Cooperative Localization of an Acoustic Sourceusing Towed Hydrophone Arrays

Aditya S. Gadre, Darren K. Maczka, Davide Spinello, Brian R. McCarter,Daniel J. Stilwell, Wayne Neu, Michael J. Roan, John B. Hennage

Virginia Polytechnic Institute and State UniversityBlacksburg, VA 24061

Email: {agadre,dmaczka,dspinell,mccarter,stilwell,neu,mroan,jhennage}@vt.edu

Abstract—We describe field experiments in which a teamof autonomous underwater vehicles cooperatively localize anacoustic source. The team implements a data fusion algorithm toenhance the localization performance of each individual vehicleand implements a decentralized motion control algorithm so thateach vehicle maneuvers to minimize the joint localization errorof the acoustic source. Each autonomous underwater vehicle isequipped with a custom-designed towed hydrophone array thatmeasures the bearing angle between the array and the acousticsource. The noise statistics of the hydrophone arrays are state-dependent, and a generalized Kalman filter that accounts for thestate-dependant measurement noise is utilized for localization.

I. INTRODUCTION

Cooperative sensing of an object is one of the manyproposed capabilities for mobile sensor networks. Mobility,coupled with data fusion algorithms and motion control al-gorithms, enables a team of mobile sensors to adjust theirrelative geometry in real-time to enhance sensing performance.These ideas have been applied to a variety of applications,including target tracking [1]–[4], formation and coveragecontrol [5]–[8], and environmental tracking and monitoring[9]–[11]. We apply these ideas to the problem of localizingan acoustic source using a team of autonomous underwatervehicles equipped with towed hydrophone arrays. Unlike in-airapplications which utilize radio-frequency communications,underwater application is often limited by severe communi-cation bandwidth limitations. We address the challenge oflimited communication by utilizing data fusion and motioncontrol algorithms that are well-suited for applications inwhich communication between sensor nodes is infrequent.

We describe field experiments in which a team of au-tonomous underwater vehicles (AUVs) cooperatively localizean acoustic source and maneuver to minimize the joint lo-calization error of the source. This is equivalent to directingthe AUVs to locations at which their geometry relative tothe acoustic source yields improved localization performance.Each AUV tows a small hydrophone array that measuresthe bearing angle between itself and the acoustic source. Toestimate the source location, we utilize a generalized extendedKalman filter for which local estimates from different sensorscan be easily fused. Motion control is accomplished using agradient descent algorithm that adjusts the relative geometryof the sensor network to locally maximize information gain.

In works that address control of sensor motion in mobilesensor networks, the estimation problem is commonly solvedby assuming that the observation noise is independent of theprocess state, see for example [1], [3], [4], [12]. However,as pointed out in [13], this assumption is not realistic whenbearing-only sensors are employed. In order to account forstate dependent measurement noise, we compute acousticsource location estimates using a generalized extended Kalmanfilter that is proposed in [14].

Cooperative object localization with limited communica-tion is addressed in [4] by coupling estimation and motioncontrol algorithms with consensus filters in order to achieveasymptotic agreement between agents. A similar idea appearsin [9] for environmental tracking applications. In this workwe address constrained underwater communication due tolow communication bandwidth by considering a new class ofdistributed motion control algorithms that were developed tooperate with only occasional communication between vehicles.To enable coordination despite limited communication, weembed a local observer on board each AUV to estimate statesof the other AUVs in the team. The local observer used hereingeneralizes the observer proposed in [15] to the case of sensornetworks with limited communication between vehicles. EachAUV implements a centralized gradient-based control law, butwith locally estimated variables in the place of state variablesthat would be communicated directly from other AUVs ina centralized implementation. Asymptotic convergence of theestimators allows for the implementation of a system which isasymptotically equivalent to the centralized one proposed in[1].

This paper is organized as follows. In section II we describehardware details of the towed array that was designed andbuilt at Virginia Tech. We discuss details of the estimation andcontrol algorithm in sections III and IV and conclude with theresults of field trials presented in section V.

II. TOWED ARRAY DESIGN ELEMENTS AND LIMITATIONS

The Virginia Tech Towed array is a uniform linear array ofeight hydrophone transducer elements enclosed in a stream-lined package. It is assumed that the distance of the acousticsource from the array is much larger than the length of thearray. This assumption allows the use of the parallel linetransmission model for acoustic signals arriving at the array

Fig. 1. The Virginia Tech Towed Array is a uniform linear array of eighthydrophone elements contained in a streamline package suitable for towingby a small AUV.

