1. Feasibility of Demonstrating PPT’s on FalconSAT-3 C1C Andrea Johnson United States Air Force Academy

2. Outline Problems encountered with PPT’s Methods of demonstrating use Spiral Transfer Attitude Model Experimental Results Recommendations

3. Problems Encountered Low Thrust 160e -6 N maximum thrust 15e -6 second pulse, 2 Hz => 4.8e -9 N average thrust Updated data indicates possibly higher average thrust (50 μN-s) Power requirements Inaccuracy of original model Uncoupled equations of motion Inaccurate disturbance torque models

4. Methods of Demonstrating Spiral Transfer One PPT yields 1.6 cm change in semimajor axis with no disturbance torques No GPS receiver

5. Methods of Demonstrating Cont. Attitude Control Z-axis only possibility for control because of small moment of inertia (1.31 versus 67.4 kg-m 2 )

6. Model Assumptions Equations of motion PPT modeling Disturbance torques Validation

7. Assumptions Simplified satellite model Small center of pressure - center of mass offset No products of inertia Constant, known PPT decay rate Negligible orbital perturbations

8. Assumptions Cont. Body Mass:35.5 kg Boom Mass (without tip mass):3.15 kg Tip Mass:7.45 kg Total Mass:46.1 kg Inertia Tensor (Stowed Boom): kg-m 2 Inertia Tensor (Deployed Boom): kg-m 2 Coefficient of Drag (Cd):2.6 Spacecraft Dipole:0.05 A-m 2 Orbit:Altitude = 560 km Semimajor axis = 6938.137 km Inclination = 35.4 o Eccentricity = 0 Right Ascension = 0 o

9. Equations of Motion

10. PPT Modeling t 15 usec 4.8 nNs 160 μN t 4.8 nN Actual Simulation 1 sec

11. Disturbance Torques Gravity Gradient Magnetic Drag Solar Pressure

12. Gravity Gradient

13. Magnetic 13 th degree, 13 th order IGRF 10 th generation model with secular terms up to 8 th degree and 8 th order

14. Magnetic Cont.

15. Magnetic cont. x y z θ r φ

16. Magnetic Cont. ECF to ECI coordinate frame conversion Precession Nutation Sidereal time Polar motion

17. Drag

18. Solar Pressure

19. Validation Integrator: Attitude and orbital energy and momentum should be constant Gravity gradient: Should match C program data Magnetic field: Should match C program data Drag and solar pressure validated using hand calculations

20. Integrator Energy and momentum constant if no external torques Attitude Orbit Normalized error

21. Integrator: Attitude Energy Momentum Maximum error: 3e -14 Maximum error: 1.5e -14

22. Integrator: Orbit EnergyMomentum Maximum error: 2.5e -14 Maximum error: 7.5e -15

23. Gravity Gradient Validation

24. Gravity Gradient Validation Cont.

25. Magnetic Field Validation Magnetic field in ECF matched C program numerical output 8 th degree, 8 th order With secular terms ECF to ECI conversion output matched C program

26. Estimation Theory Kalman filter Truncate results Statistical mean smoother Batch estimator Data used by filters comes from attitude determination Kalman filter

27. Estimation Theory Cont.

29. Batch Filter Algorithm

30. Batch Filter Algorithm Cont. If (user defined), then exit the loop. If not,

31. Experimental Results No NoiseActual PPT torque0.0000 Dipole (x)0.0000 Dipole (y)0.0500 Percent Error Kalman w/o Smoothing Percent Error Kalman w/ Smoothing Percent Error Batch N/A 3.4001E-108.6001E-100.0000E+00

32. Experimental Results Cont. 0.3E-6 on B fieldActual PPT torque0.0000 Dipole (x)0.0000 Dipole (y)0.0500 Percent Error Kalman w/o Smoothing Percent Error Kalman w/ SmoothingPercent Error Batch N/A 2.07092.23310.0318

33. Experimental Results Cont. No PPT's With PPT's NoisePercent errorNoisePercent error 0.3E-3 on w0.3794880.3E-3 on w10.07 1.33E-6 on wdot0.379488 1.33E-6 on wdot10.07

34. Experimental Results Cont. Batch filter is more accurate with and without noise for longer firing times Kalman filter converges faster for short firing times, but has comparatively poor accuracy

35. Recommendations 24 hour firing Magnetorquers and non-essential systems off Magnetometer readings are taken or IGRF data provided Attitude data for the entire firing period is taken Initialize attitude determination Kalman filter at the start of firing and provide batch filter data only after convergence

36. Questions?