Using the Kalman Filter to Estimate the state of a Maneuvering Aircraft

Prepared By:

Kevin Meier

Alok Desai

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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

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• 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 zero-mean white Gaussian random variables

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Error Covariance for Range

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Error covariance (One prediction) Error covariance (Multiple prediction)

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Error Covariance of Bearing

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Error covariance (One prediction) Error covariance (Multiple prediction)

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Bearing Angle

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Bearing Angle (One prediction) Bearing Angle (Multiple prediction)

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Vehicle Range

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Vehicle Range (One Prediction) Vehicle Range (Multiple Prediction)

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Range Error

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Range Error (One Prediction)c

Vehicle Range (Multiple Prediction)

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Bearing Rate

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Bearing ( one prediction ) Bearing (multiple prediction )

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Range

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Range (One prediction ) Range (Multiple prediction )

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Range Error and Range Ratewith correlated noise

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Range Error Range Rate

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Questions??

Thank you !

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