Cognitive Multistatic AUV Networks - IEEE Xplore
Cognitive Multistatic AUV Networks Paolo Braca∗ Ryan Goldhahn∗ Kevin D. LePage∗ Stefano Marano† Vincenzo Matta† Peter Willett‡ ∗ NATO
STO-CMRE, La Spezia, Italy, Email: {braca/goldhahn/lepage}@cmre.nato.int. of Salerno, Fisciano, Italy, Email: {marano/vmatta}@unisa.it. ‡ University of Connecticut, Storrs CT, Email: [email protected].
† University
Abstract—Autonomous underwater vehicle (AUV) platforms are low-cost devices with respect to conventional detection and tracking systems for the purpose of anti-submarine warfare (ASW). Unfortunately, the increased level of manageability is often paid in terms of capabilities, e.g., limited speed and endurance, inferior sensor payloads, and so on. This work exploits two fundamental concepts aimed at filling the consequent performance gap. First, a multistatic network of AUVs is considered, where a smart and collaborative multi-sensor data fusion allows going beyond the individual sensors limitations. Then, we focus on the cognitive paradigm, where the single AUV units optimize their future actions (i.e., their path planning) in view of the final inference purpose of the network, and based on the evidence collected up to the present. A multistatic configuration of the platforms and the corresponding acoustic model are considered, and taken into account in order to derive a proper Bayesian model. Using the information contained in the Bayesian full posterior, cognitive detection and tracking algorithms are designed. Seeing them at work in practical scenarios shows the benefits of the cognitive network paradigm. Index Terms—Data fusion, Anti-submarine warfare, multistatic active sonar, Bayesian target tracking, underwater wireless sensor networks, autonomous underwater vehicles, cognitive systems, acoustic model.
I. I NTRODUCTION Nowadays, underwater autonomous systems have a wide range of applications, see e.g. [1]. Anti-submarine warfare (ASW), though the oldest, is still perhaps the most important. The specific ASW configuration considered in this work is that of a wireless sensor network working in a multistatic setup. Multistatic sonar systems for ASW differ from conventional monostatic active systems in that there are multiple receive nodes that are not co-located with the transmitters [2]–[4]. One advantage of this deployment is that the position of the receivers is not easily discernible by the target. In a multistatic system, the judicious placement of covert receivers allows for an increased probability of a favorable geometry, taking into account aspect-dependent target strength, which can dramatically increase the SNR, as well as the effective network coverage. Recently, autonomous underwater vehicles (AUVs) have been proposed as sensor nodes in multistatic surveillance networks due to their low cost, manoeuvrability, and ability to reconfigure their positions based on the observed data and mission objectives without human intervention. They may however be inferior to conventional ASW assets with respect to array aperture, speed, and endurance. The expected performance gap may be bridged by deploying multiple sensor
platforms, so as to form a collaborative network of smart autonomous vehicles. However, real-world applications are invariably characterized by a certain degree of variability (changes in the relative target-AUV position, in the environmental conditions, and so on), so that it is often desired to guarantee an adequate level of adaptation and reconfigurability of the ASW system. The above problem can be put in the more general framework of sensor management [5], [6] and cognitive systems [7], defined as the tasking, sequencing, and scheduling of a heterogeneous network of sensors to optimally achieve a prescribed goal. There are two approaches to sensor management, as it is called in much of the literature [8]: Task- or mission-driven and information-driven sensor management. The former chooses sensor actions based on a given performance metric or error quantity directly related to the mission objectives. Informationdriven sensor management chooses sensor placements and actions that maximize a measure of the information gain. Elaborating on these concepts, in this work we propose a method to design networks of cognitive AUVs, namely, of AUVs capable of planning their (future) actions i) based upon the available (present) knowledge, and ii) in view of the final goal of the network. We consider the problem of localizing a target in the ASW environment by means of a network of AUVs. Each AUV can plan its future navigation steps by choosing among a collection of predefined local paths. The best path is selected on-line according to some criterion optimizing the network inference performance, using the information currently available at all the AUVs. The basic elements for the path planning are as follows. Given a target position, the quality of the measurements collected by the AUVs strongly depends upon their spatial configuration, and from the environmental conditions. Otherwise stated, the spatial profile of the detection probability is a function of the target and AUV positions and of the environmental conditions. A key observation is that the target state (position, velocity and even its presence/absence) is unknown, and is indeed the object of the network inference. As a result, we must consider a statistical path planning, and the target tracking algorithm becomes essential not only for the original inference goal, but also for the path planning itself. We focus on optimal Bayesian tracking, and propose workable algorithms that take into account several important aspects, such as the geometry of the network, the acoustic propagation, the difficulty for a single AUV to discriminate port from starboard contacts, and the problem of target ap-
pearance/disappearance. The paper is organized as follows. In Sec. II we formalize the ASW tracking problem using AUVs in a bistatic configuration and describe the optimal Bayesian estimation procedure. Sec. III describes the autonomous cognitive behaviours of AUVs. Numerical experiments are reported in Sec. IV, and Sec. V draws the conclusions. II. M ULTI - SENSOR BAYESIAN TRACKING OF A B ERNOULLI RFS TARGET
Fig. 1. Sketch of the port-starboard ambiguity in the bistatic geometry.
