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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.

Zac Manchester

June 16, 2015
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  1. Zac  Manchester
    Centimeter-­‐Scale  Spacecraft:
    Design,  Fabrication,  and  Deployment

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  2. Why?
    1

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  3. ”Space  Transportation  Costs:  Trends  in  Price  Per  Pound  to  Orbit.”  
    Futron Corporation.  Bethesda,  MD.  September  6,  2002.
    Expanding  Access  to  Space

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  4. • Electronics  are  improving  much  faster
    Wikipedia
    Expanding  Access  to  Space

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  5. Shrink  The  Spacecraft!
    4

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  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

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  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

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  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

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  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

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  10. Centimeter-­‐Scale  Spacecraft
    9

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  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

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  12. 11

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  13. View Slide

  14. View Slide

  15. Centimeter-­‐Scale  Spacecraft
    Returned  To  Earth  (5/18/2014):
    14

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  16. Video  Courtesy  Ben  Bishop 15
    KickSat  Mission  Concept:  
    Centimeter-­‐Scale  Spacecraft

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  17. 16
    Deployment  Testing:
    Centimeter-­‐Scale  Spacecraft

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  18. Centimeter-­‐Scale  Spacecraft
    17
    Launch  (4/18/2014):

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  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

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  20. 19
    Communication
    Clean  Signal:

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  21. 20
    Communication
    Received  Signal:

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  22. 21
    Communication
    Clean  Signal:

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  23. 22
    Communication
    Received  Signal:

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  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

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  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

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  26. 25
    Communication
    Received  Signal  After  Matched  Filter:

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  27. 26
    Communication
    Reciever:

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  28. 27
    Communication
    Testing:

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  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)

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  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

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  31. Flat-­‐Spin  Recovery
    30
    Gyrostat  Dynamics:
    J = I 1
    ˙
    h = h ⇥ J ·(h ⇢)
    h = I·! + ⇢
    I· ˙
    ! + ! ⇥ (I·! + ⇢) + ˙
    ⇢ = ⌧

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  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

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  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

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  34. Flat-­‐Spin  Recovery
    33
    Periodic  Averaging:
    V 0
    L
    = hd
    ·J ·hd
    1
    T
    Z T
    0
    hd
    ·J ·h dt

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  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

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  36. Flat-­‐Spin  Recovery
    35
    Control  Law:
    ˙
    V =
    1
    T
    Z T
    0
    (h + hd)·J ·h ⇥ J ·⇢ dt
    b(h)
    ⇢ ⇡ ⇢
    max
    sign(b)

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  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

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  38. Flat-­‐Spin  Recovery
    37
    Flat-­‐Spin  Recovery  Simulation:
    0.5
    0
    -0.5
    -1
    1
    0.5
    0
    -0.5
    0.5
    0
    -0.5
    1

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  39. Flat-­‐Spin  Recovery
    38
    Spin-­‐Inversion  Simulation:
    0.5
    0
    -0.5
    -0.5
    0
    0.5
    -1
    1
    0.8
    0.6
    0.4
    0.2
    0
    -0.2
    -0.4
    -0.6
    -0.8

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  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 = ?

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  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

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  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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 (%)

    View Slide

  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 (%)

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  51. 50
    A  Least-­‐Squares  Problem:
    Inertia  Estimation
    I· ˙
    ! + ! ⇥ (I·! + ⇢) + ˙
    ⇢ = ⌧
    H(!, ˙
    !)i = y(!, ⇢, ˙
    ⇢, ⌧)
    i =

    I11 I22 I33 I12 I13 I23
    ⇤|

    View Slide

  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)

    View Slide

  53. 52
    A  Least-­‐Squares  Problem:
    Inertia  Estimation
    minimize
    i
    kHi yk2
    2
    subject to
    (
    I >
    0
    Ijj +
    Ikk
    Ill 0

    View Slide

  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

    View Slide

  55. 54
    Schur Complement:
    Inertia  Estimation
    =)
    (
    C > 0
    A BC 1B|
    0

    A B
    B|
    C
    0

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  60. Questions?
    59

    View Slide

  61. Backup  Slides

    View Slide

  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

    View Slide

  63. • Weather/Climate
    • Ionospheric Physics
    C-­‐REX  Mission,  NASA 62
    Applications

    View Slide

  64. • Planetary  Exploration
    • Gravimetry
    Draper  Labs
    Applications
    63

    View Slide

  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

    View Slide

  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(

    ))

    View Slide

  67. 66
    KickSat

    View Slide

  68. 67

    View Slide

  69. 68

    View Slide

  70. That’s  Us!

    View Slide

  71. 70

    View Slide

  72. Centimeter-­‐Scale  Spacecraft
    Mission  Operations:

    View Slide

  73. 72
    Vibration  Testing:
    Centimeter-­‐Scale  Spacecraft

    View Slide

  74. 73
    Thermal  Vacuum  Testing:
    Centimeter-­‐Scale  Spacecraft

    View Slide

  75. 74
    Thermal  Vacuum  Testing:
    Centimeter-­‐Scale  Spacecraft

    View Slide

  76. 75
    Mass  Properties  Measurement:
    Centimeter-­‐Scale  Spacecraft

    View Slide

  77. 76
    Centimeter-­‐Scale  Spacecraft
    Integration  (12/3/2013):

    View Slide

  78. ChipSat  Spread
    77

    View Slide

  79. ChipSat  Spread
    78

    View Slide

  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)

    View Slide

  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

    View Slide

  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

    View Slide

  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);

    View Slide

  84. Communication
    83
    Number  of  Simultaneous  Users:
    P
    on
    =
    n! Dk(1 D)(
    n k
    )
    k! (n k)!

    View Slide

  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 }| {

    View Slide

  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

    View Slide

  87. 86
    Communication
    GNURadio:

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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:

    View Slide

  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:

    View Slide

  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

    View Slide

  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
    !

    View Slide

  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 =

    r✓/2

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide