Ph.D. Defense

Ph.D. Defense

The slides from the talk I gave as part of my Ph.D. defense at Cornell. It covers my work on KickSat, plus some stuff on discrete mechanics, control of spinning spacecraft, and on-orbit inertia estimation for satellites.

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Zac Manchester

June 16, 2015
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Transcript

  1. 2.
  2. 3.

    ”Space  Transportation  Costs:  Trends  in  Price  Per  Pound  to  Orbit.”

      Futron Corporation.  Bethesda,  MD.  September  6,  2002. Expanding  Access  to  Space
  3. 6.

    Contributions 5 1. Design  and  fabrication  of  a  complete  spacecraft

     on  a  printed  circuit  board 2. A  CubeSat-­‐based  deployment  system  for  centimeter-­‐scale  spacecraft 3. A  Low-­‐power,  long-­‐range  communication  system  for  centimeter-­‐scale   spacecraft 4. Feedback  control  laws  capable  of  both  flat-­‐spin  recovery  and  spin-­‐axis   inversion  of  spinning  spacecraft 5. Variational  integrators  for  spacecraft  dynamics  with  quaternion  state   variables 6. An  algorithm  for  recursive  estimation  of  spacecraft  inertia  using   semidefinite  programming
  4. 7.

    Courses 6 Methods  of  Applied  Math  I  (TAM  6100) Methods

     of  Applied  Math  II  (TAM  6110) Numerical  Analysis:  Linear  and  Nonlinear  Problems  (CS  5223) Stochastic  Systems:  Estimation  and  Control  (ECE  5555) Minor  (Applied  Math) Feedback  Control  Systems  (MAE  4780) Intermediate  Dynamics  (TAM  5700) GPS:  Theory  and  Design  (MAE  4150) Spacecraft  Dynamics  (MAE  6060) Celestial  Mechanics  (TAM  6720) Computer  Vision  (ECE  5470) Theory  of  Linear  Systems  (MAE  5210) Differential  Equations  and  Dynamical  Systems  (MATH  4200) Multivariable  Control  Theory  (MAE  6780) Engineering  Vibrations  (MAE  5770) Model-­‐Based  Estimation  (MAE  6760) Engineering
  5. 8.

    Publications  To  Date Manchester,  Z.  R.,  and  Peck,  M.  A.,

     “Quaternion  Variational  Integrators  for  Spacecraft   Dynamics,”  Journal  of  Guidance,  Control,  and  Dynamics,  2015  (To  Appear). Manchester,  Z.  R.,  Peck,  M.  A.,  and  Filo,  A..,  “KickSat:  A  Crowd-­‐Funded  Mission  To   Demonstrate  The  World’s  Smallest  Spacecraft,”  AIAA/USU  Conference  on  Small  Satellites,   Logan,  Utah,  August  12-­‐16,  2013. Manchester,  Z.  R.,  and  Peck,  M.  A.,  “Stochastic  Space  Exploration  with  Microscale Spacecraft,”     AIAA  Guidance,  Navigation,  and  Control  Conference,  Portland,  OR,  August  8-­‐11,  2011. Atchison,  J.  A.,  Manchester,  Z.  R.,  and  Peck,  M.  A.,  "Microscale Atmospheric  Reentry  Sensors,"   7th  International  Planetary  Probe  Workshop,  Barcelona,  Spain,  June  14-­‐18,  2010.   7
  6. 9.

    Publications  To  Be  Submitted  (In  Thesis): Manchester,  Z.  R.,  and

     Peck,  M.  A.,  “Recursive  Spacecraft  Inertia  Estimation  with  Semidefinite Programming,”  Journal  of  Guidance,  Control,  and  Dynamics. Manchester,  Z.  R.,  and  Peck,  M.  A.,  “Lyapunov-­‐Based  Control  for  Flat-­‐Spin  Recovery  and  Spin   Inversion  of  Spin-­‐Stabilized  Spacecraft,”  Journal  of  Guidance,  Control,  and  Dynamics. Manchester,  Z.  R.,  and  Peck,  M.  A.,  “The  Sprite:  A  Centimeter-­‐Scale   Spacecraft,”  Journal  of   Spacecraft  and  Rockets. 8
  7. 11.

