l l  in  the  slew  trajectories  appendage  same  appendage  slew  of  course,  h i s t o r i e s  of  duration,  the  It  to  effect is  and  The  attributed  of  same the  to  the  the For  to  over  give  that  in  s p e c i f i e d  motions  pitch  of  the  response.  the  in  slew  different  The  f i n a l  torque  s a t e l l i t e  variation  orbit,  r=0.00l  the  orientation  by  response  of  by  disturbance  slew  5=0.001  total  s a t e l l i t e  pitch  0.0005  sequence  governed  the  the  executing  followed  deviation  appendage.  effect  to  with  slew  l i b r a t i o n a l  significant  on  excited,  r o l l .  0=0  pitch  the  r o l l  mainly  diadic  in  motion  expected,  represent  from  the  an  orientation.  appendage  case  the  are  undergo  slew  be  different  lines  motion  is  can  appendage  the  the  As  freedom  of  to  general  appendage  S o l i d  third  motion.  the  the  slewing  leading  dynamics inertia  of  against  i . e . ,  l i t t l e  s a t e l l i t e  3.11.  same  slew  duration.  have  l i b r a t i o n a l  i s ,  the  reversed,  such  simultaneously  then  over  for  and  with  f i r s t  is  attain  considered.  appendage  motion  to  the  the  F i n a l l y ,  simultaneously  effect  contrasted  a x i s ) . free  shows to  spin  (except  is  degrees  the  the  accuracy  appendage  to  motion  orbit.  the  3.10  s a t e l l i t e  is  with  sensitive  leading are  pointing  Response  l i b r a t i o n a l  trajectories  r o l l  motion  Figures  a l l  Three  aligned  orientation.  now  and  its  instrument  presented  with  affect  an  arbitrary is  it  the of  appendage  from  the  response slew  time  longer path  on  the  45  Satellite Parameters  C Ca s  7]  r  Appendage Cubic Time History Parameters  = 0.05 = 0.06 = 0.06  =(oo  1)  T  0  = /?, = 0° = .001 o r b i t = .001 o r b i t  orbit normal  Initial Conditions f ( 0 ) = V'(0) = 0 0(0) = 0'(O) = 0 A(0) = X'(0) = 0 Eccentricity € = 0  orbit  vertical/vA . N  I  a  f » Pf  a, = -30  a, = 30 (a)  (b)  g,=30,/?,=30 o,=30,ft,=-30 g,=-30,/3i=30 g,=-30,/S,=-30  orbits Figure  3.10  System response due to appendage p i t c h slew motions: (a) pitch response; (b) response; (c) yaw r e s p o n s e .  and r o l l  r o l l  46 Satellite Parameters  Co r  = 0.06 = 0.06 = 0.06 = ( 0 0 1)  Appendage Cubic Time History Parameters  «i = /S, = 0«  = .001 o r b i t = .001 o r b i t  orbit normal  Initial Conditions f(0) = f '(0) = 0 0(0) = 0'(O) = 0 A(0)=A'(0)=0 Eccentricity E = 0  ^ | - N t A^U o  r  b  ;  vertical/v-V^ * >^ , °f » Pi v  10 (a)  g,=30,/g,=50 q =30,/3,=60 (  Q,=60,/5,=3O  orbits Figure  3.11  System response due to appendage p i t c h slew motions: (a) pitch response; (b) response; ( c ) yaw r e s p o n s e .  and r o l l  roll  47  Satellite  Appendage  Initial C o n d i t i o n s  Parameters  Cubic Time  TKO)  = V'(0)  History P a r a m e t e r s  0(0)  =  =  0.06  =  0.06  a  s  =  /Sj  = 0°  •n  =  0.06  a  f  =  /S  = 30°  V  =(0  0  1)  f  T(total) =  .001  Eccentricity orbit  = 0  0'(O) =  A(0) = V ( 0 )  orbit normal 0  •'"^ o r b i l  = 0 vertical y v -  € = 0  10 (a)  -10 10 (b)  a  and ft  a,then ft ft.then a orbits Figure  3.12  System response showing e f f e c t of path_ of appendage d u r i n g slew motion: (a) pitch response; (b) r o l l response; (c) yaw response  48  s a t e l l i t e the  gravitational  3.3  Effect Of  of  response  of  the  S a t e l l i t e  appendage.  influence The on  of  trends  are  physical  slender  slewing  relative  body  is  l i b r a t i o n mass  the  s a t e l l i t e  s a t e l l i t e  of  mass  As  the  what the  the  (Figure  3.13b).  most  c r i t i c a l  parameter  The  d e s t a b i l i z i n g through  governed  inertia  body  the  mass  effect careful  by  ratio  (TJ)  as  the  pitch  of  the  central the  gradient  of  of  appendage  play  s a t e l l i t e  selection  larger  slenderness the  more  hand,  a  assessing  the of  to  the  gravity  based  becomes  other  ratio  in  ratio  on  expect  course,  inertia  indicates  amplitude  leads  Of  body,  dynamics.  body  the  and  to  3.13  would  increases,  the  is  Figure  On  influence  central  l i b r a t i o n a l  appendage  because  the  one  slender  of  body  l i b r a t i o n a l  3.13a).  appendage  the  axial  central  Clearly,  moment  and  be  central  of  ratio.  on  with  the  (ratio  grow.  body.  compensated  would  (Figure  the  appendage/central  h i s t o r i e s .  their  s t a b i l i t y ,  s t a b i l i z i n g  their  a  design  appendage  s a t e l l i t e )  of  Geometry  with  because  before.  and  t  parameters  markedly  l i b r a t i o n s  ,  considerations.  motion  amplitude  the  consistent  (dumbbell  diminishes  of  these  in  <  pronounced  mentioned  Inertia  are  TJ,  and  more  associated  These  inertia  respectively)  as  importance  parameters  transverse  become  moment,  c r i t i c a l  inertia  w i l l  lesser  response the  the  ($ ) a  roles can  and as  be  appendage  time  49  Satellite Parameters  App. Cubic Time History Parameters  Initial Conditions V(0) = ^'(0) = 0  orbit normal  0(0) = 0'(O) = 0 a, a r  = -90° = 0° = .001 o r b i t /?. = fi, = = 0  X(0) = X'(0) = 0  r  =(oo  a  1)  T  orbit  a  f  Eccentricity € — 0  !A vertical  < = 0.06 rj - 0.06 0  77 = 0.06 77 = 0.09 77=0.12 orbits Figure  3.13  System response showing e f f e c t of: (a) central body i n e r t i a ratio; (b) appendage inertia ratio; (c) a p p e n d a g e / c e n t r a l body mass ratio.  