  Localization is an important required task for enabling vehicle autonomy for
underwater vehicles. Localization entails the determination of position of the
center of mass and orientation of a vehicle from the available measurements. In
this paper, we focus on localization by using One-Way Travel Time (OWTT)
measurements available to a vehicle from the communication of its multiple
on-board receivers with acoustic beacons, more specifically, long baseline
(LBL) beacons. Range can be inferred by multiplying the OWTT with speed of
sound; however, water conditions can change spatially and temporally resulting
in uncertainty in range measurement. The farther a beacon is from a receiver,
the larger is the uncertainty. The proposed method for localization accounts
captures this uncertainty by bounding the true distance with an increasing
(calibrating) function of the range measurement. Determination of this
calibration function is formulated as polynomial optimization problem and is a
crucial step for localization. The proposed two-step procedure for localization
is as follows: based on the range measurements specific to a receiver from the
beacons, a convex optimization problem is proposed to estimate the location of
the receiver. The estimate is essentially a center of the set of possible
locations of the receiver. In the second step, the location estimates of the
vehicle are corrected using rigid body motion constraints and the orientation
of the rigid body is thus determined. Numerical examples and experimental
results provided at the end corroborate the procedures developed in this paper.
