COORDINATED TRACKING OF AN ACOUSTIC SIGNAL BY A TEAM OF AUTONOMOUS UNDERWATER VEHICLES

COORDINATED TRACKING OF AN ACOUSTIC SIGNAL BY A TEAM OF AUTONOMOUS UNDERWATER VEHICLES Darren K. Maczka 1, Davide Spinello 1, Daniel J. Stilwell 1, Aditya S. Gadre 1, and Wayne L. Neu 2 1 Bradley Department of Electrical and Computer Engineering, Virginia Tech 2 Aerospace and Ocean Engineering Department, Virginia Tech * Phone: (540) 231-3204, Fax: (540) 231-3362, Email: stilwell@vt.edu We describe an approach to decentralized control and distributed data fusion that enables a team of autonomous underwater vehicles to cooperatively and autonomously localize and track a source of acoustic noise, and we report on field experiments that demonstrate the efficacy of our approach. A principal challenge of subsea coordination is the the extremely low-bandwidth acoustic communication channel that is available underwater. To address this challenge, and to enable coordination despite extremely limited communication, we have devised a new class of decentralized control algorithm that utilizes local models of the environment that are embedded onboard each AUV. Each AUV implements a centralized control law, but with locally estimated variables in the place of state variables that would normally be communicated between vehicles. We show both in theory and in successful field trials that surprisingly little communication is required to implement meaningful underwater vehicle coordination. INTRODUCTION Cooperative sensing of an object is one of the many proposed capabilities for mobile sensor networks. Mobility, coupled with data fusion algorithms and motion control algorithms, enables a team of mobile sensors to adjust their relative geometry in real-time to enhance sensing performance. These ideas have been applied to a variety of applications, including target tracking [1, 2, 3, 4], formation and coverage control [5, 6, 7, 8], and environmental tracking and monitoring [9, 10, 11]. We apply these ideas to the problem of tracking and maneuvering relative to an acoustic source using a team of autonomous underwater vehicles equipped with towed hydrophone arrays. Unlike in-air applications which utilize radio-frequency communications, underwater applications are challenged by the severely bandwidth-limited acoustic subsea communication channel. We address the challenge of limited communication by introducing a new class of data fusion and motion control algorithms that are well-suited for applications in which communication between sensor nodes is infrequent. We describe initial results from field trials in which a team of autonomous underwater vehicles (AUVs) cooperatively localize a source of acoustic noise and maneuver to minimize the joint localization error of the acoustic source. This is equivalent to directing 1

the AUVs to locations at which their geometry relative to the acoustic source yields improved localization performance. Each AUV tows a small hydrophone array that measures the bearing angle between itself and the acoustic source. To estimate the location of the acoustic source, we utilize a generalized extended Kalman filter for which local estimates from different sensors can be easily fused. Although similar demonstrations have been performed with system in air, we successfully demonstrate that our approach to decentralized control yields significantly reduced communication requirements and is well-suited to subsea applications. In works that address control of sensor motion in mobile sensor networks, the estimation problem is commonly solved by assuming that the observation noise is independent of the process state, see for example [1, 12, 3, 4]. However, as pointed out in [13], this assumption is not realistic when bearing-only sensors are employed. In order to account for state dependent measurement noise, we compute acoustic source location estimates using a generalized extended Kalman filter that is proposed in [14]. Cooperative object localization with limited communication is addressed in [4] by coupling estimation and motion control algorithms with consensus filters in order to achieve asymptotic agreement between agents. A similar idea appears in [9] for environmental tracking applications. In this work we address constrained underwater communication due to low communication bandwidth by considering a new class of distributed motion control algorithms that were developed to operate with only occasional communication between vehicles. To enable coordination despite limited communication, we embed a local observer on board each AUV to estimate states of the other AUVs in the team. The local observer used herein generalizes the observer proposed in [15] to the case of sensor networks with limited communication between vehicles. Each AUV simultaneously implements a centralized control law, but with locally estimated variables in the place of state variables that would be communicated directly from other AUVs in a centralized implementation. Asymptotic convergence of the estimators allows for the implementation of a system which is asymptotically equivalent to the centralized one proposed in [1]. PLATFORMS AND SENSORS Virginia Tech 475 AUV Experiments are conducted using the Virginia Tech 475 autonomous underwater vehicle (AUV). The 475 AUV is a small, low-cost, yet fully field-deployable AUV that is utilized for a wide variety of experimental activities and demonstrations. It costs only $10,000 in components and machining labor, and has hosted a variety of mission sensors and mission software modules that were developed by researchers at Virginia Tech and at other institutions. Specifications of the 475 are listed in Table 1. The 475 was designed to facilitate rapid integration of new payloads. Hardpoints on the bottom of the AUV are available for mounting external payloads, and a watertight bulkhead connector is also available to provide regulated power and a bi-directional data interface to the payload. The AUV uses the WHOI micromodem for communication and ranging. The syn- 2