50  Figure i . e . ,  the  response  3.14  appendage with  i l l u s t r a t e s plane of  on  the  the  appendage  that  consideration of  desirable  due  induces  the  planar  s a t e l l i t e  by  and  appendage  moment  pitch  variation  Note of  that plane  placement  response,  degree  of  of  show,  on  for  is the  combination  of  system  l i b r a t i o n a l  response.  -  a  must  be  and  responses location plane  (Figure due  plane  central  careful  of  the  across  slew  orbital  body  as  well  3.16).  However,  reluctance  which  the of  a  the in  lead proved  s a t e l l i t e  t h e <j>  location.  determination  may  system  (Figure  Clearly,  influences  at  a  as  plane  hinge  parameters  as  of  motion  out  directed  the  location  s i g n i f i c a n t  was  a  Figures  response  (Figure  plane  for  achieve  p a r t i c u l a r l y of  It  coupling.  r o l l  the  i n e r t i a .  hinge  respectively,  within  of  3.14c)  to  determines  thus  r o l l  of  consequence  orientation  s a t e l l i t e .  out  orbital  given  the  hinge  the  3.14(a)  the  turn,  desired  appendage  the  attention  The  and  orbital  which  freedom  for  3.14b)  location the  a  in  moment  i n e r t i a ,  motion  (Figure  hinge  stable.  minimum hinge  of  the  prescribed  system  rather  of  state  3.15,3.16  the  axis  in  on  Figure  state  which,  location, body,  location  equilibrium diadic  hinge  motion.  hinge  obtain  the  the  slew  to  order  across  Next,  central  in  motion  with  on  the  out  Figures  of  of  to  plane.  the  effect  location  show  orbital  3.15)  the  equilibrium  s a t e l l i t e  affected  of  minimum  3.14(b),(c)  the  of  inertia  orientation  axis  pitch  s a t e l l i t e ' s  s a t e l l i t e  the  placement  influence  the  suggests  the  exhibits  to  to to  of  a  unstable be  tumble  when  51 Satellite  Appendage  Initial Conditions  Porameters  Cubic Time  f(0)  History Porameters  0(0) = 0'(O) = 0 A(0) = A'(0) = 0  <t  = 0.05  a,  <  0  = 0.06  a,  77  = 0.05  T  s  = -90°  = f ' (0) = 0  orbit normal orbit  0= a  = .001 o r b i t = p = - 0 f  T  Eccentricity vertical  1=0  r  = (-101)  L_=iz-L°l) 11=1,501  r = (101)  r  = (o  -11)  r  = (0  01)  r = (0 r = (0 orbiis Figure  3.14  System motion (a) (b)  r e s p o n s e due to appendage p i t c h as a f f e c t e d by h i n g e location:  in the orbital ,(c) across the  plane; orbital  plane.  slew  -s  1)  11)  52  Satellite  App. Cubic Time  Parameters  History Parameters  r (o)  Initial Conditions = 0  orbit normal  0'(O) = 0  <r  = 0.06  <t 77  =  s  Q  = 0.06 0.06  P\  = -90°  Pi  =  P  T  a-.  0°  ver i i c a l  = .001 o r b i t a  f  =  T  orbit  A'(0) = 0  a  = 0  Eccentricity € = 0  r  =(-iop  lLz±-A.°J> r...=.(..i.p..D.. orbits  Figure  3.15  System r e s p o n s e due to appendage r o l l slew motion as a f f e c t e d by h i n g e l o c a t i o n in the orbital plane: (a) p i t c h r e s p o n s e ; (b) r o l l response.  53  Satellite  App. Cubic Time  Initio! Conditions  Parameters  History Parameters  f(0) = f'(0) = 0 0(0) = 0'(O) = 0  C Co s  v  -  0.06  =  0.06  =  0.06  P\ Pi T  P  = -90° =  0°  =  .001  a  f  =  T  X(0)  =  orbit normal  orbit  A'(0) = 0 vertical  orbit a  Eccentricity  = 0  £ = 0  r  =(Q-I-I)  ______*_  r  = (o -s D  orbits  Figure  3.16  System  response  motion  as  orbital  plane:  response.  due  affected (a)  to by  appendage hinge  pitch  r o l l  location  response;  slew across  (b)  r o l l  the  54  subjected 3.17. and  to  Even  a  disturbance  faced  with  an  the  suggests  unlikely  slew  when  geometry,  This  smooth  to  s a t e l l i t e  that  a  tumble  adverse  response  conventional  from  the  is  evident  in  combination  of  inertia  be  stable.  continues  s a t e l l i t e  action  of  a  to  would  smooth  Figure  indeed  be  slewing  maneuver.  3.4  Effect  of  Figure prescribed impulsive and  Impulsive  3.18  shows  appendage  the  i l l u s t r a t e s  inplane  impulsive in  the  i n i t i a l  v e l o c i t y .  pitch  This  is  response  influence order  of  from  s a t e l l i t e  to  disturbance plane. pitch  As  of  out  out of and  the  r o l l  show, slew plane  3.19  that by  the  a  judiciously or  negative  i n s e n s i t i v i t y  seem  to  suggesting  be  any  lower  terms.  respectively, motion and  impulse.  to  disturbances.  not  response,  by  Figure  positive  does  with  across  suggests  obtained  desirable  obtained  there  3.18a)  response  suggests  impulsive  coupling  3.20  Figure be  as  by  (Figure  pitch  reflects  plane  plane  orbital  can a  of  be  with  affected  3 . l 8 b , c ) .  appropriate  to  the  before,  with  an  as  of  This  can  s a t e l l i t e  plane  (Figures  3.18(b)  appendage  amplitude  appendage  at  a  motion  orbital  Figure  the  of  s e n s i t i v i t y  suprising  3.19  in  the  amplitude  magnitude  Figures  slew  plane  the  appendage  somewhat  response  disturbances.  pitch  slewing  of  in  orbital  3.18(a)  reduction  the  pitch  disturbance  across  Disturbance  that  an  a  also  of  impulsive  the  judiciously It  response  orbital  reduction slewing  shows  that  in the  a  55 Satellite  Appendage  Initial C o n d i t i o n s  Parameters  Cubic Time  iKo) = r(o)  = 0  4  History P a r a m e t e r s  0(0)  = 0'(O)  = 0  J-N  A(0)=A'<0)  = 0  <c  V  V  =  1.90  =  0.001  = 0,  =  0.25  = / S = 60 =  =  (0  0  1  = 0« Eccentricity  f  T  a  =  =  .001  orbi t  e  =  orbit  (Ty—  vertical/v-V^  0  « f » Pi  AAA"'  90  r  orbit normal  .  