    10 Literature  Survey:   Janson,  S.  W.,  “Mass-­‐Producible  Silicon  Spacecraft

     for  21st Century  Missions,”  AIAA  Space  Technology  Conference  and  Exposition,   Albuquerque,  NM,  Sept.  28-­‐30,  1999. Barnhart,  D.  J.,  Vladimirova,  T.,  and  Sweeting,  M.  N.,  “Satellite-­‐on-­‐a-­‐ Chip  Development  for  Future  Distributed  Space  Missions,”  CANEUS  2006   Conference,  Aug.  27-­‐Sept.  1,  2006.   Barnhart,  D.  J.,  Vladimirova,  T.,  Baker,  A.  M.,  and  Sweeting,  M.  N.,  “A  Low-­‐Cost  Femtosatellite to  Enable  Distributed  Space  Missions,”  57th International  Astronautical  Congress,  Valencia,  Spain,  Oct.  2-­‐6,  2006. Barnhart,  D.  J.,  Vladimirova,  T.,  and  Sweeting,  M.  N.,  “Very-­‐Small-­‐Satellite  Design  for  Distributed  Space  Missions,”  Journal  of  Spacecraft  and   Rockets,  Vol.  46,  No.  2,  2009,  pp.  469-­‐472. Atchison,  J.  A.,    and  Peck,  M.  A.,  “A  Millimeter-­‐Scale  Lorentz-­‐Propelled  Spacecraft,”  AIAA  Guidance,  Navigation  and  Control  Conference  and   Exhibit,  Hilton  Head,  SC,  Aug  20-­‐23,  2007. Atchison,  J.  A.,  Manchester,  Z.  R.,  and  Peck,  M.  A.,  "MicroscaleAtmospheric  Reentry  Sensors,"  7th  International  Planetary  Probe  Workshop,   Barcelona,  Spain,  June  14-­‐18,  2010. Atchison,  J.  A.,  and  Peck,  M.  A.,  “A  Passive,  Sun-­‐Pointing,  Millimeter-­‐Scale  Solar  Sail,”  Acta Astronautica,  2010,  Vol.  67,  No.  1,  pp.  108-­‐121. Atchison,  J.  A.,  and  Peck,  M.  A.,  “Length  Scaling  in  Spacecraft  Dynamics,”  Journal  of  Guidance,  Control,  and  Dynamics,  Vol.  34,  No.  1,  2011,  pp.   231-­‐246. Colombo,  C.,  and  McInnes,  C.  R.,  “Orbital  dynamics  of  Earth-­‐orbiting  ‘smart-­‐dust’  spacecraft  under  the  effects  of  solar  radiation  pressure  and   aerodynamic  drag,”  AIAA/AAS  Astrodynamics Specialist  Conference,  Toronto,  Canada,  Aug.  2-­‐5,  2010. Colombo,  C.,  and  McInnes,  C.  R.,  “Orbit  Design  for  Future  Spacechip Swarm  Missions,”  61st International  Astronautical  Congress,  Prague,   Czech  Republic,  Sept.  27-­‐Oct.  1,  2010. McInnes,  C.  R.,  Ceriotti,  M.,  Colombo,  C.,  et.  Al.,  “Micro-­‐to-­‐Macro:  Astrodynamicsat  Extremes  of  Length-­‐scale,”  Acta Futura,  Vol.  4,  2011,  pp.   81-­‐97. Colombo,  C.,  Lucking,  C.,  and  McInnes,  C.  R.,  “Orbital  dynamics  of  high  area-­‐to-­‐mass  ratio  spacecraft  with  J2  and  solar  radiation  pressure  for   novel  Earth  observation  and  communication  services,”  Acta Astronautica,  Vol.  81,  No.  1,  2012,  pp.  137-­‐150. Centimeter-­‐Scale  Spacecraft
  8. 12.

    11

  9. 13.
  10. 14.
  11. 19.

    18 Problem: How  can  we  communicate  with  hundreds  of  small

     spacecraft  simultaneously? Constraints: • Transmitter  power  is  limited  (10  mW) • Spacecraft  antenna  must  have  low  gain  due  to  lack  of  pointing  capability • Licensing  (only  one  frequency  allocation  for  the  entire  “swarm”) Communication
  12. 24.

    23 Communication Matched  Filter: Template: 1  1  1  1  1

     0  0  1  1  0  1  0   1 1  1  0  0  1  1  1  0  0  1  1  0  0  1  0  1  1  1  0  0  0 0  1  0  1  1  0  0  0  0  0  1  0  0  1  0  1  1  0   1  1  1  1  1  0  0  1  1  0  1  0   1 20 30 40
  13. 25.

    Communication 24 PRN  Sequence: • Long  deterministically-­‐generated   sequences  of

     bits • Look  like  noise  (approximate  δ-­‐function  autocorrelation) Gold  Codes: • Commonly  used  code  family  (e.g.  in  GPS) • Easily  generated  by  a  linear  feedback  shift  register  (or  a  few  lines  of  code) • Low  bounded  cross-­‐correlation  among  members  of  the  family • 511  bit  codes  used  for  KickSat  mission
  14. 29.

