Updating filter

The relative certainty of the measurements and current state estimate is an important consideration, and it is common to discuss the response of the filter in terms of the Kalman filter's gain.The Kalman gain is the relative weight given to the measurements and current state estimate, and can be "tuned" to achieve particular performance.and Peter Swerling developed a similar algorithm earlier. Bucy of the University of Southern California contributed to the theory, leading to it often being called the Kalman–Bucy filter. Schmidt is generally credited with developing the first implementation of a Kalman filter.He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. It is also used in the guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station.It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for the Apollo program leading to its incorporation in the Apollo navigation computer. This digital filter is sometimes called the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, non-linear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich.This Kalman filter was first described and partially developed in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). In fact, some of the special case linear filter's equations appeared in these papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow.With a high gain, the filter places more weight on the most recent measurements, and thus follows them more responsively.

Once the outcome of the next measurement (necessarily corrupted with some amount of error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with higher certainty. It can run in real time, using only the present input measurements and the previously calculated state and its uncertainty matrix; no additional past information is required.At the extremes, a high gain close to one will result in a more jumpy estimated trajectory, while low gain close to zero will smooth out noise but decrease the responsiveness.When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into matrices to handle the multiple dimensions involved in a single set of calculations.The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a weighted average.The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more.

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