Estimation of periodic quantities such as angles or phase values is a common problem. However, standard approaches, for example the Kalman filter and extensions thereof, have difficulties when estimating periodic quantities. To address this problem, circular filtering algorithms have been proposed but they are limited to just a single angle. In order to deal with multiple, possibly correlated angles, toroidal filtering algorithms are necessary. We have previously proposed a bivariate filtering algorithm on the torus 1 that is limited to identity system and measurement models. In this paper, we show how the algorithm can be extended to handle nonlinear system and measurement models. The novel approach relies on the bivariate wrapped normal distribution for representing the uncertainty and it makes use of a deterministic sampling scheme for the torus. We provide a thorough evaluation of the proposed method using simulations.
Kurz et al. (Sun,) studied this question.