Fig. 2. The Virginia Tech 475 AUV configured to tow the hydrophone array.

elements. Due to the spacing between transducer elements, thesignal phase sampled at each transducer is shifted incremen-tally along the array. We use the delay-and-sum beamformingtechnique in this application to extract the incremental phaseshift and compute the bearing to the acoustic source [16].

As the intent was to tow the array with the existing VirginiaTech 475 AUVs, a primary design constraint was the physicalsize and the hydrodynamic efficiency of the array. To meetthese requirements, the array was designed to be as small aspossible while still maintaining neutral buoyancy and acoustictransparency.

A. Mechanical

A six conductor twisted pair shielded cable is used totow the array, supply power from the host vehicle and relaydata between the array and the host vehicle. High gaugeconductors are selected to minimize the total diameter andweight of the cable. The electronics and transducer assemblyis housed in a 1.5 inch outer diameter polycarbonate tube.Eight hydrophones are arranged in a uniform linear arraywith a inter-hydrophone separation of 20 mm. The towingcable enters through a hemispherical nose cone and threefixed fins on the conical tail help stabilize the flight of thesensor. Both nose and tail cones are equipped with O-ringseals and can be removed for easy access to the interior.The empty volume of the tube is filled with high viscositytechnical grade mineral oil with specific gravity of 0.89. Theoil and the housing material are chosen to provide acoustic

LPF

SPIADC

PGA

x

transducer

global demodulationsinusoid

PA

SPI BUS

PSoC Microcontroller

Fig. 3. Signal path of a single channel. A high impedance pre-amplifier (PA)is the first stage of signal processing. The amplified signal is demodulated bya global sinusoid available to all channels. The PGA, LPF, ADC and SPIinterface are implemented on a Cypress PSoC mixed signal micro-controller.

transparency between the hydrophones and exterior water tominimize acoustic distortion. High density foam is added tothe assembly to achieve neutral buoyancy. Since the cableweight pulls the nose from below the horizontal, the centerof gravity is placed slightly aft of the center of buoyancy sothat the array flies level when towed.

B. Hydrophones

Since it is desirable to keep the outer diameter of thetowed array as small as possible, very compact hydrophoneswere selected. The array uses SQ35 broadband hydrophones,manufactured by Sensor Technology Limited [17]. These are13mm long cylindrical, 15mm diameter, omni directionalhydrophones, sensitive to frequencies from 5Hz up to 100kHz.Small MMCX connectors are crimped onto the ends of the hy-drophone leads to provide easy integration with the electronicsand ensure a shielded signal path from transducer elements tothe high impedance amplifier inputs.

C. Electronics

The electronics are divided into two subsystems: digitalcommunication (Ethernet interface) and analog conditioning

and data acquisition (DAQ). Each subsystem is implementedon a custom printed circuit board,

The two subsystems communicate with each other via a6Mhz SPI bus and two dedicated handshaking lines. Theprimary function of the digital communication board is toestablish and maintain an Ethernet connection to the host(AUV) and act as a bridge between the host and the DAQboard. This functionality is built around a 16-bit PIC micro-controller (PIC24HJ256GP206) and an Ethernet controller(ENC28J60). The digital communication board acquires andmaintains a unique network IP address over DHCP. In theevent of a failure to acquire an IP address over DHCP, it fallsback onto a static IP address.

The DAQ board employs a distributed design to minimizelengths of analog signal traces and reduce crosstalk betweenchannels. The signal path of each channel is shown in Fig.3. Each channel consists of an analog demodulator, a highimpedance pre-amplifier (PA), an analog multiplier, a pro-grammable gain amplifier (PGA), a switched capacitor 6-polelow pass filter (LPF) and a 14-bit Delta-Sigma analog-to-digital converter (ADC), capable of sampling at a maximumrate of 7812 samples per second. The PGA, LPF and ADCassociated with a particular channel are contained on a CypressPSoC mixed-signal array micro-controller [18], which alsoruns the firmware for sampling and data acquisition. Themicro-controller in the first channel serves as master deviceand generates a clock signal to synchronize sampling acrossall channels. The master device communicates with the othermicro-controllers via an SPI bus, aggregates raw channel dataand transfers this data to the digital communication board.Since the maximum sampling rate of the array is limited to7812 samples per second, analog demodulation is used tobring higher frequency acoustic signals into the base-band.The array hardware has the ability to accept commands fromthe host vehicle and dynamically change certain run timeparameters related to data acquisition, which include analogdemodulation frequency, PGA gain, LPF cutoff frequency andpole placement, and ADC sampling rate. By varying theseparameters, the array can be tuned to any signal frequency inthe range of 5Hz-100kHz, with a programmable bandwidth of500Hz to a theoretical maximum of 3900Hz.

