||The localization of an autonomous underwater vehicle (AUV) is a challenging and
important problem in marine robotics. In this paper we investigate the observability problem
of the process of Simultaneous Localization and Mapping (SLAM) of an AUV equipped with
inertial sensors, a depth sensor, and an acoustic ranging device that provides relative range
measurements to stationary beacons. For the case that the motion of the AUV corresponds to
constant linear and angular velocities (expressed in the body-frame), also known as trimming
or steady-state trajectories, we provide conditions under which it is possible to reconstruct
the initial state of the resulting SLAM system (and in particular the position of the AUV).
We show that the unobservable subspace UO restricted to the assumption that the position
of one of the beacons or the initial position of the AUV is known, contains only the zero
vector with exception of a particular case where the UO is composed by a finite set of
isolated points. Numerical tests that illustrate the results derived are presented and discussed.