A network of Ns AUVs monitoring a certain surveillance region is considered. The objective of the network is to estimate the absence or presence of a target and, in the latter case, its kinematic components at each time scan k. The target state Xk can be conveniently formalized as a Bernoulli random finite set (RFS) [9]–[11], where Xk = {∅} when the target is absent, or Xk = {xk } when the target is T present with xk = [xk , x˙ k , yk , y˙ k ] , where the two positions (p) T T are xk = [xk , yk ] and [x˙ k , y˙ k ] are the corresponding velocities. In this work, when the target is present, we assume a nearly constant velocity model [12]: xk = F k xk−1 + v k ,
(1)
where F k is the state transition matrix, and v k takes into account the target acceleration or unmodeled dynamics. At each time scan k a set of data, modeled as RFS, is observed by the sth AUV, defined by m
k,s Zk,s = {z k,s,i }i=1 ,
(2)
where mk,s is the number of measurements1 . Note that the current technology available for AUVs has some limitations, and one of the most significant is the port– starboard (PS) ambiguity, namely the impossibility for the AUV to discriminate port from starboard contacts. PS ambiguity complicates the detection and tracking algorithms and may severely degrade performance. In [14], [15] a Bayesian tracking approach is proposed to track the target state in presence of PS ambiguity and measurement origin uncertainty (MOU) [12]. Given the PS ambiguity problem, z k,s,i in (2) are just the contacts on the port side, since they form a sufficient statistic, because of the deterministic dependence of the starboard contacts on the port contacts [15]. In the following subsections we explain how the target state is statistically related to data collected by the network. A. Target-originated measurement with PS ambiguity In this subsection we briefly describe the target-originated measurement model with PS ambiguity. Considerthe geometry (x)
(y)
T
denote the of the PS problem in Fig. 1. Let sk = sk , sk source position at time scan k, while the sensor T array position (x) (y) and and its heading angle are denoted by pk = pk , pk hk , respectively. The sensor measures the bistatic range bk
1 In this work we use the RSF formalism for both target and data, however other equivalent choices are available, see for instance the discussion in [13].
from source to target to receiver and the bearing angle relative to the array heading θk . The non-ambiguous measurement is given by ⎡ ⎤ (p) (p) xk − pk + xk − sk + wkb bk ⎦,
=⎣ zk = y −py θk tan−1 xkk −pkx − hk + wkθ b
k2 wk 0 σb 0 , wk = ∼ N , 0 σθ2 0 wkθ where wkb and wkθ are the range and bearing noise, respectively. The PS ambiguous contacts have the same bistatic range measurement but two different bearing angles: θkP is the angle from the receiver to the target on the port side, and θkS is the angle from the receiver to the target on the starboard side. P T S S T = b , θ and z = b , θ are given by Then z P k k k k k k ⎧
yk − pyk ⎪ −1 P S P ⎪ ≥ hk , θ = θ , θ = −θ , if tan ⎪ k k k k ⎪ xk − pxk ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ θkS = θk , θkP = −θkS ,
if tan
−1
yk − pyk xk − pxk
< hk .
B. Measurement origin uncertainty model with target presence/absence Under the hypothesis of target absence (H0 ), the contacts are independent from each others and identically distributed with a known clutter distribution. The set density of the measurements under H0 , at time k and at the sth sensor can be formally written as: fk,s (Z |H0 ) = m! μc (m; λ)
m
f (c) (z i ),
(3)
i=1
where μc (m; λ) is the probability mass function (pmf) of the clutter cardinality (often assumed as a Poisson pmf with rate λ), and f (c) (z) is probability density function (pdf) of the clutter locations (typically modeled as uniform in the region of interest), see further details in [12], [13], [15]–[18]. When the target is present (hypothesis H1 ), it can be observed with a certain detection probability. This latter may vary dramatically over the surveillance region, and is in general a function of the receiver array parameters, source parameters, and environmental parameters as such as bottom scattering strength, water column and bottom sediment density and sound speed, and bottom and surface roughness and reflection loss. Numerical and/or closed-form acoustic propagation models
[19]–[21] are then used to calculate the predicted PD . We borrow from [22] an acoustic model to predict the detection probability of the target in any position in the surveillance region, taking into account the relevant environmental acoustic effects. This will be key for enhancing the performance of the tracking algorithm, as well as for the path planning optimization that is the main subject of this paper. The aforementioned acoustic model provides us with a (p) detection probability PD (xk , pk,s , sk ), which is a function (p) on the target position xk , of the receiver position pk,s and of the source position sk . In this work we assume a stationary (p) (p) source, then sk = s and PD (xk , pk,s , sk ) ≡ PD (xk , pk,s ). Since we assume that at most one target is present, all the other contacts are clutter, independent from the target’s state. Then, the set density of data under H1 is given by m
(p) f (c) (z i ) fk,s (Z |H1 ) = m! 1 − PD (xk , pk,s ) μc (m; λ) i=1
(p)
+ (m − 1)! PD (xk , pk,s )μc (m − 1; λ) m (t) × fk,s (z j ) f (c) (z i ), j=1
(4)
i=j
(t)
where fk,s (z) is the target-originated pdf with PS ambiguity, see details in [15]. In order to make explicit the relation between the target state and the data, eqs. (3) and (4) can be equivalently rewritten as: fk,s (Z |Xk = {∅} )
=
fk,s (Z |H0 ) ,
(5)
fk,s (Z |Xk = {xk } )
=
fk,s (Z |H1 ) .