During the sampling process the array uses about 12W ofpower, the majority of which is used by the analog circuitry. Toconserve power when sampling is not required, a sleep modeis implemented which switches off the analog components andreduces total power consumption to about 2.4W. Since the 28AWG conductors through the three meter towing cable supplypower to the array, the loss due to cable resistance cannot beneglected. To minimize this loss, efficient switching regulatorsare integrated into both hardware subsystems to accept aninput from +8V to +36V and supply a regulated +/-5V toanalog components and +3.3V to digital components.

No data processing occurs on the array hardware; this taskis offloaded to the host vehicle. The software written forthe host vehicle accepts raw data transmitted over Ethernetand processes it to identify a usable signal. Once a usable

signal has been identified, the beamforming algorithm extractsbearing information, which is then fed through a generalizedextended Kalman filter [19] that is discussed in Section III-A.The output of the filter is made available to other processesfor tracking and control.

III. SOURCE LOCALIZATION WITH A MOBILEBEARING-ONLY SENSOR

eE

eN

γi

ψi

βi

source

x

qi

x− qi

Fig. 4. Geometric configuration of a bearing-only sensor and acoustic sourceat time instance k

A. Generalized extended Kalman filter

The location of the acoustic source can be estimated from aseries of bearing measurements using a generalized extendedKalman filter. In this section, we briefly define the estimationproblem and present the generalized filter. As depicted inFigure 4, the position of the acoustic source is denoted xand the position of sensor i is denoted qi. The heading angleof sensor i, as might be measured by a magnetic compass,is denoted ψi. The vectors eN and eE are the north andeast basis vectors, respectively. Each sensor obtains a bearingmeasurement to the source, denoted zi and defined by

zi = h(x, qi, ψi) + vi (1)

h(x, qi, ψi) = γ(x, qi, ψi)−π

2(2)

where γ(x, qi, ψi) is the relative angle of the sensor withrespect to the source, as shown in Figure 4. The bearing anglemeasured by the sensor is zero when the acoustic source isbroadside to the sensor, thus the term −π2 in (2). In the sequelwe suppress the dependence of h and γ on states x, qi, andψi. The sensor noise vi is zero mean Gaussian,

vi ∼ N(0, σi(x, qi, ψi))

As the sensor obtains discrete-time measurements, we furtherassume that each sample vi[k] is independent. Of particularinterest is that the covariance of the measurement noise σi isdependent on the state of the sensor and the acoustic source.For a uniform linear acoustic array it is shown in [20] that

σi = E{v>i vi

}=

κg

cos2 h(3)

where κ is a constant depending on physical parameters ofthe sensor array and g is the inverse of the signal to noiseratio which is dependent on the distance between the acousticsource and the sensor, see [1]

g = a0 + a1 (a2 − ‖x− q‖)2 (4)

Parameters a0, a1, and a2 are characteristic of the sensor.The objective is to estimate the source position x[k] us-

ing time-varying measurements zi[k]. Typically this class ofsource tracking problems is solved by applying an extendedKalman filter, see for example [1] and [21], which assumesthat the measurement noise is independent from the state ofthe system. However, in our case the sensor noise statisticsare dependent on the state of the system, which violatesthe assumptions required for a Kalman filter. To correctlyaddress the state-dependent noise in the measurements, weutilize a generalized extended Kalman filter described in [14]to generate an estimate for the source position xi and thecovariance Pi.

