Review articleMultiple autonomous surface vehicle motion planning for cooperative range-based underwater target localization☆
Introduction
Autonomous underwater vehicles (AUVs) are steadily becoming an increasingly important tool to carry out a large number of scientific and commercial missions at sea. Among the systems required for reliable AUV operation, those in charge of vehicle positioning and target localization are critical for vehicle navigation and data geo-referencing. Vehicle positioning refers to the situation where a vehicle positions itself with respect to a given inertial reference frame using appropriate motion sensor suites and acoustic devices. The first typically include a Doppler Velocity Logger (DVL) and an Attitude and Heading Reference System (AHRS), while the latter may consist of devices capable of measuring ranges to a set of acoustic transponders as part of a long baseline (LBL) system, or bearing angles with respect to a platform with a known location, as is the case with ultrashort baseline (USBL) systems. Target localization refers to the case where the position of the vehicle, viewed as an underwater target, is determined by one or more external units that may include autonomous surface vehicles equipped with acoustic ranging devices and/or USBL systems. In a possible scenario, the position of the underwater target is first computed at the sea surface and transmitted to the target via an acoustic modem. Recently, with the advent of miniaturized sensors and the availability of small embedded processors, there has been tremendous interest in the development of positioning systems that are cost-effective and easy to install and operate. The latter include so-called range-only positioning systems (often referred to as range-based positioning systems), which have emerged as viable alternatives to conventional acoustic methods - such as LBL and USBL systems - in a large number of operational scenarios.
Range-based underwater position estimation is a relatively recent technique with a tremendous potential for practical applications. In its simplest form, a range-based positioning system estimates the position of an underwater vehicle from a sequence of range measurements to an acoustic transponder installed at a known location. Analyzing the conditions under which the position of an underwater vehicle can be estimated using range measurements amounts to solving an appropriately defined observability problem. This topic has been studied in depth in previous work by a number of researchers, see for instance Song (1999), Gadre and Stilwell (2005), Zhou and Roumeliotis (2008), Batista, Silvestre, and Oliveira (2011), Crasta, Bayat, Aguiar, and Pascoal (2015), Bayat, Crasta, Aguiar, and Pascoal (2016) and the references therein for the details on different approaches to system observability analysis that include geometric, algebraic, and state augmentation methods.
Position estimation techniques using a single transponder have led to several algorithms based on least-squares (Baccou, Jouvencel, 2002, Scherbatyuk, 1995), extended Kalman filter (EKF) (Gadre, Stilwell, 2005, Larsen, 2001), and multi-model adaptive minimum energy estimation (Bayat et al., 2016) techniques. Position estimation algorithms have also been devised for the case of a moving transponder, the trajectory of which is known. Representative examples include nonlinear least-mean-squares methods (McPhail & Pebody, 2009), maximum likelihood estimation algorithms (Eustice, Singh, Whitcomb, 2011, Eustice, Whitcomb, Singh, Grund, 2007), and centralized extended Kalman filters (Webster, Eustice, Singh, Whitcomb, 2009, Webster, Eustice, Singh, Whitcomb, 2012). Notwithstanding the tremendous progress done in this area, there are still open issues related to the computation of the position estimate accuracy that can possibly be achieved with an unbiased estimator.
In the present paper we take a different venue that is motivated by the dual problem of vehicle positioning, referred above as target localization. In its simplest form, the latter problem is motivated by the situation where one or more mobile surface units are required to determine the position of an underwater target using range measurements to the target. This is best done by using highly accurate synchronized clocks on-board the different platforms (target and surface units) and equipping the target with a single acoustic pinger, thus avoiding the interrogation-reply cycle that would be required if the target carried a transponder. Because range measurements are used in the estimation process, the types of trajectories that a vehicle or group of vehicles executes impact directly on the level of accuracy with which the target position can be estimated. This leads naturally to the problem of determining what types of “sufficiently exciting” maneuvers should be performed by the the surface vehicle (or vehicles) so as to maximize the range-based information available for vehicle localization. This problem bears close affinity to that of optimal sensor placement for range-based underwater target localization by resorting to fixed units at the sea surface. See Moreno-Salinas, Pascoal, and Aranda (2016b) and the references therein for a quick introduction to the problem and for an analytical result on optimal sensor placement for target localization, together with a geometrical interpretation of the solutions obtained. In the above reference, this is done by adopting an estimation theoretical setting and determining the sensor positions that maximize the determinant of an appropriately defined Fisher information matrix (FIM). This methodology lends itself naturally to extension to the setting where, instead of multiple fixed sensors, one resorts to a single or a reduced number of mobile surface units equipped with ranging devices. Once again, the problem can be tackled in terms of maximizing the determinant of a suitably defined Fisher information matrix (FIM) that is range-dependent. In this case, however, instead of simply searching over the possible sensor locations, the search is done over a number of conveniently chosen, motion-related variables that parameterize the possible trajectories that the vehicle or vehicles at the surface may undergo. The reader will find in Pedro, Moreno-Salinas, Crasta, and Pascoal (2015) and the references therein an introduction to this technique that uses the FIM as a performance index for the problem of range-based target localization and allows for the analysis of the trade-offs involved in maneuvering to maximize a FIM-related index while reducing the energy required for the vehicles’ motion. The empirical observability Gramian has also been used in the literature as an alternative performance index, but as shown in Powel and Morgansen (2015) the two are related.
