UKF

The Unscented Kalman Filter (UKF) is a recursive algorithm for estimating the state of a nonlinear dynamical system from noisy measurements. It belongs to the Kalman filter family and uses the unscented transform to propagate mean and covariance through nonlinear models without explicit linearization.

The UKF was introduced by Simon Julier and Jeffrey Uhlmann in 1997 as an alternative to the Extended Kalman Filter. It aims to achieve higher accuracy for nonlinear systems by better capturing how uncertainty propagates through nonlinear mappings.

Conceptually, the UKF represents the state as a Gaussian distribution with a mean vector and a covariance matrix. A set of sigma points is deterministically chosen around the mean and propagated through the process model and the measurement model. The transformed points are then used to reconstruct the predicted state and covariance, and a Kalman gain is computed from the cross-covariance with the measurements to perform the update.

Common implementations use an augmented state that includes process and measurement noise, generating 2n_aug + 1 sigma points, with weights determined by parameters such as alpha, beta, and kappa. In the prediction step, points are propagated through the process model to form the predicted mean and covariance. In the update step, transformed measurements yield a predicted measurement, its covariance, the cross-covariance, and the Kalman gain, which updates the state estimate and covariance.

Applications span navigation, robotics, aerospace, and other areas requiring nonlinear state estimation. Advantages over the EKF include better handling of nonlinearities and no need for explicit Jacobians. Limitations include higher computational cost and sensitivity to tuning parameters and the assumption of Gaussian noise. See also Kalman filter, Extended Kalman Filter, and particle filter.