I  ij ii ii ii ii  orbits Figure  3.17  System r e s p o n s e f o r an a d v e r s e c o m b i n a t i o n of s a t e l l i t e inertia and geometry: (a) pitch response; (b) r o l l r e s p o n s e ; ( c ) yaw r e s p o n s e .  56 Satellite  Appendage  Initial C o n d i t i o n s  Parameters  Cubic Time  f(0)  = 0  History Parameters  0(0)  = 0  <r  = 0.06  =  C  =  0.06  =  0°  77  =  0.06  =  .001  T  =(0  p\  = r  s  a  0  1)  01  -90°  A(0) = A ' ( 0 )  orbit normal  = 0  orbit  a: orbit  Eccentricity  = 0  c =  p  0  (b)  ^'(0) = 0  (c)  0'(O) = - 6 0'(O) = - 3 0'(O) = 0_ 0'(O) = 3_ 0'(O) = 6 0  2  orbits Figure  3.18  System r e s p o n s e due to appendage p i t c h slew motion as affected by an i m p u l s i v e disturbance: (a) in the o r b i t a l plane; (b) , ( c ) a c r o s s the orbital plane.  Satellite  Appendage  Initial C o n d i t i o n s  Parameters  Cubic  f(0)  Time  History  Parameters  = 0.06  = -90°  = 0.06  =  V  = 0.06  =  r  =  a  ( 0 0 1)  = 0  0(0) = 0 ' ( O ) = 0 A(0)  = A'(0) = 0  0°  vertical  .001 o r b i t f  = T  = 0  d  Eccentricity € = 0  P f \ 1/3-  •^'(0) = -6 deg/rod V-'(0) = - 3 deg/rod  ^'(0) = 0 deg/rod ^'(0) = 3 deg/rod f{0)>_= 6 deg/rod  orbits  Figure  3.19  System  response  motion  as  in  o r b i t a l  (b)  the  r o l l  due  to  affected'by plane:  response.  appendage an (a)  r o l l  impulsive p i t c h  slew  disturbance  response;  Satellite Parameters <  s  Appendage Cubic Time History Parameters = -90° Pi = 0° Pi = .001 orbit a: = O = T = 0  = 0.06  <o  = 0.06  77 F  = 0.06 = ( 0 0  1)  f  a  58  orbit normal  Initial Conditions V'(0) = 0 0(0) = 0 A(0) = A'(0) = 0  i>(o) =  orbit  vertical  Eccentricity € = 0  PK\P\  0'(O) = - 6 deg/rod ^'(0) = - 3 deg/rod ^'(0) =  0 deg/rod  <f,'(0) = 3 deg/rod #'(0) =  6 deg/rod  orbiis  Figure  3.20  System  response  motion  as  across  the  (b)  r o l l  due  affected o r b i t a l response.  to by  appendage an  plane:  r o l l  impulsive (a)  pitch  slew  disturbance response;  59  inplane r o l l  i n i t i a l  response.  disturbances On  the  other  disturbances  do  3.20).  r e l a t i v e l y ,  Thus,  affect,  amplitude  can  impulsive  disturbance.  l i b r a t i o n a l  be  can  increase  3.5  Effect  the  of  r o l l  Orbit  E c c e n t r i c i t y on  the  system,  l i b r a t i o n a l The  perigee that  the  motion  may  induce  motion  at  the  r o l l  only  In  a  with  small an  fact,  during  the  response  reduction  inplane  or  presence  appendage  impulsive  r o l l  in  out  r o l l  of  of  (Figure  plane  i n i t i a l  slew  motions  the  orbit  introduces  leading  to  a  a  forcing  worsening  of  function the  response. of  orbit  orbit  is  adversely  s i g n i f i c a n t  s l i g h t l y ,  the  amplitude.  of  as  it  shown  the  in  response affected  shows  outright  e c c e n t r i c i t y  appendage  l i b r a t i o n a l  is  transverse  on  E c c e n t r i c i t y  performing  of  effect  the  generally  effect  s a t e l l i t e  neglible  hand,  achieved  v e l o c i t i e s  have  that  tumbling  perigee  of  the  r o l l  Figure of by  of  slew 3.21.  the orbit  the  on  motion It  at  can  be  of  of  a  the seen  during  e c c e n t r i c i t y .  s a t e l l i t e  orbit.  response  s a t e l l i t e  presence a  the  slew  This  is  e c c e n t r i c i t y performing  slew  60 Satellite  Appendage  Parameters  Cubic  s  a  0.06  =  0.06  Pi  =  0.06  T  =  =  (  =(0  T  =  =  77  0  1)  orbit normal  Time  History  C <T  Initial C o n d i t i o n s  a  Parameters  V(o) =  -90°  0(0) A(0)  0° .001 f  =  T  f(0)  =  0  =  0'(O) =  0  =  X'(0) =  0  orbit d  =  0  0.0 £ - 0.1  orbits  Figure  3.21  System r e s p o n s e due to appendage r o l l slew m o t i o n a s a f f e c t e d by o r b i t e c c e n t r i c i t y : (a) p i t c h r e s p o n s e ; (b) r o l l response.  E =  0.2  8  0.3  ~  4.  4.1  Summary A  analysis useful  general  l i b r a t i o n a l  slewing of  more  insight  into  conditions.  The  conclusions  based  smooth  (ii)  on  time  may  and  The  should  to  slew  The  location  hinge  defines  the  be  the  summarized  of  long  as  and  s a t e l l i t e  inertia  Pitch  slew  motion  hinge,  l i b r a t i o n a l  i n i t i a l  and  the  follows:  with  lead  as  to  f l a t  torque  orientation  response  the  a  a  as  during  motions.  l i b r a t i o n a l  of  as  and  disturbance  which  plane.  and  duration  light  about  o r b i t a l  an  response.  state  any  a  inertia  motions  equilibrium  for  with  provides  analysis  the  motion  the  for  towards  h i s t o r i e s  l i b r a t i o n a l  minimize  appendage  between  of  step  analysis  time  here  s a t e l l i t e  f i r s t  stopping  low  appendage  be  h i s t o r i e s  substantially  influences  (iv)  features  them  starting  possible  (iii)  The  slew  presented  r i g i d  systems.  interactions  salient  a  a  geometry,  Appendage  of  represents  complex  orbit  formulation  dynamics  appendage  parameters,  (i)  REMARKS  Conclusions  relatively  studying r i g i d  of  CONCLUDING  does  not  arbitrary However, the  same  degrees  of  61  the  out  and  hence  s a t e l l i t e  of  location out  motion  freedom.  appendage  occurs.  excite  during  the  diadic  for  hinge  slew  of  For  of  plane in  plane  excites a  r o l l  the location a l l slew  the  62  motion,  irrespective  r o l l  and  yaw  with  the  earlier  l i b r a t i o n a l (v)  (no  provided  suitable  appendage Gravity  time  s a t e l l i t e causes spin  axis.  l i t t l e  to  Gravity reduce  s a t e l l i t e s . is  unable  to  appendage the  As  well,  control  slew  offset  in  caused  that,  in  are  a  plane which  s a t e l l i t e does  c y l i n d r i c a l  libration  by  of  yaw  the  gradient  does  slewing  Out  in  about  of  and  with  s t a b i l i z a t i o n  nor  be  chosen.  