(6)
C. Optimal Bayesian tracking In Bayesian tracking, the objective is to construct the posterior distribution of the target state Xk . To this aim, let us denote by Z1:k = [Z1 , Z2 , . . . , Zk ] the aggregate in time of the data up to time step k, where Zk = [Zk,1 , Zk,2 , . . . , Zk,Ns ] is in turn the aggregate, at time k, of the observations collected by all the Ns sensors. The posterior of the target state Xk can be thus written as: fk (Zk |Xk ) P (Xk |Z1:k−1 ) P (Xk |Z1:k ) = , (7) P (Zk |Z1:k−1 ) where P (Xk |Z1:k−1 ) is the prior at time k and P (Zk |Z1:k−1 ) is the scaling factor. Since the sensors are conditionally independent given the target state, the likelihood fk (Zk |Xk ) can be factorized as: fk (Zk |Xk ) =
Ns
fk,s (Zk,s |xk ) ,
(8)
s=1
where fk,s (Zk,s |Xk ) is the likelihood of the sth sensor at time k, given in eqs. (5)-(6). We assume that the AUVs have the ability to communicate to each other their local detections Zk,s and positions pk,s . Consequently each AUV has access to the joint likelihood (8). While in principle this knowledge is enough to build the posterior distribution (7), closed-form solutions are seldom
available in practice. As a consequence, one should typically resort to some form of approximation that are both accurate and computationally tractable. Specifically, in this work we focus on the powerful technique of particle filtering to obtain a numerically efficient approximation of (7), see [16], [23]– [25]. III. C OGNITIVE AUV PATH PLANNING In this section we present our solution to design cognitive AUVs. We explain how the AUVs can adapt their navigation strategy, based on the on-line incoming information Zk , in order to enhance the network tracking/localization capabilities. In order to design its path planning, the sth AUV, at time k, must decide where to go at time k + 1, namely, it must choose its next position pk+1,s . This is made by selecting a change of speed δvk,s ∈ V and a change of heading δhk,s ∈ H. The classes V and H model the constraints on the admissible policies. The AUV updated velocity and heading are accordingly given by: vk+1,s = vk,s + δvk,s , hk+1,s = hk,s + δhk,s , yielding a consequent variation in the position: cos (hk+1,s ) pk+1,s = pk,s + vk+1,s . sin (hk+1,s )
(9) (10)
(11)
As a result, the class of admissible positions at time k + 1 is a function of the starting point pk,s . This class will be denoted by S(pk,s ). By iterating this reasoning, from a starting point pk,s , a collection T (pk,s , L) of admissible trajectories τ of a given length L is generated, formally: τ = [pk+1 , pk+2 , . . . , pk+L ] ∈ T (pk,s , L). The cost associated to a given target position X and a given AUV position p will be measured by a cost function C(X, p), which is chosen in connection to the goal that the network tries to achieve. A first optimization strategy might be that of selecting the position pk+1,s that maximizes the average cost E[C(Xk+1 , pk+1,s )|Z1:k ], where the expectation has been evaluated given all the available knowledge contained in the observations collected by the network up to time k. However, this choice will typically lead to a myopic system. Indeed, since not all paths are admissible, optimizing by looking just one step ahead might then force the AUV to reach a very unfavorable region after some steps. In order to mitigate this effect, a clever choice of the position pk+1,s should consider a planning horizon L > 1. Following [7], we propose the minimization of the following additive cost-to-go function: ρ(τ |Z1:k ) = ρ(pk+1 , pk+2 , . . . , pk+L |Z1:k ) k+L = E C (Xn , pn ) Z1:k , (12) n=k+1
In the above formula, the expectation of the single summand is computed under the distribution P (Xn |Z1:k ), that is the
Accordingly, we set: ⎧ ⎨ 1 − PD x(p) , p C (X, p) = ⎩ 0
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Fig. 2. Graphical representation of the optimization space of all considered AUV navigation paths for a horizon period of 1 hour from an initial heading of 0 degrees with a constant speed of 1 m/s.
(network) posterior distribution of the target position at time n given the data up to time k. The selection of the best position pk+1,s can be now performed as follows. The sth AUV finds the optimal path τ ∗ (pk,s ) estimated at time k as the solution of the following optimization problem: min
τ ∈T (pk,s ,L)
ρ(τ |Z1:k ).
(13)
We stress that, while, for a given trajectory τ , the function ρ(τ |Z1:k ) in (12) does not depend on the particular AUV, the estimated path τ ∗ (pk,s ) does. This is due to the search space where the optimization is performed, which is dependent on the initial AUV position pk,s . After having estimated the best path at time k, the sth AUV selects as next step pk+1,s only the first point of the optimal trajectory τ ∗ (pk,s ). At time k + 1, the next position pk+2,s is then selected by repeating the overall procedure, starting from the configuration pk+1,s chosen at time k. This means that the optimized path [pk+1,s , pk+2,s , . . . , pk+L,s ], effectively followed by the AUV, will be in general different from the path τ ∗ (pk,s ) estimated at time k. A. Path planning implementation In this subsection we summarize the main implementation details of the path planning described by (12) and (13). First, we must choose a proper cost function C(X, p), which of course, depends on the application. In the ASW context it depends on the kind of mission: submarine tracking, area clearance, etc. In this work, we propose a heuristic cost function, computationally efficient, which reflects the physics of the problem. Basically, we want to place the AUV in the best position in terms of target detection – not globally (too complicated) but locally, at this sensor and at this time.
if X = {x}, (14) if X = {∅}.
The cost-to-go function in (12) can be now computed by averaging the single summands C(Xn , pn ) under the posterior distribution P(Xn |Z1:k ). In particular, in the implementation of our algorithms, the pertinent posterior distributions are efficiently evaluated by using particle filtering, so that the expectations for computing the average costs are easily carried out by a summation over the particles. It remains to solve the optimization problem in (13). However, the search for the exact solution is typically not feasible. Several approximate solutions have been proposed, among them the most important ones being the dynamic programming strategies [6], [7], [26], [27]. Unfortunately, in our context dynamic programming is computationally too expensive given that the optimized strategy must be computed on-board of the AUV in a time interval that is typically less than a minute. For this reason we propose an ad hoc solution that is computationally tractable even locally by the AUV. We constrain the optimization problem to a smaller search space. The optimization space of navigation paths is constrained to be composed of trajectories at fixed speeds, that can be selected among few values (usually around 1-5 m/s), complying with the necessity to maintain a minimum speed while towing an array. The heading changes are forced to be constant during the planning horizon. The optimization space is then constituted by several smooth arcs2 , see further details in [29] and a graphical representation in Fig. 2. IV. C OMPUTER EXPERIMENTS In this section we show the benefit of using a cognitive network of AUVs (C-NET) with respect to a non cognitive network (NC-NET) in which AUVs are forced to be in predefined places or to move along predefined paths. Furthermore we compare the performance between myopic (L = 1, referred as CM-NET) and non myopic (L > 1) implementations of the cognitive strategy described in the previous sections. The scenario under test is the same of the Co-operative LittoraL ASW Behaviour 2013 (COLLAB13) experiment, conducted by NATO-STO Centre for Maritime Research and Experimentation in the Mediterranean Sea off the coast of Porto Venere, Italy from June 29 to July 07, 2013. Note that in this scenario the sound speed depends dramatically upon the depth of the surveyed area and exhibits typical summertime features. The clutter is assumed to be uniform. The signal to reverberation plus noise ratio (SRNR) is calculated using the ARTEMIS adiabatic normal mode propagation code using the bathymetry, sound speed profile, and environmental parameters measured during the COLLAB13 experiment. No target aspect dependency is included in the 2 The navigation of AUVs along smooth paths is highly desirable in order to avoid sharp manoeuvres that would deform the shape of the linear array towed by the AUV with large systematic errors in the data, see details in [28].