    Flat-­‐Spin  Recovery 28 Problem: How  do  we  ensure  KickSat’s angular

     momentum  vector  is  aligned  with  its  +z  axis? Constraints: • Very  limited  reaction  wheel  torque  (3.8  mNm) • Very  limited  angular  momentum  storage  in  reaction  wheels  (.5  mNms)
  15. 30.

    29 Literature  Survey: Bracewell,   R.  N.,  and  Garriott,  O.

     K.,  “Rotation   of  Artificial  Earth  Satellites,”   Nature,   Vol.  182,  Sept.,   1958,  pp.  760-­‐762. Barba,  P.  M.,  Furumoto,  N.,  and  Leliakov,  I.  P.,  “Techniques   for  Flat-­‐Spin  Recovery  of  Spinning  Satellites,”   AIAA  Guidance  and  Control   Conference,  Key  Biscayne,   Florida,  Aug.  20-­‐22,  1973. Gebman,   J.  R.,  and  Mingori,  D.  L.,  “Perturbation  Solution  for  the  Flat  Spin  Recovery  of  a  Dual-­‐Spin  Spacecraft,”   AIAA  Journal,  Vol.  14,   No.  7,  July,  1976. Cronin,  D.  L.,  "Flat  Spin  Recovery  of  a  Rigid  Asymmetric   Spacecraft",   Journal  of  Guidance,  Control,  and  Dynamics,  Vol.  1,  No.  4,  1978,  pp.   281-­‐282. Beachley,   N.  H.  and  Uicker,  J.  J.,  “Wobble-­‐Spin   Technique   for  Spacecraft   Inversion  and  Earth  Photography,”  Journal  of  Spacecraft  and   Rockets,  Vol.  6,  No.  3,  March  1969,  pp.  245-­‐248. Beachley,   N.  H.,  “Inversion  of  Spin-­‐Stabilized  Spacecraft   by  Mass  Translation   – Some  Practical   Aspects,”   Journal  of  Spacecraft   and   Rockets,  Vol.   8,  No.  10,  Oct.  1971,  pp.  1078-­‐1080 Rahn,  C.  D.,  and  Barba,  P.  M.,  “Reorientation   Maneuver  for  Spinning  Spacecraft,”   Journal  of  Guidance,  Control,  and  Dynamics,   Vol.  14,   No.  4,  July  1991,  pp.  724-­‐728. Rand,  R.  H.,  and  Hall,  C.  D.,  “Spinup Dynamics   of  Axial  Dual-­‐Spin  Spacecraft,”   Journal  of  Guidance,   Control,  and  Dynamics,   Vol.   17,  No.   1,  1994,  pp.  30-­‐37. Chaturvedi,  N.  A.,  Bernstein,   D.  S.,  Ahmed,  J.,  Bacconi,  F.,  and  McClamroch,   N.  H.,  “Globally  Convergent  Adaptive  Tracking  of  Angular   Velocity   and  Inertia  Identification   for  a  3-­‐DOF  Rigid  Body,”   IEEE  Transactions  on  Control  Systems  Technology,  Vol.  14,  No.  5,  Sept.,   2006. Lawrence,   D.  A.,  and  Holden,  T.  E.,  “Essentially   Globally  Asymptotically  Stable  Nutation   Control  Using  a  Single  Reaction   Wheel,” Journal   of  Guidance,  Control,  and  Dynamics,  Vol.  30,  No.  6,  Nov.-­‐Dec.,   2007. Myung,  H.  S.,  and  Bang,   H.,  “Predictive   Nutation   and  Spin  Inversion  Control  of  Spin-­‐Stabilized  Spacecraft,”   Journal  of  Spacecraft   and   Rockets,  Vol.  47,  No.  6,  Nov.-­‐Dec.,   2010. Flat-­‐Spin  Recovery
  16. 31.

    Flat-­‐Spin  Recovery 30 Gyrostat  Dynamics: J = I 1 ˙

    h = h ⇥ J ·(h ⇢) h = I·! + ⇢ I· ˙ ! + ! ⇥ (I·! + ⇢) + ˙ ⇢ = ⌧
  17. 32.

    0 0.5 1 1.5 2 Flat-­‐Spin  Recovery 31 Lyapunov Stability:

    ˙ x = f ( x ) V ( x ) 0 ˙ V ( x ) = @V @x d x d t  0 f ( x ⇤) = 0
  18. 33.

    Flat-­‐Spin  Recovery 32 Candidate  Lyapunov Functions: VQ = 1 2

    hd ·J ·hd 1 2 h·J ·h VL = hd ·J ·hd hd ·J ·h
  19. 35.