libration  gravity  error  can  s a t e l l i t e  accuracy.  l i b r a t i o n  motions,  r o l l  s t a b i l i z a t i o n of  revolve  large  consistent to  s a t e l l i t e s  gradient  yaw  and  parameters  pointing  to  is  pitch,  s a t e l l i t e s .  gradient  results  appendage  This  pitch  s t a b i l i z e d poor  motion  the  in  location,  reference  combination  history  have  with r i g i d  gravity  gradient  appendages  of  tumbling)  with  hinge  excited.  studies  maintained a  the  always  dynamics  S t a b i l i t y  (vi)  are  of  it  s t a b i l i z a t i o n caused  by  eliminate  nonzero  fast the  equilibrium  state. (vii)  4.2  It  appears  affects  the  slewing  maneuvers.  l i b r a t i o n a l  Recommendations This  for  preliminary  insight  into  the  slewing  appendage.  general,  Future  response  However,  behaviour it  e c c e n t r i c i t y adversely  during  Work  investigation  dynamical  the  i s  has  provided  of  only  some  spacecraft a  f i r s t  step  with  a  towards  63  analysis  of  more  suggestions are  (i)  r e a l i s t i c  concerning  recorded  hence  direction  for  complex  models.  A  few  future  investigation  below:  Most  s a t e l l i t e s  damping  of  are  mechanism,  damping, be  and  to  provided in  minimize  considerable  performance  of  addition  to  l i b r a t i o n a l  interest  the  with  to  s a t e l l i t e  some  form  the  internal  response.  evaluate  due  to  of  It  would  the  the  presence  of  damping. (ii)  A  high  many  degree  s a t e l l i t e  developed the (iii)  (iv)  A  of  to  missions.  meet  presence  s t a b i l i t y  of  To  manifold  the  in  free  drag)  may  is  this  s a t e l l i t e  where  a l l  is  active  required  control  accuracy  for  should  be  c r i t e r i a ,  in  maneuvers.  end,  the  space  missions  have  An  important  phase  molecular  accuracy  pointing  slewing  study  undertaken.  Many  pointing  and  concept may  require  reaction  s i g n i f i c a n t  should  integral  prove near  useful. earth  forces  effect  be  o r b i t s ,  (aerodynamic  on  the  s a t e l l i t e  performance. 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C . , "Gear: Ordinary Differential Equation System S o l v e r , " Lawrence Livermore Laboratory, R e p o r t U C I D - 3 0 0 0 1 , R e v i s i o n 3, December 1 974.  APPENDIX  A ~  A N G U L A R MOMENTUM  VECTOR  {H}  s  'm  +  Recalling r r  - c  - c  =  - c  - c  .  -5  .  a  9t  | l a  m„  -c+c3  =  the  J  =  s  l i s  from  s  J  f m  (wxr a  With  m  a  +h  +f  x(f+a)  9t  s  of  ) ~ ot  q  5  =  ( £ x r „ ) * TT/ a o L  F  a  +a  a  d  m  (A.1)  a  that:  2.2.1  ;  domain  drn  s  =  D  J  =  s ni C J X ( C + C s o  0  .  integral  dni a  =  - c  becomes  ux(-c-c  nri  J  g  the  - c  - H  )  ;  dm„ = s  s  a  a  in  --r—  Evaluation  /  ;  7\ r*  (cox?)  since  tn  o  ^  section  +s  integral  s  J  o  (  a  o  dm, = a m =  J  m  + s ) « - c dnt  o )-c  in  the  « x ( - c - c a  +h  +f  J  c3x(g+a)  o  s ...Ah.2)  domain  +h+f+a)  D  a  i s  • (-c+3  a  as  follows:  Y( f + a ) )  dni  a  ,  • (-c+w  a J" c 3 x ( g + a ) « - c m_ a  74  a  x( f + a ) )  +[cox(g+a) •  dni  a  _ w x(f+a)] a  dm  a  75  =  J  3  - c « (_xg+-xa)+  -(_•(f+a))((g+a) =  ;  -c«_xg-c«  a , + ( _ « _  )  -(_• f+_«a) =  /  since  f  ni  a  (_  -c»_xg C'  =  =  (_•_  =  =  (_  a  where  [I  ]  =  a  ( g -f+a • f + a « g  _ _ _ _ - ( _ > • _ ) (Z3_ • g ) a  -(_•a)(_  a  • a ) dm  a  ( a « a )  a  dm a _ (_ • a ) d m a a  -m ( _ • £ ) ( _ a a  -g)  -0  -0  —  ( _ • _ ) ( g • f ) a  +  - ( £ • f ) ( _ « g ) ] a  (_•_ ) (a *a) - ( _ • _ ) ( _ • _ ) a a a. _^ _ m [-c«_xg+ (_xg)•(" xf)] m  a  din a  a  (_xa)•  m  ( _ x a ) dni a a _ . (-,xf c)+ / (_xa)• a m a  a m (_xg)« a  =  _  hand  side  in  _ (_  t h e above  a  x a ) dm . a  expression  follows:  v a ) dm_ = a  J  m a  _ • (ax(w  =  ax  =  a _ . [ I ] ^  frame  body x  a  , y  a  a  v a ) ) dm  a  x a ) dm  a  appendage to  )  (_'a)  on t h e r i g h t as  a  m [-c«_xg a  +/  (_xa)•  a  _ *g)  a  — —  m  + f  m  a  a  ( g - f ) +0 +0 +  J  J"  dm  -0 +rn (_-c3)  -f  be e v a l u a t e d  -a)  = in ( - c « _ x g )  a  can  a  becomes:  f 'm  integral  a  the integral  0  a  The  ) dni  ,  a  din a  a  _xa+(c3«_ )  •a)  a  • q+co  a  _ _ - ( _ * f ) ( -  - ( _ • £ ) ( _  However,  •_  _xa  -  ™a_ +a«a)  '(f+a))  a  _ , ^ (g•f+a•f+a•g+a«a)  m  a  (_«_ )((g+a)  a  inertia , z  a  .  diadic  with  respect  76 The  in  (wxr ) • | f  S  m  integral  a  a  m  dm  a  ot  Substituting  S  domain  ( t j x r ) • ~^rr d m = ot  H  a  equations  vector  {H}  of  with  =  c3«(-c-c  axes  f,  x~c  +m u - g x  =  £j'(m_(-c-c_)  x~c  +m  x  a  , y  a  From locates the  H  onto  a  and  , z  a  [I  to  Figure  the  o  to  the  frame  a  into  x , y , z .  (c3 x f - c )  [ I _ ]Z3_ a a 1 u>  -  + [I  ]<3 3  vector.  Let  The angular  now a p p e a r s  0  (A.l):  a.  momentum  x , y , z  a  g x ( « v f - c )  +m (-c-c +h+[M ]f)  x ( [ M  a  a  ) 3  {H}  =  momentum  as:  ] ( «  a  x f ) ~ c )  a  ]  2.3,  center  +h+[M_]f) a  respect  a  axes  ( A . 3)  ]e3  are  with  respect  to  the  appendage  body  .  