1. Satellite Passive Attitude Stabilization Using Permanent Magnets – Dynamic Model and Simulation
Darren Pais
and
Dr. Sanjay Jayaram
Parks College, Saint Louis U.

2. BillikenSat-II Antenna Payload Antenna
Introduction Dynamics Hysteresis Quaternions Conclusions

3. Attitude Control System Decision
REQUIREMENTS:
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

4. The Idea Nm Sm : Permanent Magnet / Antenna
orbit
Communication Window
Geo-Magnetic Lines of Force
: Permanent Magnet / Antenna
Introduction Dynamics Hysteresis Quaternions Conclusions

5. Reference Frames Z X Y x z y IRF MRF Transformation Matrix X x z 
Circular Polar Orbit
Introduction Dynamics Hysteresis Quaternions Conclusions

6. Reference Frames O b2 (hysteresis axis) b3 (permanent magnet axis)
BRF
Transformation Matrix
Yaw ψ
Roll Φ
Pitch 
Introduction Dynamics Hysteresis Quaternions Conclusions

7. Dynamics Equations ORBITAL DYNAMICS ATTITUDE DYNAMICS
Introduction Dynamics Hysteresis Quaternions Conclusions

8. Geo-magnetic field L-Shell Model (Wertz SMAAD): WMM 2005 Model:
Magnetic field vector in XYZ coordinates
Obtained from fitting experimental data
Introduction Dynamics Hysteresis Quaternions Conclusions

9. Simulation Parameters
INERTIA TENSOR:
Polar, Circular, 800 km altitude, starting at north pole
ORBIT:
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

10. Simulation Red: 0.01 Am2 Blue: 0.03 Am2
Introduction Dynamics Hysteresis Quaternions Conclusions

11. 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

12. Hysteresis Modeling Tangent Function: Time Dependence:
B: magnetic induction
H: external magnetizing field
Reference: Flately and Henretty, A Magnetic Hysteresis Model, NASA-GSFC Flight Mechanics Symposium 1995
Introduction Dynamics Hysteresis Quaternions Conclusions

13. Hysteresis Modeling Parameters (Transit-1B) Bo= 120 Gauss
Bm=2500 Gauss
Ho=0.035 Oe
Introduction Dynamics Hysteresis Quaternions Conclusions

14. Hysteresis Simulation
Introduction Dynamics Hysteresis Quaternions Conclusions

15. Singularities! 90o pitch! Nm Sm Nm Sm Pitch Singularities!
Introduction Dynamics Hysteresis Quaternions Conclusions

16. Quaternion Representation
Euler Angles
Quaternions
Φ, , ψ
Introduction Dynamics Hysteresis Quaternions Conclusions

17. Quaternion Simulation
Introduction Dynamics Hysteresis Quaternions Conclusions

18. 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. George
BillikenSat-II Team
Friends and Colleagues at Parks College
Introduction Dynamics Hysteresis Quaternions Conclusions

19. Appendix- No External Moments