chronous transmission feature of the modem enables an AUV to compute the range to a node in the network whenever the AUV receives a data packet from that node. For the experimental activities described herein, navigation is accomplished by ranging between AUVs using a distributed navigation filter. Parameter Length Diameter Mass Propulsion/Control CPU/Software Communications Table 1. Specification Specification of the 475 AUV. 34 inches 4.75 inches 18.3 lbs. Brushless direct-drive DC motor and four independently controlled flaps x86 compatible; LINUX OS, database server architecture utilizing TCP/IP client/server connections 900MHz RF modem; Wi-Fi with external antenna; WHOI micromodem for acoustic communication Navigation GPS, transponder-based acoustic navigation; timesynchronized acoustic navigation for AUV to AUV ranging and AUV to chase boat ranging; gyro-stabilized dead reckoning Endurance 8+ hours at 3 knots Bearing sensor A custom towed hydrophone array was designed and fabricated to support experiments in distributed data fusion and decentralized control. Shown in Figure 2, the towed array is a uniform linear array consisting of eight hydrophone transducer elements enclosed in a streamlined package. As the intent was to tow the array with the existing Virginia Tech 475 AUVs, a primary design constraint was the physical size and the hydrodynamic efficiency of the array. To meet these requirements, the array was designed to be as small as possible while still maintaining neutral buoyancy and acoustic transparency. Onboard electronics includes an Ethernet interface for communicating with the AUV, analog signal conditioning, frequency shifting(mixing), and data acquisition. No data processing occurs on the array hardware; this task is offloaded to the host vehicle. The software written for the host vehicle accepts raw data transmitted over Ethernet and processes it to identify a usable signal. Once a usable signal has been identified, a beamforming algorithm extracts bearing information, which is then fed through a generalized extended Kalman filter [16]. The output of the filter is made available to other processes for tracking and control. Bearing measurements of an acoustic source by the towed-array are most accurate when the acoustic source is broadside to the sensor, and accuracy becomes increasingly poor as the acoustic source appears closer to endfire. In other words, the noise statistics of the 3

Figure 1. Virginia Tech 475 AUV. Figure 2. Virginia Tech 475 AUV with towed hydrophone array. 4

Figure 3. Virginia Tech 475 AUV with hydrophone mount. bearing sensor are a function of the bearing angle. While this fact is well-known, it plays an important role in our work, and we briefly present a noise model for the bearing sensor. As depicted in Figure 4, the position of the acoustic source is denoted x and the position of sensor i is denoted q i. The heading angle of sensor i, as might be measured by a magnetic compass, is denoted ψ i. The vectors e N and e E are the north and east basis vectors, respectively. Each sensor obtains a bearing measurement to the source, denoted z i and defined by z i = h(x, q i, ψ i ) + v i (1) h(x, q i, ψ i ) = γ(x, q i, ψ i ) π 2 where γ(x, q i, ψ i ) is the relative angle of the sensor with respect to the source, as shown in Figure 4. The bearing angle measured by the sensor is zero when the acoustic source is broadside to the sensor, thus the term π 2 in (2). The sensor noise v i is zero mean Gaussian, v i N(0, σ i (x, q i, ψ i )) As the sensor obtains discrete-time measurements, we further assume that each sample v i [k] is independent. Of particular interest is that the covariance of the measurement noise σ i is dependent on the state of the sensor and the acoustic source. For a uniform linear acoustic array it is shown in [17] that σ i = E { v i v i } = 5 κg cos 2 h (2) (3)

