“2-second” Filter: Software Development Review

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“2-second” Filter: Software Development Review

M.Heifetz, J.Conklin

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Outline Fundamentals of 2-sec Filter

Modular Software Structure

Schedule of Tests

…

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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

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SQUID Readout Signal Model

noisebias

ttstttsttCtZ

rEWEW

rNSNSgSQUID

)})(sin()]()([))(cos()]()(){[()(

SQUIDD

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

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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

Scale Factor (Cg) – 4 Approaches

)()( tCCtC TFg

LMgg

)()()(

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

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

01

0001

00

1000

tba

tbta

tttata

tmbtmataCtC

nN

nmn

mn

m

m

nN

nn

M

mmm

LMg

cg

cg

cg

L

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4. Ideal 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

tttba

tbta

tmbtmaCtC

nmN

nmn

mn

m

m

M

mpmpm

LMg

cg

cg

L

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Gyroscope Motion: Torque Model

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

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

rrNSNSEW

EW

rrEWEWNS

NS

tctcstkrdtds

tctcstkrdtds

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

001

tmktmktkm

kM

mm

k

mnkN

nmn

mn

m

m Mmtkk

kk

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

00

2

1

2

1

c

nncN

n

mn

mn

m

m Mmtcc

cc

,...,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()(

tsttstt

NSNS

EWEW

2.

TFM

Misalignment Torque Roll-resonance TorqueRelativity

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• Explicit solution for orientation (Alex S.)

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

s

),,,,,()()(

00

r

EW

NS txftsts

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

WENSmnmnmnmnEWNSsscckkrrx

• - state vector (constant parameters)x

• No need for numerical ODE integration !

• Allows explicit computation of the Jacobian !xh

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Pointing Error Compensation (matching)

wsswssN

• 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 ),( ss

2 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

,1Tc

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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(ˆ

xxoxxxhuxhZ

xk

)( nN matrix in batch caseCompute Jacobian:

Form Innovations: ).,ˆ( uxhzz

xxhJ

ˆ|

xxx ˆDefine correction vector:

kxxJz ,...)ˆ(• Linear structure:

(1)

(2)

(3)

(4)

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Output: and),(

;ˆˆˆ

xx

new

x

new

PPfPxxx

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

x̂ xP

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

SQUID Data(GSV)

SQUID Model(GSV)

+

-

LSQ Estimator x̂

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

xh

hModuleh-Jacobian

Module

Z

- z Module IEKF

x̂

P

xCg

xs

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

ModuleTruth Model

ModuleOptimization

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List of Modules

• 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)

)(tZ

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

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• Module 4 methods (see above) Input: Cg parameters (Cg

LM, ank, bnk ) Cg

TF, polhode phase and angle Output:

Algorithms: M.Heifetz, A.Silbergleit, J.Conklin, V.Solomonik

Code: V.Solomonik, J.ConklinReadiness: 80 % for methods 1 and 2, 50% for others (4 weeks) Comments:

gC

,g

CxC

g

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) Algorithms: A.Silbergleit, M.Heifetz, J.Conklin Code: V.Solomonik Readiness: numerical integrator 100% (back-up), analytic 20% (4 weeks)

xs

,...),( txss

s

)(),( tstsEWNS


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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)

Algorithms: T.Holmes (20%), M.Heifetz, V.SolomonikCode: V.Solomonik, T.Holmes (20%)Readiness: 80% (2 weeks)

GSV

GSV

x


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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 Algorithm: M.Heifetz, A.Silbergleit, V.Solomonik, J. Conklin Code: V.Solomonik Readiness: 50% (3 weeks)

)(th

xh

,xs

,xC

g

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)


),( txhxh

x̂

• 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)

),( txhxh

x̂

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• 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)

)(tZPx̂

),( txh

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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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• Module Truth Model Algorithm: M.Heifetz, KACST Code: KACST Readiness: 0%


• 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%

GSI

GSI

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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.• Module Residuals Analysis Goodness-of-fit tests, Residual structure identification Algorithms: T.Holmes, M.Heifetz, KACST Code: KACST Readiness: 0%

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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 TestsSchedule of Tests

• Phase 1: Test of baseline configuration March - 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);

gC

GSI gC

• Phase 2: Test of extended baseline configuration April-June - Data: Segment 5 + 6- Module : Mode 2 (Estimated parameters);- Module : Initial profile, no iterative update;- Matching: estimation of telescope scale factorsg

Cg

CGSI

• Phase 3: Full Mission Analysis Test July - August

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Modules where KACST can contribute• Module Residuals Analysis Goodness-of-fit tests, Residual model identification Algorithms: T.Holmes, M.Heifetz, KACST Code: KACST

• 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

GSI

GSI

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• Module Truth 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.ConklinCode: K. Stahl, KACST

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• Module SPF (for Phase 3) Investigate alternative nonlinear estimation techniques: Sigma-point filters

Algorithms: T.Holmes, M.Heifetz, J. Conklin, KACST

Code: KACST

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“2-second” Filter: Software Development Review