instantaneous  (-c-c  body  u  [I ]  angular  a  (i3 x f c )  a  O  the  0  u> 3  iri ( u x g ) • a a  )  s  where  (A.3)  m  m (-c-c )x-c a  +  and  =  respect  +[M ]  (A.2)  ni « x ( - c - c ) « - c s o  represents  projection  becomes:  a  S  where  thus  = m (c3xg) • (c3 x f - c ) +c3- [ I  a  from  D  of  it  is  apparent  central  center  locates  of the  instantaneous  body  mass  C,  center center  that  mass  O  vector  (-c-c  with  respect  g  and that of of  to  vector  appendage mass  )  C.  mass  0_ a  with  Note  that  The x , y , z  {H}  =  d  = - c .  s  angular  momentum  c a n now b e w r i t t e n  m  s  d  s  x d  s  +  m  a  d  a  vector  with  respect  to  body  axes  as  x  ( [ M j  (_ xf) a  +  d  g  )  +  [ M  a  ] [ I  a  ] _  a  .  APPENDIX  The  s a t e l l i t e  =  [I ] s  +  f  a  m [  d  s  ]  SATELLITE  matrix  theorems  +  M  -  inertia  p a r a l l e l - a x i s  [I]  B  [  a  l  H  s  a  M  . d  ]  for  T  +  [I]  - d a  m  is  moments  [ E ]  s  INERTIA  d  s  V  [  computed and  [I]  using  products  the  of  inertia,  ]  T s  a  5  MATRIX  [  E  ^  ]  a  V  '  1  where:  [I]  s a t e l l i t e body  [l ] g  inertia  axes  central  body  x  a  , y  a  , z  unit  d  position  position Figure  m m  g  to  s a t e l l i t e  diadic  with  respect  to  diadic  with  respect  to  coordinate  ;  axes  matrix x a  ' y  a  relating '  z a  body  axes  x,y,z  ?  matrix;  Figure d^  a  transformation  [E]  respect  x „ , y „ , z : s s s  inertia  coordinate  s  inertia  axes  appendage axes  [ M l a  with  x , y , z ;  coordinate  [I_]  diadic  vector  from  C  to  0 ,  from  C  to  0  - c - c  g  Q  ,  2.3; vector 2.3;  mass  of  central  body;  mass  of  appendage.  78  .  - c - c  +h+[M ]f, =  to  APPENDIX  From  where  The  f o r the system  c^ =  and  U  = c^/2  =  e  +  T  ;  ;  .  3 C  angular  [M]  T  T  potential  by  {u} {H}  {/} [I]{/}  and  {q}  velocity  +  {N}  vector  c a n be w r i t t e n  as  6  c l e a r l y ,  {a>}  where and  = I{O>} [IHO>}  3M/R  {"}  are given  T^  s a t e l l i t e  OF MOTION  2.2.1 a n d 2.2.2, t h e k i n e t i c  sections  energies  C ~ EQUATIONS  [M]  7  = {q}  a n d {N}  T  [M]  T  +6  represent  6 on t h e s a t e l l i t e  body  {N}  T  respective axes  79  projections  x , y , z .  of  {q}  80  The  governing  the  Lagrange  equations  {q}  / 8 T \  a n d {Q}  coordinates  and forces,  o  )  T  {  i  n  o  +  /  c  2  9 U  J  +  f  M  of  generalized  and potential  aTqT  w } T [ I ]  u  { / } T [ I ] { / }  _  the vectors  + i{ } [i]-£j{ }  w  9lqT  /  2  {  /  }  T  [  I  alfl  +  { c j } T { H }  + - - i - ^ ^ H }  u  c  { w }  energies:  ]  alfl  {  /  }  =  {  Q  }  '  i . e . ,  df< ^ { } [ I H ] T  T  -( +  W  U  ^-i-fco} !!]!.}  +  y r f j ^ ^ H }  + __| { } {H}  7  c  )  T  T  )  W  IT{/} [I]{/} = {Q} , M 9lqT T  or dt  {  -  y r l j C " }  +  % 3TqT  But  9TqT  {  a  ,  }  T  =  [ ^ M  )  +{H}}  { / } T [ l ] { / }  T  (H}}  +  {[I]{w}  1  using  respectively.  ^  +  }  T  T  9 T  f o r the kinetic  df h j h -( i _ - | { _ } [ i ] { } i  _  represent  Substituting  {  c a n now b e o b t a i n e d  equation  _d dt  where  of motion  T  " Q) ' {  and a{q)i^}  1  c  a  n be w r i t t e n  as  ]  )  81  [q]  where:  S i m i l a r l y ,  +  [M]  for  j r | y { u }  [ M ] { [ l ] { w } + {H}} H q T  {  /  {[l]{u}  T  M  q  ]  -  3 T q 7  [  N  q  ]  -  8 l q 7  C  {  T  {0}  T  {q}  {0}  T  {0}  M  N  ]  }  T  { O f  T  {q} _ T  T  ;  '  T  }  T  t  l  H  /  -  }  ,  T  the  equations  of  motion  ) - - | { } {[I]{ } +  T  %  [  {0} '  T  let  Substituting  gf(  {0}  =  T  I  {  Q  T  W  U  {H}}  ]  + {H}} + [ M ]  {[i]{u>]  T  + [I]{w} +  {H}}  - - ^ { w } ^ [!]{„} + {H}} + c ^ 9 T | j U } [ l ] { / } =  {Q}  ([M] -  +  T  T  ^ ^ { c o }  + [M] [I]{^} T  At  this  1  )  {[l]{co}  +  point,  {co}  + {H}} + [ M ]  T  - | { / } [ i ] { / } . = {Q} T  T  {d>}  T  must  = [M]  {q}  become:  be  evaluated:  +  {X}6  ;  .  {H}}  82  {to} =  If  [M]  i s  [M]{q}  written  in  +[M]{q}  the  + {N}0  form  [M]  =  m  y  m  then  + {N}0  [M]  =  z  m  X  Note  that  m A  c a n be w r i t t e n  m  where m  x  <  m  x  v  x  =  {q} ^  represents  Applying  T  m xv  vector  similar  of  procedure  =  [q]  [M ] y  a  x  ° y  a  y  o_ z  a„ z.  o  a  X  m  y  ° y  m  z  ° z  X  as  , '  c o e f f i c i e n t s to  [M],  [M]  °x  ,  the  for  remaining  {q}  within  elements  of  where  C l e a r l y ,  [M  [M]  can  be  ]  v  written  [M]  T  =  m xv  o  m yv  o  m zv  o  =  [q]  [M  {N}  {N}  is  written  -f  n.  a y v  yv  d  a  zv  zv  in  the  v  t  ]  ,  m xv  m yv  m  °xv  °yv  °zv  a  if  xv  as  where  S i m i l a r l y ,  a  xv  xv  form  a  y v  a  zv  z v  84  Noting  that  ri  c a n be w r i t t e n  ri  where ri  ,  ri*  =  n  represents  xv  {N}  x  {N}  [N  where  v  ]  {_}  {"}  =  [M ] y  {q}  Substituting  for  equatio'ns  motion  of  +  [ M j  for  {q}  within  as  ,  XV  T  as  [M]  T  c o e f f i c i e n t s  T  c a n be w r i t t e n  [q]  of  v  -  n  , '  J  [N ]{q}  " z v  Now  M  vector  =  =  {q}  T  xv  c a n be w r i t t e n  as  ,  {q}  -J-jico} , 3TqT  become:  1  +  [N ]{q}0 y  {_},  + {N}0  and  3 -rr^ril  .  t.tf  }'  ,  the  85  ( [ q ] [ M „ ]  -[q][M ]  t  +[M] [I] M  [L  On  g  =  v  ]-[M  t  f o r {_},  ])  g  +[M]  +[ N H q } 0  -0[N ]) Q  the equation  takes  {[I]{[M]  +{N}0}  {q}  { [ M ] { q } + {N}<?} +{H}} + [ M ] [ I ] {  +  +[N ]  [ M ] {q}  [q]([M  v  t  ]  T  {q}  Y  [ I ] [ M ] {q}  T  +[M]  T  [ M ] {q}  [I]  {N}  6 +C  +[M]  +([M] -e +c  [N ] y  The  [ L  g  ( [ i ]  g  y  [ M ] {q} y  {N}  e  form  +{H}}  [q] [M ] y  {q}  v f c  ]  " [ M ])  ( [ i ]  {N}  ,  ( [ I ] {N} 6  + {H})  6  6 +{H})  [ I ] [M]  {q}  +[M]  T  [I] [ N  y  ] {q} 6  = {Q} .  