Cramér–Rao Bounds and Optimal Design Metrics for Pose-Graph SLAM

Two-dimensional (2-D)/3-D pose-graph simultaneous localization and mapping (SLAM) is a problem of estimating a set of poses based on noisy measurements of relative rotations and translations. This article focuses on the relation between the graphical structure of pose-graph SLAM and Fisher information matrix (FIM), Cramér-Rao lower bounds (CRLB), and its optimal design metrics (T-optimality and D-optimality). As a main contribution, based on the assumption of isotropic Langevin noise for rotation and block-isotropic Gaussian noise for translation, the FIM and CRLB are derived and shown to be closely related to the graph structure, in particular, the weighted Laplacian matrix. We also prove that total node degree and weighted number of spanning trees, as two graph connectivity metrics, are, respectively, closely related to the trace and determinant of the FIM. The discussions show that, compared with the D-optimality metric, the T-optimality metric is more easily computed but less effective. We also present upper and lower bounds for the D-optimality metric, which can be efficiently computed and are almost independent of the estimation results. The results are verified with several well-known datasets, such as Intel, KITTI, sphere, and so on.