1. LTP Barcelona_26_27_06_2007 S. Vitale1 Master Plan Analyses DA#6

2. LTP Barcelona_26_27_06_2007 S. Vitale2 X-dynamics update

3. LTP Barcelona_26_27_06_2007 S. Vitale3 The changes

4. LTP Barcelona_26_27_06_2007 S. Vitale4 The effective dynamics 2-inputs/2-outputs system (IFO signals x 1, x  ) Multi parameter dynamical model o i,1 o1o1 o i,  oo

5. LTP Barcelona_26_27_06_2007 S. Vitale5 Basic dynamics 2-body difference of force drives the two differential coordinates. Only two inputs are needed Force on x 1 can be applied either via o i1 (Thrusters) or via g x1 (Electrostatic) If g x1 =-H df o i1 formulas stay the same but cross-talk coefficient changes meaning. Careful: to apply g x1 must increase stiffness.

6. LTP Barcelona_26_27_06_2007 S. Vitale6 Signal amplitude adjusted to compensate for gain and to avoid force saturation

7. LTP Barcelona_26_27_06_2007 S. Vitale7 Blue:  x. Red: x 1

8. LTP Barcelona_26_27_06_2007 S. Vitale8 Real displacement

9. LTP Barcelona_26_27_06_2007 S. Vitale9 Results

10. LTP Barcelona_26_27_06_2007 S. Vitale10 y-x Cross-Talk Same approach: response linear in imperfections Imperfections unknown amplitudes to be extracted by Wiener-Kolmogorov (vectorial) theory Stimuli also applied as forces to distinguish actuation cross-talk

11. LTP Barcelona_26_27_06_2007 S. Vitale11 Starting point

12. LTP Barcelona_26_27_06_2007 S. Vitale12 Linearisation Force input“Displacement” input

13. LTP Barcelona_26_27_06_2007 S. Vitale13

14. LTP Barcelona_26_27_06_2007 S. Vitale14 Dynamics 2: y-  nominal Acting on TM for S/C attitude TM attitude irrelevant to first order

15. LTP Barcelona_26_27_06_2007 S. Vitale15 Dynamics 2: Solving for S/C acceleration

16. LTP Barcelona_26_27_06_2007 S. Vitale16 Dynamics 3: x and imperfections No inputs on x No imperfections on x Nominal response = 0

17. LTP Barcelona_26_27_06_2007 S. Vitale17

18. LTP Barcelona_26_27_06_2007 S. Vitale18 Various Matrices

19. LTP Barcelona_26_27_06_2007 S. Vitale19 Dynamics and control imperfections

20. LTP Barcelona_26_27_06_2007 S. Vitale20 Signal and actuation imperfections

21. LTP Barcelona_26_27_06_2007 S. Vitale21 Numerical Calculation Two cases: –Displacement command within drag-free loop Excites thrusters Probes thrusters cross-talk Electrostatic TM cross-talk only excited through very low frequency attitude control –Force command on TM1 and TM2 Probes electrostatics

22. LTP Barcelona_26_27_06_2007 S. Vitale22 Acceleration noise on x axis. Noise cross-talk neglected Channel 1 (S/C-TM1) Channel  (TM2-TM1) |Cross-spectrum|

23. LTP Barcelona_26_27_06_2007 S. Vitale23 Case 1: displacement excitation y1 y2 Force limited < 10 µN

24. LTP Barcelona_26_27_06_2007 S. Vitale24 Excitation

25. LTP Barcelona_26_27_06_2007 S. Vitale25 Response

26. LTP Barcelona_26_27_06_2007 S. Vitale26 Response, absolute attitude

27. LTP Barcelona_26_27_06_2007 S. Vitale27 Results (Preliminary): statistically independent combinations Signal cross-talk y-to-  measured with high resolution If ifo sensitivity calibrated on ground, thruster cross-talk and cross- stiffness can be disentangled

28. LTP Barcelona_26_27_06_2007 S. Vitale28 Case 2: force excitation (<1µm displacement) y1 y2

29. LTP Barcelona_26_27_06_2007 S. Vitale29 Force excitation

30. LTP Barcelona_26_27_06_2007 S. Vitale30 Displacement response

31. LTP Barcelona_26_27_06_2007 S. Vitale31 Results (preliminary): statistically independent imperfections Actuation cross-talk measured with high resolution Signal cross-talk y-to-  also measured scaled for smaller excitation

32. LTP Barcelona_26_27_06_2007 S. Vitale32 Actuation cross-talk channel 1

33. LTP Barcelona_26_27_06_2007 S. Vitale33 Actuation cross-talk channel 

34. LTP Barcelona_26_27_06_2007 S. Vitale34 Signal cross-talk channel 

35. LTP Barcelona_26_27_06_2007 S. Vitale35 Acceleration PSD Key element: converting data from displacement to acceleration

36. LTP Barcelona_26_27_06_2007 S. Vitale36 Getting the filter Time domain operation more transparent Double derivative required Non-casual filter preferred Currently in torsion pendulum: 2 nd order fit to 5 points

37. LTP Barcelona_26_27_06_2007 S. Vitale37

38. LTP Barcelona_26_27_06_2007 S. Vitale38 Planning Next steps –Complete technical note –Complete analysis for acceleration PSD –Get the number for z –  -x cross-talk –  -x cross-talk –More difficult  -x cross- talk

39. LTP Barcelona_26_27_06_2007 S. Vitale39