gives  t  [N  T  the  [ L ] [ I ] {/} = {Q}  {N}  ] [ I ] {/}  + [ q ] ( [ M „ ]  [ I ] [M] -6  T  [ L  t h e terms  [I] [q]  T  +[M]  y  m  -8 [I]  T  [q]  [ I ] [ M ] {q}  T  {q}  +[M] (  [I]  Rearranging  [M]  + c  [N ]  +  +[M]  0}  [ I ] [M]  -6  T  +{N}  +[q]([M  + {H}) [M]  6  ~ [ M ]) [ I ] [ M ] {q} [N ]  } +  + {N}0  v  +[M] {[I] T  { [ i ] { _ } + {H}}  T  {Q} .  substituting  ([q]([M  +[M]{q}  v  ][l]{/}  { [ l ] { _ } + {H}}  Q  { [q][M ]{q}  T  C  -0[N ])  g  g  "[M ]) g  +[q]([M  ] [I]  v  t  ]  -[M ])([l]  [M] +[M]  +[M] ([i]  +{H})  T  [ M ] {q}  [ I ]  T  {N}  {N}  [ I ] [ N „ ] 6) e  +[i]  {N}  6 +{H}) {q} e  +{H})  ] [ I ] {/} = {Q} .  equations  multiplication  of motion by  c a nbe nondimensionalized  through  86  where  I  a  r  j  i s an a r b i t r a r y  3  nondimensionalize  Noting  the  that  inertia  the equations  {q}  =  {q}  0  {q}  =  {q}  6  nondimensionalization  {q}/6  =  (q}/0  of  to  motion.  ,  +{q}  2  of  {q} =  2  chosen  0  {q}  ,  a n d {q}  results  in  ,  {q}  +  {q}  f  ,  £  where: f  =  e  e  The  equations  of  multiplication  [M]  [I*]  T  +[M]  [I*]  T  +([M] -[N +  c  [M]  T  ]([I*]  e  [L  where:  ]  {N} [I*]  motion  {q}  y  [N  +{H*}) {/}  [I*]  =  nondimensionalized  r  2  [M ] -  are  f  =  ]  c  )  [I*] +[M]  {Q*}  T  r  K  t  ]  y  t  ]  +[M]  ([I*]  {N}  ;  >[M  - [ M  [M]  ,  a  v  by  giving  +[q]([M  +[q]([M  [ l ] / l  ;  e c c e n t r i c i t y .  ^/^-^vh^ ^  ({q}+{q}  [M]  = -2esin0 1+CCOS0  2  = orbit  with  [q]  [1*]  0 / 0  q  +  [I*]  ] ) ( [ l * ]  [I*]  T  ])  [I*]  [N  {N} ])  {N}  [M]  {q}  +  {H*})  {q} f  e  +{H*})  87  Rearranging  {H*}  = !H)/(i  li*)  - • t n / ( i .  r  b  {H*}  =  f H ) / d  a  r  {Q*}  -  fe>/<i  c  =  e  the terms  governing  equations  [M]  [M]  +[M] +  [I*]  T  [I*]  T  (H*})  +[M] +{H*}  f  £  {q}  [q]  +[([M] ) ]  {q}  +[!*]  /6  c  y  [I*]  -[N  {N}  f  )  e);  b  »); b  ^>;  a r b  3M/(R  f i n a l l y  of  {q}  t  ]  ]  [ L  2  -[M ])  v f c  ]  [I*])  g  )  [I*]  -[M  [M]  +{H*}) ]  in  the  nondimensional  as  [I*]  g  {N}  f  0  appearing  +[q]([M  -[N  + c  v  3  results  motion  ]([I*]  e  r  =  2  +[q]([M  [M ] T  a  [M] ])  {q}  ([I*]  +[M]  T  {N}  [I*]  ([N ] v  +[M] ([I*]  {N}  T  {/}  =  {Q*}  .  APPENDIX  The  l i b r a t i o n a l  sequence  {_}  =  of  [M]  \p  {q}  terms  are  (pitch),  +  D  -  LIBRATIONAL  evaluated  <j> ( r o l l ) ,  X  for  TERMS  the  rotation  (yaw).  6  {N}  where:  {q}  =  [M]  {N}  (  ^  4>  X  )  T  ;  m  x  °x  a  x  m  y  °y  a  y  m  z  °z  =  a  c#sX  c  -s<t>  z-  c0sX  n.  c^cX  -s<p  [M]  T  "c0sX  cX  c0cX  ~s<p'  -sX  88  0c X  cX  - s X  [M ] g  =  4  [  M  )  T  "  -s0sX  -s#cX  c^cX  -c#sX  -sX  -cX  -c<j>  0  90  {N}  T  =  (  c0sX  n~~  c<t>c\  -s<f> )  ;  __{N} " T  4  f  N  )  T  -s0sX  - s ^ c X - c <f>  c0cX  -c^sX 0  -si//cX + c<//s0sX  sy>sX +  ci//s0cX  C\JJC<I>  -~{/} " T  -c^cX -si//s0sX  ci//sX -si/>s0cX  -Sl/>C0  ct//c0sX  c^c0cX  -C^S0  s^sX +c^s0cX  s\//cX -c\//s0sX  [M] = g2[M]  =  [M ]  m  v  [M ] ;  [q]  y  o  xv  m yv  U _  The  m  x  m  elements  =  d f  (  m  x  )  o  o  2 V  of  =  therefore  [M ]  a  yv  a  zv  are  v  a  xv  xv  yv  zv  obtained  as  follows:  i// (0) + ^ (-s0sX) + X (c^cX) ,  m. xv  •s#sX  c^cX  S i m i l a r l y ,  for  the  remaining  elements  of  [M ]: y  zv  yv  s0cX  -c0sX  -c<t>  92  0  0  0  -stfsX  0  0  c0cX  -sX  0  0  0  0  -s<t>c\  0  0  -C0SX  -cX  0  0  0  0  -c0  0  0  0  0  0  -s<t>s\  -s#cX  -c<$>  c^cX  -c0sX  0  0  0  0  0  0  0  -sX  - c X  0  0  0  0  APPENDIX  E  ~  Transformation  =  [ M J  TRANSFORMATION TERMS  matrix  [M  xa  m  za  cosa  -sina  tM ] a  =  ]  a  [M ]/6 a  =  m  and  xa  ya  ya  za  za  sinasin/3  sinacos/3  cos/3  -sin/3  cosasin/3  cosacosj3  xa  °xa  a  o  a  za  m  ffi  ya  m o  a  xa  za  xa  -  ya  za  m  za  =  -a  : '  =  a  m xa o  za  ; ' +  B a  H  96  xa  ya  za  *  0  SLEW  derivatives  xa  m ya  where:  its  FOR  xa  : '  MANEUVERS  [M  a  ],  [M  a  ]  97  0  y a  5  o  =  ya  -a  za  =  xa  a  a  *  [ M l  =  [ M j / < ?  a  - a  o m  =  2  a  "  '  °xa  '  t  a  ~  xa  °za  P  0  '  0  xa  xa  xa  ya  o O.y a  ya  o  o m  U_  *  z a  a  •  o  —  z a  i  xa  z a  a  = a  ' +  o  o O  za  o a za  za  where:  tl  o  0  (a  :x a o m ya T O za m m  0  aa  +  - U  (  a  +  =  and  f  (a  = e  ' e  +  -(a  o a xa o a .y a o a za  f  a  £  a  t  +i  O.z a  m  e  2  + a m.  a  za  ;  ;  o O.x a o Oya o  f )  +  f )  a "  f ) e'  o  e  + a • '  a  f  f ) e>  J ^  f  -(a  +  a  f  appears  =  a  e  +  2  m  x  °za  )  -{0  6/d  )  e  e  e'  v  a  ) )  as  "  a  «  +  y  x  a  a z a  +  -2esinfl 1+ecos0  a  "  w  K  ^  (  a  J  *  +  ( i  o ya a  l  "  e  f  x a  where  i  +  )  a  x a  f ) e' e  " *  +  +  xa  F  '  f  e  a  z  a  xa  )  fi  +  - 0 0 K  ; xa  '  ' "<i  consequence  ,  *  +  ;  a  x a x  ;  a  +  y  o  x  z a  a  -  ya  'xa  a  a " '  m  z a  J  -  a  a  5  +  x a  o  a  +  of  e  i  f  e>  0  i za  - / J o K  za  '  ;  nondimensionalization  =  orbit  e c c e n t r i c i t y .  as  The  appendage  and  its  relative  derivative  _  *  angular can  be  velocity written  vector  as:  _  3  *  APPENDIX  Determination center  of  F i r s t ,  the  of  F  -  CENTER  p o s i t i o n ,  OF  velocity  and  dimensions  of  the  two  are  established.  two  are  determined  cylinders  parameters cylinder  $  is  hinge  vector The  .  TJ .  