Inertial-Sensor Bias Estimation from Brightness/Depth Images and Based on $SO(3)$-Invariant Integro/Partial Differential Equations on the Unit Sphere

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 nonlinear integro/partial differential system where the two independent scalar variables indexing the pixels live on ${\mathbb S}^2$. The observer design and analysis are strongly simplified by coordinate-free differential calculus on ${\mathbb S}^2$ 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 the Ascoli--Arzela theorem and precompactness of the observer trajectories. It is proved that the estimated biases converge towards the true ones if and only if the scene admits no cyl...