It is against this background of ideas that in the present paper we address the most general problem of range-based target localization, where the objective is to estimate the position of one or more underwater vehicles/targets from a sequence of range measurements obtained by single or multiple acoustic surface vehicles, henceforth called the trackers. Optimal vehicle trajectories can again be computed by maximizing the determinants of related FIMs, subject to inter-vehicle collision avoidance and vehicle maneuvering constraints. The paper derives general expressions for the FIMs involved and clarifies their role in the computation of optimal tracker trajectories using numerical and analytic methods. Analytical solutions are obtained first in the case of one tracker and one target, when the latter undergoes trajectories that are straight lines, pieces of arcs, or a combination thereof. The theoretical analysis is complemented with a practical experiment, where the objective is to estimate the position of a moving underwater target by using range measurements between the target and the tracker moving along a trajectory that is measured on-line. The experimental set-up includes a surface and an autonomous underwater vehicle, both of the Medusa-class, that play the roles of tracker and target, respectively. In the methodology adopted for system implementation, the tracker runs three key algorithms simultaneously, over a sliding time window: (i) tracker motion planning, (ii) tracker motion control, and (iii) target motion estimation based on range data acquired on-line.
A key observation that emerges from the experimental result described is that even in the simple case where the target undergoes straight line motions, the tracker is required to execute quite demanding maneuvers “encircling” the target. Furthermore, it may take considerable time in order to reduce the uncertainty associated with target motion estimates.
In view of the above considerations, it is natural to study the possible advantages of using more than one vehicle for cooperative, range-based underwater target localization. Although this may seem interesting to pursue in a very general setting, from a practical standpoint the consideration of a large number of tracking vehicles meets with two key difficulties. The first stems from the added complexity involved in the cooperative operation of multiple vehicles equipped with acoustic ranging devices. The second is tied with the fact that range-based localization algorithms, which have been inspired by the so-called single beacon navigation algorithms in the related problem of underwater vehicle positioning, were first introduced to overcome the problems inherent to classical localization solutions using multiple, fixed acoustic ranging sensors. Increasing the number of tracking vehicles would clearly defeat the purpose of attempting to decrease the number of acoustic ranging devices. For this reason, we consider two tracking vehicles. The rationale for our study stems from the desire to reduce the complexity of the maneuvers that may have to be performed by a single tracker, and also to reduce the time required to obtain the location of the target with adequate precision.
With this brief background, in addition to the single-tracker single-target problem, we study three cases that involve three vehicles called the tracker, the tracker’s companion, and the target to be localized. In the first case, two vehicles independently measure their ranges to the target and exchange information on their positions. In the second case, the companion shares its range to a fixed target with the tracker and the tracker knows the position of the companion. For these two cases, we show that the optimal position vectors of each vehicle are such that the relative position vectors with respect to the target position are always orthogonal. In the third case, the companion shares its range to a fixed target with the tracker but the companion position is unknown to the tracker. This case amounts to the localization of two targets and can only be solved by resorting to numerical methods.
To relate our problem with previous work reported in the literature, in Zhou and Roumeliotis (2008), a single target localization problem using a group of vehicles was addressed using range-only information in a centralized architecture. The key assumptions were that each vehicle knows the positions of all the other vehicles and the optimization variables were body-speed and heading. As we will see soon, among the three cases that we consider, this corresponds to the first case only. In addition, our optimization variables are speed and heading rate as opposed to heading.
