Abstract-Distributions in position and orientation are central to many problems in robot localization. To increase efficiency, a majority of algorithms for planar mobile robots use Gaussians defined on positional Cartesian coordinates and heading. However, the distribution of poses for a noisy two-wheeled robot moving in the plane has been observed by many to be a "bananashaped" distribution, which is clearly not Gaussian/normal in these coordinates. As uncertainty increases, many localization algorithms therefore become "inconsistent" due to the normality assumption breaking down. We observe that this is because the combination of Cartesian coordinates and heading is not the most appropriate set of coordinates to use, and that the banana distribution can be described in closed form as a Gaussian in an alternative set of coordinates via the so-called exponential map.With this formulation, we can derive closed-form expressions for propagating the mean and covariance of the Gaussian in these exponential coordinates for a differential-drive car moving along a trajectory constructed from sections of straight segments and arcs of constant curvature. In addition, we detail how to fuse two or more Gaussians in exponential coordinates together with given relative pose measurements between robots moving in formation. These propagation and fusion formulas utilized here reduce uncertainty in localization better than when using traditional methods. We demonstrate with numerical examples dramatic improvements in the estimated pose of three robots moving in formation when compared to classical Cartesiancoordinate-based Gaussian fusion methods.

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

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