Inertial-sensor bias estimation from brightness/depth images and based on SO(3)-invariant integro/partial-differential equations on the unit sphere

TL;DR

This paper introduces a Lyapunov-based observer for estimating constant biases in linear and angular velocities using brightness and depth images, leveraging SO(3)-invariance and differential calculus on the sphere, with proven convergence under certain scene conditions.

Contribution

It develops a novel SO(3)-invariant integro/partial differential equation-based observer for bias estimation from visual data, simplifying analysis via coordinate-free calculus.

Findings

Biases converge to true values if scene has no cylindrical symmetry.

Observer is robust to noise around 10% in synthetic data.

Design can be adapted for partial sphere data.

Abstract

Constant biases associated to measured linear and angular velocities of a moving object can be estimated from measurements of a static scene by embedded brightness and depth sensors. We propose here a Lyapunov-based observer taking advantage of the SO(3)-invariance of the partial differential equations satisfied by the measured brightness and depth fields. The resulting asymptotic observer is governed by a non-linear integro/partial differential system where the two independent scalar variables indexing the pixels live on the unit sphere of the 3D Euclidian space. The observer design and analysis are strongly simplified by coordinate-free differential calculus on the unit sphere equipped with its natural Riemannian structure. The observer convergence is investigated under C^1 regularity assumptions on the object motion and its scene. It relies on Ascoli-Arzela theorem and pre-compactness of the observer trajectories. It is proved that the estimated biases converge towards the true ones, if and only if, the scene admits no cylindrical symmetry. The observer design can be adapted to realistic sensors where brightness and depth data are only available on a subset of the unit sphere. Preliminary simulations with synthetic brightness and depth images (corrupted by noise around 10%) indicate that such Lyapunov-based observers should be robust and convergent for much weaker regularity assumptions.