Presentation

```Using the Kalman Filter to
Estimate the state of a
Maneuvering Aircraft
ECEn -670 Stochastic Process
Prepared By:
Kevin Meier
Alok Desai
Instructor:
Dr. Brian Mazzeo
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Outlines
• Kalman filter
• Correlation Between the Process and
Measurement Noise
• Application of KF for estimating Bearing and
Range
• Simulation results
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Kalman Filter
• Purpose: It is to use measurements observed over
time, containing noise (random variations) and
other inaccuracies, and produce values that tend
to be closer to the true values of the
measurements and their associated calculated
values.
• When system model and measurement model
equations are linear, then to estimate the state
vector recursively.
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Estimating States
• System dynamic model:
• Measurement model:
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Kalman Filter Estimation
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Kalman Filter (Cont.)
• State estimation:
• Error covariance (a priori):
• Kalman Gain:
• Error covariance update (a posteriori):
• State estimate update:
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Correlation Between the Process and
Measurement Noise
• Correlation be given by
• Prediction equation remain unchanged.
• Measurement equation
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Range and Bearing Estimation
• Radars are used to track aircraft.
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• Range = ct/2
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How the Kalman filter applies to Radar
• Radar is used to track the state of an aircraft
• The state is the range, range rate, bearing and
bearing rate
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How to model the aircraft with no
acceleration data
• Model the acceleration as a uniform random
variable using the singer model. Where the
acceleration is correlated from sample to
sample
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How the Kalman filter applies to Radar
• The radar uses sensors to measure the Range
and Bearing angle. In this process there is
sensor measurement noise
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How the Kalman filter applies to Radar
• The process and measurement noise are zeromean white Gaussian random variables
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Error Covariance for Range
Error covariance (One prediction)
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Error covariance (Multiple prediction)
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Error Covariance of Bearing
Error covariance (One prediction)
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Error covariance (Multiple prediction)
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Bearing Angle
Bearing Angle (One prediction)
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Bearing Angle (Multiple prediction)
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Vehicle Range
Vehicle Range (One Prediction)
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Vehicle Range (Multiple Prediction)
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Range Error
Range Error (One Prediction)
c
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Vehicle Range (Multiple Prediction)
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Bearing Rate
Bearing ( one prediction )
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Bearing (multiple prediction )
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Range
Range (One prediction )
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Range (Multiple prediction )
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Range Error and Range Rate
with correlated noise
Range Rate
Range Error
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Questions??
Thank you !
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