Dimensionality reduction for point feature SLAM problems with spherical covariance matrices

Abstract

a b s t r a c tThe main contribution of this paper is the dimensionality reduction for multiple-step 2D point feature based Simultaneous Localization and Mapping (SLAM), which is an extension of our previous work on one-step SLAM (Wang et al., 2013). It has been proved that SLAM with multiple robot poses and a number of point feature positions as variables is equivalent to an optimization problem with only the robot orientations as variables, when the associated uncertainties can be described using spherical covariance matrices. This reduces the dimension of original problem from 3m + 2n to m only (where m is the number of poses and n is the number of features). The optimization problem after dimensionality reduction can be solved numerically using the unconstrained optimization algorithms. While dimensionality reduction may not provide computational saving for all nonlinear optimization problems, for some SLAM problems we can achieve benefits such as improvement on time consumption and convergence. For the special case of two-step SLAM when the orientation information from odometry is not incorporated, an algorithm that can guarantee to obtain the globally optimal solution (in the maximum likelihood sense) is derived. Simulation and experimental datasets are used to verify the equivalence between the reduced nonlinear optimization problem and the original full optimization problem, as well as the proposed new algorithm for obtaining the globally optimal solution for two-step SLAM.

A critique of current developments in simultaneous localization and mapping

“…Some recent research work by Wang et al has revealed the partially linear structure of SLAM and provides a necessary and sufficient condition for the existence of only one minimum for one-step SLAM, as well as a numerical method for obtaining the global minimum of two-step SLAM, assuming spherical covariance matrices. 20,51 In a paper by Carlone, a conservative estimate of the region of attraction of the global minimum for a Gauss-Newton algorithm is provided for 2D pose-graph SLAM. 48 Furthermore, Carlone et al provide a method to verify whether the global minimum solution is obtained by using Lagrange duality.…”

Exploiting the Separable Structure of SLAM

“…In our previous work [29] linear variables of 2D featurebased problems with spherical noise are explicitly eliminated to obtain a smaller optimization problem over θ. This approach is similar to Golub and Pereyra's VP [12], but with numerical differentiation and Newton iterations.…”

Linear SLAM: Linearising the SLAM problems using submap joining

“…Due to the spherical assumption on the covariance matrices, it can efficiently optimize pose-chains by exploiting their Lie group structure. The spherical covariance assumption significantly reduces the complexity of the SLAM problems, as shown in recent works [21,22,25]. In [17], a relative formulation of the relationship between multiple pose graphs is proposed to avoid the initialization problem and thus lead to an efficient solution; the anchor nodes are closely related to the base nodes in [18].…”