Example Application, Technical
Consider a truck on perfectly frictionless, infinitely long straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random acceleration. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocity is. We show here how we derive the model from which we create our Kalman filter.
Since F, H, R and Q are constant, their time indices are dropped.
The position and velocity of the truck are described by the linear state space
where is the velocity, that is, the derivative of position with respect to time.
We assume that between the (k−1) and k timestep the truck undergoes a constant acceleration of ak that is normally distributed, with mean 0 and standard deviation σa. From Newton's laws of motion we conclude that
(note that there is no term since we have no known control inputs) where
and
so that
where and
At each time step, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also normally distributed, with mean 0 and standard deviation σz.
where
and
We know the initial starting state of the truck with perfect precision, so we initialize
and to tell the filter that we know the exact position, we give it a zero covariance matrix:
If the initial position and velocity are not known perfectly the covariance matrix should be initialized with a suitably large number, say L, on its diagonal.
The filter will then prefer the information from the first measurements over the information already in the model.
Read more about this topic: Kalman Filter
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