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Advanced 2-Second Filter Development for GP-B Data Analysis and Estimation Techniques

This document outlines the core principles and methodologies behind the development of the 2-second filter utilized in the Gravity Probe B (GP-B) experiment. It explores the modular software structure necessary for effective data analysis, detailing various estimation algorithms such as the Iterative Extended Kalman Filter (IEKF) and Sigma Point Filter (SPF). The paper also discusses gyroscope motion characterization, including torque models and data pointing error compensation, while emphasizing the importance of establishing reliable models for nonlinear filtering in spacecraft attitude estimation.

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Advanced 2-Second Filter Development for GP-B Data Analysis and Estimation Techniques

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  1. GP-B “2-second” Filter: Data Analysis Development 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. Guide Star Apparent Guide Star aberration θ Gyroscope Readout System μ • SQUID signal: • Proportional to (t-s) • Scaled by magnetic flux (Cg) • Modulated by spacecraft rotation

  5. 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

  6. I3  s polhode 14 Nov 2004  s 6 Sept 2004 I2 I1 Polhode Evolution and Cg Determination Cg 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 Cg &p Sept. 6, 2004 Nov. 14, 2004

  7. I3  s polhode 14 Nov 2004  s 6 Sept 2004 I2 I1 1. Ideal Cg Approach: Exact Polhode Phase • Cg Model using exact polhode phase p Algebraic filter will estimate CgLM, update TFM estimates of amn, bmn

  8. Cg – 3 Additional 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

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

  10. - 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

  11. 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) s+ s- s+ s- • - part of state vector • (per gyro, per telescope side)

  12. 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

  13. - 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)

  14. 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

  15. 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

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

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

  19. 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) Additional Modules(possible future KACST involvment)

  20. List of Modules – cont. • Module • 4 methods (see above) • Input: Cg parameters (CgLM, ank, bnk ) • CgTF, polhode phase and angle • Output: • 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

  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-moduleMisalignment torque (MT) • Misalignment torque model(s) • Sub-moduleRoll-resonance torque (RT) • Roll-resonance torque model(s) • Readiness: numerical integrator 100% (back-up), analytic 20% (4 weeks) List of Modules – cont.

  22. 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) • Readiness: 80% (2 weeks)

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

  24. 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) • ModuleTruth Model • Algorithm: M.Heifetz, KACST • Code: KACST • Readiness: 0%

  25. List of Modules – cont. • 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%

  26. 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)

  27. 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

  28. Schedule of Initial Tests • Phase 1: Test of baseline configurationApril - 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 configurationJune - 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

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