satellite passive attitude stabilization using permanent magnets – dynamic model and simulation...
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Satellite Passive Attitude Stabilization Using Permanent Magnets – Dynamic
Model and Simulation
Darren Pais and Dr. Sanjay JayaramParks College, Saint Louis U.
Introduction Dynamics Hysteresis Quaternions Conclusions
Payload
Antenna
BillikenSat-II
Antenna
Introduction Dynamics Hysteresis Quaternions Conclusions
Attitude Control System DecisionREQUIREMENTS:
• Orient omni-directional antennas parallel to Earth’s surface
• Stability in flight (mitigate large amplitude oscillation/angular rates)
• Payload has no pointing requirements
CONSTRAINTS:
• Fail-safe design (control system is NOT an experiment)
• Inexpensive in terms of cost, size & weight and computation, simple design
DECISION:
Completely passive control system using
permanent magnets and hysteresis dampers
Introduction Dynamics Hysteresis Quaternions Conclusions
The Idea
Nm
Smorbit
Communication Window
Nm
Smorbit
Geo-Magnetic Lines of Force
: Permanent Magnet / Antenna
Introduction Dynamics Hysteresis Quaternions Conclusions
Reference FramesZ
X
Y
xz
y
y
xz
IRF
MRF
X
x
x
z
z
Circular Polar Orbit
cos 0 sin
0 1 0
sin 0 cos
x X
y Y
z Z
Transformation Matrix
Introduction Dynamics Hysteresis Quaternions Conclusions
Reference Frames
O
b2 (hysteresis axis)
b3 (permanent magnet axis)
b1 (hysteresis axis)
BRFO
Transformation Matrix M BT
Roll Φ
Pitch
Yaw ψ
1
2
3
cos cos sin sin sin cos sin sin cos sin cos sin
cos sin sin sin cos cos cos sin sin sin cos cos
cos sin sin cos cos
b x
b y
zb
Introduction Dynamics Hysteresis Quaternions Conclusions
Dynamics EquationsORBITAL DYNAMICS
ATTITUDE DYNAMICS
1/, / /
B IB ext B B I B B I
dI M I
dt
1 2
1 2
1 2 3
1sin cos
cos
cos sin
tan sin cos
d
dt
d
dtd
dt
1
2 /
3
0
1
0
M BB I n T
Introduction Dynamics Hysteresis Quaternions Conclusions
Geo-magnetic field
L-Shell Model (Wertz SMAAD):
2cosR L
WMM 2005 Model:
• Magnetic field vector in XYZ coordinates
• Obtained from fitting experimental data
,perm B B BM B
Introduction Dynamics Hysteresis Quaternions Conclusions
Simulation Parameters
2B
0.00182 0 0
I = 0 0.00185 0 kg.m
0 0 0.00220
INERTIA TENSOR:
ORBIT: Polar, Circular, 800 km altitude, starting at north pole
INITIAL ATTITUDE: Roll, pitch and yaw set to 00
METHOD: Numerical integration of differential equations at discrete time-steps
PARAMETERS OF INTEREST: B-offset, tumbling at pole, stability
Introduction Dynamics Hysteresis Quaternions Conclusions
SimulationRed: 0.01 Am2
Blue: 0.03 Am2
Introduction Dynamics Hysteresis Quaternions Conclusions
Magnetic Hysteresis
Hysteresis Materials: Realignment of internal dipoles under low external fields Frictional heat dissipation
Modeling Hysteresis
Ref: Levesque, J-F, Passive Magnetic Attitude Stabilization using Hysteresis Materials, U. of Sherbrooke
Introduction Dynamics Hysteresis Quaternions Conclusions
Hysteresis Modeling
1 00
0
2 1tan tan ( )
2m
m
B BB H H
H B
Tangent Function:
Time Dependence:
.dB dB dH
dt dH dt
B: magnetic inductionH: external magnetizing field
Reference: Flately and Henretty, A Magnetic Hysteresis Model, NASA-GSFC Flight Mechanics Symposium 1995
Introduction Dynamics Hysteresis Quaternions Conclusions
Hysteresis Modeling
Parameters (Transit-1B)
Bo= 120 GaussBm=2500 GaussHo=0.035 Oe
Introduction Dynamics Hysteresis Quaternions Conclusions
Hysteresis Simulation
Introduction Dynamics Hysteresis Quaternions Conclusions
Singularities!
Nm
Sm
Nm
Sm
Pitch Singularities!
90o pitch!
Introduction Dynamics Hysteresis Quaternions Conclusions
Quaternion Representation
0 1 2 3ˆˆ ˆq q i q j q k qΦ, , ψ
cos sinu
q
QuaternionsEuler Angles
2 2 2 21 2 3 0 1 2 0 3 1 3 0 2
2 2 2 21 2 0 3 1 2 3 0 2 3 0 1
2 2 2 21 3 0 2 2 3 0 1 1 2 3 0
2( ) 2( )
2( ) 2( )
2( ) 2( )
M B
q q q q q q q q q q q q
T q q q q q q q q q q q q
q q q q q q q q q q q q
cos cos sin sin sin cos sin sin cos sin cos sin
cos sin sin sin cos cos cos sin sin sin cos cos
cos sin sin cos cos
1 3 2 1 1
2 3 1 2 2
3 2 1 3 3
0 1 2 3 0
0
01
02
0
q q
q qd
q qdt
q q
1 2
1 2
1 2 3
1sin cos
cos
cos sin
tan sin cos
d
dt
d
dtd
dt
Introduction Dynamics Hysteresis Quaternions Conclusions
Quaternion Simulation
Introduction Dynamics Hysteresis Quaternions Conclusions
Conclusions• Passive control system using magnets is efficient, fail-safe and inexpensive
• Dynamic Magnetic Hysteresis modeling using tangent functions is a uniquely good representation for sizing hysteresis material for nano-satellites
• Quaternion-based attitude representation provides a non-singular attitude representation
• Optimal solution is a tradeoff between Hysteresis Damping and Permanent Magnet Strengths
Dynamics + Quaternions + Tangent Hysteresis = Representative Dynamic Model
Thank You: Dr. Jayaram, Dr. Ravindra and Dr. GeorgeBillikenSat-II TeamFriends and Colleagues at Parks College
Appendix- No External Moments