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Fig. 3. Target detection probability, PD [x, y] , pk,s , at time step k. AUV positions pk,s (white square ’’), source position (white diamond ’♦’) current target position (white ’×’). The AUV positions are given for the case of C-NET.
model, however it would be straightforward to take into account the target aspect. We are now ready to illustrate the results of the numerical experiments. Fig. 3 shows the target detection probability maps and positions of source, target and AUVs given by the C-NET behaviour. At the beginning, k = 1, the AUVs are located close to the source with a large coverage, in terms of detection probability, of the surveillance region. However, the target is sailing (from south to north) in the bottom corner, far away from AUVs and source, where the detection probability is low, and then, is well hidden. In particular, the target will be hidden for all its trajectory if the AUVs will not change their positions (as in the case of the NC-NET). In order to detect the target, the AUVs shall try to get close to it. Now, the crucial point is that they have only an imperfect information about the target navigation trajectory, embodied in the Bayesian posterior. The behaviour of the distributed Bayesian tracking for the case of C-NET, CM-NET and NC-NET, is reported in Figs. 4 and 5. In particular, we show the posterior of the target position in Fig. 4 and the posterior of the target presence in Fig. 5. The path decided by AUVs is reported in Fig. 6 as well as the target trajectory. The fact that the target is hiding matches with the behaviour of the algorithm, indeed from an initial half chance of having a target, see Fig. 5, the target presence probability decreases. However, the Bayesian tracking algorithm is able to recognize that if the target is present then it would be located in the most
west region, see Fig. 4 at k = 25, 50 for all the strategies. At this point we are able to recognize the effects of using the cognitive paradigm. AUVs in both C-NET and CM-NET cases try to explore and move towards the regions where the target could be located, or in other words they want to be active and hunt the target. This goal is accomplished and it is easy to recognize the benefit of using a cognitive strategy. When C-NET is in force the target presence is correctly declared around k = 50 − 70, earlier, as expected, than CM-NET. Interestingly, the NC-NET is not able to detect the target because the target always sails in the region where the PD , seen by AUVs in fixed positions (see Fig. 3(a) and 3(d)), is very low. V. C ONCLUSIONS AUVs offer low-cost ASW capability. However, their full potential is not currently available, as their superior multilateration requires intelligent positioning, meaning path-planning. The implied global “cognitive” problem is easy to pose but not really feasible in light of limited AUV computational resource and coordination ability. Hence we here propose an alternative that is local, light in algorithmic footprint – and seems to perform very well. A multistatic configuration of the platforms and the corresponding acoustic model are considered, and taken into account in a proper Bayesian model. A smart and collaborative multi-sensor data fusion is proposed. The focus is on the cognitive paradigm, where the single AUV units optimize their
(p)
Fig. 4. Posterior distribution of the target locations xk (red dots ’·’) and true target position (blue ’×’). The first row is referred to the C-NET, the second to CM-NET and the third to NC-NET. The distribution is numerically computed by using a particle filtering strategy.
path planning in view of the final inference purpose: Hunting the enemy submarine. The key element of this work is the cognitive path planning, based on the evidences collected up to the current time. Using the information contained in the Bayesian posterior, we show the benefit of using the cognitive paradigm in a practical scenario. Future work can include the study of a multi-target scenario, in which AUVs want to optimize their paths to detect and track more than a single target by using multi-objective optimization tools. Other interesting future directions involve the problem of collision/obstacle avoidance in the AUV cognitive algorithm, and to the interdependency of the target/AUV velocities.
VI. ACKNOWLEDGEMENTS This work has been funded by the NATO Allied Command Transformation (NATO-ACT) under the projects System Concepts for Littoral Surveillance and Maritime Situational Awareness. Peter Willett was supported by the Office of Naval Research under contract N000014- 13-1-0231.
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[11] C-NET, L=15 CM-NET, L=1 NC-NET
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Fig. 6. AUV trajectories for the case of C-NET (blue square ’’), CM-NET (red square ’’), and AUV positions of NC-NET (black square ’’). Source position (yellow diamond ’♦’). Target trajectory (black dashed line ’−−’). Arrows indicate the directions of AUVs and target.
