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In this technical note, we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that the error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which is reminiscent of the linear stationary case. Index Terms--Inertial navigation, invariance, Lie group, nonlinear asymptotic observer, symmetry.

The recently introduced matrix group $\mathbf {SE_2(3)}$ provides a 5 $\mathbf {\times }$ 5 matrix representation for the orientation, velocity, and position of an object in the 3-D space, a triplet we call “extended pose.” In this article, we build on this group to develop a theory to associate uncertainty with extended poses represented by 5 $\mathbf {\times }$ 5 matrices. Our approach is particularly suited to describe how uncertainty propagates when the extended pose represents the state of an inertial measurement unit (IMU). In particular, it allows revisiting the theory of IMU preintegration on manifold and reaching a further theoretic level in this field. Exact preintegration formulas that account for rotating earth, that is, centrifugal force and Coriolis force, are derived as a byproduct, and the factors are shown to be more accurate. The approach is validated through extensive simulations and applied to sensor fusion where a loosely coupled fixed-lag smoother fuses IMU and LiDAR on 1-h-long experiments using our experimental car. It shows how handling rotating earth may be beneficial for long-term navigation within incremental smoothing algorithms. [ABSTRACT FROM AUTHOR]

This paper presents the theory of invariant observers, i.e, symmetry-preserving observers. We consider an observer to consist of a copy of the system and a correction term, and we propose a constructive method (based on the Cartan moving-frame method) to find all the symmetry-preserving correction terms. The construction relies on an invariant frame (a classical notion) and on an invariant output-error, a less standard notion precisely defined here. Using the theory we build three non-linear observers for three examples of engineering interest: a non-holonomic car, a chemical reactor, and an inertial navigation system. For each example, the design is based on physical symmetries and the convergence analysis relies on the use of invariant state-errors, a symmetry-preserving way to define the estimation error. Index Terms--Inertial navigation, moving frame, non-linear observer, symmetries.

We consider the problem of localizing a manned, semiautonomous, or autonomous vehicle in the environment using information coming from the vehicle’s sensors, a problem known as navigation or simultaneous localization and mapping (SLAM) depending on the context. To infer knowledge from sensors’ measurements, while drawing on a priori knowledge about the vehicle’s dynamics, modern approaches solve an optimization problem to compute the most likely trajectory given all past observations, an approach known as smoothing. Improving smoothing solvers is an active field of research in the SLAM community. Most work is focused on reducing computation load by inverting the involved linear system while preserving its sparsity. This article raises an issue that, to the best of our knowledge, has not been addressed yet: standard smoothing solvers require explicitly using the inverse of sensor noise covariance matrices. This means the parameters that reflect the noise magnitude must be sufficiently large for the smoother to properly function. When matrices are close to singular, which is the case when using high-precision modern inertial measurement units (IMUs), numerical issues necessarily arise, especially with 32-bit implementation demanded by most industrial aerospace applications. We discuss these issues and propose a solution that builds upon the Kalman filter to improve smoothing algorithms. We then leverage the results to devise a localization algorithm based on the fusion of IMU and vision sensors. Successful real experiments using an actual car equipped with a tactical grade high-performance IMU and a LiDAR illustrate the relevance of the approach to the field of autonomous vehicles. [ABSTRACT FROM AUTHOR]

The extended Kalman filter (EKF) is both the historical algorithm for multi-sensor fusion and still state of the art in numerous industrial applications. However, it may prove inconsistent in the presence of unobservability under a group of transformations. In this paper, we first build an alternative EKF based on an alternative nonlinear state error. This EKF is intimately related to the theory of the invariant EKF. Then, under a simple compatibility assumption between the error and the transformation group, we prove that the linearized model of the alternative EKF automatically captures the unobservable directions, and many desirable properties of the linear case then directly follow. This provides a novel fundamental result in filtering theory. We apply the theory to multi-sensor fusion for navigation, when all the sensors are attached to the vehicle and do not have access to absolute information, as typically occurs in GPS-denied environments. In the context of simultaneous localization and mapping (SLAM), Monte Carlo runs and comparisons to OC-EKF, robocentric EKF, and optimization-based smoothing algorithms (iSAM) illustrate the results. The proposed EKF is also proved to outperform standard EKF and to achieve comparable performance to iSAM on a publicly available real dataset for multi-robot SLAM. [ABSTRACT FROM AUTHOR]

Localization in indoor environments is a technique that estimates the robot’s pose by fusing data from onboard motion sensors with readings of the environment, in our case obtained by scan matching point clouds captured by a low-cost Kinect depth camera. We develop both an invariant extended Kalman filter (IEKF)-based and a multiplicative extended Kalman filter-based solution to this problem. The two designs are successfully validated in experiments and demonstrate the advantage of the IEKF design. [ABSTRACT FROM AUTHOR]