1. Task Overview • Create a dynamic model of the spacecraft • Account for the environmental disturbances • Evaluate the dynamic response of the spacecraft • Observe roll, pitch, yaw in time and frequency domains • Interface with the other teams to meet mission requirements

2. Status at PDR • Initial point mass dynamics completed • J2, atmospheric drag, and solar pressure quantified • Orbital elements calculated • Two-mass model of body created • Determined stiffness, damping m2 m1

3. RFAs From PDR • Provide a detailed analysis of loads from two mass model (Prof. Pellegrino) • Directive: switched to rigid body model, now a structural task • Account for J3 and solar activity if long term orbital elements are important (Dr. Watkins) • Accepted for advisory: Insufficient time to analyze complete effect • Deliver accelerations to predict loads to structures, and attitude history for thermal (Prof. Quadrelli) • Action item: accelerations given to structures, attitude history not incorporated into thermal model

4. Receivables • Project: • 300 microradian control • 120 microradian knowledge • Orbit description • Structures: • Moments of inertia for response • Modal frequencies for controller design • Metrology: • Asserted that the mission requirements were sufficient

5. Baseline Approach for Modeling Disturbances • Disturbances Modeled • Solar pressure, Atmospheric drag, and Earth’s Oblateness. • Formulation • AGI-Satellite Kit-3rd party software used to predict Keplerian Orbital parameters. • Matlab used for • Independent verification of AGI-Satellite Kit results. • Integrate the disturbance models with the rigid body dynamics.

6. Reference Frame • Earth Centered Inertial Frame (XYZ) • S/C starts at Perigee • Convert the initial condition from orbit to ECI frame • [Ixyz]=[R][Iorbit]

7. Input Parameters • Physical Data • Satellite Mass ~ 330kg • Drag Cross-Sectional Area ~ 10m2 • Solar Radiation Pressure Cross-Sectional Area ~ 10m2 • Area-to-Mass Ratio β ~ 0.019 m2/kg • Ballistic Coefficient (B = M/Cd*A) ~ 15 (kg/m2) • Drag Coefficient Cd ~ 2.2

8. AGI-Satellite Kit • Assumptions • Orbit Start : 1st July 2012 • Orbit Stop : 2nd July 2014 • Time Step between calculated Ephemeris - 50sec • J2 Coordinate System with Keplerian Orbit Elements • Long-Term-Orbit-Predictor (LOP) • Force Models • Atmospheric Drag • Exponential Density Model • Osculating Altitude • Earth’s Oblateness i.e J2 effect • Solar Radiation Pressure • Perfectly Black Body assumption (Solar Radiation Constant Cp = 1)

9. MATLAB • Algorithm • ODE45 Solver (Runge-Kutta method) • Forces • Solar Pressure • Neglected seasonal variations • Nominal Solar Pressure Radiation constant fp = 4.5*10-6 (N/m2) • Reflectivity ref = 0 ( worst case i.e Black Body) • Atmospheric Drag • Exponential Model for the Density • Earth’s Oblateness – J2 effect

10. Orbital Elements Results 10

11. Orbital Elements Results 11

12. Orbital Elements Results 12

13. Orbital Lifetime Results (STK) 27 Deg Inclined Orbit 5 Deg Inclined Orbit 13

14. Attitude Model • Goal: Predictive capability of attitude dynamics, and provide framework for developing ACS • Assumptions • Circular orbit • J2 and gravity forces are only system torques • Model is linearized about the stable pointing (Jyy towards Earth) • White noise on angle rates (10-7 rad/s) • PD controllers • Gains tuned manually • Goal is 250s settle time from 1 degree offset for pitch, or 1 degree yaw/0.1 degree roll for roll-yaw • Reaction wheels saturate at 0.01 Nm torque and 20 Nms angular momentum • Two wheels in roll direction 14

15. Coordinate System • Yaw about x-axis, Pitch about y-axis (out of plane), Roll about z-axis

16. Attitude Model Equations are for RPY angles and rates. Source: Tewari, Section 14.7.2 16

17. Verification • Theoretical pitch frequency calculated from Wertz, p.610: • Model libration frequency is very close to the theoretical • Model: 0.0018 rad/s ~ 0.28 mHz • Theoretical: 0.0019 rad/s ~ 0.30 mHz 17

18. Verification continued

19. Integrated Model • 6 DOF model of attitude and orbit data • Orbit should be independent of attitude to first order • Attitude does not affect orbit, but orbit data is input to attitude dynamics • Integrated model uses same assumptions as independent attitude and orbit models • Check librations and controllers with simulated orbit data • Compare to stand-alone attitude model • White noise on rates is disturbance • 10-7 rad/s standard deviation 19

20. Integrated Model Results Yaw is not stable because Jxx = Jyy. 20

21. Integrated Model Results (2)

22. Settling Times • Settling times calculated for both independent attitude dynamics model and integrated model • Noise included in model • Settling times calculated separately for pitch; roll and yaw calculated together • Initial conditions • For pitch model: • 0 degree roll, 1 degree pitch, 0 degree yaw • No angle rates • For roll-yaw model: • 0.1 degree roll, 0 degree pitch, 1 degree yaw • No angle rates 22

23. Controller tuning Manually tuned PD controllers to achieve fastest settling times accounting for saturation Settling Times 23

24. Deliverables • Project: • Coded model • PD controller • Structures: • Accelerations from torques • Controller frequencies • Metrology: • Performance of controller (long-term) • Relation of controllability and observability (long-term)

25. Incorporating cross-sectional area Changes over time Incorporate 3D model Evaluating reaction wheels trade Torque vs. extra mass Performance of known options Optimizing controller 100s settling time from 1 degree Open Issues and Concerns 25

26. Summary • Created dynamics model • Predicts orbital elements with disturbances • Verified with STK • Created attitude model • Predicts controller performance in linear region • Verified with pitch frequency • Integrated the two models • Driven by white noise • 256 second pitch settling time

27. Back-up Material

28. Integrated Model Results (2) 28

29. Review Protocol Requests for Action (RFAs) will be categorized as follows: • Action Items (AIs) • Formal write-up required to close • Results presented at CDR • Directives • Recommendations for design improvements or investigations • Informal write-up required to close • Summary of all directive results presented at CDR • Advisories • Recommendations for investigations or identification of problems • No formal closures

30. Functional Requirements Summarize your subsystem functional requirements derived from Level 1s : • 300 microradians axis pointing accuracy • .0075 radion rotation about z-axis • Response time? • Torque required? 30

31. Level 1 Requirements Summarize Level 1s from Project relevant to your subsystem: • Pointing accuracy • Stability • Orbital requirements