where κ is a constant depending on physical parameters of the sensor array and g is the inverse of the signal to noise ratio which is dependent on the distance between the acoustic source and the sensor, see [1] source x q i γ i β i ψ i x e E q i Figure 4. Geometric configuration of a bearing-only sensor and acoustic source at time instance k e N Generalized extended Kalman filter for state-dependent sensor noise The objective is to estimate the source position x[k] using time-varying measurements z i [k]. Typically this class of source tracking problems is solved by applying an extended Kalman filter, see for example [1] and [18], which assumes that the measurement noise is independent from the state of the system. However, in our case the sensor noise statistics are dependent on the state of the system, which violates the assumptions required for the Kalman filter. To correctly address the state-dependent noise in the measurements, we utilize a generalized extended Kalman filter described in [14] to generate an estimate ˆx i and the covariance P i. As with the extended Kalman filter, the modified algorithm consists of prediction and update steps. The prediction steps are identical to a standard Kalman filter. The local state and covariance update equations are ˆx i [k k] = ˆx i [k k 1] [ P 1 i [k k 1] + R i [k] ] 1 si [k] [ s i [k] = ζ i x h i + 1 ( ) ] 1 ζ2 i x σ i σ i 2σ σ i [ 1 R i [k] = x h i x h i σ i + ζ i ( ) x h i x σ i + x σ i x h i 2σ 2 i + 1 4σ 2 i ˆx i [k k 1] ( ζ 2 i + 1 ) ] x σ i x σ i σ i ln σ i ˆx i [k k 1] (4a) (4b) (4c) 6

P 1 i [k k] = P 1 i [k k 1] + U i [k] (4d) [ 1 U i [k] = x h i x h i + 1 ] σ i 2σi 2 x σ i x σ i (4e) ˆx i [k k 1] where ζ i = z i h i. Note that if σ i did not depend on x then the gradient term x σ would be zero and (4) would reduce to the standard extended Kalman filter update equations. Since our nonlinear control formulation involves computing gradients of the covariance with respect to the system state, using a filter that correctly treats state-dependent noise is shown to make a significant difference in the performance of the closed-loop system. Data Fusion Note that a generalized extended Kalman filter (4) is implemented on each sensor. Whenever a sensor receives data from another sensor, the external information is fused to obtain a source position estimate that accounts for the shared data. In [16] data fusion equations are derived by considering the joint probability distribution of measurements taken from different sensors. Using the maximum likelihood approach one obtains equations analogous to (4) that account for shared measurements and predictions. In practice, unknown and unequal biases in the sensors measurements, along with long periods of no communication, restrict the utility of a rigorously developed approach to data fusion based on joint likelihood functions. Instead, we have found that consensus algorithms work very well in practice for the data fusion problem addressed herein. Consensus algorithms represent a wide class of weighted averaging protocols whose asymptotic behaviors are well-known in both deterministic and stochastic settings (see, for example, [19, 20]). Indeed, we find that a true average of estimates is appropriate for our measurements. Let I i [k] be the set of indices of all sensors that communicate with vehicle i at time k, and I i [k] the cardinality of the set I i [k]. Also note that i I i [k], for all k. The fused estimate of the source state for vehicle i is obtained as ˆx f i[k k] = 1 I i [k] j I i [k] ˆx j [k k] (5) Note that since i I i [k] the operation on the right-hand side of (5) is always well defined. In particular, if at time k vehicle i does not receive any estimate then ˆx f i[k k] = ˆx i [k k], where ˆx i [k k] is given by (4). Time-varying network topology Since communication between vehicles is sparse, the corresponding network topology is always disconnected. Networks that are not connected in a frozen-time sense pose technical challenges for analysis and design of decentralized control systems. To address this challenge, we have shown that it is sufficient that a formally defined average network be connected if the network switches sufficiently fast among topologies. The required switching rate defines our notion of a network time-constant. For a given mission specification, 7