[29]
uncertainty,” Signal Processing, IEEE Transactions on, vol. 60, no. 5, pp. 2138–2151, May 2012. B. Ristic, B.-T. Vo, B.-N. Vo, and A. Farina, “A tutorial on Bernoulli filters: Theory, implementation and applications,” Signal Processing, IEEE Transactions on, vol. 61, no. 13, pp. 3406–3430, July 2013. Y. Bar-Shalom, P. Willett, and X. Tian, Tracking and Data Fusion: A Handbook of Algorithms. Storrs, CT: YBS Publishing, 2011. P. Braca, S. Marano, V. Matta, and P. Willett, “Asymptotic efficiency of the PHD in multitarget/multisensor estimation,” IEEE J. Sel. Topics Signal Process., vol. 7, no. 3, pp. 553–564, 2013. P. Braca, K. LePage, P. Willett, S. Marano, and V. Matta, “Particle filtering approach to multistatic underwater sensor networks with leftright ambiguity,” in Proc. of the 16th Intern. Conf. on Inform. Fusion (FUSION), Istanbul, 2013. P. Braca, P. Willett, K. LePage, S. Marano, and V. Matta, “Bayesian tracking in underwater wireless sensor networks with port-starboard ambiguity,” IEEE Trans. Signal Process., 2014. P. Braca, S. Marano, V. Matta, and P. Willett, “A linear complexity particle approach to the exact multi-sensor PHD,” in in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), May 2013. ——, “Multitarget-multisensor ML and PHD: Some asymptotics,” in Proc. of the 15th Intern. Conf. on Inform. Fusion (FUSION), Singapore, 2012. P. Braca, M. Guerriero, S. Marano, V. Matta, and P. Willett, “Selective measurement transmission in distributed estimation with data association,” IEEE Trans. Signal Process., vol. 58, no. 8, pp. 4311–4321, Aug. 2010. C. Harrison, “Closed-form expressions for ocean reverberation and signal excess with mode stripping and Lambert’s law,” J. Acoust. Soc. Am., vol. 114, no. 5, pp. 2744–2756, 2003. ——, “Fast bistatic signal-to-reverberation-ratio calculation,” J. Comp. Acoust., vol. 13, no. 2, pp. 317–340, 2005. ——, “Target time smearing with short transmissions and multipath propagation,” J. Acoust. Soc. Am., vol. 130, no. 3, pp. 1282–1286, 2011. R. Goldhahn, P. Braca, K. LePage, P. Willett, S. Marano, and V. Matta, “Environmentally sensitive particle filter tracking in multistatic AUV networks with port-starboard ambiguity,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), Florence, 2014. M. R. Morelande, C. M. Kreucher, and K. Kastella, “A Bayesian approach to multiple target detection and tracking,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 1589–1604, 2007. C. Kreucher, K. Kastella, and A. O. Hero, “Multitarget tracking using the joint multitarget probability density,” IEEE Trans. Aerosp. Electron. Syst., vol. 21, no. 4, pp. 1224–1245, 2005. B.-N. Vo, S. Singh, and A. Doucet, “Sequential Monte Carlo methods for multitarget filtering with random finite sets,” IEEE Trans. Aerosp. Electron. Syst., vol. 21, no. 4, pp. 1224–1245, 2005. D. P. Bertsekas, Dynamic Programming and Optimal Control, vol. 1, third edition. Athena Scientific, 2005. ——, Dynamic Programming and Optimal Control, vol. 2, third edition. Athena Scientific, 2007. A. Maguer, R. Dymond, M. Mazzi, S. Biagini, and S. Fioravanti, “SLITA: a new slim towed array for AUV applications,” in Acoustics’08, 2008, pp. 141–146. R. Goldhahn, P. Braca, and K. LePage, “Environmentally sensitive behaviours for collaborating autonomous underwater vehicles in multistatic surveillance networks,” NATO-STO Centre for Maritime Research and Experimentation, Tech. Rep. CMRE-FR-2013-012, Nov. 2013 (NATO UNCLASSIFIED).
STO-CMRE, La Spezia, Italy, Email: {braca/goldhahn/lepage}@cmre.nato.int. of Salerno, Fisciano, Italy, Email: {marano/vmatta}@unisa.it. ‡ University of Connecticut, Storrs CT, Email: [email protected].
† University
Abstract—Autonomous underwater vehicle (AUV) platforms are low-cost devices with respect to conventional detection and tracking systems for the purpose of anti-submarine warfare (ASW). Unfortunately, the increased level of manageability is often paid in terms of capabilities, e.g., limited speed and endurance, inferior sensor payloads, and so on. This work exploits two fundamental concepts aimed at filling the consequent performance gap. First, a multistatic network of AUVs is considered, where a smart and collaborative multi-sensor data fusion allows going beyond the individual sensors limitations. Then, we focus on the cognitive paradigm, where the single AUV units optimize their future actions (i.e., their path planning) in view of the final inference purpose of the network, and based on the evidence collected up to the present. A multistatic configuration of the platforms and the corresponding acoustic model are considered, and taken into account in order to derive a proper Bayesian model. Using the information contained in the Bayesian full posterior, cognitive detection and tracking algorithms are designed. Seeing them at work in practical scenarios shows the benefits of the cognitive network paradigm. Index Terms—Data fusion, Anti-submarine warfare, multistatic active sonar, Bayesian target tracking, underwater wireless sensor networks, autonomous underwater vehicles, cognitive systems, acoustic model.
I. I NTRODUCTION Nowadays, underwater autonomous systems have a wide range of applications, see e.g. [1]. Anti-submarine warfare (ASW), though the oldest, is still perhaps the most important. The specific ASW configuration considered in this work is that of a wireless sensor network working in a multistatic setup. Multistatic sonar systems for ASW differ from conventional monostatic active systems in that there are multiple receive nodes that are not co-located with the transmitters [2]–[4]. One advantage of this deployment is that the position of the receivers is not easily discernible by the target. In a multistatic system, the judicious placement of covert receivers allows for an increased probability of a favorable geometry, taking into account aspect-dependent target strength, which can dramatically increase the SNR, as well as the effective network coverage. Recently, autonomous underwater vehicles (AUVs) have been proposed as sensor nodes in multistatic surveillance networks due to their low cost, manoeuvrability, and ability to reconfigure their positions based on the observed data and mission objectives without human intervention. They may however be inferior to conventional ASW assets with respect to array aperture, speed, and endurance. The expected performance gap may be bridged by deploying multiple sensor
platforms, so as to form a collaborative network of smart autonomous vehicles. However, real-world applications are invariably characterized by a certain degree of variability (changes in the relative target-AUV position, in the environmental conditions, and so on), so that it is often desired to guarantee an adequate level of adaptation and reconfigurability of the ASW system. The above problem can be put in the more general framework of sensor management [5], [6] and cognitive systems [7], defined as the tasking, sequencing, and scheduling of a heterogeneous network of sensors to optimally achieve a prescribed goal. There are two approaches to sensor management, as it is called in much of the literature [8]: Task- or mission-driven and information-driven sensor management. The former chooses sensor actions based on a given performance metric or error quantity directly related to the mission objectives. Informationdriven sensor management chooses sensor placements and actions that maximize a measure of the information gain. Elaborating on these concepts, in this work we propose a method to design networks of cognitive AUVs, namely, of AUVs capable of planning their (future) actions i) based upon the available (present) knowledge, and ii) in view of the final goal of the network. We consider the problem of localizing a target in the ASW environment by means of a network of AUVs. Each AUV can plan its future navigation steps by choosing among a collection of predefined local paths. The best path is selected on-line according to some criterion optimizing the network inference performance, using the information currently available at all the AUVs. The basic elements for the path planning are as follows. Given a target position, the quality of the measurements collected by the AUVs strongly depends upon their spatial configuration, and from the environmental conditions. Otherwise stated, the spatial profile of the detection probability is a function of the target and AUV positions and of the environmental conditions. A key observation is that the target state (position, velocity and even its presence/absence) is unknown, and is indeed the object of the network inference. As a result, we must consider a statistical path planning, and the target tracking algorithm becomes essential not only for the original inference goal, but also for the path planning itself. We focus on optimal Bayesian tracking, and propose workable algorithms that take into account several important aspects, such as the geometry of the network, the acoustic propagation, the difficulty for a single AUV to discriminate port from starboard contacts, and the problem of target ap-
pearance/disappearance. The paper is organized as follows. In Sec. II we formalize the ASW tracking problem using AUVs in a bistatic configuration and describe the optimal Bayesian estimation procedure. Sec. III describes the autonomous cognitive behaviours of AUVs. Numerical experiments are reported in Sec. IV, and Sec. V draws the conclusions. II. M ULTI - SENSOR BAYESIAN TRACKING OF A B ERNOULLI RFS TARGET
Fig. 1. Sketch of the port-starboard ambiguity in the bistatic geometry.
