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In this paper, we focus on extended Kalman filtering (EKF), in the difficult case where a function of the state has been perfectly observed, and is thus known with certainty, while the full state still has unobserved degrees of freedom. In the linear case, the Kalman filter seamlessly handles such constraints, which result in the state being in an affine subspace. Yet, in the nonlinear case, the EKF poorly handles such type of constraints. As a remedy, we propose a novel general methodology of EKF based on an (arbitrary) nonlinear error $e$. And we prove that under compatibility of the error $e$ with the constraints, the EKF based on $e$ seamlessly handles the constraints. Furthermore, when the state space is a Lie group, we prove the EKF based on invariant errors is exactly the invariant EKF (IEKF), and we prove further properties. The theory is applied to the problem of simultaneous localization and mapping, where the IEKF is shown to perfectly handle some partial deterministic information about the map. As a byproduct, the theory is also shown to readily allow devising EKFs on state spaces defined by a class of equality constraints. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper proposes a probabilistic approach to the problem of intrinsic filtering of a system on a matrix Lie group with invariance properties. The problem of an invariant continuous-time model with discrete-time measurements is cast into a rigorous stochastic and geometric framework. Building upon the theory of continuous-time invariant observers, we introduce a class of simple filters and study their properties (without addressing the optimal filtering problem). We show that, akin to the Kalman filter for linear systems, the error equation is a Markov chain that does not depend on the state estimate. Thus, when the filter's gains are held fixed, the noisy error's distribution is proved to converge to a stationary distribution, under some convergence properties of the filter with noise turned off. We also introduce two novel tools of engineering interest: the discrete-time invariant extended Kalman filter, for which the trusted covariance matrix is shown to converge, and the invariant ensemble Kalman filter. The methods are applied to attitude estimation, allowing to derive novel theoretical results in this field, and illustrated through simulations on synthetic data. [ABSTRACT FROM AUTHOR]

The contraction properties of the extended Kalman filter, viewed as a deterministic observer for nonlinear systems, are analyzed. The approach relies on the study of an auxiliary “virtual” dynamical system. Some conditions under which exponential convergence of the state error can be guaranteed are derived. Moreover, contraction provides a simple formalism to study some robustness properties of the filter, especially with respect to measurement errors, as illustrated by a simplified inertial navigation example. This technical note sheds another light on the theoretical properties of this popular observer. [ABSTRACT FROM AUTHOR]

Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically. [ABSTRACT FROM AUTHOR]

Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The (position and velocity) observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose an intrinsic reduced order (velocity) observer based on the Jacobi metric, which can be initialized such that it converges exponentially for any initial true velocity. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced observer provides velocity estimation for geodesic flow with position measurements. Thus it can be (formally) used as a fluid flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the geodesic flow is sensitive to initial conditions. Surprisingly in this case we have global exponential convergence and the more unstable the flow is, faster is the convergence. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]