“2-second” Filter: Software Development Review

1. “2-second” Filter: Software Development Review M.Heifetz, J.Conklin

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

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

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

5. Scale Factor (Cg) – 4 Approaches • Use TFM scale factor variations as is (simplest) • Algebraic filter will estimate constant CgLM only • Use Cgmodel without TFM prior information (symmetric phase) • Algebraic filter estimates full set of Cgcoefficients ank, bnk and CgLM • Use TFM scale factor and estimate correction via Cg • Algebraic filter estimates subset of Cgcoefficients amn, bmn, and CgLM

6. 4. Ideal Approach: Exact Polhode Phase • Cg Model using exact polhode phase p • Algebraic filter will estimate CgLM, update TFM estimates of amn, bmn

7. Models for : 1. 2. Gyroscope Motion: Torque Model Relativity Misalignment Torque Roll-resonance Torque TFM

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

9. 2 Telescope sides (A,B) 2 axes (x,y) 2 axes (x,y) • Normalized Pointing signal (per axis, per telescope side) 2 signals / axis 2 signals / axis • Pointing Error ( per axis / per telescope side): matching model • Gyroscopes 1 and 3: • Gyroscopes 2 and 4: Pointing Error Compensation (matching) • - part of state vector • (per gyro, per telescope side)

10. Noise statistics • number of data points SQUIDData Model: Nonlinear in x GP-B Data Analysis: Nonlinear Filtering Problem 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

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

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

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

14. Module • Data preparation: • - Calibration signal removal • - Grades • - Bandpass filter (roll ± orbit) • Input: SQUID signal (sampling rate: 2sec) • Data grades • Output: SQUID signal • Algorithms: T.Holmes (30%), K.Stahl (30%) • Code: K.Stahl • Readiness: 100% (for current set of Data Grades) List of Modules

15. List of Modules – cont. • Module • 4 methods (see above) • Input: Cg parameters (CgLM, ank, bnk ) • CgTF, polhode phase and angle • Output: • Algorithms: M.Heifetz, A.Silbergleit,J.Conklin, V.Solomonik • Code: V.Solomonik, J.Conklin • Readiness: 80 % for methods 1 and 2, 50% for others (4 weeks) • Comments: • Code for all methods exist and have been vetted • Must be packaged into a single function with option to select method • For Cg with exact polhode phase (method 4), p, p should be written to L3 (and L3 speedread) to drastically reduce execution time

16. 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-moduleMisalignment torque (MT) • Misalignment torque model(s) • Sub-moduleRoll-resonance torque (RT) • Roll-resonance torque model(s) • Algorithms: A.Silbergleit, M.Heifetz, J.Conklin • Code: V.Solomonik • Readiness: numerical integrator 100% (back-up), analytic 20% (4 weeks) List of Modules – cont.

17. List of Modules – cont. • 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) • Algorithms: T.Holmes (20%), M.Heifetz, V.Solomonik • Code: V.Solomonik, T.Holmes (20%) • Readiness: 80% (2 weeks)

18. Moduleh-Jacobian • Input: • - • - as a part of the state vector • - Parts of Jacobian (from corresponding modules): • Output: • - Model • - Jacobian • Algorithm: M.Heifetz, A.Silbergleit, V.Solomonik, J. Conklin • Code: V.Solomonik • Readiness: 50% (3 weeks) List of Modules – cont.

19. Module Optimization • Input: Z(t), , Output: State vector estimate, • Method: Nonlinear least-squares fit • Algorithm: A. Bradley (Stanford Optimization Lab) • Code: K. Stahl, KACST • Readiness: 0% (3 months) List of Modules – cont. • 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)

20. Module SPF (for Phase 3) • Input: , (no Jacobian required) • Output: State vector estimate , covariance matrix • Method: Sigma-point filter • Algorithm: T.Holmes (20%), M.Heifetz, J. Conklin, KACST • Code: V.Solomonik, KACST • Readiness: 0% (4 months) List of Modules – cont.

21. 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 Two interwoven loops • Guide Star Valid Data Loop (full mission) • State vector parameters estimation: • Relativity (rNS, rEW) • Gyro scale factor coefficients (CgLM, ank , bnk) • Roll phase offset (δ) • Telescope scale factor coeffs. (Gyro/Telescope Matching) (cTi) • Roll-resonance torque parameters (c±1mn, c±2mn) • Misalignment torque parameters (k1mn, k2mn) • Initial orientation (sNS0, sWE0)

22. Module • Compute and update based on SQUID data (GSI) and estimated parameters • Initial estimate from B. Clarke, J. Conklin exists • Algorithms: M.Heifetz, T.Holmes, J.Conklin, M.Adams, KACST • Code: KACST, M.Adams • Readiness: 0% List of Modules – cont. • ModuleTruth Model • Algorithm: M.Heifetz, KACST • Code: KACST • Readiness: 0%

23. List of Modules – cont. • ModuleResiduals Analysis • Goodness-of-fit tests, Residual structure identification • Algorithms: T.Holmes, M.Heifetz, KACST • Code: KACST • Readiness: 0% • ModuleGeometric 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%

24. Segments to analyze first Data Segmentation 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) 307 days of science data available

25. Schedule of Tests • Phase 1: Test of baseline configurationMarch - April - Data: Segment 5 (or 6) - Module : Mode 1 ( from TFM); - Module : Initial profile, no iterative update; - Matching based on known telescope scale factors (no update); • Phase 2: Test of extended baseline configurationApril-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 TestJuly - August

26. Module • Compute and update spacecraft pointing during GSI based on SQUID data • and estimated parameters • Algorithms: M.Heifetz, T.Holmes, J.Conklin, M.Adams, KACST • Code: KACST, M.Adams Modules where KACST can contribute • ModuleResiduals Analysis • Goodness-of-fit tests, Residual model identification • Algorithms: T.Holmes, M.Heifetz, KACST • Code: KACST

27. ModuleTruth Model • Simulate SQUID data and test Estimation Methods • Algorithms: M.Heifetz, 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: A. Bradley (Stanford Optimization Lab), J.Conklin • Code: K. Stahl, KACST

28. Module SPF (for Phase 3) • Investigate alternative nonlinear estimation techniques: Sigma-point filters • Algorithms: T.Holmes, M.Heifetz, J. Conklin, KACST • Code: KACST