KALMAN FILTER WITHOUT TEARS

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ne of the greatest theories postulated in the twentieth century is the Kalman Filter. It is not actually a filter, but a mathematical estimator, which uses a series of measurements observed over a period of time consisting of random variations ( noise) and other inaccuracies and produces estimates that tend to be more precise than what would be based on a single measurement.

The Kalman filter has numerous applications in technology. A common application is for guidance, navigation and control of vehicles, particularly aircraft and spacecraft.

Take for example the readings from a GPS receiver or the outputs of an IRS. The outputs are not always predictable and contain some errors which may be classified as random and systemic. The simplest solution that comes to mind is to take average of a series of consequent samples. This simple approach doesn’t work for most problems. We need a more sophisticated approach.

Kalman filter, also known as linear quadratic estimation, operates recursively on streams of noisy input data to produce a statistically optimal estimate of the underlying system state.

The algorithm works in a two-step process: in the prediction step, the Kalman filter produces estimates of the current state variables, along with their uncertainties. 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. Because of the algorithm's recursive nature, it can run in real time using only the present input measurements and the previously calculated state; no additional past information is required.

The autopilot in a modern large aircraft typically reads its position and the aircraft's attitude from an inertial guidance system. Inertial guidance systems accumulate errors over time. They will incorporate error reduction systems such as the carousel system that rotates once a minute so that any errors are dissipated in different directions and have an overall nulling effect. Error in gyroscopes is known as drift. This is due to physical properties within the system, be it mechanical or laser guided, that corrupt positional data. The disagreements between the two are resolved with digital signal processing, most often a six-dimensional Kalman filter. The six dimensions are usually roll, pitch, yaw, altitude, latitude, and longitude. Aircraft may fly routes that have a required performance factor; therefore the amount of error or actual performance factor must be monitored in order to fly those particular routes. The longer the flight, the more error accumulates within the system. Radio aids such as DME, DME updates, and GPS may be used to correct the aircraft position.

A Kalman filter is essentially an optimal estimator. It infers parameters of interest from indirect, uncertain and inaccurate observations and is most optimal where the error distribution is Gaussian in nature. It is also recursive, in that new measurements are processed as they arrive to minimise the output error.