The  by  £  s p e c i f i e d  location  at  the  (h+f ) 3  p o s i t i o n ,  on  the  The  in  center locates  of  the  0  with  a  and  instantaneous  center  mass  Dimensions  the  s  's V  s  w  a  'a V  a  P  From taken  of  terms  be  and  of  of  the  hinge  location  =  £  (£  £ y  )  central  a x i a l  and  then  ,  by  to  0  of  ;  =  length of  central  cylinder  ;  =  volume  of  central  cylinder  ;  =  radius  of  appendage  cylinder  ;  =  length of  appendage  cylinder  ;  =  volume  appendage  cylinder  ;  =  density  symmetric  appendage  central about  body  their  99  f  .  The  2.3).  cylinders  cylinder  the  gives  the  central  2.5.1  h.  determined.  appendage  of  giving  (Figure  g  of  ratio  central  constraint,  radius  of  the  thus  which  acceleration is  the  s a t e l l i t e on  T  of  z  end,  respect  s a t e l l i t e  lengths  the  =  section to  of  body  radii  appendage - i s ,  v e l o c i t y ,  w  acceleration  cylinder  x  located  TERMS  mass  configuration  The  MASS  and  to  central  the  respective  body  appendage axis  z  are and  z  .  100  The  moments  s,z  =  of  2 m w * s s  inertia  are:  axial  2  moment  central I  =  v  TT  =  m  c(3w  +  2  /  2  )  I_  _  =  4  m w  axial  2  =  T T ni ( 3 w + •* a a  /  2  ' ) a  appendage  The  volumes  of  the  cylinders  v s  V  From  section  a  a  =  it  =  t  =  a  (F.3), can  w  2  w  I  s  /  2  a  s,z  a , z  = m  (F.4)  a  /I  / m  and  1  s  given  of  inertia  for  cylinder; moment  of  inertia  for  cylinder.  by:  (F.2)  parameters  s,x  a,x  are  given  (F.5)  as:  (F.3)  '  (F.4)  '  .  rewritten  for  .  /T 7  inertia  . . . . ( F . 1 )  a  '  of  i  inertia  T A  be  l  s  the  K *s  r?  substitutions  *  2.5.2,  s  Equations  "  are  moment  moment  transverse  2  for  cylinder;  appendage ^-» v Of A  inertia  cylinder;  transverse central  of  (F.5)  after as:  appropriate  101 S  6  =  s  3  w  w  +  2  (F.1)  Equation  /  and  with  =  can  =  s  V  3  w  P  V  be  (  substitution  = V V  /_  Substitution  (F.6)  ;  (F.7)  5 w  2  =  +  2  a  /  a  /  V  w  TT  2  a  (F.8)  .  s  /  c  as  )  2 s  (F.9)  ,  (F.8),  from  (F.9)  of  ; 2  rearranged  /  s  /  S  6 ta  2  5  (  it w  p  )  2  (F.6)  into  (F.2)  can  be  rewritten  .  (F.  (F.10)  and  into  10)  (F.7)  gives:  i  w„  ((  =  s  w_  ( (  =  a  V  The  effect  of  the  s a t e l l i t e  V  g  is  V  3  3  to  s  TT2  V^ ff  2  2  *s  2  \  2  (  2 - 5  2  „2  s  )  t  ?  )  —*• p  2  impose  -configuration.  \  2 - 5  (  a The  6  N  *a  (F.11)  .  )'  )  constraint numerical  (F.12)  •  on  the  value  size of  V  g  of would  102  play  a  s i g n i f i c a n t  surface drag. the  area  As  of  does  numerical  value  of  set  considers  density  ratio  and  1  p  =  (F.12)  s P  /„  =  w^ a  =  s  of  known  also  with  be  and  rewritten  TT  — ( 2  2  s  3  equal  (  7T 2  effect,  is  therefore  unity.  equations  this  2.5.1), For  V  (F.11)  equations  the  = °1 and (F.9)  ;  (F.13)  J  )  (F.14)  ;  a  \  2  to  since  (section  to  of  )  r (  density  respectively,  )  —  due  aerodynamic and  the  as:  K  -  where  response  Furthermore,  set  (F.10),  study  arbitrary  substitutions  {  7 , 3  becomes  g  its  consider  bodies  be  3  not  unity.  ( V  affects  to  (F.9)  can  V  aerodynamic  equal  can  and  into  (F.12)  =  ,  p  an  s a t e l l i t e  study  study  w  in  this  conveniently  to  a  role  — t  )  6  )  ;  (F.15)  Cl  ij/  =  ,3 (  (  2  -  $  ) v —  J  •  (F.16)  103  Determination  of  h  hinge  locates  system  x  the ,y_,z  position  vector  position  (Figure  (h  +  P  with  P  along  f.,)  respect  to  coordinate  2.5),  where:  f  locates  0  coordinate  location of  the  with axes  on  a x i a l  the  x  respect  a  .y_,z a a  appendage  end,  to  (Figure  2.5).  cylinder  is  the  appendage  Since  the  confined  to  hinge  the  center  1 04  where:  f  Recalling  instant, same  as  that  a a  (h  +  that  f ) 3  ,y_,z — t o o  projection projection  where:  Note  x  (h  f  3x  "  f  3y  -  f  3z  '  +  f-) «3  i s  of  p a r a l l e l  f  on  on  the  the  axes  a  x a  (/  /2)  a  y a  (l /2)  a  z a  a  a  locates  0  <  *x  w  s  +  <  «  w  s  +  y  <  *  ' s  /  z  2  V  2  (  V  2  (  /  ,z  a  )  a  >  /  2  )  to  >  y a  >  a  z a  is  ,  x a  a  at  x , y , z  respect  (  +  axes  ,y  with  3  x , y , z  body x  </ /2> a  to  >  0  5  1  5  &  ,  any  the  105  Position, of  velocity  and  acceleration  of  instantaneous  mass  Let  the  the  body  position axes  position axes  x , y , z  vector  x , y , z ,  vector  be  be  locating denoted  locating denoted  0_  by  0  by  with  d  .  with d  .  to  Similarly,  respect  These  respect  are  to the  vectors  From  as  mentioned  Figure  C along same  d  position  +  E  =  ( m  p  s  a  of  + c ) ( m  o  in  Appendix  s  =  - c  - c _ o  ;  =  - c  - c  +h  C,  + m  o  +f,  B.  measured  a  )  =  0  +  from  *  a ' < "s "a " <  h  .  3  m  +  s  £ 3  Therefore:  3  a " ilT  ( E  +  h  »  •  0  ,  i s  given  + ( h + f , ) m 3  »•  by  a  along the  the  body  position  2.3:  d  The  e a r l i e r  C  let  a  ( c  center  ,  So  d  g  where:  and  d  g  appear  sx  as:  =  sy  w  =  (  K J l  2  ay az  =  F T  *x  w  (  *  w  (  y  *z  a  xa  >  a„_  ) )  )  za  i  ax  *  s  +  +  s  V  >  /„/2  (  +  az  {  2  ( IJ2  +  a  F T 1 F T  V  (  s  d  ax  +  x  <  TJ+1  sz  *  (  a  l  (  /2  < ' /2 a  2  +  <  >x a  )  )  a _ ) ya )  za  107  Let by  the v_ 5  v e l o c i t y  and  v"  derivatives anomaly  ,  a  the  9.  where:  and  0  with  0  respectively.  