Jethro's Braindump

Kalman Filter

tags
Extended Kalman Filter, Gaussian Filter

The Kalman filter is a technique for belief computation in linear systems. It implements belief computation over continuous states, and is not suitable for discrete or hybrid state spaces.

It uses the moments representation. At time t, the belief is represented by mean μt and covariance Σt. It makes the following assumptions:

  1. The Markovian Assumption (Markovian Assumption)
  2. The next state probability p(xt|ut,xt−1) is a linear function in its arguments with added Gaussian noise:

xt=Atxt−1+Btut+ϵt

xt and \ut are column vectors. This assumption defines the state transition probability p(xt|ut,xt−1) by substituting the mean Atxt−1+Btut and covariance Rt in the multi-variate normal distribution formula.

  1. The measurement probability p(zt|xt) is linear in its arguments:

zt=Ctxt+δt

for some multivariate Gaussian noise δt with 0 mean and co-variance Qt.

  1. The initial belief bel(x0) is normally distributed, with initial belief μ0 and covariance Σ0

These 4 assumptions give rise to the Kalman Filter algorithm.

Unknown environment 'algorithm'

Cons

The linearity assumptions are often unfulfilled in practice. For example, circular trajectories cannot be described with linear state transitions. To overcome this difficulty, people use the Extended Kalman Filter.

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