As with the extended Kalman filter, the modified algorithmconsists of prediction and update steps. The source is assumedto be stationary and thus the prediction step is

xi[k|k − 1] = xi[k − 1|k − 1]Pi[k|k − 1] = Pi[k − 1|k − 1]

The local state and covariance update equations are

xi[k|k] = xi[k|k − 1] (5a)

−[P−1i [k|k − 1] +Ri[k]

]−1si[k]

si[k] =[− ζiσi∇>x hi +

12σ

(1− ζ2

i

σi

)∇>x σi

]xi[k|k−1]

(5b)

Ri[k] =[

1σi∇>x hi∇xhi (5c)

+ζi

2σ2i

(∇>x hi∇xσi +∇>x σi∇xhi

)+

14σ2

i

(ζ2i

σi+

1lnσi

)∇>x σi∇xσi

]xi[k|k−1]

P−1i [k|k] = P−1

i [k|k − 1] + Ui[k] (5d)

Ui[k] =[

1σi∇>x hi∇xhi +

12σ2

i

∇>x σi∇xσi]xi[k|k−1]

(5e)

where ζi = zi − hi. The terms in (5) are defined in theAppendix. Note that if σi did not depend on x then the gradientterm ∇xσ would be zero and (5) would reduce to the standardextended Kalman filter update equations.

B. Data Fusion

Note that a generalized extended Kalman filter (5) isimplemented on each sensor. Whenever a sensor receivesdata from another sensor, the external information is fusedto obtain a source position estimate that accounts for theshared data. In [19] data fusion equations are derived by

considering the joint probability distribution of measurementstaken from different sensors. Using the maximum likelihoodapproach one obtains equations analogous to (5) that accountfor shared measurements and predictions. In practice, unknownand unequal biases in the sensors’ measurements, along withlong periods of no communication, restrict the utility of arigorously developed approach to data fusion based on jointlikelihood functions. Instead, we have found that consensusalgorithms work very well in practice for the data fusionproblem addressed herein. Consensus algorithms represent awide class of weighted averaging protocols whose asymptoticbehaviors are well-known in both deterministic and stochasticsettings (see, for example, [22], [23]). Indeed, we find that atrue average of estimates is appropriate for our measurements.

Let Ii[k] be the set of indices of all sensors that commu-nicate with vehicle i at time k, and |Ii[k]| the cardinality ofthe set Ii[k]. Also note that i ∈ Ii[k], for all k. The fusedestimate of the source state for vehicle i is obtained as

xfi[k|k] =

1|Ii[k]|

∑j∈Ii[k]

xj [k|k] (6)

Note that since i ∈ Ii[k] the operation on the right-hand sideof (6) is always well defined. In particular, if at time k vehiclei does not receive any estimate then xf

i[k|k] = xi[k|k], wherexi[k|k] is given by (5).

IV. GRADIENT DESCENT CONTROL

A. General Formulation

To maximize estimation performance, the sensors’ states,defined by position qi and yaw ψi, must be moved to valuesthat maximize the total information

U [k] =∑j

Uj [k] (7)

This is equivalent to driving the sensors so that the geometry oftheir positions with respect to the acoustic source maximizeslocalization performance. Note that U is the sum over all j, notjust j ∈ Ii. Thus we first address the motion control problemby assuming that all vehicles can share data. In Section IV-B,we propose a method for addressing the practical realitiesof underwater communication in which vehicles share datainfrequently.

By inspecting the gradients that make up the informationterm Ui defined in (5), we see that the parameters affectingthe information are the relative distance between source andsensor ri = ‖x− qi‖, the relative angle of the sensor withrespect to the source γi, and the absolute bearing to the sourceβi (see Figure 4 and the Appendix). A standard approach tothis class of problems is to apply a gradient descent algorithmto actively control these parameters so that a suitable costfunction is minimized, such as in [21]. Following [2] and[4], we implement a gradient-based control system to locallyminimize the cost function

J [k] = det U−1[k] (8)

For convergence properties of the gradient-descent algorithmsee [24].

Assume that at time k the states qi[k] and ψi[k] are known.The state at time k + 1 is updated according to the followingkinematics

qi[k + 1] = qi[k] + TVi[k] (eN cosψi[k] + eE sinψi[k]) (9a)ψi[k + 1] = ψi[k] + Tωi[k] (9b)

where T is the updating time interval, and Vi and ωi are controlinputs representing the speed and the yaw rate, respectively.The absolute and the relative bearings βi and γi are updatedaccording to

βi[k + 1] = βi[k] + βi[k]γi[k + 1] = γi[k] + γi[k]

(10)

Inputs βi and γi are generated through gradient-descent lawsto minimize (8)

γi = −TKγ∂J

∂γi

βi = −TKβ∂J

∂βi

(11)

where Kγ > 0 and Kβ > 0 are control gains. Using (5) andthe following equalities from matrix calculus

