Kalman Filter - Example Application, Technical

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 a_k that is normally distributed, with mean 0 and standard deviation σ_a. 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 v_k is also normally distributed, with mean 0 and standard deviation σ_z.

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.