1. 1 GP-B “2-second” Filter: Data Analysis Development M.Heifetz, J.Conklin

2. 2 Outline  Fundamentals of 2-sec Filter  Modular Software Structure  Schedule of Tests

3. 3 Four Cornerstones of Filter Development Estimation Algorithms: Numerical Techniques Estimation Theory Gyroscope Motion: Torque Model(s) SQUID Readout Signal Structure: Measurement Model(s) Algebraic Method Machinery: Development and Experience GP-B Data Analysis Experience

4. 4  Gyroscope Readout System θ Apparent Guide Star aberration Guide Star SQUID signal: – Proportional to (  -s) – Scaled by magnetic flux (Cg) – Modulated by spacecraft rotation μ

5. 5  SQUID Readout Signal Model SQUID Data Estimation performed for the data collected during Guide Star Valid (GSV) mode Pointing Orbital data Earth Ephemerides known Estimated (?) T orb = 24.648770 days Pointing Error Compensation: Telescope data + scale factor matching

6. 6  Polhode Evolution and C g Determination C g is sum of LM (tied to gyro spin axis) & trapped magnetic flux (tied to body) The polhode path evolves over the mission Requires more sophisticated estimation Trapped flux mapping provides continuous C g &  p I3I3 I2I2 I1I1 s s  s s  6 Sept 2004 14 Nov 2004 polhode Sept. 6, 2004 Nov. 14, 2004

7. 7 I3I3 I2I2 I1I1 s s  s s  6 Sept 2004 14 Nov 2004 polhode 1. Ideal C g Approach: Exact Polhode Phase  C g Model using exact polhode phase  p Algebraic filter will estimate C g LM, update TFM estimates of a mn, b mn

8. 8 1.Use TFM scale factor variations as is (simplest) Algebraic filter will estimate constant C g LM only 2.Use  C g model without TFM prior information (symmetric phase) Algebraic filter estimates full set of  C g coefficients a nk, b nk and C g LM 3.Use TFM scale factor and estimate correction via  C g Algebraic filter estimates subset of  C g coefficients a mn, b mn, and C g LM  C g – 3 Additional Approaches

9. 9  Gyroscope Motion: Torque Model Models for : 1. 2. TFM Misalignment TorqueRoll-resonance TorqueRelativity

10. 10 Explicit solution for orientation Explicit computation of as a part of Jacobian computation ! - state vector (constant parameters) No need for numerical ODE integration ! Allows explicit computation of the Jacobian !

11. 11  Pointing Error Compensation (matching) Normalized Pointing signal (per axis, per telescope side) Pointing Error ( per axis / per telescope side): matching model 2 Telescope sides (A,B) 2 axes (x,y) 2 signals / axis Gyroscopes 1 and 3: Gyroscopes 2 and 4: - part of state vector (per gyro, per telescope side) s + s - s+ s-

12. 12 GP-B Data Analysis: Nonlinear Filtering Problem - number of data points SQUID Data Model: Nonlinear in x Noise statistics Two main approaches: Iterative Extended Kalman Filter (IEKF) - widely used in post-flight data analysis - drawbacks: linearization and potentially biased state-vector estimate Sigma Point Filter (SPF) - recently developed by the aero-astro community for spacecraft attitude estimation, nonlinear aerodynamic parameter estimation, and tracking applications - claims that performance is better than EKF/IEKF - drawbacks: more computationally intensive than EKF

13. 13 Iterative Extended Kalman Filter (IEKF) Iterative linearization process - Current estimate of the state-vector and its covariance matrix Linearization about current estimate: matrix in batch case Compute Jacobian: Form Innovations: Define correction vector: Linear structure: (1) (2) (3) (4)

14. 14 Output: and Apply linear least-squares estimator (e.g. square-root information filter): Iteration process repeats until the cost function reaches plateau (or ) SQUID Data (GSV) SQUID Model (GSV) + - LSQ Estimator Jacobian Difficulty: Jacobian computation - analytic - numerical Analytic solution for clears the way for the analytic Jacobian computation

15. 15 Module-based Functional Block Diagram -state vector Module h-Jacobian Module - Module IEKF Module Relativity Estimate TFM Data SQUID Data Telescope Data Aberration Data Relativity Estimate uncertainty Roll Phase Data Module Residual Analysis - KACST Module Truth Model Module Optimization

16. 16 Modules where KACST can contribute Module Residuals Analysis Goodness-of-fit tests, Residual model identification Algorithms: Stanford/KACST Code: KACST Module Compute and update spacecraft pointing during GSI based on SQUID data and estimated parameters Algorithms: Stanford/KACST Code: KACST/Stanford