∂ detA∂τ

= detA tr[A−1 ∂A

∂τ

]∂A−1

∂τ= −A−1 ∂A

∂τA−1

we can derive expressions for the partial derivatives in (11)

∂J

∂γi= −det U−1 tr

[∂U

∂γiU−1

]∂J

∂βi= −det U−1 tr

[∂U

∂βiU−1

]where

∂U

∂γi= − 1

σ2i

∂σi∂γi

(∇>x hi∇xhi +

1σi∇>x σi∇xσi

)+

12σ2

i

(∂∇>x σi∂γi

∇xσi +∇>x σi∂∇xσi∂γi

)∂U

∂βi=

1σi

(∂∇>x hi∂βi

∇xhi +∇>x hi∂∇xhi∂βi

)+

12σ2

i

(∂∇>x σi∂βi

∇xσi +∇>x σi∂∇xσi∂βi

)(12)

Expressions for derivatives appearing in (12) are given in theAppendix.

The next step is to express Vi and ωi in terms of thegeometry and the control inputs βi and γi. Consider thekinematic relations, see Figure 4

βi = arctan(

(x− qi)>eE(x− qi)>eN

)(13a)

x− qi = ri (eN cosβi + eE sinβi) (13b)ri = ‖x− qi‖ (13c)

Taking the time derivative of the first equation and substitutingfrom (9) with the assumption of stationary source so that x =0, we obtain

βi =1ri

(Vi cosψi sinβi − Vi cosβi sinψi)

Rearranging terms and approximating the time derivative asin (10) we express Vi as

Vi =βiri

cosψi sinβi − cosβi sinψi

=βiri

sin γi

(14)

where we used the relation, see Figure 4

γi = βi − ψi (15)

Taking the time derivative of (15) and substituting from (10)we express ωi as

ωi = βi − γi (16)

B. Local Observer

Note that U incorporates information from all sensors ateach time step, not just from those sensors that communicateduring that time step. Since communication only occurs inter-mittently, in general sensor i does not have all the necessaryinformation to compute Uj for j 6= i. To overcome thisproblem, we embed a local observer on each sensor to maintaina local estimate of external sensor states. In general, observer-based decentralized control is fraught with difficulties sincethere is no separation principle for decentralized systems.However, in the case of gradient-based control, one can showthat local observers asymptotically agree, while at the sametime, local control policies asymptotically agree with thedesired global control policy.

Through direct communication vehicles share local stateestimates xi[k|k], positions qi[k] and heading angles ψi[k].A vehicle’s output is

y[k] =(x[k|k] q[k] ψ[k]

)>(17)

By using (13) and (15) the derivatives of the output withrespect to γ and β are given by

∂y

∂γ=(0 r (eN sin(γ + ψ)− eE cos(γ + ψ)) −1

)>(18a)

∂y

∂β=(0 r (eN sinβ − eE cosβ) 1

)>(18b)

Each sensor i computes control values for every sensor j inthe networkγij [k + 1] = γij [k]

− TKγ

(∂J

∂γij+ aij [k]

∂y>ij [k]∂γij

(yij [k]− yj [k])

)βij [k + 1] = βij [k]

− TKβ

(∂J

∂βij+ aij [k]

∂y>ij [k]

∂βij(yij [k]− yj [k])

)

−200 −150 −100 −50 0 50 100 150 200

−100

−50

0

50

100

150

200

East/West (m)

Nor

th/S

outh

(m

)

Fig. 5. Vehicle trajectories; triangles denotes initial condition, squaresdenotes location of vehicles when bearing angle is first measured, dotted linesand corresponding squares indicate relative positions of both vehicles every30 seconds, circle indicates true position of acoustic source throughout theexperiment.

where

aij [k] =

{1 if j ∈ Ii[k]0 if j /∈ Ii[k]

and yij and yj equal, respectively, the function y in (17)evaluated at the local estimates xi[k|k], qij , ψij , and at thecommunicated quantities xj [k|k], qj , ψj . Control parametersVij and ωij are estimated using the relationships (14) and(16). The observer updates the local position and yaw angleestimates of sensor j using

qij [k + 1] = qij [k] + T Vij [k](eN cos ψij [k] + eE sin ψij [k]

)ψij [k + 1] = ψij [k] + T ωij [k]

where qij and ψij are, respectively, the estimate of the positionand the estimate of the yaw angle of sensor j maintained bysensor i for all j 6= i. The fused information matrix U iscomputed using the estimated sensor positions qij . From (5e)we see that computation of U also requires the acoustic sourcestate estimate. Each sensor uses its local fused state estimatexi computed through (6).