    Flat-­‐Spin  Recovery 34 Periodic  Averaging: V = 3 2 hd

    ·J ·hd 1 T Z T 0 hd ·J ·h + 1 2 h·J ·h dt
  20. 36.

    Flat-­‐Spin  Recovery 35 Control  Law: ˙ V = 1 T

    Z T 0 (h + hd)·J ·h ⇥ J ·⇢ dt b(h) ⇢ ⇡ ⇢ max sign(b)
  21. 37.

    Flat-­‐Spin  Recovery 36 Control  Law: ⇢ = ⇢ max tanh(↵b)

    x -5 0 5 tanh(x) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
  22. 40.

    Discrete  Variational  Mechanics 39 Motivation: • Variational  integrators  have  many

     advantages  over  e.g.  Runge-­‐Kutta  schemes • Well  suited  for  use  in  estimation  and  control  applications • Theory  is  based  on  Lagrangian  and/or  Hamiltonian  Mechanics ⇒ Need  a  Lagrangian  formulation  of  attitude  dynamics! L = T V = 1 2 !·I·! S = Z tf t0 1 2 !·I·! dt = ?
  23. 41.

    40 Literature  Survey: Moser,  J.,  and  Veselov,   A.  P.,

     “Discrete  Versions   of  Some  Classical   Integrable  Systems  and  Factorizations  of  Matrix   Polynomials,”   Communications   in  Mathematical  Physics,   Vol.  139,  1991,  pp.  217-­‐243. McLachlan,  R.  I.,  “Explicit  Lie-­‐Poisson   Integration  and  the  Euler  Equations,”   Physical  Review  Letters,  Vol.  71,  1993,  pp.  3043-­‐ 3046. Reich,  S.  “Numerical  Integration  of  the  Generalized  Euler  Equations,”   Technical  Report  93-­‐20,   Department  of  Computer   Science,   University  of  British  Columbia,   1993. Marsden,  J.  E.,  and  West,  M.,  “Discrete  mechanics  and  variational  integrators,”  Acta Numerica,  Vol.  10,  2001,  pp.  357-­‐514. Lee,  T.,  Leok,  M.,  and  McClamroch,   N.  H.,  “A  Lie  Group  Variational  Integrator  for  the  Attitude  Dynamics  of  a  Rigid  Body  with   Applications   to  the  3D  Pendulum,”   IEEE  Conference  on  Control  Applications,   Toronto,  Canada,  Aug.  28-­‐31,  2005. Hussein,   I.  I.,  Leok,  M.,  Sanyal,  A.  K.,  and  Block,  A.  M.,  “A  Discrete  Variational  Integrator  for  Optimal  Control  Problems   on   SO(3),”   IEEE  Conference  on  Decision  &  Control,  San  Diego,  CA,  Dec.  13015,  2006. Lee,  T.,  Leok,  M.,  and  McClamroch,   N.  H.,  “Lie  group  variational  integrators  for  the  full  body  problem,”   Computer  methods  in   applied  mechanics  and  engineering,  Vol.  196,  2007,  pp.  2907-­‐2924. Krysl,  P.,  “Direct  time  integration  of  rigid  body  motion  with  discrete-­‐impulse   midpoint   approximation:  Explicit  Newmark algorithms,”  Communications   in  Numerical  Methods  in  Engineering,   Vol.  22,  2006,  pp.  441-­‐451. Saccon,  A.,  “Midpoint  rule  for  variational  integrators  on  Lie  groups,”  International  Journal  for  Numerical  Methods  in   Engineering,   Vol.  78,  2009,  pp.  1345-­‐1364. Discrete  Variational  Mechanics
  24. 42.

    41 Quaternions: Discrete  Variational  Mechanics r ✓ q = 2

    6 6 4 q1 q2 q3 q4 3 7 7 5 = 2 6 6 4 qv qs 3 7 7 5 = 2 6 6 4 r sin( ✓/ 2) cos( ✓/ 2) 3 7 7 5
  25. 43.

    42 Hamilton’s  Principle: Discrete  Variational  Mechanics =) I· ˙ !

    + ! ⇥ (I·!) = 0 S = d d✏ ✏=0 Z tf t0 1 2 ✏ ˆ !· ˆ I·✏ ˆ ! dt = 0
  26. 44.

    43 Discrete  Action  Sum: Discrete  Variational  Mechanics S = Z

    tf t0 1 2 ˆ !· ˆ I·ˆ ! dt = N X k=0 Z tk+1 tk 1 2 ˆ !· ˆ I·ˆ ! dt S ⇡ Sd = N X k=0 2 ⌧ fk · ˆ I · fk
  27. 45.

