Manifold Filtering ProblemLockheed Martin

Jarett HailesJonathan Wiersma

Richard VanWeeldenJuly 21, 2003

Outline

• Pod problem description

• Signal / Observation Model

• SERP Simulation

• Parameter Details

• Results

• IDEX Implementation

• Sketch of progress towards proof

Problem Description

Pod Sensor

Second Sensor

Signal Description

233

33

:

:x

h

Process Wiener D2: tW

tttt dWXdtXhdX )()( Stratonovich SDE

3

2

1

x

x

x

X t 3 Dimensional State

Signal Constraints

2

0 0

0 )()()(k

tk

sskt dWXfAXfXf

3

10 )()())((

iii xfxhxfA

3

1

1,2 j )()())((i

iijj xfxxfA

0fAk

2

3

2

2

2

1)( xxxXf

Signal Implementation

tttt dWXdtXhdX )()( Stratonovich SDE

dtXdWXdtXhdX ttttt )()()( Ito SDE

)(~

)(cos)( 2

1

tt XhxXh

)(tansin

)(tancos

)(~

1

31

2

2

3

2

1

1

31

2

x

xx

xx

x

xx

Xh t

))(

~(

))(~

(

))(~

(

0)(

3

2

1

1

3

2

t

t

t

t

Xh

Xh

Xh

x

x

X

3

2

1

5.0)(

x

x

x

X t

Pod Observation Model

ktk VXYk

)( proj31 ,

31, : basis of plane normal to r

r : resting position of pod sensor <0,1,0>

31,

2

4

3

1) Resample Particles

2) Evolve Particles

3) Update Weights given Yk

If W(ξti) < ρ W(ξt

j):

)()(

)(jt

it

jt

WW

W

Prob:

22

)()(

)(jt

it

it

WW

W

)( j

tW )( i

tW

2

)()( it

jt WW

Filtering Using SERP

Simulation Parameters

001.1z1010

600007.0

6.04.0

93

V

Count Particle

e

de

02

2cos

21Error

Function:

<0,0,0>

r = 1

r = 1

eSignal

Estimate

Simulation

Future Directions

- Workable explicit solution-Eliminate approximation errors in SERP particle evolve

- Use IDEX as filter

- More realistic manifolds- Cantilever equations

- Enhance signal motion- Damped Harmonic Motion

θ

φ

Other Observation Model

Filtering with IDEX

• Goal: prove explicit solutions exist

• IDEX provides:

• Faster computation

• No inherent approximation error

Background

• From Kouritzin and Remillard (2000):

txt

xt

xt dWtXdttXhdX ),(),(

t

uxx

t dWuUtX0

))(,(

m are 1-step nilpotent, h constraint holds

Problem Description

• Two dimensional manifold in three space

x

tYtYtXx

xxt

)0,0,0(

))(),(,( 21

t

sxs

t xs

xt dWXdsXhxX

00)()(

Conditions

t

si

i

i

t

s

dWsYsUtY

dWsUtY

0

2

1

1,2

2

0

1

1

)()()(

)()(

abledifferentily continuous ,, 2,21,21 UUU

Equivalency

• By Ito’s formula and martingale theory:

),(),,,(

)(),,,()),,,((

121

2121

2

1

ysAyysx

sAyysxyysx

y

y

)0,0(,,0,

),,,()),,,((21

2121

NyysDx

yysxyysxh s

(1)

Results

• Conjecture: Φ exists iff (2)

mymymm DDhh )()()()( 21

))()((),(

))(()(2

1

1,21

1

sYsUysD

sUsD

ii

i

msm

msm

where

2,1,~,))()(())()(( ~~~~

kmmmmxmmxkxkmmxmmxx

(2)

Sketch of Proof

• Assuming that Φ exists, then (2) is equivalent to two-step nilpotentcy

• Apply chain rule

• Simplify equation

Sketch of Proof Cont.

• If (2) holds, and all σm are two step nilpotent, then Φ exists

• Idea: find satisfying),( )23(51 C

hAAmyms 2)))()((( 1

(3)

(4)

(5)

(6)

(7)

)(1 hmyms

myyAA

m

21 )))()((( 1

)))()((()))()((( 1

2

1

1 2

1

2

2

AAAAyy

12 11

ys

Sketch of Proof Cont.

• Let Φ be such that

• Get dΦ is exact by (3-7)

• Future: construct (4-6), show they converge

1y