Advances on a null-space-based approach to range-only underwater steering and positioning
Daniela De Palma, Giovanni Indiveri (Dipartimento Ingegneria Innovazione, University of Salento - ISME Node, Italy; António M. Pascoal, Laboratory of Robotics and Systems in Engineering and Science (LARSyS), Instituto Superior Tecnico (IST), University of Lisbon, Portugal
The latest years have seen an increasing development of Autonomous Underwater Vehicles (AUVs). In marine environments the most common applications include ocean monitoring, bathymetric data collection, harbour patrolling, geophysics and geotechnical surveying, just to name a few. In all the mentioned applications, autonomy and mission success are strongly based on the capability of a vehicle to localise itself and navigate with the necessary accuracy. As a result, there is widespread interest in equipping AUVs with simple, reliable, and easy to operate positioning systems. These have been the subject of intensive research and development work over the years . Recently, a novel methodology for AUV positioning has come to the fore that has the potential to substantially reduce the costs of the sensors required and drastically simplify the tasks of system installation and operation: single beacon underwater positioning. In this set-up, the AUV performs dead reckoning and obtains measurements of its successive ranges to a single transponder deployed at a fixed position. By imparting persistently exciting manoeuvres to the AUV it becomes possible, under certain conditions, to compute its position in a given earth-fixed reference frame. In this context, several approaches in the literature try to identify trajectories able to maximise the range-related information for vehicle positioning . In practice, however, an AUV is supposed to do useful work such as moving from an initial to a desired final point. Hence, the need of tradeoffs between these competing objectives.
This paper addresses the problem of driving an underwater vehicle to a desired position while optimising the performance of a range based localisation system. To this effect, we avail ourselves of previous work in the area of task prioritization using a null-space-based projection technique . This allows us to decouple the motion objectives from those of maximizing an observability based criterion related to the minimum singular value of a properly defined FIM (Fisher Information Matrix). In particular, prior to being implemented, the lower priority task commands (observability task) are projected in the null space of the higher priority ones (guidance task). The prioritized task oriented technique is also combined with a saturation management algorithm to cope with velocity command saturations . The stability of the resulting control law is analysed using a Lyapunov-based approach. Moreover robustness against uncertainty in the initial position of the vehicle is also handled.
Preliminary results on this topic have been presented in : here we extend them including a careful singularity analysis of the null space projector operator and a new strategy to handle the uncertainty in the initial position of the vehicle. Indeed, the derivation of the observability task command would require exact knowledge of the initial position of the vehicle. In practice, the initial position is not known, but it can be assumed to lie in a region of uncertainty. Thus, the problem becomes that of finding the optimal control input to reach the final destination given that the unknown vehicle’s initial position can be anywhere inside the uncertainty region. Such a problem is addressed comparing two alternative procedures: the former  aims at optimising the worst case scenario, that is, maximising the minimum of the observability metric inside the uncertainty region; the latter, novel approach, aims at optimising the most frequent case inside the uncertainty region. Whatever optimisation strategy is adopted, the computation of the guidance control law requires knowledge about the position error: in practice this could be computed based on a vehicle position estimate. Hence, the work is completed combining the guidance algorithm with an observer for the estimation of the vehicle position and its uncertainty. Numerical simulations of different scenarios have shown the effectiveness of the proposed strategies.
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