    44 Discrete  Hamilton’s  Principle: Discrete  Variational  Mechanics Sd = d

    d✏ ✏=0 N X k=0 1 2 ✏fk · ˆ I·✏fk = 0 =) p 1 k · k I· k k ⇥ I· k = p 1 k+1 · k+1 I· k+1 + k+1 ⇥ I· k+1 fk = 2 6 6 4 k p 1 k · k 3 7 7 5
  28. 46.

    45 Free  Rigid  Body  Simulation: Discrete  Variational  Mechanics 0 2

    4 6 8 10 0 5 10 15 20 25 Time (min) Error (%) Variational Midpoint ODE45 Momentum Error (%)
  29. 47.

    46 Free  Rigid  Body  Simulation: Discrete  Variational  Mechanics 0 2

    4 6 8 10 0 2 4 6 8 Time (min) Error (%) Variational Midpoint ODE45 Energy Error (%)
  30. 48.

    47 Kalman  Filtering: Discrete  Variational  Mechanics 10−1 100 101 102

    0 10 20 30 40 50 Sample Rate (Hz) RMS Error (deg) Standard MEKF Variational MEKF
  31. 49.

    Inertia  Estimation 48 Problem: • Inertia  knowledge  is  needed  for

     control  of  KickSat’s spin  axis • Accurate  ground-­‐based  inertia  measurements  require  expensive  specialized   equipment
  32. 50.

    49 Literature  Survey: Bergmann,  E.  V.,  Walker,  B.  K.,  and

     Levy,  D.  R.,  “Mass  Property  Estimation  for  Control  of  Asymmetrical  Satellites,”  Journal   of  Guidance,  Control,  and  Dynamics,  Vol.  10,  No.  5,  1987,  pp.  483-­‐491. Tanygin,  S.  and  Williams,  T.,  “Mass  Property  Estimation  Using  Coasting  Maneuvers,”  Journal  of  Guidance,  Control,  and   Dynamics,  Vol.  20,  No.  4,  1997,  pp.  625-­‐632. Ahmed,  J.,  Coppola,  V.  T.,  and  Bernstein,  D.  S.,  “Adaptive  Asymptotic  Tracking  of  Spacecraft  Attitude  Motion  with  Inertia   Matrix  Identification,”  Journal  of  Guidance,  Control,  and  Dynamics,  Vol.  21,  No.  5,  1998,  pp.  684-­‐691. Psiaki,  M.  L.,  “Estimation  of  a  Spacecraft’s  Attitude  Dynamics  Parameters  by  Using  Flight  Data,”  Journal  of  Guidance,   Control,  and  Dynamics,  Vol.  28,  No.  4,  2005,  pp.  594-­‐603. Peck,  M.  A.,  “Uncertainty  Models  for  Physically  Realizable  Inertia  Dyadics,”  Journal  of  the  AstronauticalSciences,  Vol.  54,   No.  1,  2006,  pp.  1-­‐16. Norman,  M.  C.,  Peck,  M.  A.,  and  O’Shaghnessy,  D.  J.,  “In-­‐orbit  Estimation  of  Inertia  and  Momentum-­‐Actuator  Alignment   Parameters,”  Journal  of  Guidance,  Control,  and  Dynamics,  Vol.  34,  No.  6,  2011,  pp.  1798-­‐1814. Keim,  J.  A.,  Acikmese,  A.,  and  Shields,  J.  F.,  “Spacecraft  Inertia  Estimation  via  Constrained  Least  Squares,”  Proceedings  of   the  IEEE  Aerospace  Conference,  2006. Inertia  Estimation
  33. 51.

    50 A  Least-­‐Squares  Problem: Inertia  Estimation I· ˙ ! +

    ! ⇥ (I·! + ⇢) + ˙ ⇢ = ⌧ H(!, ˙ !)i = y(!, ⇢, ˙ ⇢, ⌧) i = ⇥ I11 I22 I33 I12 I13 I23 ⇤|
  34. 52.

    51 A  Least-­‐Squares  Problem: Inertia  Estimation q 1 k+1 ·

    k+1 (I· k+1 + h 2 ⇢k+1 ) + k+1 ⇥ (I· k+1 + h 2 ⇢k+1 ) + h2 2 ⌧k+1 = p 1 k · k (I· k + h 2 ⇢k ) k ⇥ (I· k + h 2 ⇢k ) H( k , k+1 )i = y( k , k+1 , ⇢k , ⇢k+1 , ⌧k+1)
  35. 54.

    53 Semidefinite Programming: Inertia  Estimation minimize x c | x

    subject to F0 + n X i =1 xiFi 0 Fi 2 Sn F 0 =) v| Fv 0 8 v 2 Rn
  36. 56.