we can compute the network time-constant as a function of the system dynamics and determine how fast agents must communicate. These ideas are discussed more fully and applied to multi-vehicle coordination problems in [20, 21], among others. OBSERVER-BASED DECENTRALIZED CONTROL To enable coordination with limited communication, we embed a local observer onboard each AUV to estimate the location, sensor output, and target estimate of every other AUV. Then each AUV independently implements the same centralized control law, but with estimated variables in the place of state variables that would be communicated directly from other vehicles in a centralized implementation. In general, this approach presents significant technical challenges because there is no separation principle for decentralized systems. In other words, the closed-loop system composed of observers and control laws embedded on each AUV may not be stable even if the control law and the observer error dynamics are each individually stable. Thus our principal contribution is to develop methods for simultaneously designing control laws and distributed observers with guaranteed performance properties (e.g., stability). We consider an acoustic source modeled by the dynamics x[k + 1] = f s (x[k]) + v(k) where v(k) N(0, σ). The team of N mobile sensors (e.g., AUVs) is modeled by the dynamics q i [k + 1] = f i (q i [k], u i [k]), i {1,..., N} where u i is the control signal for sensor i. We make no assumptions on the fidelity of the sensor (AUV) motion model within our general framework, although we have found that a simple point-mass model where the states consist of position variables and perhaps velocity variables is useful. We have also employed a kinematic model that explicitly accounts for the heading of the sensor for situations where heading is important. The state of the entire system, consisting of N sensors and an acoustic source, is represented q = [q s, q 1,..., q N] Each mobile sensor maintains an estimate of the entire system state, denoted ˆq i. Note that ˆq i is an estimate of q and not an estimate of q i. For notational convenience, we collect all of the state estimates, the outputs, and the inputs (control signals) into vectors, ˆq = [ˆq 1,..., ˆq N] g(ˆq) = [g(ˆq 1 ),..., g(ˆq N ) ] u(ˆq) = [u 1 (ˆq 1 ),..., u(ˆq N ) ] 8

Our task is now to implement control signals u i (ˆq i ) on each mobile sensor so that the system behaves as desired, and to implement observers on each mobile sensor so that each estimate ˆq i asymptotically agrees with all other estimates and the true system state q. Our approach is to simultaneously minimize an objective function J(q) corresponding to the desired behavior of the team of sensors and an additional quadratic term corresponding to the observers estimation error. Thus we seek approaches that minimize the functional F (q, ˆq, k) = J(ˆq) + 1 2 (g(ˆq) g(q)) A[k](g(ˆq) g(q)) (6) The matrix A[k] models the intermittent communication that occurs between mobile sensors. When there is no communication between two sensors, the corresponding terms in A is zero and right-most term in (6) has no effect on the corresponding components of ˆq. When there is communication, the corresponding terms in A are unity. The desired behavior of the team is encoded in the functional J(q). A variety of behaviors are possible, including trajectory (or location) estimate error minimization, formation flying, and multiple source discrimination, among many others. In this paper, we consider minimization of the trajectory estimation error and formation flying relative to the estimated location of the acoustic source. Gradient descent Minimization of (6) using gradient descent is discussed in detail in [22], and we provide only an outline here. Each mobile sensor computes the control signal u(q) = Γ q J(q) using its estimate of the system state ˆq, where Γ is diagonal matrix of control gains. Sensor i uses only the component u i (q) of u(q) for its local control decision, but uses the remaining components of u(q) to update its estimate of the state of other mobile sensors in the absence of communication from other sensors. When communicated information from other sensors is available, the same gradient approach yields ˆq[k + 1] = ˆq[k] + T (u(q[k]) + K(ˆq[k])A[k](g(ˆq[k]) 1 N g(q[k]))) where the observer gain K(q) is derived in [22], T is the period between control updates, and 1 N is the N-vector with all entries unity. Note that the gradient approach yields a standard observer structure when information from other sensors is available. Receding horizon nonlinear optimization Gradient descent works well in many circumstances and has a small computational requirement. However, state trajectories resulting from gradient descent can become trapped in local minima of the objective function F, and these local minima may represent physically undesirable solutions. To address the phenomena of local minima, we adopt a receding horizon control (RHC) approach [23]. RHC is a control technique in which the 9

