Continuous-Discrete von Mises-Fisher Filtering on S² for Reference Vector Tracking

Abstract

This paper is concerned with tracking of reference vectors in the continuous-discrete-time setting. For this end, an Itô stochastic differential equation, using the gyroscope as input, is formulated that explicitly accounts for the geometry of the problem. The filtering problem is solved by restricting the prediction and filtering distributions to the von Mises-Fisher class, resulting in ordinary differential equations for the parameters. A strategy for approximating Bayesian updates and marginal likelihoods is developed for the class of conditionally spherical measurement distributions, which is realistic for sensors such as accelerometers and magnetometers, and includes robust likelihoods. Furthermore, computationally efficient and numerically robust implementations are presented. The method is compared to other state-of-the-art filters in simulation experiments involving tracking of the local gravity vector. Additionally, the methodology is demonstrated in the calibration of a smartphone's accelerometer and magnetometer. Lastly, the method is compared to state-of-the-art in gravity vector tracking for smartphones in two use cases, where it is shown to be more robust to unmodeled accelerations.

A fuzzy adaptive extended Kalman filter exploiting the student's t distribution for mobile robot tracking

“…where x is the mean vector, P is the symmetric matrix, Γ(r) = ´∞ 0 e −t t r−1 dr is the Gamma function, and ν is used to determine the tail behavior of the density. One thing that needs special attention is that the covariance matrix P is not the covariance matrix of the random vector x, and when ν > 2 the covariance matrix of the random vector is related as follows [23],…”

“…This filter has been widely used in many applications including trajectory tracking [19], positioning, signal processing and navigation. Because the KF is designed for a linear system, multiple variations of the KF have been developed to resolve the state estimation for nonlinear systems such as the extended Kalman filter (EKF) [20], cubature Kalman filter (CKF) [21,22], unscented Kalman filter (UKF) [23] and extended Kalman filter with Student's t distribution (STEKF) [24]. In statistical filtering, more accurate noise models are being used, allowing better estimation accuracy to be achieved.…”

Score matching filters for Gaussian Markov random fields with a linear model of the precision matrix

“…Suppose now that (33) holds with t − 1 in place of t for some t ≥ 2. Then, by the continuous mapping theorem applied to the mapping (µ f t−1 , Σ f t−1 ) → (µ a t−1 , Σ a t−1 ) defined by (30,31), we obtain the convergence…”

“…The score matching method has been already used for estimating parameters in von Mises -Fisher filter on a unit sphere [3,30]. However, its usage in ensemble filtering algorithms for Gaussian Markov random fields seems to be new.…”