    55 Semidefinite  Program: Inertia  Estimation minimize ⇥ · · ·

    0 · · · 1 ⇤  i s subject to 8 > > > > > > > > < > > > > > > > > :  s ( Hi y ) | ( Hi y ) 1 0 I > 0 I11 + I22 I33 0 I11 + I33 I22 0 I22 + I33 I11 0
  37. 57.

    56 Simulation: Inertia  Estimation 0 10 20 30 40 50

    60 ! 1 -0.2 0 0.2 0 10 20 30 40 50 60 ! 2 -0.2 -0.1 0 0.1 Time (s) 0 10 20 30 40 50 60 ! 3 0.76 0.77 0.78 0.79 0 10 20 30 40 50 60 ; 1 -0.05 0 0.05 0 10 20 30 40 50 60 ; 2 -0.05 0 0.05 Time (s) 0 10 20 30 40 50 60 ; 3 -0.05 0 0.05
  38. 58.

    57 Simulation: Inertia  Estimation 0 10 20 30 40 50

    60 J 11 10-8 10-6 10-4 10-2 100 Momentum Based SDP Based 0 10 20 30 40 50 60 J 22 10-8 10-6 10-4 10-2 100 Time (s) 0 10 20 30 40 50 60 J 33 10-8 10-6 10-4 10-2 100 0 10 20 30 40 50 60 J 12 10-8 10-6 10-4 10-2 100 0 10 20 30 40 50 60 J 13 10-8 10-6 10-4 10-2 100 Time (s) 0 10 20 30 40 50 60 J 23 10-8 10-6 10-4 10-2 100
  39. 59.

    Acknowledgements Becky  Manchester Justin  Atchison Phillipe Tosi Matt  Reyes Andy

     Filo Pete  Worden John  Hines Harry  Partridge Elwood  Agasid Jasper  Wolfe Cedric  Priscal Oriol Tintore Gazulla Alberto  Guillen Salas Ken  Oyadomari Watson  Attai Justin  Manchester Scott  Higginbotham Ryan  Nugent
  40. 62.

    • Launch  costs  are  not  dropping  very  quickly: 1995: •

    Average:  $36,200/kg  (Futron*,  inflation  adjusted) 2015: • Atlas  V:  $35,000/kg  (ULA  Website) • Falcon  9:  $12,619/kg  (SpaceX Website) ”Space  Transportation  Costs:  Trends  in  Price  Per  Pound  to  Orbit.”   Futron Corporation.  Bethesda,  MD.  September  6,  2002. * Expanding  Access  to  Space
  41. 65.

    Solar  Cells Microcontroller Radio Chip  Antenna Energy  Storage Capacitor DC-­‐DC

     Power Converter The  Sprite  Spacecraft  (2009) Atchison,  J.  A.,  Manchester,  Z.  R.,  and  Peck,  M.  A.,  "MicroscaleAtmospheric  Reentry  Sensors,"  7th  International   Planetary  Probe  Workshop,  Barcelona,  Spain,  June  14-­‐18,  2010.   64
  42. 66.

    Bifilar  Pendulum 65 I ¨ ✓ + mgr2 sin( ✓

    ) p L2 2 r2 (1 cos( ✓ )) ⇡ I ¨ ✓ + mgr2 L ✓ = 0 !2 = mgr2 IL =) I = mgr2 !2L I, m L 2r L s h s r r θ h = p L2 2 r2 (1 cos( ✓ )) s2 = 2 r2 (1 cos( ✓ )) L = 1 2 I ˙ ✓2 + mg p L2 2 r2 (1 cos( ✓ ))
  43. 68.

    67

  44. 69.

    68

  45. 71.

    70

  46. 80.

    Orbital  Lifetime 79 0 5 10 15 20 25 50

    100 150 200 250 300 350 Altitude vs. Time With No Attitude Control Time (days) Altitude (km)
  47. 81.

