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Collective Motion, Sensor Networks and Ocean Sampling

Naomi Ehrich Leonard, Derek Paley, Francois LekienMechanical and Aerospace Engineering

Princeton UniversityPrinceton, NJ 08544, USA

{naomi,dpaley,lekien }@princeton.edu

Rodolphe SepulchreElectrical Engineering & Computer Science

Universite de LiegeInstitut Montefiore B28, B-4000 Liege, Belgium

r.sepulchre@ulg.ac.be

David M. FratantoniWoods Hole Oceanographic Institution

Physical Oceanography Department, MS#21Woods Hole, MA 02543, USAdfratantoni@whoi.edu

Abstract— This paper addresses the design of mobilesensor networks for optimal data collection. The develop-ment is strongly motivated by the application to adaptiveocean sampling for an autonomous ocean observing andprediction system. A performance metric, used to deriveoptimal paths for the network of mobile sensors, definesthe optimal data set as one which minimizes error in amodel estimate of the sampled field. Feedback control lawsare presented that stably coordinate sensors on structuredtracks that have been optimized over a minimal set ofparameters. Optimal, closed-loop solutions are computedin a number of low-dimensional cases to illustrate themethodology. Robustness of the performance to the in-fluence of a steady flow field on relatively slow-movingmobile sensors is also explored.

I. I NTRODUCTION

The coupled physical and biological dynamics [1], [2]of the oceans have a major impact on the environment,from marine ecosystems to the global climate. In or-der to understand, model and predict these dynamics,oceanographers and ecologists seek measurements oftemperature, salinity, flow and biological variables acrossa range of spatial and temporal scales [3], [4], [5].

Small spatial and temporal scales drive the need for amobile sensor network rather than a static sensor array.For example, a static sensor network designed to measurean eddy that is localized and moving will necessarilybe very refined and require many sensors. On the otherhand, mobile sensor networks, comprised of sensor-equipped autonomous vehicles, can exploit their mobilityto follow features and/or monitor large areas with time-varying, spatially distributed fields, assuming that the

Corresponding Author: Naomi Leonard, MAE Department, En-gineering Quadrangle, Olden Street, Princeton University, Prince-ton, NJ 08544, Tel: +1 609 258 5129, Fax: +1 609 258 6109,naomi@princeton.edu

number of vehicles and their speed and endurance arewell matched to the speeds and scales of interest [6].

Our goal is to design a mobile sampling networkto take measurements of scalar and vector fields1 andcollect the “best” data set. A cost function, or samplingmetric must be defined in order to give meaning to theterm “optimal data set”. For example, the performancemetric that we consider in this paper defines an optimaldata set as one in which uncertainty in a linear modelestimate of the sampled field is minimized. A comple-mentary approach to defining a synoptic performancemetric is presented in [9]. Alternate metrics emphasizethe sampling of regions of highest dynamic variabilityor focus on areas of high economical or strategicalimportance. Clearly the coordination of the sensors inthe network is critical to maintain optimal data col-lection, independent of the metric chosen. Accordingly,coordination and collective motion play a central role inthe development here. We note further that the fields tobe sampled are three-dimensional, but it is reasonableto consider two-dimensional surfaces as we do in thispaper. Justification for this choice is discussed further inSectionIV-B.

One effective way to enable a mobile sensor networkto track and sample features in a field is to use coordi-nated gradient climbing strategies. For instance, in oceansampling problems, the sensor network could be used toestimate and track maximal changes in the magnitude ofthe gradient in order to find thermal fronts or boundariesof phytoplankton patches. Such feature-tracking strate-gies are particularly useful for sampling at relativelysmall spatial scales. Boundary tracking algorithms are

1The results and methods in this paper focus on a single scalarfield but can be applied to multivariate fields by using appropriateweights in the cost function [7], [8].

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developed, for example, in [10], [11], [12].On the other hand, strategies best suited for larger

spatial scales are those that direct mobile sensors toprovide synoptic coverage. Typically, the goal is tocontrol the sensor network so that error in the estimate ofthe field of interest is minimized over the region in spaceand time. In this case, sensors should not cluster elsethey take redundant measurements. Coordinated vehicletrajectories should be designed according to the spatialand temporal variability in the field in order to keep thesensor measurements appropriately distributed in spaceand time.

In SectionII we motivate the ocean sampling problemand state our central objective. This objective, aimed atcollecting the richest possible data set with a mobilesensor network, is representative of sampling objectivesin an number of domains. We describe some of thechallenges that distinguish adaptive sampling networksin the ocean from networks on land, in the air or inspace.