finite-time optimal controls are computed for the finite-time horizon H. The optimal control is usually recomputed periodically with a period that is much shorter than H. The objective function is defined V H (x, u, k) = k+h i=k L(x[i], u[i]) + Q(x[k + H]) (7) where L is the integral cost and Q defines the terminal cost. At each control time k an optimality problem is solved which generates a control sequence u = {u [k],, u [k + H]} that minimizes the objective function (7) and obtains the solution V H(x, u) = min {V H (x, u) u U, x X } (8) where U and X are sets that describe constraints on the input and state, respectively. This finite-time constrained optimization problem can be solved numerically using a variety of methods, including the Nonmonotone Spectral Projected Gradient Method [24] RESULTS Field Experiments Gradient descent was demonstrated in a field trail at Claytor Lake, a 4,500 acre hydroelectric impoundment of the New River near Dublin, VA. For this initial demonstration, conducted summer 2008, a static acoustic source was utilized, and the objective of the mobile sensors was to maneuver so that the localization error of the acoustic source was minimized. That is, we chose J(q) in (6) to be J(q) = det U 1 (q) (9) where U 1 is defined in (4e) and q consists of position variables since it is assumed that the acoustic source is stationary. Two Virginia Tech 475 AUVs were equipped with towed array sensors to measure the relative bearing to an acoustic source located on a support craft. The vehicles were able to communicate with one another using a WHOI micromodem. The goal of the experiments was to demonstrate that with only intermittent communication the two vehicles could control trajectories to jointly minimize the cost function (6). Intuition along with inspection of the cost function suggest that at steady-state, both vehicles should circle the estimated location of the acoustic source so that their sensors are broadside to the acoustic source and so that both vehicles are separated by π/2 around the circle. The results of the demonstration are shown in Figure 5. The acoustic source was mounted on a chase boat that moved throughout the experiment due to wind. Its position was within the area enclosed by the solid circle for the duration of the mission. The two vehicles start at the position denoted by the solid triangle and are commanded on an initial straight line path. At the position denoted by the solid square the towed array sensor reports an 10

initial bearing measurement to the acoustic source, and the gradient descent controller is activated. The curved path around the source position results from the controller driving the relative bearing of each vehicle to π/2. The AUVs choose to do this because the bearing sensor works best when the acoustic source is broadside to the sensor. The dotted lines are visual aids that show same-time positions for the AUVs. It is evident that the velocity of red AUV increases while the velocity of blue AUV decreases, and that the relative separation between vehicles increases over the length of the mission. The commanded speed of each vehicle is plotted in Figure 6. The dots in Figure 6 denote times when communication allowed a fusion event to occur. Blue dots indicate that blue AUV received a data packet, while red dots indicate that red AUV receives a data packet. Speed commands are in the range of 0.8 m/s to 1.5 m/s. Several fusion events occur in the beginning the mission, which cause each local observer to achieve a reasonable level of agreement. There are few fusion events in the middle of the mission, but each local observer maintains an estimate of the evolving state of the other vehicle, and the combined actions of each vehicle proceed as expected. It is interesting to note that while no fusion events occur towards the end of the mission, the commanded speeds begin fluctuating to slow the sensor separation based on the information inferred by the local observers which indicated that the vehicles are approaching their optimal positions. The magnitude of the acoustic source estimation error is plotted in Figure 7, which shows that the local position estimates converge when fusion events occur. During a period of no communication in the middle of the experiment the two estimates diverge from each other which is expected due to unmodeled biases present in each sensor. The vehicles achieve a desired relative bearing and geometry with respect to the target that minimizes the joint localization error of the acoustic source. Importantly, we do not explicitly command one vehicle to speed up while the other slows down. The behavior of each vehicle is an emergent consequence of making joint decisions, aided by a local observer when no communication occurs, to minimize the cost function (6). 11

Figure 5 Vehicle trajectories; triangles denotes initial condition, squares denotes location of vehicles when bearing angle is first measured, the black circle indicates true position of acoustic source throughout the experiment Figure 6. Speed of AUVs during demonstration shown in Figure 5. 12