The paper is organized as follows. Section 2 introduces the basic notation that will be used in the sequel. We present the process model in Section 3 followed by the problem formulation in Section 4. Section 5 deals with the computation of the FIM for the problem at hand, while Section 6 describes the optimal FIM that can be obtained. Section 7 describes the analytical construction of the optimal vehicle trajectories for one tracker and one target case and the experimental set-up adopted for target localization, which validates the efficacy of the method proposed. Section 8 describes the cooperative target localization and details three possible operational scenarios, together with the computation of optimal tracker trajectories that maximize the corresponding FIMs. Finally, Section 9 contains the main conclusions and Section 10 highlights some challenging open problems.
Section snippets
Preliminaries
For with m < n, we let We denote by In the identity matrix of size n and by 0m × n the zero matrix of size m × n. A positive definite matrix is simply written A≻0. We further let and trace(·) denote the determinant and the trace of a square matrix, respectively. Given denotes the column vector obtained by stacking the columns of the matrix A on top of one another. Given denotes the diagonal matrix whose
Process modelling
In this paper we consider a general set-up of multiple trackers and multiple targets. In what follows we develop a continuous-time kinematic model for the trackers and a discrete-time measurement model of the ranges between trackers and targets.
Problem formulation
To simplify the notation, for each and we let and
Then, for each Eq. (2) can be written asWe make the following assumption for the trackers. Assumption 1 For simplicity of analysis, we assume that the speed and course rate of each tracker are piecewise constant functions
Fisher information matrix
In this section we derive the FIM for the model described before, consisting of the continuous-time tracker kinematics given by Eq. (1) and the discrete-time measurements given by Eq. (5).
We start by recalling the concept of Fisher information in a general set-up. Consider the problem of estimating an unknown but fixed parameter using a sequence of measurements zk, 1 ≤ k ≤ N, corrupted by additive noise ηk, 1 ≤ k ≤ N, according to the measurement modelwhere is a
Optimal Fisher information matrix
We are now ready to investigate the optimality of the FIM. With this objective in mind, in this section we look for the optimum value of the cost function described byobtained by noting thatFrom the above, it follows that the maximum values of the determinant of the overall FIM cannot exceed the product of the maximum obtained for each individual FIM, possibly with distinct strategies for . Remark 2 In the set-up
Single-tracker and single-target configuration
The objective of this section is to provide a constructive method for the generation of a family of feasible optimal trajectories that will maximize the FIM for the single tracker-single target case In the sequel, all the optimal values of the variables are denoted by the superscript “ * ” and for the sake of simplicity we drop all the other super- and sub-scripts.
Let and denote the initial position of the tracker and target, respectively. Further, suppose be the
Cooperative target localization
We now consider the problem of cooperative, single target localization where, in addition to ranges to the target, the tracker has some additional source of information about the target through a companion vehicle. The set-up consists of a tracker, a companion, and a stationary/moving target, the motion of which must be estimated. We consider the following three cases.
Conclusions
The paper derived a general theoretical framework for the computation of the optimal trajectories that should be imparted to one or more surface tracking vehicles with a view to accurate range-based multiple undwerwater target localization. Building on classical estimation theory, the methodology adopted amounts to maximizing the determinant of an appropriately defined FIM subjected to the tracker maneuvering constraints and collision avoidance among the trackers. Theoretical and experimental
Future directions
With the new paradigm of internet of underwater things (IoUT), the future will witness the operation of multiple assets at sea, involving the coordinated operation of surface and underwater autonomous vehicles for a vast number of scientific and commercial applications. The latter will require the use of large groups of heterogeneous vehicles equipped with vision systems, acoustic sources, and sensors for seismic surveying (geophysical and geotechnical), 3D mapping of unstructured environments,
Acknowledgments
The authors are indebted to J. Botelho, P. Góis, J. Ribeiro, M. Rufino, M. Ribeiro, L. Sebastiáo, and H. Silva for their untiring support and collaboration on the planning and execution of sea trials with the Medusa vehicles.
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Cited by (0)
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A preliminary version of this paper appeared in the proceedings of the 10th IFAC conference on Control Applications in Marine Systems 2016, Moreno-Salinas et al. (2016a), and the 10th IFAC World Congress 2017, Crasta et al. (2017).
- 1
The work of the first and third authors was supported in part by the H2020 EU Marine Robotics Research Infrastructure Network (Project ID 731103) and the Portuguese FCT Project UID/EEA/5009/2013.
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The second author is grateful to the “Ministerio de Educación, Cultura y Deporte” for support under “Programa Estatal de Promoción del Talento y su Empleabilidad en I+D+i, Subprograma Estatal de Movilidad, del Plan Estatal de Investigación Cientfica y Técnica y de Innovación 2013-2016” for the “Jose Castillejo 2015” grant under number CAS15/00341.