Ph.D. Defense

Zac Manchester
Centimeter-‐Scale Spacecraft:
Design, Fabrication, and Deployment

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Why?
1

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”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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• Electronics are improving much faster
Wikipedia

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Shrink The Spacecraft!
4

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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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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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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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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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Centimeter-‐Scale Spacecraft
9

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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.

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11

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Returned To Earth (5/18/2014):
14

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Video Courtesy Ben Bishop 15
KickSat Mission Concept:

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16
Deployment Testing:

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17
Launch (4/18/2014):

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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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19
Communication
Clean Signal:

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20
Communication
Received Signal:

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21
Communication
Clean Signal:

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22
Communication
Received Signal:

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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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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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25
Communication
Received Signal After Matched Filter:

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26
Communication
Reciever:

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27
Communication
Testing:

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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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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.

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30
Gyrostat Dynamics:
J = I 1
˙
h = h ⇥ J ·(h ⇢)
h = I·! + ⇢
I· ˙
! + ! ⇥ (I·! + ⇢) + ˙
⇢ = ⌧

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0
0.5
1
1.5
2
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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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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33
Periodic Averaging:
V 0
L
= hd
·J ·hd
1
T
Z T
0
hd
·J ·h dt

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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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35
Control Law:
˙
V =
1
T
Z T
0
(h + hd)·J ·h ⇥ J ·⇢ dt
b(h)
⇢ ⇡ ⇢
max
sign(b)

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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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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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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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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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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.

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41
Quaternions:
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

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42
Hamilton’s Principle:
=) I· ˙
! + ! ⇥ (I·!) = 0
S =
d
d✏
✏=0
Z tf
t0
1
2
✏ ˆ
!· ˆ
I·✏ ˆ
! dt = 0

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43
Discrete Action Sum:
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

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44
Discrete Hamilton’s Principle:
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

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45
Free Rigid Body Simulation:
0 2 4 6 8 10
0
5
10
15
20
25
Time (min)
Error (%)
Variational
Midpoint
ODE45
Momentum Error (%)

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46
Free Rigid Body Simulation:
0 2 4 6 8 10
0
2
4
6
8
Time (min)
Error (%)
Variational
Midpoint
ODE45
Energy Error (%)

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47
Kalman Filtering:
10−1
100
101
102
0
10
20
30
40
50
Sample Rate (Hz)
RMS Error (deg)
Standard MEKF
Variational MEKF

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Inertia Estimation
48
Problem:
• Inertia knowledge is needed for control of KickSat’s spin axis
• Accurate ground-‐based inertia measurements require expensive specialized
equipment

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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

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50
A Least-‐Squares Problem:
Inertia Estimation
I· ˙
! + ! ⇥ (I·! + ⇢) + ˙
⇢ = ⌧
H(!, ˙
!)i = y(!, ⇢, ˙
⇢, ⌧)
i =
⇥
I11 I22 I33 I12 I13 I23
⇤|

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51
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)

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52
Inertia Estimation
minimize
i
kHi yk2
2
subject to
(
I >
0
Ijj +
Ikk
Ill 0

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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

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

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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

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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

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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

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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

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Questions?
59

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Backup Slides

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• 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.
*

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

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• Planetary Exploration
• Gravimetry
Draper Labs
Applications
63

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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

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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(
✓
))

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66
KickSat

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67

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That’s Us!

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70

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Mission Operations:

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72
Vibration Testing:

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73
Thermal Vacuum Testing:

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74
Thermal Vacuum Testing:

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75
Mass Properties Measurement:

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76
Integration (12/3/2013):

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ChipSat Spread
77

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ChipSat Spread
78

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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)

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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

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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
◆

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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);

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

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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 }| {

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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

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86
Communication
GNURadio:

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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

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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

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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

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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
91
Flat-‐Spin Recovery Simulation:

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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
92
Spin-‐Inversion Simulation:

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93
Quaternions:
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

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94
Quaternions:
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
!

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95
Quaternion Exponential:
eq
=
eqs
2
6
6
4
qv
kqv
k sin(
kqv
k
)
cos(
kqv
k
)
3
7
7
5 =
eˆ
r✓/2

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96
Hamilton’s Principle:
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
⌘

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97
Simulation with Kane Damper:
10−1
100
101
102
0
10
20
30
40
50
60
Damping Constant
Running Time (sec)
ODE45
Variational
!
!d

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98
!
!d
0 100 200 300
1.22
1.225
1.23
1.235
1.24
1.245
Time (sec)
Energy
ODE45
Variational

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99
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

Ph.D. Defense

Ph.D. Defense

Zac Manchester

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