Second-Order-Optimal Minimum-Energy Filters on Lie Groups

Abstract

Abstract-Systems on Lie groups naturally appear as models for physical systems with full symmetry. We consider the state estimation problem for such systems where both input and output measurements are corrupted by unknown disturbances. We provide an explicit formula for the second-order-optimal nonlinear filter on a general Lie group where optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). The resulting filter depends on the choice of affine connection which encodes the nonlinear geometry of the state space. As an example, we look at attitude estimation, where we are given a second order mechanical system on the tangent bundle of the special orthogonal group SO(3), namely the rigid body kinematics together with the Euler equation. When we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a perturbed matrix Riccati differential equation that updates the filter gain. This example demonstrates how to construct a matrix representation of the abstract general filter formula.

Joint Recursive Monocular Filtering of Camera Motion and Disparity Map

“…We provided a sound mathematical filtering framework for monocular scene reconstruction based on novel minimum energy filters, extending the classical quadratic energy function from Saccon et al [25] to a generalized Charbonnier Figure 2: Best viewed in color. Reconstruction of the disparity maps; left column: ground truth from the KITTI stereo benchmark, middle column: monocular method of Becker et al [1], right column: reconstruction with our monocular method.…”

“…Moreover, other state-of-the-art non-linear filters, such as particle filters [11], that can be applied to specific Lie groups [17], cannot be easily extended to high dimensional problems [8]. Due to these mathematical problems we will use the novel minimum energy filter on compact Lie groups [25] that minimizes a quadratic energy function to penalize deviations of the filtering equations by means of optimal control theory. This filter was shown to be superior to extended Kalman filters on the low dimensional Lie group SE 3 [3].…”

“…Zamani et al [29] introduces so-called minimum energy filters for linear filtering problems for compact Lie groups based on optimal control theory and the recursive filtering principle of Mortensen [18]. This approach was generalized to (non-)compact Lie groups in [25] and applied to a non-linear filtering problem on SE 3 for camera motion estimation [4].…”

“…• to provide a generalized Charbonnier energy function instead of a quadratic energy function [25], which results in robustness against outliers .…”

A novel in‐motion alignment method for underwater SINS using a state‐dependent Lie group filter

“…Additionally, rigid‐body rotation space is a Lie group (Chaturvedi, Sanyal, & Mcclamroch, 2011), which is compact and contains the full attitude of the rigid‐body. Thus, much effort has been paid to the development of attitude representation and estimation using a Lie group (Barrau & Bonnabel, 2015; De Ruiter, 2014; Markley, 2006; Saccon, Trumpf, Mahony, & Pedro Aguiar, 2016). Motivated by above observations, this paper is going to develop a novel in‐motion alignment method for underwater vehicle applications using a Lie group to represent the attitude.…”