The Invariant Extended Kalman Filter as a Stable Observer
TL;DR: In this article, the authors analyzed the convergence aspects of the invariant extended Kalman filter (IEKF) when the latter is used as a deterministic nonlinear observer on Lie groups, for continuous-time systems with discrete observations.
Abstract: We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic nonlinear observer on Lie groups, for continuous-time systems with discrete observations. One of the main features of invariant observers for left-invariant systems on Lie groups is that the estimation error is autonomous. In this paper we first generalize this result by characterizing the (much broader) class of systems for which this property holds. For those systems, the Lie logarithm of the error turns out to obey a linear differential equation. Then, we leverage this “log-linear” property of the error evolution, to prove for those systems the local stability of the IEKF around any trajectory, under the standard conditions of the linear case. One mobile robotics example and one inertial navigation example illustrate the interest of the approach. Simulations evidence the fact that the EKF is capable of diverging in some challenging situations, where the IEKF with identical tuning keeps converging.
Citations
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TL;DR: This work develops a contact-aided invariant extended Kalman filter (InEKF) using the theory of Lie groups and invariant observer design and demonstrates how to include IMU biases, add/remove contacts, and formulate both world-centric and robo-centric versions.
Abstract: Legged robots require knowledge of pose and velocity in order to maintain stability and execute walking paths. Current solutions either rely on vision data, which is susceptible to environmental an...
145 citations
13 Mar 2020
TL;DR: This paper proposes a novel accurate method for dead-reckoning of wheeled vehicles based only on an Inertial Measurement Unit (IMU) using the Kalman filter and the use of deep neural networks to dynamically adapt the noise parameters of the filter.
Abstract: In this paper, we propose a novel accurate method for dead-reckoning of wheeled vehicles based only on an Inertial Measurement Unit (IMU). In the context of intelligent vehicles, robust and accurate dead-reckoning based on the IMU may prove useful to correlate feeds from imaging sensors, to safely navigate through obstructions, or for safe emergency stops in the extreme case of exteroceptive sensors failure. The key components of the method are the Kalman filter and the use of deep neural networks to dynamically adapt the noise parameters of the filter. The method is tested on the KITTI odometry dataset, and our dead-reckoning inertial method based only on the IMU accurately estimates 3D position, velocity, orientation of the vehicle and self-calibrates the IMU biases. We achieve on average a 1.10% translational error and the algorithm competes with top-ranked methods which, by contrast, use LiDAR or stereo vision.
140 citations
TL;DR: A systematic introduction to the Bayesian state estimation framework is offered and various Kalman filtering U+0028 KF U-0029 techniques are reviewed, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KFFor nonlinear systems.
Abstract: This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics U+0028 e.g., mean and covariance U+0029 conditioned on a system U+02BC s measurement data. This article offers a systematic introduction to the Bayesian state estimation framework and reviews various Kalman filtering U+0028 KF U+0029 techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation to more complicated problems such as simultaneous state and parameter U+002F input estimation.
115 citations
Posted Content•
TL;DR: This paper addresses the inconsistency of the EKF-based SLAM algorithm that stems from non-observability of the origin and orientation of the global reference frame and proves on the non-linear two-dimensional problem with point landmarks that this type of inconsistency is remedied using the Invariant EKf.
Abstract: In this paper we address the inconsistency of the EKF-based SLAM algorithm that stems from non-observability of the origin and orientation of the global reference frame. We prove on the non-linear two-dimensional problem with point landmarks observed that this type of inconsistency is remedied using the Invariant EKF, a recently introduced variant ot the EKF meant to account for the symmetries of the state space. Extensive Monte-Carlo runs illustrate the theoretical results.
105 citations
TL;DR: A novel visual-inertial navigation algorithm for low-cost and computationally constrained vehicle in global positioning system denied environments is presented by modeling the state space as the matrix Lie group (LG), based on the recent theory of the invariant Kalman filter.
