arXiv subjects

Axel Barrau

Publications and source records attributed to Axel Barrau.

28 records · Page 2Linked to original sources

arXiv2020

Generalized mean shift with triangular kernel profile

The mean shift algorithm is a popular way to find modes of some probability density functions taking a specific kernel-based shape, used for clustering or visual tracking. Since its introduction, it underwent several practical improvements and generalizations, as well as deep theoretical analysis mainly focused on its convergence properties. In spite of encouraging results, this question has not received a clear general answer yet. In this paper we focus on a specific class of kernels, adapted in particular to the distributions clustering applications which motivated this work. We show that a novel Mean Shift variant adapted to them can be derived, and proved to converge after a finite number of iterations. In order to situate this new class of methods in the general picture of the Mean Shift theory, we alo give a synthetic exposure of existing results of this field.

cs.LG↗

arXiv2019

AI-IMU Dead-Reckoning

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. We make our implementation open-source at: https://github.com/mbrossar/ai-imu-dr

cs.RO↗

arXiv2019

Exploiting Symmetries to Design EKFs with Consistency Properties for Navigation and SLAM

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 (IEKF). Then, under a simple compatibility assumption between the error and the transformation group, we prove 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.

cs.RO↗

arXiv2018

Symmetry reduction for dynamic programming

We present a method of exploiting symmetries of discrete-time optimal control problems to reduce the dimensionality of dynamic programming iterations. The results are derived for systems with continuous state variables, and can be applied to systems with continuous or discrete symmetry groups. We prove that symmetries of the state update equation and stage costs induce corresponding symmetries of the optimal cost function and the optimal policies. We then provide a general framework for computing the optimal cost function based on gridding a space of lower dimension than the original state space. This method does not require algebraic manipulation of the state update equations; it only requires knowledge of the symmetries that the state update equations possess. Since the method can be performed without any knowledge of the state update map beyond being able to evaluate it and verify its symmetries, this enables the method to be applied in a wide range of application problems. We illustrate these results on two six-dimensional optimal control problems that are computationally difficult to solve by dynamic programming without symmetry reduction.

eess.SY↗

arXiv2018

Invariant Smoothing on Lie Groups

In this paper we propose a (non-linear) smoothing algorithm for group-affine observation systems, a recently introduced class of estimation problems on Lie groups that bear a particular structure. As most non-linear smoothing methods, the proposed algorithm is based on a maximum a posteriori estimator, determined by optimization. But owing to the specific properties of the considered class of problems, the involved linearizations are proved to have a form of independence with respect to the current estimates, leveraged to avoid (partially or sometimes totally) the need to relinearize. The method is validated on a robot localization example, both in simulations and on real experimental data.

cs.RO↗

arXiv2016

An EKF-SLAM algorithm with consistency properties

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.

cs.RO↗

arXiv2015

The invariant extended Kalman filter as a stable observer

We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic non-linear 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. Then, we leverage the result 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.

eess.SY↗

arXiv2015

An intrinsic Cramér-Rao bound on SO(3) for (dynamic) attitude filtering

In this note an intrinsic version of the Cramér-Rao bound on estimation accuracy is established on the Special Orthogonal group $SO(3)$. It is intrinsic in the sense that it does not rely on a specific choice of coordinates on $SO(3)$: the result is derived using rotation matrices, but remains valid when using other parameterizations, such as quaternions. For any estimator $\hat R$ of $R\in SO(3)$ we give indeed a lower bound on the quantity $E(\log(R\hat R^T))$, that is, the estimation error expressed in terms of group multiplication, whereas the usual estimation error $E(\hat R-R)$ is meaningless on $SO(3)$. The result is first applied to Whaba's problem. Then, we consider the problem of a continuous-time nonlinear deterministic system on $SO(3)$ with discrete measurements subject to additive isotropic Gaussian noise, and we derive a lower bound to the estimation error covariance matrix. We prove the intrinsic Cramér-Rao bound coincides with the covariance matrix returned by the Invariant EKF, and thus can be computed online. This is in sharp contrast with the general case, where the bound can only be computed if the true trajectory of the system is known.

math.OC↗

arXiv2015

An intrinsic Cramér-Rao bound on Lie groups

In his 2005 paper, S.T. Smith proposed an intrinsic Cramér-Rao bound on the variance of estimators of a parameter defined on a Riemannian manifold. In the present technical note, we consider the special case where the parameter lives in a Lie group. In this case, by choosing, e.g., the right invariant metric, parallel transport becomes very simple, which allows a more straightforward and natural derivation of the bound in terms of Lie bracket, albeit for a slightly different definition of the estimation error. For bi-invariant metrics, the Lie group exponential map we use to define the estimation error, and the Riemannian exponential map used by S.T. Smith coincide, and we prove in this case that both results are identical indeed.

eess.SY↗

arXiv2014

Intrinsic filtering on Lie groups with applications to attitude estimation

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 show that, as in the linear case, the error equation is a Markov chain that does not depend on the state estimate. Thus, when the filter's gains are held fixed, and the filter admits almost-global convergence properties with noise turned off, the noisy error's distribution is proved to converge to a stationary distribution, providing insight into the mathematical theory of filtering on Lie groups. For engineering purposes we also introduce the discrete-time Invariant Extended Kalman Filter, for which the trusted covariance matrix is shown to asymptotically converge, and some numerically more involved sample-based methods as well to compute the Kalman gains. The methods are applied to attitude estimation, allowing to derive novel theoretical results in this field, and illustrated through simulations on synthetic data.

eess.SY↗