Kalman Filtering Derivation

1
Kalman FilteringDerivation Application
Slides by Damien Jade Duff
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Outline

• Motivation
• Dynamic Observation Models
• Derivation
• Bayesian Tracking Relation
• Linearity Normality
• Prediction Update Procedure
• Worked Example
• Tracking a balloons' position

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Conceptual flowof State (general case)
ut
xt-1
F
xt
f
f
wt
Q
S
h
S
h
zt-1
zt
vt
vt
R
R
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Conceptual flowof State (Linear, Gaussian system)
ut
B


xt-1
F
xt
F
F


wt
Q
S
H
S
H


zt-1
zt
vt
vt
R
R
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Conceptual flowof Estimated State Variance
ut
B




xt-1




xt
F
xt-1
xt
F
F








St-1
St
St
St-1
wt
Q
K
K
S
H
S
H
zt-1

zt

zt-1


zt
vt
vt
St-1
St
R
R
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Worked Example(actual simulation)

• t 0 1 2

w
x
v
z
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1
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Worked Example(actual simulation)

• t 0 1 2

w
x 100 0T
v
z
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1
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Worked Example(actual simulation)

• t 0 1 2

w 0.25 0.625T
x 100 0T 100.25 0.625T
v -0.25
z 100
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1
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Worked Example(actual simulation)

• t 0 1 2

w 0.25 0.625T 0.375 2.875T
x 100 0T 100.25 0.625T 101.25 3.5T

v -0.25 0
z 100 101.25
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1
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Worked Example(actual simulation)

• t 2 3 4

w
X 101.25 3.5T
v
Z
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1
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Worked Example(actual simulation)

• t 2 3 4

w 0.25 0.25T
X 101.25 3.5T 105.0 3.75T
v 0.375
Z 105.375
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1
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Worked Example(actual simulation)

• t 2 3 4

w 0.25 0.25T -1.25 -0.875T
X 101.25 3.5T 105.0 3.75T 107.5
2.875T
v 0.375 1.25
Z 105.375 108.75
F 1 1 Q ½ ½ 0 1
½ 1½ H 1 0 R 1