Abstract: In this paper, we present a novel visual-inertial navigation algorithm for low-cost and computationally constrained vehicle in global positioning system denied environments by modeling the state space as the matrix Lie group (LG), based on the recent theory of the invariant Kalman filter. The multistate constraint Kalman filter (MSCKF) is a well-known visual-inertial odometry algorithm that performs the fusion of the visual and inertial information by constraining each other through the stochastically cloned pose within a sliding window. However, conventional MSCKF (MSCKF-Conv) suffers from the inconsistent state estimates caused by the spurious gain along the unobservable directions, resulting in large estimation errors. To tackle this problem, we extend the concepts of the state and noise of the MSCKF from Euclidean space to matrix LG. We model the state of the MSCKF as the element of the specially customized matrix LG and use the noise uncertainty modeling with the corresponding Lie algebra. The detailed derivation and observability analysis of the proposed filter are provided to prove that the proposed filter is more consistent than the MSCKF-Conv. The proposed MSCKF on matrix LG naturally enforces the state vector to exist in the state space that maintains the unobservability characteristics without any artificial remedies. The performance of the proposed filter is validated through the Monte-Carlo simulation and the real-world experimental dataset.
85 citations
References
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TL;DR: In this article, the authors present a review of the methods of Kalman filtering in attitude estimation and their development over the last two decades, focusing on three-axis gyros and attitude sensors.
Abstract: HIS report reviews the methods of Kalman filtering in attitude estimation and their development over the last two decades. This review is not intended to be complete but is limited to algorithms suitable for spacecraft equipped with three-axis gyros as well as attitude sensors. These are the systems to which we feel that Kalman filtering is most ap- plicable. The Kalman filter uses a dynamical model for the time development of the system and a model of the sensor measurements to obtain the most accurate estimate possible of the system state using a linear estimator based on present and past measurements. It is, thus, ideally suited to both ground-based and on-board attitude determination. However, the applicability of the Kalman filtering technique rests on the availability of an accurate dynamical model. The dynamic equations for the spacecraft attitude pose many difficulties in the filter modeling. In particular, the external torques and the distribution of momentum internally due to the use of rotating or rastering instruments lead to significant uncertainties in the modeling. For autonomous spacecraft the use of inertial reference units as a model replacement permits the circumvention of these problems. In this representation the angular velocity of the spacecraft is obtained from the gyro data. The kinematic equations are used to obtain the attitude state and this is augmented by means of additional state-vector components for the gyro biases. Thus, gyro data are not treated as observations and the gyro noise appears as state noise rather than as observation noise. It is theoretically possible that a spacecraft is three-axis stabilized with such rigidity that the time development of the system can be described accurately without gyro information, or that it is one-axis stabilized so that only a single gyro is needed to provide information on the time history of the system. The modification of the algorithms presented here in order to apply to those cases is slight. However, this is of little practical importance because a control system capable of such
1,266 citations
01 Jan 1998
550 citations
12 Dec 2005
TL;DR: In this article, two different non-linear complementary filters are proposed: Direct complementary filter and Passive nonlinear complementary filter, which evolve explicity on the special orthogonal group SO(3) and can be expressed in quaternion form for easy implementation.
Abstract: This paper considers the problem of obtaining high quality attitude extraction and gyros bias estimation from typical low cost intertial measurement units for applications in control of unmanned aerial vehiccles. Two different non-linear complementary filters are proposed: Direct complementary filter and Passive non-linear complementary filter. Both filters evolve explicity on the special orthogonal group SO(3) and can be expressed in quaternion form for easy implementation. An extension to the passive ocmplementary filter is proposed to provide adaptive gyro bias estimation.
290 citations
TL;DR: Convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented and it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly.
Abstract: In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly. The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems.
284 citations
TL;DR: Using the theory of invariant observers, i.e, symmetry-preserving observers, three non-linear observers are built for three examples of engineering interest: a non-holonomic car, a chemical reactor, and an inertial navigation system.
Abstract: 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.
256 citations