Tag Archives: Lie Groups

Faster long-term SLAM through direct use of Lie groups in filtering

Kruno Lenac, Josip Ćesić, Ivan Marković, and Ivan Petrović, Exactly sparse delayed state filter on Lie groups for long-term pose graph SLAM, The International Journal of Robotics Research Vol 37, Issue 6, pp. 585 – 610 DOI: 10.1177/0278364918767756.

In this paper we propose a simultaneous localization and mapping (SLAM) back-end solution called the exactly sparse delayed state filter on Lie groups (LG-ESDSF). We derive LG-ESDSF and demonstrate that it retains all the good characteristics of the classic Euclidean ESDSF, the main advantage being the exact sparsity of the information matrix. The key advantage of LG-ESDSF in comparison with the classic ESDSF lies in the ability to respect the state space geometry by negotiating uncertainties and employing filtering equations directly on Lie groups. We also exploit the special structure of the information matrix in order to allow long-term operation while the robot is moving repeatedly through the same environment. To prove the effectiveness of the proposed SLAM solution, we conducted extensive experiments on two different publicly available datasets, namely the KITTI and EuRoC datasets, using two front-ends: one based on the stereo camera and the other on the 3D LIDAR. We compare LG-ESDSF with the general graph optimization framework (g2o) when coupled with the same front-ends. Similarly to g2o the proposed LG-ESDSF is front-end agnostic and the comparison demonstrates that our solution can match the accuracy of g2o, while maintaining faster computation times. Furthermore, the proposed back-end coupled with the stereo camera front-end forms a complete visual SLAM solution dubbed LG-SLAM. Finally, we evaluated LG-SLAM using the online KITTI protocol and at the time of writing it achieved the second best result among the stereo odometry solutions and the best result among the tested SLAM algorithms.

Extending bayesian fusion from Euclidean spaces to Lie groups

Kevin C. Wolfe, Michael Mashner, Gregory S. Chirikjian, Bayesian Fusion on Lie Groups, JOURNAL OF ALGEBRAIC STATISTICS Vol. 2, No. 1, 2011, 75-97, DOI: 10.18409/jas.v2i1.11.

An increasing number of real-world problems involve the measurement of data, and the computation of estimates, on Lie groups. Moreover, establishing confidence in the resulting estimates is important. This paper therefore seeks to contribute to a larger theoretical framework that generalizes classical multivariate statistical analysis from Euclidean space to the setting of Lie groups. The particular focus here is on extending Bayesian fusion, based on exponential families of probability densities, from the Euclidean setting to Lie groups. The definition and properties of a new kind of Gaussian distribution for connected unimodular Lie groups are articulated, and explicit formulas and algorithms are given for finding the mean and covariance of the fusion model based on the means and covariances of the constituent probability densities. The Lie groups that find 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 problems including satellite attitude, molecular spectroscopy, and global geological studies. In robotics and manufacturing, quantifying errors in the position and orientation of tools and parts are important for task performance and quality control. Developing a general way to handle problems on Lie groups can be applied to all of these problems. In particular, we study the issue of how to ‘fuse’ two such Gaussians and how to obtain a new Gaussian of the same form that is ‘close to’ the fused density.This is done at two levels of approximation that result from truncating the Baker-Campbell-Hausdorff formula with different numbers of terms. Algorithms are developed and numerical results are presented that are shown to generate the equivalent fused density with good accuracy