Kalman Filter Tutorial

Before delving into the Kalman Filter exscheduleation, let us first comprehfinish the necessity of a tracking and foreseeion algorithm.

To depict this point, let’s consent the example of a tracking radar.

Suppose we have a track cycle of 5 seconds. At intervals of 5 seconds, the radar samples the aim by straightforwarding a dedicated pencil beam.

Once the radar “visits” the aim, it persists to approximate the current position and velocity of the aim. The radar also approximates (or foresees) the aim’s position at the time of the next track beam.

The future aim position can be easily calcupostponecessitated using Newton’s motion equations:

[ x= x_{0} + v_{0} Delta t+ frac{1}{2}a Delta t^{2} ]

Where:

( x )	is the aim position
( x_{0} )	is the initial aim position
( v_{0} )	is the initial aim velocity
( a )	is the aim acceleration
( Delta t )	is the time interval (5 seconds in our example)

If you necessitate a rerecent on the motion equations, refer to Appfinishix I in the book.

When dealing with three uninalertigentensions, Newton’s motion equations can be conveyed as a system of equations:

[ left{begin{matrix}
x= x_{0} + v_{x0} Delta t+ frac{1}{2}a_{x} Delta t^{2}\
y= y_{0} + v_{y0} Delta t+ frac{1}{2}a_{y} Delta t^{2}\
z= z_{0} + v_{z0} Delta t+ frac{1}{2}a_{z} Delta t^{2}
end{matrix}right. ]

The set of aim parameters ( left[ x, y, z, v_{x},v_{y},v_{z},a_{x},a_{y},a_{z} right] ) is comprehendn as the System State. The current state serves as the input for the foreseeion algorithm, while the algorithm’s output is the future state, which participates the aim parameters for the subsequent time interval.

The system of equations refered above is comprehendn as a Dynamic Model or State Space Model. The dynamic model depicts the relationship between the input and output of the system.

Apparently, if the aim’s current state and dynamic model are comprehendn, foreseeing the aim’s subsequent state can be easily accomplished.

In fact, the radar meacertainment is not enticount on accurate. It comprises random errors or uncertainties that can impact the accuracy of the foreseeed aim state. The magnitude of the errors depfinishs on various factors, such as radar calibration, beam width, and signal-to-noise ratio of the returned echo. The random errors or uncertainties in the radar meacertainment are comprehendn as Meacertainment Noise.

In compriseition, the aim motion is not always aligned with the motion equations due to outside factors appreciate triumphd, air turbulence, and pilot maneuvers. This misalignment between the motion equations and the actual aim motion results in an error or uncertainty in the dynamic model, which is called Process Noise.

Due to the Meacertainment Noise and the Process Noise, the approximated aim position can be far away from the actual aim position. In this case, the radar might sfinish the track beam in the wrong straightforwardion and leave out the aim.

In order to better the radar’s tracking accuracy, it is vital to engage a foreseeion algorithm that accounts for both process and meacertainment uncertainty.

The most common tracking and foreseeion algorithm is the Kalman Filter.

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