On the Probabilistic Coupling Between Rotation and Translation in State Estimation

Object tracking is an essential part of autonomous systems and deals with the estimation of the states of all dynamic objects within the area of surveillance. An appropriate state representation is highly important to capture the associated uncertainties correctly and Lie groups have been investigated with respect to this aspect over the last years. In this paper, we evaluate the performance of an extended Kalman filter on Lie groups employing two different state space representations: the direct and the semi-direct product of the two-dimensional special orthogonal group and a two-dimensional Euclidean vector capturing the rotational and translational component, respectively. Focus of our study is to investigate if the intrinsic relationship between rotation and translation is appropriately captured by means of the covariance matrix or if an encapsulation into the state space is beneficial. Statistical evaluation shows that a representation using the semi-direct product of the two components yields a better performance in terms of the Wasserstein distance and the root-mean-square errors of position and orientation.