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

摘要

Constant biases associated to measured linear and angular velocities of amoving object can be estimated from measurements of a static scene by embeddedbrightness and depth sensors. We propose here a Lyapunov-based observer takingadvantage of the SO(3)-invariance of the partial differential equationssatisfied by the measured brightness and depth fields. The resulting asymptoticobserver is governed by a non-linear integro/partial differential system wherethe two independent scalar variables indexing the pixels live on the unitsphere of the 3D Euclidian space. The observer design and analysis are stronglysimplified by coordinate-free differential calculus on the unit sphere equippedwith its natural Riemannian structure. The observer convergence is investigatedunder C^1 regularity assumptions on the object motion and its scene. It relieson Ascoli-Arzela theorem and pre-compactness of the observer trajectories. Itis proved that the estimated biases converge towards the true ones, if and onlyif, the scene admits no cylindrical symmetry. The observer design can beadapted to realistic sensors where brightness and depth data are only availableon a subset of the unit sphere. Preliminary simulations with syntheticbrightness and depth images (corrupted by noise around 10%) indicate that suchLyapunov-based observers should be robust and convergent for much weakerregularity assumptions.