    80 Literature  Survey: Ageev,  D.  V.,  “Bases  of  the  Theory

     of  Linear  Selection.  Code  Demultiplexing,”  Proceedings  of  the  Leningrad   Experimental  Institute  of  Communication,  Vol.  3,  No.  35,  1935. Shannon,  C.  E.,    Weaver,  W.,  The  Mathematical  Theory  of  Communication,  University  of  Illinois  Press,  1949. Gold,  R.,  “Optimal  binary  sequences  for  spread  spectrum  multiplexing,”  IEEE  Transactions  on  Information  Theory,   Vol.  13,  No.  4,  Oct.  1967,  pp.  619-­‐621. Welch,  L.R.,  “Lower  Bounds  on  the  Maximum  Cross  Correlation  of  Signals,”  IEEE  Transactions  on  Information   Theory,  Vol.  20,  No.  3,  May  1974,  pp.  397–399. Gilhousen,  K.  S.,  Jacobs,  I.  M.,  Padovani,  R.,  et.  Al.,  “On  the  Capacity  of  a  Cellular  CDMA  System,”  IEEE   Transactions  on  Vehicular  Technology,  Vol.  40,  No.  2,  May,  1991. Kohno,  R.,  Meidan,  R.,  and  Milstein,  L.  B.,  “Spread  Spectrum  Access  Methods  for  Wireless  Communications,”  IEEE   Communications  Magazine,  Jan.  1995. Viterbi,  A.  J.,  CDMA:  Principles  of  Spread  Spectrum  Communication,  Addison-­‐Wesley,  1995. Misra,  P.,  Enge,  P.,  Global  Positioning  System:  Signals,  Measurements,  and  Performance,  Ganga-­‐Jamuna Press,   Lincoln,  MA,  2001. Blossom,  E.,  “GNU  Radio:  Tools  for  Exploring  the  Radio  Frequency  Spectrum,”  Linux  Journal,  URL:   http://www.linuxjournal.com/article/7319  [cited  11  February  2013]. Communication
  48. 82.

    81 Communication Shannon-­‐Hartley  Theorem: • Channel  is  assumed  to  be

     corrupted  with  Gaussian  white  noise • B  =  Bandwidth • S  =  Signal  Power • N  =  Noise  Power • C  =  Channel  Capacity  (bits/sec) C = B Log2 ✓ 1 + S N ◆
  49. 83.

    Communication 82 CDMA: • Assign  each  user  a  PRN  code

    • For  every  data  bit,  transmit  a  PRN  code • Receiver  separates  signals  using  matched  filters • All  users  can  transmit  simultaneously  without  coordination Caveats: • Must  handle  frequency  offset  between  transmitter  and  receiver  (e.g.  Doppler) • Inter-­‐user  interference  leads  to  upper  bound  on  number  of  simultaneous  users //Transmit  data  byte byte  &  BIT7  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT6  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT5  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT4  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT3  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT2  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT1  ?  transmitRaw(prn1)   :  transmitRaw(prn0); byte  &  BIT0  ?  transmitRaw(prn1)   :  transmitRaw(prn0);
  50. 85.

    Communication 84 Forward  Error  Correction: • We  don’t  have  a

     “back  channel”  to  request  re-­‐transmission  if  a  byte  is  corrupted • Add  redundant  information  to  the  message  to  help  detect  and  correct  errors Linear  Block  Codes: • Simplest  forward  error  correction  technique • Multiply  message  vector  by  code’s  generator  matrix  to  create  code  vector ⇥ · · · · · · c · · · · · · ⇤ 1⇥16 = ⇥ · · · m · · · ⇤ 1⇥8 2 6 6 6 6 6 6 6 6 6 6 4 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 7 5 16⇥8 G z }| {
  51. 86.

    Communication 85 Hamming  Distance: • Minimum  1-­‐norm  distance  between  code

     vectors • Measure  of  code’s  error  correcting  power Maximum  Likelihood  Soft  Decoder: • Take  dot  product  of  received  code  vector  against  all  possible  code  vectors  in  Rn • Output  is  message  vector  corresponding  to  largest  dot  product • Scales  poorly  as  message  length  increases c1 c2 c3 r
  52. 88.

    Flat-­‐Spin  Recovery 87 Lyapunov Stability: V = 1 2m ˙

    x 2 + 1 2kx 2  ˙ x ¨ x =  0 1 k/m c/m  x ˙ x ˙ V = c ˙ x 2 m ¨ x + c ˙ x + kx = 0
  53. 89.

    Flat-­‐Spin  Recovery 88 Control  Law: hk = h(t0 + k

    t) ⇢k = ⇢(t0 + k t) xk = 2 6 4 hk . . . hk+N 1 3 7 5 uk = 2 6 4 ⇢k . . . ⇢k+N 1 3 7 5 1 T Z T 0 hd · J · h + 1 2 h · J · h d t ⇡ 1 N x | d ¯ J xk + 1 2 N x | k ¯ J xk xd = 2 6 4 hd . . . hd 3 7 5 ¯ J = 2 6 6 6 4 J 0 · · · 0 0 J · · · 0 . . . . . . ... . . . 0 0 · · · J 3 7 7 7 5
  54. 90.

