1. Active magnetic attitude control system providing three-axis inertial attitude M. Ovchinnikov, V. Penkov, D. Roldugin, A. Guerman Keldysh Institute of Applied Mathematics of RAS University of Beira Interior

2. Contents Main goal Problem statement – Geomagnetic field vectors – Analysis methods Analysis – Fast and moderate rotations – Slow rotation without gravity – Optimal control gains – Gravity influence Conclusion Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 2

3. Main goal To overcome two principal MACS problems – Underactuation, that is the torque is perpendicular to the geomagnetic induction vector – Torque value limit due to the limited power and dimension capabilities To assess gravity effect on the inertial attitude 3Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems

4. Problem statement Euler and osculating equations are used Satellite is equipped with three magnetorquers Attitude is known Control torque or control and gravitational torques are taken into account Orbit is a Keplerian one Averaged and IGRF models are used Control torque is based on the PD-controller Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 4

5. Averaged: Geomagnetic induction vector evenly rotates on the cone surface Geomagnetic field models IGRF 5Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems

6. Analysis methods System in arbitrary motion is often analyzed using numerical methods. Not enough assumptions can be made to simplify the system Floquet theory Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 6 Transient motion Multiple scales method Arbitrary motion Numerical analysis Steady-state motion Poincare method

7. Fast/moderate rotation Control gains are comparable Mean influence of the positional control part and gravitational torque is equal to zero The control is identical to the “-Bdot” one and to the eddy currents The angular velocity is exponentially damped to the orbital velocity value 7Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems

8. Slow rotation. Averaged equations Averaged linearized equations without gravity Damped oscillations for each angle Linearized averaged equations are asymptotically stable, so initial equations allow asymptotically stable limit cycle. The solution is in the vicinity of the averaged equations equilibrium, so the averaging is valid on the unlimited time interval 8Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems

9. Control gains 9Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems

10. Floquet theory: more accurate results Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 10

11. Poincare method: damping effect Three-axis control with gravity, orbital plane motion Oscillations without damping Damping-driven modulus change for discrete frequency values Approximate solution Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 11

12. Periodic solutions Periodic solutions are found numerically Approximate formulas for the amplitude depending on satellite parameters are found Simple formula may be used in the vicinity of necessary attitude Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 12

13. Floquet theory: stability area change Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 13

14. Floquet theory: stability area change Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 14

15. Floquet theory: stability area change Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 15

16. Floquet theory: stability area change Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 16

17. Periodic solutions amplitude Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 17

18. Results Three-axis attitude is proven to be accessible Simple formulas for optimal control gains are found More accurate Floquet-based optimal parameters are found Gravity effect is found to be controversial, acting as control or disturbing torque in general Mar 24-26, 2014 2nd IAA Conference on Dynamics and Control of Space Systems 18

19. Acknowledgment 2nd IAA Conference on Dynamics and Control of Space Systems 19 The work was supported by RFBR grants №№ 12-01-33045, 13-01-00665, 14-01-31313 project Odyssea Mar 24-26, 2014