Figure 7 Agreement between AUVs on the location estimate of the acoustic source during demonstration shown in Figure 5. 13

Receding horizon control example Receding horizon optimal control has been implemented in simulation with the dual objectives of estimating the trajectory of a moving acoustic source and formation flying relative to the acoustic source. Initially, two AUVs utilize a similar objective function as in (9). The functionals in the receding horizon objective function (7) are L(q, u) = c 1 det U 1 (q) (10) Q(q) = c 2 det U 1 (q) (11) where c 1 and c 2 are constants. Once an initial maneuver to reduce trajectory estimation error is complete, the AUVs utilize a new objective function that enables them to perform formation flying relative to the moving acoustic source. Distributed trajectory estimation continues during formation flying, although the AUVs no longer maneuver explicitly to decrease estimation error. During formation flying, the functionals in the receding horizon objective functionals encode the distance between two AUVs, the orientation of the formation with respect to the velocity vector of the acoustic source, and the relative angle between the average velocity of the AUVs and the velocity of the acoustic source. The simulation results show the trajectory of two sensors and a single acoustic source over a time period of three minutes. During the first 60 seconds of the simulation the sensors use the estimation covariance objective function to improve estimation performance. The determinant of estimation error covariance is shown in Figure 8(b). The initial maneuver, in which the AUVs maneuver to decrease estimation error, appears in Figure 8(a). We see that the sensors maneuver to simultaneously increase their relative spacing with respect to the acoustic source and orient themselves so that the acoustic source is broadside to the sensor. After 60 seconds the controller switches to the formation objective as shown in Figure 9. The objectives are (1) the distance between two AUVs is 20m, the orientation of the formation with respect to the velocity of the acoustic source is 0, and the relative angle between the average velocity of the AUVs and the velocity of the acoustic source is π/2. The choices causes the AUVs to maneuver away at a right angle from the estimated path of the acoustic source. For the purposes of illustration, the acoustic sources changes direction during the time period between Figures 9 (b) and (c). Throughout the simulation, we assume that each AUV successfully transmits a data packet ever 20 seconds. 14

200 150 source sensor start location 1 0.9 0.8 0.7 0.6 y (m) 100 50 0.5 0.4 0.3 0 0 50 100 150 200 250 300 350 x (m) 0.2 0.1 0 0 10 20 30 40 50 60 Figure 8 Simulation of two AUVs maneuvering relative to a moving acoustic source; (a) initial maneuver to minimize trajectory estimation error; (b) determinant of estimation error covariance with respect to time. 15

200 source sensor sensor start location 200 150 150 y (m) 100 y (m) 100 50 50 0 0 0 50 100 150 200 250 300 350 x (m) (a) 0 50 100 150 200 250 300 350 x (m) (b) 200 200 150 150 y (m) 100 y (m) 100 50 50 0 0 0 50 100 150 200 250 300 350 x (m) (c) 0 50 100 150 200 250 300 350 x (m) (d) Figure 9 Continuation of simulation shown in Figure 8 where two AUVs have switched to formation control relative to a moving acoustic source; (a) t = 60 seconds, (b) t = 90 seconds, (c) t = 120 seconds, (d) t = 150 seconds. 16