A network of Ns AUVs monitoring a certain surveillance region is considered. The objective of the network is to estimate the absence or presence of a target and, in the latter case, its kinematic components at each time scan k. The target state Xk can be conveniently formalized as a Bernoulli random finite set (RFS) [9]–[11], where Xk = {∅} when the target is absent, or Xk = {xk } when the target is T present with xk = [xk , x˙ k , yk , y˙ k ] , where the two positions (p) T T are xk = [xk , yk ] and [x˙ k , y˙ k ] are the corresponding velocities. In this work, when the target is present, we assume a nearly constant velocity model [12]: xk = F k xk−1 + v k ,
(1)
where F k is the state transition matrix, and v k takes into account the target acceleration or unmodeled dynamics. At each time scan k a set of data, modeled as RFS, is observed by the sth AUV, defined by m
k,s Zk,s = {z k,s,i }i=1 ,
(2)
where mk,s is the number of measurements1 . Note that the current technology available for AUVs has some limitations, and one of the most significant is the port– starboard (PS) ambiguity, namely the impossibility for the AUV to discriminate port from starboard contacts. PS ambiguity complicates the detection and tracking algorithms and may severely degrade performance. In [14], [15] a Bayesian tracking approach is proposed to track the target state in presence of PS ambiguity and measurement origin uncertainty (MOU) [12]. Given the PS ambiguity problem, z k,s,i in (2) are just the contacts on the port side, since they form a sufficient statistic, because of the deterministic dependence of the starboard contacts on the port contacts [15]. In the following subsections we explain how the target state is statistically related to data collected by the network. A. Target-originated measurement with PS ambiguity In this subsection we briefly describe the target-originated measurement model with PS ambiguity. Considerthe geometry (x)
(y)
T
denote the of the PS problem in Fig. 1. Let sk = sk , sk source position at time scan k, while the sensor T array position (x) (y) and and its heading angle are denoted by pk = pk , pk hk , respectively. The sensor measures the bistatic range bk
1 In this work we use the RSF formalism for both target and data, however other equivalent choices are available, see for instance the discussion in [13].
from source to target to receiver and the bearing angle relative to the array heading θk . The non-ambiguous measurement is given by ⎡ ⎤ (p) (p) xk − pk + xk − sk + wkb bk ⎦,
=⎣ zk = y −py θk tan−1 xkk −pkx − hk + wkθ b
k2 wk 0 σb 0 , wk = ∼ N , 0 σθ2 0 wkθ where wkb and wkθ are the range and bearing noise, respectively. The PS ambiguous contacts have the same bistatic range measurement but two different bearing angles: θkP is the angle from the receiver to the target on the port side, and θkS is the angle from the receiver to the target on the starboard side. P T S S T = b , θ and z = b , θ are given by Then z P k k k k k k ⎧
yk − pyk ⎪ −1 P S P ⎪ ≥ hk , θ = θ , θ = −θ , if tan ⎪ k k k k ⎪ xk − pxk ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ θkS = θk , θkP = −θkS ,
if tan
−1
yk − pyk xk − pxk
< hk .
B. Measurement origin uncertainty model with target presence/absence Under the hypothesis of target absence (H0 ), the contacts are independent from each others and identically distributed with a known clutter distribution. The set density of the measurements under H0 , at time k and at the sth sensor can be formally written as: fk,s (Z |H0 ) = m! μc (m; λ)
m
f (c) (z i ),
(3)
i=1
where μc (m; λ) is the probability mass function (pmf) of the clutter cardinality (often assumed as a Poisson pmf with rate λ), and f (c) (z) is probability density function (pdf) of the clutter locations (typically modeled as uniform in the region of interest), see further details in [12], [13], [15]–[18]. When the target is present (hypothesis H1 ), it can be observed with a certain detection probability. This latter may vary dramatically over the surveillance region, and is in general a function of the receiver array parameters, source parameters, and environmental parameters as such as bottom scattering strength, water column and bottom sediment density and sound speed, and bottom and surface roughness and reflection loss. Numerical and/or closed-form acoustic propagation models
[19]–[21] are then used to calculate the predicted PD . We borrow from [22] an acoustic model to predict the detection probability of the target in any position in the surveillance region, taking into account the relevant environmental acoustic effects. This will be key for enhancing the performance of the tracking algorithm, as well as for the path planning optimization that is the main subject of this paper. The aforementioned acoustic model provides us with a (p) detection probability PD (xk , pk,s , sk ), which is a function (p) on the target position xk , of the receiver position pk,s and of the source position sk . In this work we assume a stationary (p) (p) source, then sk = s and PD (xk , pk,s , sk ) ≡ PD (xk , pk,s ). Since we assume that at most one target is present, all the other contacts are clutter, independent from the target’s state. Then, the set density of data under H1 is given by m
(p) f (c) (z i ) fk,s (Z |H1 ) = m! 1 − PD (xk , pk,s ) μc (m; λ) i=1
(p)
+ (m − 1)! PD (xk , pk,s )μc (m − 1; λ) m (t) × fk,s (z j ) f (c) (z i ), j=1
(4)
i=j
(t)
where fk,s (z) is the target-originated pdf with PS ambiguity, see details in [15]. In order to make explicit the relation between the target state and the data, eqs. (3) and (4) can be equivalently rewritten as: fk,s (Z |Xk = {∅} )
=
fk,s (Z |H0 ) ,
(5)
fk,s (Z |Xk = {xk } )
=
fk,s (Z |H1 ) .
