Range-based Underwater Target Localization using an Autonomous Surface Vehicle: Observability Analysis
N. Crasta, Laboratory of Robotics and Systems in Engineering and Science (LARSyS), ISR/IST, University of Lisbon, Portugal; D. Moreno-Salinas, Department of Computer Science and Automatic Control, National Distance Education University, Spain; B. Bayat, BioRobotics Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland; A. M. Pascoal, LARSyS), ISR/IST, University of Lisbon, Portugal; J. Aranda, Department of Computer Science and Automatic Control, National Distance Education University, Spain
In the area of marine robotics, range-based navigation systems [1,2] have been successfully implemented to solve the problem of vehicle positioning, i.e. to compute the position of one or more underwater vehicles by measuring ranges to a set of either fixed or moving beacons, the inertial positions of which are known functions of time. The literature on this problem is vast and defies a simple summary. In a nutshell, the solution to the problem involves a number of key components: modeling , qualitative observability analysis [4,5,6], quantitative observability analysis yielding a measure of observability (using for example the Fisher information matrix  or the observability Gramian ), and estimator design [1,4]. Among these, qualitative observability analysis is a theoretically challenging task that has been studied by many authors by resorting to a number of methods that include linearization, and state augmentation techniques, as well as algebraic and geometric methods. Once a measure of observability has been defined, it is natural to consider the problem of optimal vehicle motion planning, aimed at maximizing the range-related information available for vehicle positioning. The solution to this problem plays a central role in the development of systems for single vehicle positioning using a single beacon (also referred to as single-beacon navigation systems), which implement a three steps process over a sliding time-window : i) vehicle motion planning, ii) vehicle motion control, and iii) vehicle state estimation using the range data acquired on-line .
Inspired by various complex and challenging scientific and commercial mission scenarios, there arises another type of problem wherein a group of surface vehicles, henceforth referred as trackers, equipped with global positioning systems (GPSs) are required to localize another group of underwater vehicles, referred as targets, using range information. In the literature, this problem is often known as target localization problem or target motion analysis . In particular, the range-based target localization problem has received considerable impetus in recent years motivated by the need for higher position accuracy , with a growing interest in exploring trajectories for the trackers that can maximize the range-related information for multiple target localization. This has been explored in an estimation theoretic sense, using the so-called Fisher information matrix (FIM), whose non-singularity ensures the observability of the model underlying the target localization problem. However, the types of motions to be imparted to the trackers in order to ensure observability of the parameters of interest of the targets, such as linear and angular speeds, course angle, position, etc. are still largely not clear, except for some simple cases. From a practical standpoint, it is useful to consider the target moving along lines or arcs of circumferences of a circle with constant linear and angular speeds. At the outset, this seems to be a dual problem of the vehicle positioning, which is not true. However, in the specific case of single tracker and a single target, this is a dual of the single-beacon problem, where the vehicle and beacon play the roles of tracker and target, respectively.
With this background, in this paper we consider the (qualitative) observability analysis of the range-based single target localization problem. The set-up consists of a tracker and a target, where the latter is restricted to move along a straight line or an arc of a circumference with constant linear and angular speeds, respectively, on a given observation window. In addition, the tracker receives ranges to the target. Notice that for this restricted class of target motions, the corresponding trajectories obtained are fully determined by a small number of initial conditions which we refer to as target parameters: i) position, linear speed, and course angle in the case of straight lines and ii) position, linear speed, course angle, and course angle rate in the case of circumferences. We recall that in the context of vehicle positioning, for the simple case of unknown initial vehicle position, it has been shown  that the range-based system is locally observable and observable for the straight line and circular motion, respectively. By duality, similar conclusions can be drawn about the observability properties of the range-based target localization problem with unknown initial target position.
Apart from the above simple case, the conditions under which one can estimate any combination of the three (for straight line motion) or four (for circular motion) target parameters using range-only information has not been fully explored yet. To make the problem mathematically tractable, we consider simple kinematics models for the tracker and target with planar position and course angle as state variables. With this set-up and the given assumptions, for all the possible target parameter combinations, we derive sufficient conditions on the inputs of the tracker that render range-based target localization problem observable. In particular, when the target is moving along a straight line, for most of the unknown target parameter combinations, we show that observability can be achieved by a nonzero constant course rate, which is a simple condition. In the case where the target is moving along a circular path, the sufficient conditions on the tracker’s input to achieve observability are more demanding and less straightforward, and do not lend themselves to a simple geometrical explanation.
 M. Bayat, N. Crasta, A. P. Aguiar, and A. M. Pascoal. “Range-based underwater vehicle localization in the presence of unknown ocean currents: Theory and experiments,” IEEE Transactions on Control Systems Technology, 24(1), pp. 122-139, 2016.
 A. Bahr, J. J. Leonard, and M. F. Fallon. “Cooperative localization for autonomous underwater vehicles,” The International Journal of Robotics Research, 28(6), pp. 714-728, 2009.
 Thor I. Fossen. Handbook of Marine craft Hydrodynamics and Motion Control. John Wiley & Sons, 2011.
 P. Batista, C. Silvestre, and P. Oliveira. “Single beacon navigation: Observability analysis and filter design,” IEEE American Control Conference, pp. 6191-6196, 2010.
 G. Indiveri, D. De Palma, and G. Parlangel. “Single Range Localization in 3-D: Observability and Robustness Issues,” IEEE Transaction on Control Systems Technology, 24(5), pp. 1853-1860, 2016.
 N. Crasta, M. Bayat, A. P. Aguiar, and A. M. Pascoal. “Observability analysis of 3D AUV trimming trajectories in the presence of ocean currents using range and depth measurements,” Annual Reviews in Control, 40, pp. 142-156, 2015.
 D. Moreno-Salinas, N. Crasta, M. Ribeiro, B. Bayat, A. M. Pascoal, and J. Aranda. “Integrated motion planning, control, and estimation for range-based marine vehicle positioning and target localization,” 10th IFAC Conference on Control Applications in Marine Systems (CAMS), 49(23), pp. 34-40, 2016.
 A. Filippo, G. Antonelli, A. Pedro Aguiar, and A. M. Pascoal. “An observability metric for underwater vehicle localization using range measurements,” Sensors, 13(12), pp. 16191-16215, 2013.
 B. Ristic, S. Arulampalam and J. McCarth. “Target motion analysis using range-only measurements: Algorithms, performance and application to Ingara ISAR data,” Technical Report.
 D. Moreno-Salinas, A. Pascoal and J. Aranda. “Optimal sensor placement for acoustic underwater target positioning with range-only measurements,” IEEE Journal of Oceanic Engineering, 41(3), 2016.
 N. Crasta, M. Bayat, A. A. Pedro, and A. M. Pascoal, “Observability analysis of 2D single beacon navigation in the presence of constant currents for two classes of maneuvers,” 19th IFAC Conference on Control Applications in Marine Systems, Osaka, Japan, September 17-20, 2013.