17. 17 Module Truth Model Simulate SQUID data and test Estimation Methods Algorithms: Stanford/KACST Code: KACST Module Optimization Interface between optimization package and GP-B data analysis software Study optimization package that will be used as a part of estimation process; This package exploits subroutines written in C and/or Fortran, and GP-B analysis software is written in Matlab: therefore some interface is needed for communication between various modules Algorithm: Stanford Code: KACST/Stanford

18. 18 Module SPF (for Phase 3) Investigate alternative nonlinear estimation techniques: Sigma-point filters Input:, (no Jacobian required) Output: State vector estimate, covariance matrix Method: Sigma-point filter Algorithm: Stanford/KACST Code: KACST/Stanford Readiness: 0% (4 months)

19. 19 Additional Modules (possible future KACST involvment) Module Data preparation: - Calibration signal removal - Grades - Bandpass filter (roll ± orbit) Input: SQUID signal (sampling rate: 2sec) Data grades Output: SQUID signal Readiness: 100% (for current set of Data Grades)

20. 20 Module 4 methods (see above) Input: C g parameters (C g LM, a nk, b nk ) C g TF, polhode phase and angle Output: Readiness: 80 % for methods 1 and 2, 50% for others (4 weeks) Comments: List of Modules – cont. Code for all methods exist and have been vetted Must be packaged into a single function with option to select method For C g with exact polhode phase (method 4),  p,  p should be written to L3 (and L3 speedread) to drastically reduce execution time

21. 21 Module Input: s-parameters – part of state vector (relativity, torque coefficients) Pointing (both GSV and GSI) Roll Phase, Polhode Phase and Angle Output: orientation Jacobian Method: Explicit solution Numerical integration (back-up) Sub-module Misalignment torque (MT) Misalignment torque model(s) Sub-module Roll-resonance torque (RT) Roll-resonance torque model(s) Readiness: numerical integrator 100% (back-up), analytic 20% (4 weeks) List of Modules – cont.

22. 22 Module Input: - Aberrations (orbital, annual), starlight bending, parallax; - Telescope signals; - Telescope scale factor coefficients (part of state vector) Output: - Pointing - Jacobian - Pointing error estimate (Gyro/Telescope matching) Readiness: 80% (2 weeks) List of Modules – cont.

23. 23 List of Modules – cont. Module h - Jacobian Input: - - as a part of the state vector - Parts of Jacobian (from corresponding modules): Output: - Model - Jacobian Readiness: 50% (3 weeks)

24. 24 Module IEKF (Primary method) Input: Z(t),, Output: State vector estimate, covariance matrix, P Method: IEKF (uses Bierman library) Algorithm: T.Holmes (20%), V.Solomonik, M.Heifetz, J. Conklin Code: V.Solomonik Readiness: 0% (1 month) List of Modules – cont. Module Truth Model Algorithm: M.Heifetz, KACST Code: KACST Readiness: 0%

25. 25 Module Geometric Method Integration Purpose: Apply Geometric Method to s(t) with Roll-Resonance torque removed Algorithm: M.Keiser, J.Conklin, K. Stahl Code: K. Stahl Readiness: 0% List of Modules – cont.

26. 26 Two interwoven loops Guide Star Valid Data Loop (full mission) –State vector parameters estimation: Relativity (r NS, r EW ) Gyro scale factor coefficients (C g LM, a nk, b nk ) Roll phase offset ( δ  ) Telescope scale factor coeffs. (Gyro/Telescope Matching) (c T i ) Roll-resonance torque parameters (c ± 1mn, c ± 2mn ) Misalignment torque parameters (k 1mn, k 2mn ) Initial orientation (s NS0, s WE0 ) Guide Star Invalid Data Loop (full mission) –Pointing determination Pointing is needed for s-propagation Advantage of redundancy: 4 sources of information (4 Gyros) for determining 2 components

27. 27 10 Data Segments interrupted by anomalous events 1) September 13, 2004 – September 23 (11 days) 2) September 25 – November 10 (47 days) 3) November 12 – December 04 (23 days) 4) December 05 – December 09 (5 days) 5) December 10 – January 20, 2005 (42 days) 6) January 21 – March 04 (43 days) 7) March 07 – March 15 (9 days) 8) March 16 – March 18 (3 days) 9) March 19 – May 27 (70 days) 10) May 31 – July 23 (54days) Data Segmentation 307 days of science data available Segments to analyze first

28. 28 Schedule of Initial Tests Phase 1: Test of baseline configuration April - Data: Segment 5 (or 6) - Module : Mode 1 ( from TFM); - Module : Initial profile, no iterative update; - Matching with known telescope scale factors Phase 2: Test of extended baseline configuration June - Data: Segment 5 + 6 - Module : Mode 2 (Estimated parameters); - Module : Initial profile, no iterative update; - Matching: estimation of telescope scale factors Phase 3: Full Mission Analysis Test July