control//state estimation//Kalman filter//unscented Kalman filter

2026-10-01

The unscented Kalman filter handles nonlinear models without linearizing them: instead of approximating the function with a tangent, it pushes a few carefully chosen points through the exact function and measures where they land. It is used where the extended Kalman filter struggles, with strongly curved models (orientation, ranges and bearings at short distance) or where Jacobians are tedious or impossible to write.

The idea is that a distribution is easier to approximate than a function. From the current estimate and its covariance PPP, the filter builds 2n+12n+12n+1 sigma points for an nnn-dimensional state: the mean itself, and a pair on either side of it along each principal direction of PPP, at a distance set by its spread. Each point goes through the true nonlinear model, and the mean and covariance of the transformed points, with fixed weights, become the prediction. The same is done through the sensor function to get the predicted reading and its covariance, and the correction is the ordinary Kalman update.

It captures the mean and covariance more accurately than linearization when the function is curved, because it samples the function across the spread of the uncertainty instead of at a single point. No derivatives are needed: the model can be a black-box simulation.

Its cost is of the same order as the extended filter's, about 2n+12n+12n+1 evaluations of the model per step plus a matrix square root of PPP. For the state sizes of navigation (tens of variables) the difference rarely matters.

It is still a Gaussian filter. It describes the belief by a mean and a covariance, so a belief with several peaks is beyond it as much as beyond the extended filter (particle filter).

The spread of the sigma points is tuned by a few scaling parameters, and a bad choice can produce a covariance that is not positive definite; square-root formulations of the filter avoid that (estimate covariance).