The Rauch-Tung-Striebel (RTS) smoother is an algorithm for computing state estimates in time series using noisy measurements from all time steps. The RTS smoother works by first filtering the state when measurements arrive and then using a backward pass to obtain the smoothed state estimates for all time steps that use measurement information from all time steps as well. The backward pass goes through all time steps for which a filtering estimate were obtained in backward order. We propose a smoother that does the backward pass using only a fraction of the time steps and provides the same results as the conventional RTS smoother for these time steps. This reduces computational complexity and required memory significantly as only data for these interesting time steps need to be stored. We also propose the extension of the reduced sampling-rate smoother for non-linear systems. We show an example application involving position estimation using a state space model that uses a high filtering rate, but where it is suitable to present the final smoothed route with a considerably lower rate. In a second example, we show how the reduced rate smoother works in a nonlinear case.

The Kalman Filter (KF) is a classical algorithm that was developed for estimating a state that evolves in time based on noisy measurements by assuming linear state transition and measurements models. There exist various KF extensions for non-linear situations, but they are not exact and provide different linearization errors. The Iterated Posterior Linearization Filter (IPLF) does the linearizations iteratively to achieve better linearizations. However, it is possible that some measurements cannot be well linearized using the current knowledge, but their linearization may be better after more measurements are available. Thus, we propose an algorithm that can store the older state elements and measurements when their linearization error is high. The resulting algorithm, the Stacked Iterated Posterior Linearization Filter (S-IPLF), is based on linear dynamic models and uses information from multiple time instances to make the linearization of the measurement function. Results show that the proposed algorithm outperforms traditional KF extensions when some of the measurements cannot be well linearized with the current knowledge, but can be when future information is available.

In this paper, we present a method for device localization based on the fusion of location data from Global Navigation Satellite System and data from inertial sensors. We use a Kalman filter as well as its non-linear variants for realtime position estimation, and corresponding smoothers for offline position estimation. In all filters we use information about changes of user's heading, which are computed from the acceleration and gyroscope data. Models used with Extended and Unscented Kalman filters also take into account information about step length, whereas Kalman Filter does not, because the measurement is non-linear. In order to overcome this shortcoming, we introduce a modified Kalman Filter which adjusts the state vector according to the step length measurements. Our experiments show that use of step length information does not significantly improve performance when location measurements are constantly available. However, in real situations, when location data is partially unavailable, information about step length and its appropriate integration into the filter design is important, and improve localization accuracy considerably.

In this letter, we propose an iterative Kalman type algorithm based on posterior linearization. The proposed algorithm uses a nested loop structure to optimize the mean of the estimate in the inner loop and update the covariance, which is a computationally more expensive operation, only in the outer loop. The optimization of the mean update is done using a damped algorithm to avoid divergence. Our simulations show that the proposed algorithm is more accurate than existing iterative Kalman filters.

In the Gaussian mixture approach a Bayesian posterior probability distribution function is approximated using a weighted sum of Gaussians. This work presents a novel method for generating a Gaussian mixture by splitting the prior taking the direction of maximum nonlinearity into account. The proposed method is computationally feasible and does not require analytical differentiation. Tests show that the method approximates the posterior better with fewer Gaussian components than existing methods.