1. An ISO 9001:2008 Registered Company
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Kalman Filter
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DP-FM-016, Rev 2
Effective Date: 22 February 2012

2. Kalman Filter Facts
 Dr. Rudolf Kalman is alive and well today (82 years old)
 Important and used everywhere: GPS (predict update
location), surface to air missiles (hit target), machine
vision (track targets), brain computer interface
 Not really a filter, it is an optimal estimator (infers
parameters of interest from indirect, noise
measurements)
 It is recursive – so when a new measurement arrives it is
processed and you get a new estimate
 Performs Data Fusion usually between measured and
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estimated states
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3. Conceptual Overview – Example Definition
y
 Lost on the 1-dimensional line, boat is not moving
 Imagine that you are guessing your position by looking at
the stars using sextant
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 Position function of time: y(t)
 Assume Gaussian distributed measurements (errors)
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4. Conceptual Overview - Prediction
0.16
 Sextant Measurement 0.14
at t1: Mean = z1 and 0.12
Variance = z1 0.1
probability
ŷ(t1) = z1
 Optimal estimate of 0.08 Predicted Position
position is: ŷ(t1) = z1 0.06
 Variance of error 0.04
[y(t1) - ŷ(t1)] estimate: 0.02
2 (t ) = 2 0
0 10 20 30 40 50 60 70 80 90 100
e 1 z1 z
 Boat in same position
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at time t2 - Predicted  What if we also had a
position is z1 GPS unit?
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5. Conceptual Overview - Measurement
0.16
0.14 prediction ŷ-(t2)
State (by looking
0.12 at the stars at t2)
0.1
0.08 Measurement
using GPS z(t2)
0.06
0.04
0.02
0
0 10 20 30 40 50 60 70 80 90 100
• So we have the prediction ŷ-(t2)
• GPS Measurement at t2: Mean = z2 and Variance = z2
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• Need to correct the prediction by Sextant due to
measurement to get ŷ(t2)
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6. Conceptual Overview – Data Fusion
0.16  Kalman filter: fuse
corrected optimal
0.14
estimate ŷ(t2) measurement and
0.12 prediction ŷ-(t2) prediction based on
0.1
confidence
0.08 measurement
z(t2)  Corrected mean is
0.06
the new optimal
0.04
0.02
estimate of position
0
0 10 20 30 40 50 60 70 80 90 100
 New variance is
smaller than either
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What if the boat is of the previous two
moving? variances
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7. Conceptual Overview – Prediction Model
0.16
ŷ(t2)
 At time t3, boat
0.14
moves with velocity
0.12
Naïve Prediction
dy/dt=u
0.1 (sextant) ŷ-(t3)
 Naïve approach:
0.08
Shift probability to
0.06
0.04
the right to predict
0.02
 This would work if
0
0 10 20 30 40 50 60 70 80 90 100
we knew the velocity
exactly (perfect
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Try and predict where model)
it winds up.
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8. Conceptual Overview – Prediction Model
0.16
ŷ(t2)
 But you may not be
0.14
so sure about the
0.12
Naïve Prediction
exact velocity
0.1 (sextant) ŷ-(t3)
 Better to assume
0.08
Prediction ŷ-(t3)
imperfect model by
0.06
0.04
adding Gaussian
0.02
noise
0
0 10 20 30 40 50 60 70 80 90 100
 dy/dt = u + w
 Distribution for
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Assumptions: prediction is prediction moves
linear, noise is Gaussian and spreads out
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9. Conceptual Overview – Update
0.16
Corrected optimal estimate ŷ(t3)
• Now we take a
0.14 Updated Sextant position using
GPS
measurement (GPS)
0.12
at t3
0.1 Measurement z(t3) GPS
• Need to once again
0.08
correct the
0.06
Prediction ŷ-(t3) Sextant
0.04
prediction (fusion)
0.02
• Recursive – rinse
0
0 10 20 30 40 50 60 70 80 90 100
and repeat as time
goes on
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Update, recursively
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10. Conceptual Overview
 Optimal estimator only if:
 Prediction model is linear (function of measurements)
 All error (noise) is Gaussian: model error, measurement
error
 Why is Kalman Filter so popular
 Good results in practice due to optimality and structure.
 Convenient form for online real time processing.
 Easy to formulate and implement given a basic
understanding.
 Measurement equations need not be inverted.
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11. State Space Equations
Estimated Estimated Control
State State Input
(now) (before)
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Observed
Measurement How do you find A,B,H?
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AWGN?
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12. Input
Output
Update Equations Place holder
Description Equation
State Prediction
Where do we end up
Covariance Prediction
When we get there, how much error
Innovation
Compare Reality to Prediction
Innovation Covariance
Compare real error to predicted error
Kalman Gain
What do you trust more?
State Update
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New estimate of where we are
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Covariance Update
New estimate of error
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13. Algorithm
Correction (Measurement Update)
Prediction (Time Update)
(1) Compute the Kalman Gain
(1) Project the state ahead
(2) Update estimate with measurement zk
(2) Project the error covariance ahead
(3) Update Error Covariance
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14. Measuring Constant Voltage (Classic Example 1)
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15. Predicting Trajectory of Projectile (Angry Bird)
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16. Equations
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17. Simulation Results
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18. Modified TWS example
State : y  
{ x , y , x , y}
Cov : Q E [ ww *]
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19. Derivation

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20. What if Assumptions don’t hold

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