    Flat-­‐Spin  Recovery 89 Control  Law: xk+1 = Axk + Bkuk

    A = 2 6 6 6 6 6 6 4 0 13⇥3 0 · · · 0 0 0 13⇥3 . . . 0 . . . . . . ... ... . . . 0 0 · · · 0 13⇥3 13⇥3 0 · · · 0 0 3 7 7 7 7 7 7 5 Bk = t 2 6 6 6 6 4 h⇥ k J 0 · · · 0 0 h⇥ k+1 J 0 . . . . . . ... ... 0 0 · · · 0 h⇥ k+N 1 J 3 7 7 7 7 5
  55. 91.

    Flat-­‐Spin  Recovery 90 Control  Law: V = 1 N x

    | d ¯ J xk + 1 2 N x | k ¯ J xk 1 N x | d ¯ J xk+1 1 2 N x | k+1 ¯ J xk+1 = 1 N x | d ¯ J Bk uk 1 N x | k A | ¯ J Bk uk 1 2 N u | k B | k ¯ J Bk uk ˙ V = lim N!1 V t = 1 T Z t0+T t0 (h + hd) · J · h ⇥ J · ⇢ dt
  56. 92.

    0 2 4 6 8 _ ;1 -0.1 0 0.1

    0 2 4 6 8 _ ;2 -0.1 0 0.1 Time (min) 0 2 4 6 8 _ ;3 -0.02 0 0.02 0 2 4 6 8 ; 1 -0.01 0 0.01 0 2 4 6 8 ; 2 -0.01 0 0.01 Time (min) 0 2 4 6 8 ; 3 -0.01 0 0.01 Flat-­‐Spin  Recovery 91 Flat-­‐Spin  Recovery  Simulation:
  57. 93.

    0 1 2 3 4 5 6 _ ;1 -0.1

    0 0.1 0 1 2 3 4 5 6 _ ;2 -0.1 0 0.1 Time (min) 0 1 2 3 4 5 6 _ ;3 -0.02 0 0.02 0 1 2 3 4 5 6 ; 1 -0.01 0 0.01 0 1 2 3 4 5 6 ; 2 -0.01 0 0.01 Time (min) 0 1 2 3 4 5 6 ; 3 -0.01 0 0.01 Flat-­‐Spin  Recovery 92 Spin-­‐Inversion  Simulation:
  58. 94.

    93 Quaternions: Discrete  Variational  Mechanics pq = 2 6 6

    4 psqv + qspv + pv ⇥ qv psqs pv · qv 3 7 7 5 q† = 2 6 6 4 qv qs 3 7 7 5 q = 2 6 6 4 r sin( ✓/ 2) cos( ✓/ 2) 3 7 7 5 v0 = qvq† ˆ v = 2 6 6 4 v 0 3 7 7 5
  59. 95.

    94 Quaternions: Discrete  Variational  Mechanics ˆ ! = 2q† ˙

    q = 2 6 6 4 ! 0 3 7 7 5 q† = 2 6 6 4 qv qs 3 7 7 5 ˆ I = 2 6 6 4 I11 I12 I13 0 I21 I22 I23 0 I31 I32 I33 0 0 0 0 0 3 7 7 5 L = 1 2 ˆ !· ˆ I·ˆ !
  60. 96.

    95 Quaternion  Exponential: Discrete  Variational  Mechanics eq = eqs 2

    6 6 4 qv kqv k sin( kqv k ) cos( kqv k ) 3 7 7 5 = eˆ r✓/2
  61. 97.

    96 Hamilton’s  Principle: Discrete  Variational  Mechanics ˆ ! = 2q†

    ˙ q =) ✏ ˆ ! = e ✏ˆ ⌘ ˆ !e✏ˆ ⌘ + 2✏ ˙ ˆ ⌘ =) I· ˙ ! + ! ⇥ (I·!) = 0 S = d d✏ ✏=0 Z tf t0 1 2 ✏ ˆ !· ˆ I·✏ ˆ ! dt ✏ ˙ q = ˙ qe✏ˆ ⌘ + ✏q✏ ˙ ˆ ⌘ ✏q = qe✏ˆ ⌘
  62. 98.

    97 Simulation  with  Kane  Damper: Discrete  Variational  Mechanics 10−1 100

    101 102 0 10 20 30 40 50 60 Damping Constant Running Time (sec) ODE45 Variational ! !d
  63. 99.

    98 Simulation  with  Kane  Damper: Discrete  Variational  Mechanics ! !d

    0 100 200 300 1.22 1.225 1.23 1.235 1.24 1.245 Time (sec) Energy ODE45 Variational
  64. 100.

    99 Simulation  with  Kane  Damper: Discrete  Variational  Mechanics 0 100

    200 300 1.22 1.23 1.24 1.25 1.26 1.27 1.28 Time (sec) Energy ODE15s Variational ! !d