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

BillikenSat-II Antenna Payload Antenna Introduction Dynamics Hysteresis Quaternions Conclusions

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

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

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

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

Dynamics Equations ORBITAL DYNAMICS ATTITUDE DYNAMICS Introduction Dynamics Hysteresis Quaternions Conclusions

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

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

Simulation Red: 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 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

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

Hysteresis Simulation Introduction Dynamics Hysteresis Quaternions Conclusions

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

Quaternion Representation Euler Angles Quaternions Φ, , ψ 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. George BillikenSat-II Team Friends and Colleagues at Parks College Introduction Dynamics Hysteresis Quaternions Conclusions

Appendix- No External Moments