Bayesian Fusion on Lie Groups

Abstract

An increasing number of real-world problems involve the measurement of data, andthe computation of estimates, on Lie groups. Moreover, establishing confidence in the resultingestimates is important. This paper therefore seeks to contribute to a larger theoretical frameworkthat generalizes classical multivariate statistical analysis from Euclidean space to the setting of Liegroups. The particular focus here is on extending Bayesian fusion, based on exponential familiesof probability densities, from the Euclidean setting to Lie groups. The definition and properties ofa new kind of Gaussian distribution for connected unimodular Lie groups are articulated, and ex-plicit formulas and algorithms are given for finding the mean and covariance of the fusion modelbased on the means and covariances of the constituent probability densities. The Lie groups thatfind the most applications in engineering are rotation groups and groups of rigid-body motions.Orientational (rotation-group) data and associated algorithms for estimation arise in problemsincluding satellite attitude, molecular spectroscopy, and global geological studies. In robotics andmanufacturing, quantifying errors in the position and orientation of tools and parts are impor-tant for task performance and quality control. Developing a general way to handle problemson Lie groups can be applied to all of these problems. In particular, we study the issue of howto ‘fuse’ two such Gaussians and how to obtain a new Gaussian of the same form that is ‘closeto’ the fused density.This is done at two levels of approximation that result from truncating theBaker-Campbell-Hausdorff formula with different numbers of terms. Algorithms are developedand numerical results are presented that are shown to generate the equivalent fused density withgood accuracy

Gaussian approximation of non-linear measurement models on Lie groups

“…And therefore, previously developed methods for fusion of pdfs in [7], [18] can be used for nonlinear measurement models.…”

“…And even outside of the context of attitude estimation, the group SO(3) arises in other applications [15]. In mobile robot localization [30], the groups of rigid-body motions of the plane and of 3D space, SE(2) and SE (3), are of interest [8], [7], [18], [21]. Other recent works on Lie-group filtering in a more abstract settings and for other Lie groups include [27], [17].…”

Kalman filtering with a class of geometric state equality constraints

“…Thus Equations (19) and (20) lead to the following Kalman gain and covariance updates, wherê N denotes the covariance matrix of the observation noiseV n :…”

“…However, due to the nonlinear nature of the navigation equations, and in particular to the fact the orientation of the aircraft (i.e., the attitude) does not live in a vector space, the EKF may have some shortcomings. This has motivated the development of alternative filters, especially for attitude estimation, see e.g., [12], [19], [6], [15], [13], [10], [16], [20], [7].…”

Mixture Reduction on Matrix Lie Groups

“…Furthermore, attitude estimation arises naturally on the SO(3) group [15]. In [18] a feedback particle filter on matrix Lie groups was proposed, while in [19], [20] authors proposed an extended Kalman filter on matrix Lie groups (LG-EKF), building the theory upon the concentrated Gaussian distribution (CGD) on matrix Lie groups [21].…”