of  rate  of  position  The  vectors,  Hence,  77+1  sy  77+1 ( /a /2  )  a  ZZL  )  a  a  a'  xa  x 1  =  sz  /  6  77+1  (  /  a  /  v  =  2  ax  ay  v  V  ay  TT  V  az  az  1 V 77+1 1 77+1 < ' a / 1 (  ax  TT  V  velocity  sx  d  respect  2  77+I < ' a ^  ya za  1  J  k  » xa S  2  > ya  2  » za  J  S  to  C  be  vectors  divided  by  denoted are  the  time  true  108  Let a  g  d  g  the .  ,  It  acceleration is  divided  a  second by  d  of  O  time  with  g  derivative  s  6*  /  a  s x  1  a  s y  3  sy  ,  sz  =  =  IJL. T?+1  (  /  Z2L n+1  (  /  a  = IJ-( / TJ+1  /  a  a  /  k  sz  2  2  /2  )  a  )  a  xa ya  ) a  za  to the  2  sx L  of  6,  a  where:  respect  ;  '  C  be  denoted  position  by  vector  APPENDIX  [I]  was  G  given  ~  NONDIMENSIONAL  in  [I*]  Appendix  s, x  '  (  m  s  /  J  s , x  +  (  m  a  /  X  s , x  +  + +  s  INERTIA  MATRIX  .  '  +  t l  B  SATELLITE  >  }  [  [  h  '  a  *  5  s  S  a  a  [  [  E  -  ]  E  ]  "  5  s  a  5  V  V  ]  ]  '  * ] k  s  [  d  s  [MJ  [ I / ]  k  [  d  /  I  a  a  •  d  s  [E]  [M ]  -  d  s  d  s  -  d  a  d  a  T  •]  T  ]  T  a  •  d  a  [E]  ,  where:  [I  a  ]  =  tl  a  ]  Tjk. s,x  rjk,  0  S 7?k a  109  2  m  /  s  3  w  (  3  s  T  s , x  *  w  a  +  /  2  +  =  1  2  k  l  ; '  2  s  /  /  2 a  )  /  k  APPENDIX  {H}  was  {H*}  =  {H}  '  (  given  m  /  (  m  where:  s  I  a  [  5  M  a  s k  +  [M  a  >  s , x  0  x  d  )  ]  ,  X  (  s  -  (  /  a  V  d  a  0  I  MOMENTUM  VECTOR  /  s  0)  * )  /  ANGULAR  A.  )  5  ( 5 a  X  [M ] d  f  a  s , x  < S  )  + /  a  ( »  t  c  S  /  0  )  )  > »  * s d  a  s  /  x  +  NONDIMENSIONAL  Appendix  i  [ M l d  +  k  (  s  (  =  in  /  +  -  H  ]  a  (  x  [I  [l_*] cl  a  [M ]  *]  5  a  k  s  k  a  [M ].f  x  a  +  v  s  )  , '  /  I-  d  =  g  *  *  a  =  w  a  v  £  a  v  (Appendix  G)  a A r  /  0  (Appendix  E)  /  0  (Appendix  F)  a  =  =  =  d  m  m  s  s  /  I  a  ^  I  s  f  X  s , x  11 1  (Appendix  G)  (Appendix  G)  APPENDIX  I  ~  NONDIMENSIONAL  DERIVATIVE  OF  SATELLITE  MATRIX  Recalling  [i*]  from  =  [ i ] / d  =  k  s  [  S  r  2  [M ]  +  k  a  X  d  +  a  section  * >  s  ;  •  v  i i * ]  [ 2 d  2.6.2,  a  [E]  s  [M  - v  a  -  ]  T  v  +  -  s  [ M l  [ E ] - v  1 12  a  d  [ I  d a  T  v  s  * ]  -  d  ]  T s  [fi]  a  v  T  a  T  ]  .  INERTIA  APPENDIX  J  ~  NONDIMENSIONAL  DERIVATIVE  OF  ANGULAR  MOMENTUM  VECTOR  Recalling  {H*}  from  = *H}/(I =  k  (  s  +  k  +  Noting  {H*}  a  a  k  [M ]  +  [M  k  s  +  k  s  x  v  v  a  X  (  k  (  d v  \  v  s  a  Z*  +  [ M l  [I  a  x  s  a  (  i  a  v  *  =  g  *]  t  +  M  a  ,  {H  }  )  s  >  1/6  a  a  v  " a *  ]  [M ] [I  f  x  )  M  +  5 *  *]  [  a  a  3  1  }  s  .  a  reduces  to  )  s  5 /  a  x  +  [ M l a  0  f  a  f  +  [M ]  {  [M ]  " a *  3  f l  a  )  s  X  [M ]  S  a  {  [M ]  f  (  x  X  x  [  *]  x  +  a  (  g  d  a  x  a  ,  [M ]  {  [I  =  +  s  a  f  a  x  2.6.2  ;  w * ]  a  a  e')  v  a  that  +  S f X  ^a  +  =  section  [  f  f  a  " a *  ]  [M ]  £  a  5*  +  [M ]  x  £  a  [ M  +  +  a  113  ]  .  V  v  )  s  ]  x  }  s  [ M l  a  +  [I  *]  5 * .  [  M  a  ]  1  APPENDIX  The  K  appendage  Aa  =  a-  of  Ao -  Sinusoidal  last  =  time  of  of  •  Aa *  -  (a  (a  T  f  £  a  a  APPENDAGE  slewing  TIME  maneuver  HISTORIES  is  given  by  ,  change  r  in  appendage  of  the  p i t c h .  appendage  Aa  is  pitch  .  a  h i s t o r i e s  with  f i n i t e  i n i t i a l  velocites  history:  (  1  -  exp(  9 /  -20  T  FL  )) .  history:  a^)  history  95%  Aa  ,  - oj)  f  (a^  time  5%  time  (o  FOR  s p e c i f i c a t i o n  time  Exponential  Aa  the  ,  Equations  f i r s t  +  through  parameters  Ramp  during  represents  generated  Aa  EQUATIONS  pitch  a  where  -  t  ),  )  :  -  a ^  - a.)  it  sin(  is  6 /  composed  followed  2(0.95)r  (  A  d  2  of  by  (  + B  T  2  a  a  A  FL  a  linear  function  (during  quadratic  function  (during  +  6 + C  114  ) .  B  ) )  6 ,  ,  for  6  <  0.95  T  for  6 >  0.95  r  115  where  A  =  B  = -2  C  =  Equations  Cubic  1  =  of  (a  =  (a  Ramp  =  (a  time  /  -  f  a  i  r  (0.95 -1)  )  (0.95 -D  )  2  2  a  /  2  (0.95 -1) 2  h i s t o r i e s  )  ((  time  -  i  f  i  with  zero  i n i t i a l  velocites  (  time  f  -  i  history  of  followed  6  /  2  r  2 a  ) -  (  2  9  /  T  6  r  /  3  ))  3 f l  1  -  exp(  -20  2  ))  2  a  history:  a )  f i r s t  3  history:  a.)  (during ,  (  2  a  history:  Sinusoidal  Aa  T  time  Exponential  Aa  (  0.95  time  Aa  /  5%  (  1 + sin(  is  • n & / T  composed  of  r ),  by  a  a  a  of  a  linear  quadratic  a  - i r / 2 ) ) / 2 .  quadratic  function  function  function  (during  (during  next  last  90%  5%  of  a  Aa  =  -  a.)  D  6,  Aa  =  -  a.)  (  E  0  Aa  =  -  a-) l  (  A  6,  2  + 2  F  ),  +  B  6  +  C  for  B  for  0.05r  ),  <  0.05r ; a  a  for  <  <• 0 . 9 5 T ;  6 0  a  ^  0.95T  . a  1 16  Here:  A  1  B  -2  C  The  2  a  (  T  D  1  (  E  ( A  /  r  (  0.95 (  0.95  (  D  T  +  B  0.95  =  i  (0.95-1)  (1-0.05+0.95)  )  (0.95-1)  (1-0.05+0.95)  )  T  2  r  2  a -  a  r  +  a  during  =  +  B  {  through  a  )  r o l l  appendage  replacing  (1-0.05+0.95)  a  f  +  B  0.95  0.05  O )  a  2  /  a  -  +  a  )  C  ; -  D  0.05  r  2  2  a  ;  0.05  (0.95  slewing  change  use of  T  T  (A  a -  a  0.95  0.05  maneuver  r  is  2  a  T  2  a  )  given  the  in  appendage  r o l l .  same  equations  that  parameters »  r  r  )  .  by  A/3 ,  the  r o l l »  (1-0.05+0.95)  (0.95)0.05  3  a  A/3 r e p r e s e n t s  generated  (0.95-1)  0.05  2 a  /  B  with  /  r  1 /  appendage  where  (  1 +  F  =  /  T a  •  ,  /3  £  ,  A/3 i s gave  Aa but  appropriately  

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