Locating and determining the orientation of underwater research equipment: Acoustic range and range rate data

A formulation is presented for evaluating the performance of acoustic data systems to determine the location and orientation of underwater research apparatus. The variables to describe the position of the underwater research apparatus are referenced to a surface ship, and represent the straight-line distance from the stern of the ship to a point on the apparatus (
R
), the angular distance down from the sea surface (
\Phi
), and the angular distance from the direction of motion of the ship (
\Theta
). The three orientation variables on the apparatus are the angle between the
z
and
z'
axes (
i
), the angle between the
x
and
x'
axes in the
x'-y'
plane (
\Omega
), and the angle which locates the
x'
axis in the
x'-y'
plane (
\omega
). A simple model for the sound velocity variation with depth is included in the range data analysis, while the Doppler data are shown not to need that further complication in the analysis. An error model is constructed and applied to three geometries which represent common underwater research devices. Accuracy goals for possible applications of these devices are discussed, and performance requirements for an acoustic system which would meet the goals are derived.