ACKNOWLEDGMENT The authors are extremely grateful support of the Office of Naval Research via grant N000140710434. REFERENCES [1] T. H. Chung, J. W. Burdick, and R. M. Murray, Decentralized motion control of mobile sensing agents in a network, in Proceedings of the IEEE International Conference on Robotics and Automation, Orlando, Florida, May 2006. [2] T. H. Chung, V. Gupta, J. W. Burdick, and R. M. Murray, On a decentralized active sensing strategy using mobile sensor platforms in a network, in Proceedings of the IEEE conference on Decision and Control, Paradise Island, Bahamas, December 2004. [3] S. Martínez and F. Bullo, Optimal sensor placement and motion coordination for target tracking, Automatica, vol. 42, no. 4, pp. 661 668, 2006. [4] P. Yang, R. A. Freeman, and K. M. Lynch, Distributed cooperative active sensing using consensus filters, in Proceedings of the IEEE International Conference on Robotics and Automation, Roma, Italy, Feb. 2007. [5] C. Belta and V. Kumar, Abstraction and control for groups of robots, IEEE Transactions on Robotics, vol. 20, no. 5, pp. 865 875, October 2004. [6] J. Cortés, S. Martínez, T. Karatas, and F. Bullo, Coverage control for mobile sensing networks, IEEE Transactions on Robotics and Automation, vol. 20, no. 2, pp. 243 255, 2004. [7] J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465 1476, September 2004. [8] R. A. Freeman, P. Yang, and K. M. Lynch, Distributed estimation and control of swarm formation statistics, in Proceedings of the American Control Conference, Minneapolis, Minnesota USA, June 14-16 2006, pp. 749 755. [9] M. Porfiri, D. G. Roberson, and D. J. Stilwell, Tracking and formation control of multiple autonomous agents: A two-level consensus approach, Automatica, vol. 43, no. 8, pp. 1318 1328, 2007. [10] S. Simic and S. Sastry, Distributed environmental monitoring using random sensor networks, in Proceeding of the 2nd International Workshop on Information Processing in Sensor Networks, Palo Alto, CA, 2003, pp. 582 592. [11] S. Susca, S. Martínez, and F. Bullo, Monitoring environmental boundaries with a robotic sensor network, in Proceedings of the American Control Conference, 2006, pp. 2072 2077. 17

[12] A. Farina, Target tracking with bearings-only measurements, Signal Processing, vol. 78, pp. 61 78, 1999. [13] A. Logothetis, A. Isaksson, and R. J. Evans, An information theoretic approach to observer path design for bearings-only tracking, in Proceedings of the 36th Conference on Decision and Control, San Diego, California, Dec. 1997, pp. 3132 3137. [14] D. Spinello and D. J. Stilwell, Nonlinear estimation with state-dependent Gaussian observation noise, submitted for publication. [15] K. Shimizu, Nonlinear state observers by the gradient descent method, Anchorage, Alaska, USA, pp. 616 622, September 25-27 2000. [16] D. Spinello and D. J. Stilwell, Nonlinear estimation with state-dependent gaussian observation noise, Virginia Polytechnic Institute and State University, Tech. Rep., 2008. [Online]. Available: http://www.unmanned.vt.edu/discovery/reports.html [17] A. Gadre, M. Roan, and D. J. Stilwell, Sensor error model for a uniform linear array, Virginia Polytechnic Institute and State University, Tech. Rep., 2008. [Online]. Available: http://www.unmanned.vt.edu/discovery/reports.html [18] P. Yang, R. Freeman, and K. Lynch, Distributed cooperative active sensing using consensus filters, in Proc. IEEE International Conference on Robotics and Automation, 2007, pp. 405 410. [19] W. Ren, R. Beard, and E. Atkins, A survey of consensus problems in multi-agent coordination, in Proc. American Control Conference the 2005, 2005, pp. 1859 1864 vol. 3. [20] M. Porfiri and D. Stilwell, Consensus seeking over random weighted directed graphs, vol. 52, no. 9, pp. 1767 1773, 2007. [21] M. Porfiri, D. G. Roberson, and D. J. Stilwell, Fast switching analysis of linear switched systems using exponential splitting, SIAM Journal of Control and Optimization, vol. 47, no. 5, p. 2582 2597, 2008. [22] A. S. Gadre, D. K. Maczka, D. Spinello, B. R. McCarter, D. J. Stilwell, W. L. Neu, M. J. Roan, and J. H. Hennage, Cooperative localization of an acoustic source using towed hydrophone arrays, in Proc. IEEE Workshop on Autonomous Underwater Vehicles, 2008. [23] H. Michalska and D. Q. Mayne, Robust receding horizon control of constrained nonlinear systems, vol. 38, no. 11, pp. 1623 1633, 1993. [24] E. G. Birgin, J. M. Martínez, and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, vol. 10, no. 4, pp. 1196 1211, 2000. [Online]. Available: http://link.aip.org/link/?sje/10/1196/1 18