  To fuse information from inertial measurement units (IMU) with other sensors
one needs an accurate model for IMU error propagation in terms of position,
velocity and orientation, a triplet we call extended pose. In this paper we
leverage a nontrivial result, namely log-linearity of inertial navigation
equations based on the recently introduced Lie group $SE_2(3)$, to transpose
the recent methodology of Barfoot and Furgale for associating uncertainty with
poses (position, orientation) of $SE(3)$ when using noisy wheel speeds, to the
case of extended poses (position, velocity, orientation) of $SE_2(3)$ when
using noisy IMUs. Besides, our approach to extended poses combined with
log-linearity property allows revisiting the theory of preintegration on
manifolds and reaching a further theoretic level in this field. We show exact
preintegration formulas that account for rotating Earth, that is, centrifugal
force and Coriolis effect, may be derived as a byproduct.