Before developing our ideas further, we next describein Section III an ocean sampling network field ex-periment. The intention is both to provide inspirationfor future possibilities and to illustrate a number ofthe practical challenges. Coordinated control strategiesand gradient estimation for small-scale problems (ap-proximately 3 kilometers) were tested on a group ofautonomous underwater gliders in Monterey Bay, Cali-fornia in August 2003 as part of the Autonomous OceanSampling Network (AOSN) project [13]. The method,based on artificial potentials and virtual bodies, provedsuccessful despite limitations in communication, controland computing and challenges associated with strongcurrents and great uncertainty in the relatively harshocean environment. We present results from this effortand discuss some of the operational constraints particularto this kind of ocean sampling network.

In a field experiment planned for August 2006 inMonterey Bay, a larger fleet of underwater gliders withsimilar operational constraints as those from 2003, willbe controlled to maintain synoptic coverage of a fixedregion. One primary ocean science objective is to un-derstand the dynamics of three-dimensional cold waterupwelling centers. In the remainder of this paper, weexamine robust, optimal broad-scale coverage perfor-mance that we consider integral to achieving this andother science objectives. Our effort focuses on designof coordinated, mobile sensor trajectories, optimized forsampling, and stabilization of the collective to thesetrajectories.

In SectionIV we catalog general and significant issuesand challenges in sensor networks, collective motion

and ocean sampling. We then summarize the issues andoutline the problem addressed in this paper.

In SectionV we derive and define a sampling metricbased on the classical objective mapping error [14],[15], [16]. This sampling metric can be used to evaluatethe sampling performance of a mobile sensor network.Likewise it can be used to derive sensor platform trajec-tories that optimize sampling performance. We considercoordinated patterns that arenear optimal with respectto the sampling metric; that is, we select a parametrizedfamily of solutions and define a near-optimal solutionas one which optimizes the sampling metric over theparameters. In SectionV we present a parametrizationof solutions consisting of sensors moving in a coordi-nated fashion around closed curves. We parametrize therelative position of the sensors (and thus thecoordinatedmotion of the sensors) using the relative phases of thesensors. Here the phase of a sensor refers to its angle,relative to a reference, around the closed curve on whichit moves. This choice of parametrization motivates ourapproach to stabilization of collective motion which istightly connected to coupled phase oscillator dynamics.

In SectionVI we present models for collective motionbased on a planar group of self-propelled vehicles (ourmobile sensors) with steering control. We exploit phasemodels of coupled oscillators to stabilize and controlcollective motion patterns where vehicles move aroundcircles and other closed curves, with prescribed relativespacing. We then discuss in SectionVII the performanceof these coordinated patterns with respect to the samplingmetric. We express our sampling metric as a functionof non-dimensionalsampling numbers(parameters thatdetermine the size, shape and scales in the field ofinterest in space and time, the speed of the vehicles andthe level of measurement noise), and we determine thesmallest set of parameters needed for the optimal sam-pling problem. We present results on optimal solutions inthe case of a single vehicle moving around an ellipticaltrajectory in a rectangular field and in the case of twovehicles, each moving around its own ellipse. In the caseof two vehicles we study the optimal sampling solutionin the presence of a steady flow field with (and without)the coordinated feedback control laws of SectionVI . Weconclude in SectionVIII and provide some discussion ofongoing and future directions.

II. CENTRAL OBJECTIVE

Developing models and tools to better understandocean dynamics is central to a number of importantopen problems. These include predicting and possiblyhelping to manage marine ecosystems or the globalclimate and predicting and preparing for events such as

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red tides or El Nino. For example, phytoplankton areat the bottom of the marine food chain and are there-fore major actors in marine ecosystems. They impactthe global climate because they absorb enough carbondioxide to reduce the regional temperature [17]. El Ninodisrupts conditions in the ocean and atmosphere whichin turn affect phytoplankton dynamics [18]. Therefore,phytoplankton can be viewed as indicators of changein the ocean and atmosphere. However, the dynamicsof phytoplankton are inherently coupled to the physicalocean dynamics [19]. For example, upwelling events inthe ocean bring nutrient-rich, cold water from the seabottom to the surface where phytoplankton, which needto consume iron but also need the sun for photosynthesis,can gather and grow. Accordingly, understanding thephysical oceanography and how it couples with thebiological dynamics is necessary for tackling a numberof important open problems [1], [2].

At present there are many effective ways to collectdata on the surface of the ocean. These include, forinstance, sea surface temperature measurements fromsatellite (or airplanes) using thermal infrared sensors,surface current measurements using high frequency radarand temperature and salinity measurements from surfacedrifters carrying CTD (conductivity-temperature-depth)sensors. Limited measurements under the sea surfacecan be made with stationary moorings or with floats thatmove up and down in the water column and drift withthe currents. Ships that tow sensor arrays can also beused to collect data under the surface.

