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

Four Cornerstones of Filter Development

Estimation Algorithms: Numerical Techniques

EstimationTheory

Gyroscope Motion: Torque Model(s)

SQUID Readout Signal Structure: Measurement Model(s)

Algebraic Method Machinery: Development and Experience

GP-B Data Analysis

Experience

4

Gyroscope Readout System

θ

Apparent

Guide Star

aberration

Guide Star

GSe

EWeNSe

s

• SQUID signal: – Proportional to (-s)– Scaled by magnetic flux (Cg) – Modulated by spacecraft rotation

μ

5

SQUID Readout Signal Model

noisebias

ttst

ttsttCtZ

rEWEW

rNSNSgSQUID

)})(sin()]()([

))(cos()]()(){[()(

SQUIDD

ataSQUIDD

ata

EW

NS

• Estimation performed for the data collected during Guide Star Valid (GSV) mode• Pointing

intpoorbGSparbend

ann

Aber

orb

Aber

Orbital data

Earth Ephemerides

known Estimated (?)

Torb= 24.648770 days

Pointing Error Compensation:

Telescope data + scale factor matching

6

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 estimationTrapped flux mapping provides continuous Cg & p

I3

I2I1

s

s

6 Sept 2004

14 Nov 2004

polhode

Sept. 6, 2004 Nov. 14, 2004

7

I3

I2I1

s

s

6 Sept 2004

14 Nov 2004

polhode

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

)2/)(tan()(),()(

)(

,])(sin()(cos(1[)(

2

0

0

tttb

a

tb

ta

tmbtmaCtC

nmN

nmn

mn

m

m

M

mpmpm

LM

g

cg

cg

L

8

1. Use TFM scale factor variations as is (simplest)

• Algebraic filter will estimate constant CgLM only

2. Use Cg model without TFM prior information (symmetric phase)

• Algebraic filter estimates full set of Cg coefficients ank, bnk and CgLM

3. Use TFM scale factor and estimate correction via Cg

• Algebraic filter estimates subset of Cg coefficients amn, bmn, and CgLM

Cg – 3 Additional Approaches

)()( tCCtC TFg

LMgg

)()(

)(

)2/)(tan()(),()(

,])(sin()(cos()(1[)(

01

0001

00

1000

tb

a

tb

ta

tttata

tmbtmataCtC

nN

nmn

mn

m

m

nN

nn

M

mmm

LM

g

cg

cg

cg

L

9

Gyroscope Motion: Torque Model

)]cos()()sin()([))((

)]sin()()cos()([))((

rrNSNSEW

EW

rrEWEWNS

NS

tctcstkrdt

ds

tctcstkrdt

ds

)](sin)()(cos)([)(0020

001

tmktmktkm

kM

mm

k

mnkN

nmn

mn

m

m Mmtk

k

k

k,...,1,0),(01

00

2

1

2

1

c

nncN

n

mn

mn

m

m Mmtc

c

c

c,...,1,0),(01

00

2

1

2

1

)](sin)()(cos)([)( 0

02

0

001

tmctmctcpmp

cM

mm

)2/tan(00

Models for :

;2/ 1.

))()(

)()(arctan()(

tst

tstt

NSNS

EWEW

2.

TFM

Misalignment Torque Roll-resonance TorqueRelativity

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• Explicit solution for orientation

• Explicit computation of as a part of Jacobian computation !x

s

s

),,,,,()(

)(00

r

EW

NS txfts

ts

,...]),,(),,(),,(,,[002121

WENSmnmnmnmnEWNSsscckkrrx

• - state vector (constant parameters)x

• No need for numerical ODE integration !

• Allows explicit computation of the Jacobian !x

h

11

Pointing Error Compensation (matching)

wss

wssN

• Normalized Pointing signal (per axis, per telescope side)

• Pointing Error ( per axis / per telescope side): matching model

])(1[ 2

3,NcNc TTp

yx

2 Telescope sides (A,B)

2 axes (x,y) 2 axes (x,y)

2 signals / axis

),( ss2 signals / axis

),( ss

• Gyroscopes 1 and 3:

Gyroscopes 2 and 4:x

y

• - part of state vector

(per gyro, per telescope side)

),,(3

Tcwc T

s+ s-

s+

s-

12

GP-B Data Analysis: Nonlinear Filtering Problem

Nk ,...2,1 - number of data points

kkkk uxhz ),(SQUIDData

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

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Iterative Extended Kalman Filter (IEKF)

• Iterative linearization process

xPx,ˆ - Current estimate of the state-vector and its covariance matrix)1(~ nx

Linearization about current estimate:

)ˆ()ˆ(|),ˆ(

ˆxxoxx

x

huxhZ

xk

)( nN matrix in batch caseCompute Jacobian:

Form Innovations: ).,ˆ( uxhzz

xx

hJ

ˆ|

xxx ˆDefine correction vector:

kxxJz ,...)ˆ(• Linear structure:

(1)

(2)

(3)

(4)

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Output: and),(

;ˆˆˆ

xx

new

x

new

PPfP

xxx

• Apply linear least-squares estimator (e.g. square-root information filter):

x xP

Iteration process repeats until the cost function reaches plateau (or ) 0ˆx

SQUID Data(GSV)

SQUID Model(GSV)

+

-

LSQ Estimator

x

xxxnew

ˆˆˆ Jacobian

• Difficulty: Jacobian computation- analytic- numerical

• Analytic solution for clears the way for the analytic Jacobian computation

s

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Module-based Functional Block Diagram

-state vector

Module

gC

Module

sModule

GSV

x

h

hModuleh-Jacobian

Module

Z

- z Module IEKF

x

P

x

Cg

x

s

x

gC s

x

GSVZ

Module

GSI

GSIZ

Relativity Estimate

GSI

GSITFM

Data

SQUID Data

Telescope Data

Aberration Data

ˆ

Relativity Estimate

uncertainty

Roll Phase Data

Module Residual Analysis

- KACST

Module

Truth Model

Module

Optimization

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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

GSI

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• 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

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• 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)

)(tZPx

),( txh

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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)

)(tZ

Z ,...),,()( sChtZgGSV

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• 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:

gC

,g

Cx

Cg

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 Cg with exact polhode phase (method 4), p, p should be written to L3 (and L3 speedread) to drastically reduce execution time

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• 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)

x

s

,...),( txss

s

)(),( tstsEWNS

List of Modules – cont.

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• 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)

GSV

GSV

x

List of Modules – cont.

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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)

)(th

x

h

,x

s

,x

Cg

x

),(tCg

)(),(),(),(),( ttttstsrEWNSEWNS

ˆ

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• 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.

),( txhx

h

x

• Module Truth Model Algorithm: M.Heifetz, KACST

Code: KACST

Readiness: 0%

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• 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

Two interwoven loops• Guide Star Valid Data Loop (full mission)

– State vector parameters estimation:• Relativity (rNS, rEW)• Gyro scale factor coefficients (Cg

LM, ank , bnk)• Roll phase offset (δ)• Telescope scale factor coeffs. (Gyro/Telescope Matching) (cT

i)• Roll-resonance torque parameters (c±

1mn, c±2mn)

• Misalignment torque parameters (k1mn, k2mn) • Initial orientation (sNS0, sWE0)

• 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

EWNS ,

),(EWNS

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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

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Schedule of Initial TestsSchedule 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

gC

GSI g

C

• 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

gC

gC

GSI

• Phase 3: Full Mission Analysis Test July