Observability analysis of 3D AUV trimming trajectories in the presence of ocean currents using range and depth measurements

We analyze the observability properties of the kinematic model of an autonomous underwater vehicle (AUV) moving in 3D, under the influence of ocean currents, using range and depth measurements. The results obtained shed light into the types of trajectories that an AUV may be requested to undergo in order to ensure observability, which is a crucial step in the design of single or multiple beacon positioning systems. We assume that the AUV is equipped with two sensor suites: the first computes the distance (range) of the AUV to single or multiple fixed transponders, while the second measures the vehicle's depth. In both situations, the vehicle has access to its heading angle. We further assume that the AUV undergoes maneuvers commonly known as trimming trajectories, that are naturally obtained when the inputs (thruster rpms and control surface deflections) are held constant. This is done for two main reasons: (i) the class of trajectories thus generated is sufficiently rich for a vast number of applications and (ii) from an observability-analysis standpoint they lead to mathematical tractability and allow for an intuitive physical interpretation. These facts stand in sharp contrast to common approaches adopted in the literature, where the characterization of trajectories that yield observability is only implicit and defies a simple interpretation. In the set-up adopted, the trimming trajectories are completely characterized by three variables: (a) linear body speed parallel to v parallel to; (b) flight-path angle gamma; and (c) yaw rate psi. We assume that parallel to v parallel to > 0, gamma, and psi are constant but otherwise arbitrary (within the constraints of the vehicle capabilities) and examine the observability of the resulting system with the two above mentioned sensor suites. We adopt definitions of observability and weak observability that seek inspiration from those proposed by Herman and Krener (1977) but reflect the fact that we consider specific kinds of maneuvers in 3D. We start with the single transponder case. For range measurements only, we show that in the absence of ocean currents the 3D kinematic model of an AUV undergoing trimming trajectories with nonzero flight-path angle and yaw rate is observable. In the case of non-zero but known ocean currents, identical results apply subject to the condition that the flight-path angle satisfies a current-related constraint. However, if the current is non-zero and unknown, the model is only weakly observable. The situation changes completely when both range and depth measurements are available. In this case, under the assumption that the yaw rate is different from zero, observability is obtained even when the flight-path angle is zero (vehicle moving in a horizontal plane) and there are non-zero unknown currents. These obvious advantages are lost if yaw rate is equal to zero, for in this case the model is only weakly observable. In all situations where the model is weakly observable we give a complete characterization of the sets of states that are indistinguishable from a given initial state. Finally, we show that the extended model that is obtained by considering multiple (at least two) transponders is observable in all situations if the yaw rate is different from zero. (C) 2015 International Federation of Automatic Control. Published by Elsevier Ltd. All rights reserved.