1. Dynamics Modeling and First Design of Drag-Free Controller for ASTROD I Hongyin Li, W.-T. Ni Purple Mountain Observatory, Chinese Academy of Sciences S. Theil, L. Pettazzi, M. S. Guilherme ZARM University of Bremen, Germany

2. ASTROD 2006 2006/07/15 Outline  Introduction of the Mission  Simulator  Controller Development  Conclusion  Outlook

3. ASTROD 2006 2006/07/15 Mission Introduction  Science goal Solar-system gravity mapping & relativistic gravity test  Orbit Heliocentric orbit, 0.5AU-1AU Orbit launch in 2015   SC configuration Cylinder : 2.5 (diameter)m ×2 m covered by solar panels. FEEP(Field Emission Electric Propulsion) Star Sensor, Gyroscope (??) EPS( Electrostatic Positioning/ Measurement System ) 1 test mass

4. ASTROD 2006 2006/07/15 Simulator  Equations of Motion  Disturbance & Environment

5. ASTROD 2006 2006/07/15 Coordinates I Heliocentric inertial frameCenter of mass of suni Earth-fixedCenter of earthe Spacecraft body-fixed frameCenter of mass of spacecraftb Sensor frame for test mass Geometrical center of sensor sens Body-fixed frame for test mass Center of mass of test masstm

6. ASTROD 2006 2006/07/15 Coordinates II FrameBase pointx directionz direction Heliocentric inertial frame Center of mass of sunAries Normal of ecliptic Spacecraft body- fixed frame Center of mass of spacecraft Parallel to symmetry axis of telescope Symmetrical axis Sensor frame for test mass Geometrical center of sensor Normal of the surface of the house nearby the telescope Normal of the upper surface of the house Body-fixed frame for test mass Center of mass of test mass Normal of the surface nearby the telescope Normal of the upper surface of test mass

7. ASTROD 2006 2006/07/15 Equations of Motion SC Translatio n SC Rotation TM Translatio n TM Rotation

8. ASTROD 2006 2006/07/15 Disturbance & Environment  Gravity Field (first order--spherical) Elements of

9. ASTROD 2006 2006/07/15 Disturbance & Environment  Gravity Field (second order… ) One of the Science goal of ASTROD I is to test the gravity field parameter of the solar gravity. So, It must be included in the further model.

10. ASTROD 2006 2006/07/15 Disturbance & Environment Coefficient of reflection Unit vector Effective area of SC Solar mean momentum flux / Solar pressure Solar Pressure

11. ASTROD 2006 2006/07/15 Solar Pressure deterministic part Solar pressure at 1AU Solar pressure at 0.5 AU is 4 times that of 1AU A low-pass filter is used to get the stochastic part with a white noise passed

12. ASTROD 2006 2006/07/15 Coupling between SC&TM Translation coupling Rotation coupling

13. ASTROD 2006 2006/07/15 Sensors and actuators  Star-Sensor  EPS-position measurement  EPS-attitude measurement  EPS-attitude suspension  FEEP-force  FEEP-torque White noise  shaped noise Need to consider sample frequency, nonlinearity in the future model….

14. ASTROD 2006 2006/07/15 Simulator

15. ASTROD 2006 2006/07/15 Controller Design  Structure of Controller  Requirements of controller  Synthesis  Closed loop Analysis

16. ASTROD 2006 2006/07/15 Structure of Controller DFACS 3 Sub-system SensorActuator SC-Attitude (3DOF) Star-Sensor Gyroscope (??) FEEP Drag-Free (3DOF-translation) EPSFEEP TM-Suspension (3DOF-rotation) EPS （ Electrostatic Positioning/ Measurement System ） EPS

17. ASTROD 2006 2006/07/15 Requirements of Controller SC-Attitude (3DOF) Drag-Free (3DOF-translation) TM-Suspension (3DOF-rotation) (????) It depends on the cross-talking between rotation and translation DOFs of test mass. It also need to meet the requirements of laser interferometer(not too big anglar velocity).

18. ASTROD 2006 2006/07/15 Synthesis I With the LQR method we can derive the optimal gain matrix K such that the state-feedback law minimizes the quadratic cost function Given following linear system : Just as the kalman filter we can get the minimum steady-state error covariance LQR: Linear-Quadratic-Regulator

19. ASTROD 2006 2006/07/15 Synthesis II  LQG ＝ LQR ＋ KF Linear-Quadratic-Gaussian ＝ Linear-quadratic-regulator + kalman filter(Linear-quadratic-filter)  With DC disturbance need feed forward DC disturbance

20. ASTROD 2006 2006/07/15 LQG Design Plant

21. ASTROD 2006 2006/07/15 Controller Standard LQR used here is MIMO PD controller. It ’ s feedback is proportion (position, angle ) and derivative (velocity and angular velocity) of outputs of system. Feed forward part can cancel the static error.

22. ASTROD 2006 2006/07/15 Synthesis III  What does feed forward do to the frequency Response of the controller? Improve the performance in low frequency range.(integral ) but reduce the stability of the closed loop,need to trade-off …  Stability of the closed loop system? w ith some weight gains, there is still some poles on the right phase because the negative stiffness. So we need to chose the weight gain carefully. Try and tuning to get better performance as well as a stable system.

23. ASTROD 2006 2006/07/15 Closed loop analysis  Transfer function & PSD of Simulation Data  Pointing accuracy :

24. ASTROD 2006 2006/07/15 Test mass position

25. ASTROD 2006 2006/07/15 Transfer function analysis

26. ASTROD 2006 2006/07/15 PSD analysis

27. ASTROD 2006 2006/07/15 Conclusion  A model for ASTROD I is built. However it is simple, the structure is established and each subsystem can be extended to get a more detailed model.  LQG (LQR+KF) was developed which can meet the requirements of the mission.

28. ASTROD 2006 2006/07/15 Outlook  Detailed model Based on SC design –Inertial sensor ： Cross-coupling between DOFs –Star sensor ： Sample frequency ， data fusion of two sensors. –Gyroscope to help the attitude estimate. –FEEP nonlinearity ， frequency response.Location of FEEP clusters.  Advanced control strategy –Loop-Shaping with weighted LQR –Decoupling of controllers and DOFs –Robust Control method