An Equivariant Observer Design for Visual Localisation and Mapping

This paper builds on recent work on Simultaneous Localisation and Mapping (SLAM) in the non-linear observer community, by framing the visual localisation and mapping problem as a continuous-time equivariant observer design problem on the symmetry group of a kinematic system. The state-space is a quotient of the robot pose expressed on SE(3) and multiple copies of real projective space, used to represent both points in space and bearings in a single unified framework. An observer with decoupled Riccati-gains for each landmark is derived and we show that its error system is almost globally asymptotically stable and exponentially stable in-the-large.Comment: 12 pages, 2 figures, published in 2019 IEEE CD

A Global Asymptotic Convergent Observer for SLAM

This paper examines the global convergence problem of SLAM algorithms, an issue that faces topological obstructions. This is because the state-space of attitude dynamics is defined on a non-contractible manifold: the special orthogonal group of order three SO(3). Therefore, this paper presents a novel, gradient-based hybrid observer to overcome these topological obstacles. The Lyapunov stability theorem is used to prove the globally asymptotic convergence of the proposed algorithm. Finally, comparative analyses of two simulations were conducted to evaluate the performance of the proposed scheme and to demonstrate the superiority of the proposed hybrid observer to a smooth observer.Comment: 7 pages, 8 figures, conferenc

The geometry of low-rank Kalman filters

An important property of the Kalman filter is that the underlying Riccati flow is a contraction for the natural metric of the cone of symmetric positive definite matrices. The present paper studies the geometry of a low-rank version of the Kalman filter. The underlying Riccati flow evolves on the manifold of fixed rank symmetric positive semidefinite matrices. Contraction properties of the low-rank flow are studied by means of a suitable metric recently introduced by the authors.Comment: Final version published in Matrix Information Geometry, pp53-68, Springer Verlag, 201