  In this paper we address smoothing-that is, optimisation-based-estimation
techniques for localisation problems in the case where motion sensors are very
accurate. Our mathematical analysis focuses on the difficult limit case where
motion sensors are infinitely precise, resulting in the absence of process
noise. Then the formulation degenerates, as the dynamical model that serves as
a soft constraint becomes an equality constraint, and conventional smoothing
methods are not able to fully respect it. By contrast, once an appropriate Lie
group embedding has been found, we prove theoretically that invariant smoothing
gracefully accommodates this limit case in that the estimates tend to be
consistent with the induced constraints when the noise tends to zero.
Simulations on the important problem of initial alignement in inertial
navigation show that, in a low noise setting, invariant smoothing may favorably
compare to state-of-the-art smoothers when using precise inertial measurements
units (IMU).
