EcEn 670 December 5 2013 A Comparison between Analytical and Simulated Results The Kalman Filter A Study of Covariances Kalman Overview Common Applications 1 Inertial Navigation IMU GPS ID: 682412
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David WheelerKyle IngersollEcEn 670
December 5, 2013
A Comparison between Analytical and Simulated Results
The Kalman Filter: A Study of CovariancesSlide2
Kalman Overview:
Common Applications
1
:
Inertial Navigation (IMU + GPS)
Global Navigation Satellite Systems
Estimating Constants in the Presence of Noise
Simultaneous Localization and Mapping (SLAM)
Object Tracking In Computer Vision
Economics
Predict
(P)
Forward One StepUpdate
(U)Use Measurements If Available
P
P
P
P
P
P
P
P
U
U
U
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Kalman Intuition: Predict Using Underlying Model
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Kalman Intuition: Predict Using Underlying Model
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Kalman Intuition: Update by Weighing Measurement and Model
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Measurement,
Model Estimate,
Residual
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Kalman Intuition: Update by Weighing Measurement and Model
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Measurement Covariance,
State Covariance,
Kalman
Gain,
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Kalman Intuition: Summary
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5
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Kalman
Gain,
Predict Step
Predict state
forward one step.
Predict covariance
forward one step.
Update Step
Determine Kalman Gain
(optimal weighting between
and
).
Update state
using Kalman gain and residual.
Update state covariance
.
7Slide8
Prediction Derivation:Linear:
Prediction Step: Linear Example
Current State
Recent State
Process Noise
Recent Input
k=1
k=2
Example 1
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Update:
Measurement:
Update Step: Linear Example
Measurement
Model’s Guess for Measurement
Noise
Residual
Weighting
9Slide10
Results: Linear Example
Ten Steps“Predict" Only:
500 runs
10 time steps
10Slide11
Results: Linear Example
Experimental covariance
(Cyan dots) MATLAB cov commandAnalytical covariance
(Red solid line)
Individual
runs
(Magenta dots)
(Dark blue dots)
11Slide12
Results: Linear Example
Update Step:
500 runs
0.01
12Slide13
Results: Linear Example
Experimental covariance
(Green dots)
MATLAB cov command
Analytical covariance(Magenta solid
line)
Individual runs
(Dark blue dots)
13Slide14
Linear Example: Comparing Covariance Trends
Experimental Covariance
(Blue)
Analytical Covariance (Red)
14Slide15
Linear Example: Convergence of Covariances
15Slide16
Process
Non-Linear Example
Example 2
16Slide17
Results: Non-linear Example30 Time Steps500 runsInput:
Input
Noise is Gaussian, ±5
%
(known to start at origin)
Analytical Covariance
(Cyan Ellipse)Beacon Location(Red Circle)
17Slide18
Results: Non-linear ExampleBeacon Location(Red Circle)Measurement (7/500)
(Green Lines)Gaussian Noise on Measurement
(Red Xs)Covariance (before update)Analytical
(Thin Cyan)Experimental
(Thick Cyan)
18Slide19
Results: Non-linear ExampleCovarianceBefore updateAnalytical (Thin
Cyan)Experimental (Thick
Cyan)After update
Analytical (Thin Magenta)
Experimental (Thick Magenta)
Note – the update step reduces the uncertainty in the direction of the measurement only!
19Slide20
Under certain conditions, a Kalman filter causes the covariance to convergeAnalytical and simulated covariances match closelyAnalytical and simulated covariances converge quickly if seeded with different valuesIndividual measurements can significantly reduce the covariance of the state estimateConclusion
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Questions & Discussion21