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

Authors: N. Zarrouati-Vissiere, K. Beauchard, P. Rouchon, Siam J. Control Optim., Vol 52 No 6, pp. 3463–3495, 2014

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 S2. The observer design and analysis are strongly simplified by coordinate-free differential calculus on S2 equipped with its natural Riemannian structure. The observer convergence is investigated under C1 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 S2. 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.

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BibTeX:

@Article{2015-01-30,

author = {K. Beauchard N. Zarrouati-Vissiere, P. Rouchon},

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

journal = {Siam J. Control Optim.},

volume = {52},

number = {6},

pages = {3463-3495},

year = {2014},

}

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 S2. The observer design and analysis are strongly simplified by coordinate-free differential calculus on S2 equipped with its natural Riemannian structure. The observer convergence is investigated under C1 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 S2. 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.

Download PDF

BibTeX:

@Article{2015-01-30,

author = {K. Beauchard N. Zarrouati-Vissiere, P. Rouchon},

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

journal = {Siam J. Control Optim.},

volume = {52},

number = {6},

pages = {3463-3495},

year = {2014},

}