(6)
C. Optimal Bayesian tracking In Bayesian tracking, the objective is to construct the posterior distribution of the target state Xk . To this aim, let us denote by Z1:k = [Z1 , Z2 , . . . , Zk ] the aggregate in time of the data up to time step k, where Zk = [Zk,1 , Zk,2 , . . . , Zk,Ns ] is in turn the aggregate, at time k, of the observations collected by all the Ns sensors. The posterior of the target state Xk can be thus written as: fk (Zk |Xk ) P (Xk |Z1:k−1 ) P (Xk |Z1:k ) = , (7) P (Zk |Z1:k−1 ) where P (Xk |Z1:k−1 ) is the prior at time k and P (Zk |Z1:k−1 ) is the scaling factor. Since the sensors are conditionally independent given the target state, the likelihood fk (Zk |Xk ) can be factorized as: fk (Zk |Xk ) =
Ns
fk,s (Zk,s |xk ) ,
(8)
s=1
where fk,s (Zk,s |Xk ) is the likelihood of the sth sensor at time k, given in eqs. (5)-(6). We assume that the AUVs have the ability to communicate to each other their local detections Zk,s and positions pk,s . Consequently each AUV has access to the joint likelihood (8). While in principle this knowledge is enough to build the posterior distribution (7), closed-form solutions are seldom
available in practice. As a consequence, one should typically resort to some form of approximation that are both accurate and computationally tractable. Specifically, in this work we focus on the powerful technique of particle filtering to obtain a numerically efficient approximation of (7), see [16], [23]– [25]. III. C OGNITIVE AUV PATH PLANNING In this section we present our solution to design cognitive AUVs. We explain how the AUVs can adapt their navigation strategy, based on the on-line incoming information Zk , in order to enhance the network tracking/localization capabilities. In order to design its path planning, the sth AUV, at time k, must decide where to go at time k + 1, namely, it must choose its next position pk+1,s . This is made by selecting a change of speed δvk,s ∈ V and a change of heading δhk,s ∈ H. The classes V and H model the constraints on the admissible policies. The AUV updated velocity and heading are accordingly given by: vk+1,s = vk,s + δvk,s , hk+1,s = hk,s + δhk,s , yielding a consequent variation in the position: cos (hk+1,s ) pk+1,s = pk,s + vk+1,s . sin (hk+1,s )
(9) (10)
(11)
As a result, the class of admissible positions at time k + 1 is a function of the starting point pk,s . This class will be denoted by S(pk,s ). By iterating this reasoning, from a starting point pk,s , a collection T (pk,s , L) of admissible trajectories τ of a given length L is generated, formally: τ = [pk+1 , pk+2 , . . . , pk+L ] ∈ T (pk,s , L). The cost associated to a given target position X and a given AUV position p will be measured by a cost function C(X, p), which is chosen in connection to the goal that the network tries to achieve. A first optimization strategy might be that of selecting the position pk+1,s that maximizes the average cost E[C(Xk+1 , pk+1,s )|Z1:k ], where the expectation has been evaluated given all the available knowledge contained in the observations collected by the network up to time k. However, this choice will typically lead to a myopic system. Indeed, since not all paths are admissible, optimizing by looking just one step ahead might then force the AUV to reach a very unfavorable region after some steps. In order to mitigate this effect, a clever choice of the position pk+1,s should consider a planning horizon L > 1. Following [7], we propose the minimization of the following additive cost-to-go function: ρ(τ |Z1:k ) = ρ(pk+1 , pk+2 , . . . , pk+L |Z1:k ) k+L = E C (Xn , pn ) Z1:k , (12) n=k+1
In the above formula, the expectation of the single summand is computed under the distribution P (Xn |Z1:k ), that is the
Accordingly, we set: ⎧ ⎨ 1 − PD x(p) , p C (X, p) = ⎩ 0
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(network) posterior distribution of the target position at time n given the data up to time k. The selection of the best position pk+1,s can be now performed as follows. The sth AUV finds the optimal path τ ∗ (pk,s ) estimated at time k as the solution of the following optimization problem: min
τ ∈T (pk,s ,L)
ρ(τ |Z1:k ).
(13)
We stress that, while, for a given trajectory τ , the function ρ(τ |Z1:k ) in (12) does not depend on the particular AUV, the estimated path τ ∗ (pk,s ) does. This is due to the search space where the optimization is performed, which is dependent on the initial AUV position pk,s . After having estimated the best path at time k, the sth AUV selects as next step pk+1,s only the first point of the optimal trajectory τ ∗ (pk,s ). At time k + 1, the next position pk+2,s is then selected by repeating the overall procedure, starting from the configuration pk+1,s chosen at time k. This means that the optimized path [pk+1,s , pk+2,s , . . . , pk+L,s ], effectively followed by the AUV, will be in general different from the path τ ∗ (pk,s ) estimated at time k. A. Path planning implementation In this subsection we summarize the main implementation details of the path planning described by (12) and (13). First, we must choose a proper cost function C(X, p), which of course, depends on the application. In the ASW context it depends on the kind of mission: submarine tracking, area clearance, etc. In this work, we propose a heuristic cost function, computationally efficient, which reflects the physics of the problem. Basically, we want to place the AUV in the best position in terms of target detection – not globally (too complicated) but locally, at this sensor and at this time.
if X = {x}, (14) if X = {∅}.
