17-07-2013, 03:24 PM
THE KALMAN FILTER
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Abstract
This paper provides a gentle introduction to the Kalman filter,
a numerical method that can be used for sensor fusion or for calculation of
trajectories. First, we consider the Kalman filter for a one-dimensional system.
The main idea is that the Kalman filter is simply a linear weighted average of
two sensor values. Then, we show that the general case has a similar structure
and that the mathematical formulation is quite similar.
n example of data filtering
The Kalman filter is widely used in aeronautics and engineering for two main
purposes: for combining measurements of the same variables but from different
sensors, and for combining an inexact forecast of a system’s state with an inexact
measurement of the state. The Kalman filter has also applications in statistics and
function approximation.
When dealing with a time series of data points x1 , x2 , . . . , xn , a forecaster com-
putes the best guess for the point xn+1 . A smoother looks back at the data, and
computes the best possible xi taking into account the points before and after xi . A
filter provides a correction for xn+1 taking into account all the points x1 , x2 , . . . , xn
and an inexact measurement of xn+1 .
The one-dimensional Kalman Filter
The example above showed how to update a statistical quantity once more in-
formation becomes available. Assume now that we are dealing with two different
instruments that provide a reading for some quantity of interest x. We call x 1 the
reading from the first instrument and x2 the reading from the second instrument.
We know that the first instrument has an error modelled by a Gaussian with stan-
dard deviation σ1 . The error of the second instrument is also normally distributed
around zero with standard deviation σ2 . We would like to combine both readings
into a single estimation.
Forecast and measurement of different dimension
The equations printed in books for the Kalman filter are more general than the
expressions we have derived, because they handle the more general case in which
the measurement can have a different dimension than the system’s state. But it is
easy to derive the more general form, all is needed is to “move” all calculations to
measurement space.