A Discussion of Simultaneous Localization and Mapping

TLDR: A formal proof for approximate sparsity of so-called information matrices occurring in SLAM is sketched and supports the structure of uncertainty of an estimated map and provides a foundation for algorithms based on sparse informationMatrices.

Abstract: This paper aims at a discussion of the structure of the SLAM problem. The analysis is not strictly formal but based both on informal studies and mathematical derivation. The first part highlights the structure of uncertainty of an estimated map with the key result being "Certainty of Relations despite Uncertainty of Positions". A formal proof for approximate sparsity of so-called information matrices occurring in SLAM is sketched. It supports the above mentioned characterization and provides a foundation for algorithms based on sparse information matrices. Further, issues of nonlinearity and the duality between information and covariance matrices are discussed and related to common methods for solving SLAM. Finally, three requirements concerning map quality, storage space and computation time an ideal SLAM solution should have are proposed. The current state of the art is discussed with respect to these requirements including a formal specification of the term "map quality".