The cost-to-go function in (12) can be now computed by averaging the single summands C(Xn , pn ) under the posterior distribution P(Xn |Z1:k ). In particular, in the implementation of our algorithms, the pertinent posterior distributions are efficiently evaluated by using particle filtering, so that the expectations for computing the average costs are easily carried out by a summation over the particles. It remains to solve the optimization problem in (13). However, the search for the exact solution is typically not feasible. Several approximate solutions have been proposed, among them the most important ones being the dynamic programming strategies [6], [7], [26], [27]. Unfortunately, in our context dynamic programming is computationally too expensive given that the optimized strategy must be computed on-board of the AUV in a time interval that is typically less than a minute. For this reason we propose an ad hoc solution that is computationally tractable even locally by the AUV. We constrain the optimization problem to a smaller search space. The optimization space of navigation paths is constrained to be composed of trajectories at fixed speeds, that can be selected among few values (usually around 1-5 m/s), complying with the necessity to maintain a minimum speed while towing an array. The heading changes are forced to be constant during the planning horizon. The optimization space is then constituted by several smooth arcs2 , see further details in [29] and a graphical representation in Fig. 2. IV. C OMPUTER EXPERIMENTS In this section we show the benefit of using a cognitive network of AUVs (C-NET) with respect to a non cognitive network (NC-NET) in which AUVs are forced to be in predefined places or to move along predefined paths. Furthermore we compare the performance between myopic (L = 1, referred as CM-NET) and non myopic (L > 1) implementations of the cognitive strategy described in the previous sections. The scenario under test is the same of the Co-operative LittoraL ASW Behaviour 2013 (COLLAB13) experiment, conducted by NATO-STO Centre for Maritime Research and Experimentation in the Mediterranean Sea off the coast of Porto Venere, Italy from June 29 to July 07, 2013. Note that in this scenario the sound speed depends dramatically upon the depth of the surveyed area and exhibits typical summertime features. The clutter is assumed to be uniform. The signal to reverberation plus noise ratio (SRNR) is calculated using the ARTEMIS adiabatic normal mode propagation code using the bathymetry, sound speed profile, and environmental parameters measured during the COLLAB13 experiment. No target aspect dependency is included in the 2 The navigation of AUVs along smooth paths is highly desirable in order to avoid sharp manoeuvres that would deform the shape of the linear array towed by the AUV with large systematic errors in the data, see details in [28].
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Fig. 3. Target detection probability, PD [x, y] , pk,s , at time step k. AUV positions pk,s (white square ’’), source position (white diamond ’♦’) current target position (white ’×’). The AUV positions are given for the case of C-NET.
model, however it would be straightforward to take into account the target aspect. We are now ready to illustrate the results of the numerical experiments. Fig. 3 shows the target detection probability maps and positions of source, target and AUVs given by the C-NET behaviour. At the beginning, k = 1, the AUVs are located close to the source with a large coverage, in terms of detection probability, of the surveillance region. However, the target is sailing (from south to north) in the bottom corner, far away from AUVs and source, where the detection probability is low, and then, is well hidden. In particular, the target will be hidden for all its trajectory if the AUVs will not change their positions (as in the case of the NC-NET). In order to detect the target, the AUVs shall try to get close to it. Now, the crucial point is that they have only an imperfect information about the target navigation trajectory, embodied in the Bayesian posterior. The behaviour of the distributed Bayesian tracking for the case of C-NET, CM-NET and NC-NET, is reported in Figs. 4 and 5. In particular, we show the posterior of the target position in Fig. 4 and the posterior of the target presence in Fig. 5. The path decided by AUVs is reported in Fig. 6 as well as the target trajectory. The fact that the target is hiding matches with the behaviour of the algorithm, indeed from an initial half chance of having a target, see Fig. 5, the target presence probability decreases. However, the Bayesian tracking algorithm is able to recognize that if the target is present then it would be located in the most
west region, see Fig. 4 at k = 25, 50 for all the strategies. At this point we are able to recognize the effects of using the cognitive paradigm. AUVs in both C-NET and CM-NET cases try to explore and move towards the regions where the target could be located, or in other words they want to be active and hunt the target. This goal is accomplished and it is easy to recognize the benefit of using a cognitive strategy. When C-NET is in force the target presence is correctly declared around k = 50 − 70, earlier, as expected, than CM-NET. Interestingly, the NC-NET is not able to detect the target because the target always sails in the region where the PD , seen by AUVs in fixed positions (see Fig. 3(a) and 3(d)), is very low. V. C ONCLUSIONS AUVs offer low-cost ASW capability. However, their full potential is not currently available, as their superior multilateration requires intelligent positioning, meaning path-planning. The implied global “cognitive” problem is easy to pose but not really feasible in light of limited AUV computational resource and coordination ability. Hence we here propose an alternative that is local, light in algorithmic footprint – and seems to perform very well. A multistatic configuration of the platforms and the corresponding acoustic model are considered, and taken into account in a proper Bayesian model. A smart and collaborative multi-sensor data fusion is proposed. The focus is on the cognitive paradigm, where the single AUV units optimize their
(p)
Fig. 4. Posterior distribution of the target locations xk (red dots ’·’) and true target position (blue ’×’). The first row is referred to the C-NET, the second to CM-NET and the third to NC-NET. The distribution is numerically computed by using a particle filtering strategy.
path planning in view of the final inference purpose: Hunting the enemy submarine. The key element of this work is the cognitive path planning, based on the evidences collected up to the current time. Using the information contained in the Bayesian posterior, we show the benefit of using the cognitive paradigm in a practical scenario. Future work can include the study of a multi-target scenario, in which AUVs want to optimize their paths to detect and track more than a single target by using multi-objective optimization tools. Other interesting future directions involve the problem of collision/obstacle avoidance in the AUV cognitive algorithm, and to the interdependency of the target/AUV velocities.
VI. ACKNOWLEDGEMENTS This work has been funded by the NATO Allied Command Transformation (NATO-ACT) under the projects System Concepts for Littoral Surveillance and Maritime Situational Awareness. Peter Willett was supported by the Office of Naval Research under contract N000014- 13-1-0231.
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Fig. 5. Posterior probability of the target presence at time k for the case of C-NET, CM-NET and NC-NET.
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Fig. 6. AUV trajectories for the case of C-NET (blue square ’’), CM-NET (red square ’’), and AUV positions of NC-NET (black square ’’). Source position (yellow diamond ’♦’). Target trajectory (black dashed line ’−−’). Arrows indicate the directions of AUVs and target.
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