On the Observability and Observability Analysis of SLAM

Abstract

Simultaneous localization and mapping problem for mobile robots has received considerable attention over the last decade. The widely used formulation of the SLAM problem has been the augmented state approach in an estimation theoretic framework. Although, many related issues of SLAM such as computational complexity, loop closing and data association have received much attention, the observability issue has largely remained ignored. System observability is an important aspect in any state estimation problem. Observability analysis provides for understanding of the fundamental limits of the solution obtainable, regardless of process and measurement noises. The standard world-centric SLAM formulation is a highly non-linear system. Thus the direct use of linear observability tools and criteria in the analysis of its observability yields incorrect and inconsistent results. In this paper an appropriate method of analysis of the observability of non-linear systems is applied to investigate the properties of the standard SLAM formulation. Contrary to popular belief, it is shown through theoretical analysis that the standard 2D planar world-centric SLAM formulation involving odometry inputs for robot speed and heading, and range/bearing measurements to features in the environment is unobservable. It is also shown that for the system to be observable, it requires at least two absolutely known feature point positions, thus questioning the very meaning implied by SLAM. The analytical results thus established are verified through simulations.

Observability Analysis of Opportunistic Navigation with Pseudorange Measurements

“…With a fully nonlinear observability analysis, Lee et al subsequently disproved this result and showed that at least two anchor landmarks with known positions are required for local weak observability. 9 Later analysis of the SLAM problem's Fisher information matrix confirmed the result of the nonlinear analysis. 10 However, an apparent discrepancy between linear and nonlinear SLAM observability analyses re-emerged in the work of Perera and Nettleton, 11 where it was shown that a linear analysis based on piecewise constant system (PWCS) theory 15 again predicted global planar-SLAM observability in the case of a single known anchor landmark, whereas a nonlinear analysis in the same paper indicated that two known anchor landmarks were required for local observability.…”

“…The confusion arising from the conclusions achieved by this theory, demonstrated in this simple example, are similar to the ones encountered in the SLAM observability analysis. [6][7][8][9][10][11] The reason behind the encountered discrepancies is that we cannot simply take the time segment j to coincide with the discretization instant t k , since each time segment should contain n measurement samples, while the Jacobian matrices F and H are held constant within such segment. Therefore, in the forthcoming analyses we will only employ the nonlinear observability test and the linear observability test through the l-step observability matrix.…”

H∞ Filter-Based SLAM with the Observation on an a priori Known Landmark

“…We already know that the estimation accuracy is improved when a known landmark is included in the observations [12]. In order to prove it theoretically, in this subsection we will show that in this case the determinant of the state error covariance matrix becomes small.…”

“…However, as described in [12]- [14], a system composed of observations of only unknown landmarks is only partially observable. If we add a known landmark to the observation matrix, we obtain the following lemma for the observability.…”

“…While the system of the usual SLAM problem handles only unknown landmarks, the system of our proposed method handles unknown landmarks and one landmark that is known a priori. The SLAM problem with the observation of a landmark of which the location is known has already been proposed [12]- [14]. These papers show that including an observation of a known landmark satisfies observability and improves the estimation accuracy.…”