V. FIELD EXPERIMENTS

Field trials were conducted at Claytor Lake, a 4,500 acrehydroelectric impoundment of the New River near Dublin, VA.Two Virginia Tech 475 AUVs were equipped with towed arraysensors to measure the relative bearing to an acoustic sourcelocated on a support craft. The vehicles were able to communi-cate with one another using a WHOI micromodem. The goal ofthe experiments was to demonstrate that with only intermittentcommunication the two vehicles could control their geometry,as defined in Figure 4, to minimize the cost function (8).Intuition along with inspection of (3) and (6) suggest that at

steady-state, both vehicles should circle the estimated locationof the acoustic source so that their sensors are broadside to theacoustic source (relative bearing γi is π/2), and so that bothvehicles are separated by π/2 around the circle (differencein absolute bearings βi is π/2). The sensor parameters usedfor the experiments, derived for our array configuration, areκ = 0.0054, a0 = 0.0305, a1 = 0.0015, a2 = 40 used in(3) and (4). In Figure 5, the vehicle trajectories are indicatedwith dark and light lines. The acoustic source was mountedon a chase boat that moved throughout the experiment due towind. It’s position was within the area enclosed by the solidcircle for the duration of the mission. The two vehicles start atthe position denoted by the solid triangle and are commandedon an initial straight line path. At the position denoted by thesolid square the towed array sensor reports an initial bearingmeasurement to the acoustic source, and the gradient descentcontroller is activated. The curved path around the sourceposition results from the controller driving the relative bearingγi of each vehicle to π/2. The dotted lines are visual aidsdrawn at 30 second time intervals to show how the relativeseparation between vehicles increases over the length of themission. The commanded speed of each vehicle is plotted inFigure 6. The dots denote times when communication alloweda fusion event to occur. Black and gray indicate which vehiclesent and which vehicle received the data packet. Black dotsindicate that the black vehicle received a packet sent by thegray vehicle, while the gray dots indicate the opposite. Speedcommands are in the range of 0.8 m/s to 1.5 m/s. Severalfusion events occur in the beginning the mission, which causeeach 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 evolvingstate of the other vehicle, and the combined actions of eachvehicle proceed as expected. It is interesting to note thatwhile no fusion events occur towards the end of the mission,the commanded speeds begin fluctuating to slow the sensorseparation based on the information inferred by the localobservers which indicated that the vehicles are approachingtheir optimal positions. The magnitude of the acoustic sourceestimation error is plotted in Figure 7, which shows that thelocal position estimates converge when fusion events occur.During a period of no communication in the middle of theexperiment the two estimates diverge from each other whichis expected due to unmodeled biases present in each sensor.

The vehicles achieve a desired relative bearing and geometrywith respect to the target that minimizes the joint localizationerror of the acoustic source. Importantly, we do not explicitlycommand one vehicle to speed up while the other slows down.The behavior of each vehicle is an emergent consequence ofmaking joint decisions, aided by a local observer when nocommunication occurs, to minimize the cost function (8).

APPENDIX

A. Equation Details

Gradients with respect to x of functions h, σ, and g are

∇xh =1

‖x− q‖2[−(x− q)>eEe>N + (x− q)>eNe>E

]=

1r

(−e>N sinβ + e>E cosβ

)∇xσ = σ

(2 tanh∇xh+

1g∇xg

)∇xg = −2a1(a2 − r)

(e>N cosβ + e>E sinβ

)(19)

150 200 250 300 350 400 450 500 550 600 6500.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

time (s)

spee

d (m

/s)

Fig. 6. Commanded vehicle speeds; dots denote times a data fusion eventoccurs.

150 200 250 300 350 400 450 500 550 600 6500

20

40

60

80

100

120

erro

r (m

)

time (s)

Fig. 7. Acoustic source estimation error magnitude; dots denote times a datafusion event occurs.

Derivatives with respect to β and γ in (12) are

∂σ

∂γ= 2σ tanh

∂∇xσ∂γ

=∂σ

∂γ

(2 tanh∇xh+

1g∇xg

)+

2σcos2 h

∇xh

∂∇xh∂β

= −1r

(e>N cosβ + e>E sinβ

)∂∇xσ∂β

= 2σ tanh∂∇xh∂β

(20)

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of theOffice of Naval Research via grant N00014-07-1-0434.

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