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

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.