Autonomous underwater vehicles (AUVs), equippedwith sensors for measuring the environment, are amongthe newest available underwater, oceanographic samplingtools [20]. With AUVs come compelling new opportu-nities for significantly improved ocean sensing; recentadvances in technology have made it possible to imaginenetworks of such sensor platforms scouring the oceandepths for data [21]. Underwater gliders, described inSection III , are a class of endurance AUVs designedexplicitly for collecting such data continuously overperiods of weeks or even months [22], [23], [24].

What makes AUVs particularly appealing in this con-text is their ability to control their own motion. Usingfeedback control, AUVs can be made to perform as anintelligent data-gathering collective, changing their pathsin response to measurements of their own state andmeasurements of the sampled environment. A reactiveapproach to data gathering such as this is often referredto asadaptive sampling. Naturally, with new resourcesand opportunities come new research questions. Of par-ticular importance here is the question of how to usethe mobility and adaptability of the network to greatest

advantage.Our central objective is to design and prove effective

and reliable a mobile sensor network for collecting therichest data set in an uncertain environment given limitedresources.This is a representative objective for mobilesensor networks and adaptive sampling problems overa number of domains. One such domain is the Earth’satmosphere where airplanes, balloons, satellites and net-works of radars are used to collect data for weatherobservation and prediction. In space, clusters of satelliteswith telescopes can be used to measure characteristics ofplanets in distant solar systems. Sensor networks are alsobeing developed in numerous environmental monitoringsettings such as animal habitats and river systems [25].Many of these networks use stationary sensors, althougheven if not mobile, the sensors can be made reactive,as in the network that was tested in Australia for soilmoisture sensing and evaluation of dynamic response torainfall events [26].

An ocean observing mobile sensor network is dis-tinguished from many of these other applications bytwo significant factors. The first factor is the difficultyin communicating in the ocean. On land or in theair, it is relatively easy to communicate using radiofrequency. However, radio frequency communication isnot possible underwater, and it is not yet practical touse underwater acoustic communication in the settings ofinterest, where underwater mobile sensor platforms maybe tens of kilometers apart. Communication is possiblewhen underwater vehicles surface, which they typicallydo at regular intervals to get GPS updates and to relaydata. However, the intervals between surfacings can belong and therefore challenging for the navigation of asingle vehicle and the control of the networked system.

A second distinguishing factor is the influence of theocean currents on the mobile sensor platforms. In thecase of gliders which move at approximately constantspeed relative to the flow, ocean currents can sometimesreach or even exceed the speed of the gliders. Unlike anairplane which typically has sufficient thrust to maintaincourse despite winds, a glider trying to move in the direc-tion of a strong current will make no forward progress.Since the ocean currents vary in space and in time,the problem of coordinating mobile sensors becomeschallenging. For instance, two sensors that should staysufficiently far apart may be pushed toward each otherleading to less than ideal sampling conditions.

III. A F IELD EXPERIMENT IN MONTEREY BAY

The goal of the Autonomous Ocean Sampling Net-work (AOSN) project is to develop a sustainable,

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portable, adaptive ocean observing and prediction sys-tem for use in coastal environments [21]. The projectuses autonomous underwater vehicles carrying sensorsto measure the physics and biology in the ocean togetherwith advanced ocean models in an effort to improveour ability to observe and predict coupled biologicaland physical ocean dynamics. Critical to this researchare reliable, efficient and adaptive control strategies thatensure mobile sensor platforms collect data of greatestvalue.

A. AOSN Field Experiment

In summer 2003, a multi-disciplinary research groupproduced an unprecedented in situ observational capa-bility for studying upwelling features in Monterey Bayover the course of a month-long field experiment [27].A highlight was the simultaneous deployment of morethan a dozen, sensor-equipped, autonomous underwatergliders [28], including five Spray gliders operated byRuss Davis of Scripps Institution of Oceanography andup to ten Slocum gliders operated by David Fratantoniof Woods Hole Oceanographic Institution (Figure1).

Autonomous underwater gliders are buoyancy-driven,endurance vehicles. They control their volume (Spray)or mass (Slocum) to change their net buoyancy so thatthey can move up and down in the ocean. Fixed wingsand tail give them lift and help them to follow sawtoothtrajectories in the longitudinal plane. Gliders can activelyredistribute internal mass to control attitude. For headingcontrol, they shift mass to roll, bank and turn (Spray)or use a rudder (Slocum). During the field experimentthe gliders were configured to maintain a fixed velocityrelative to the flow. Their effective forward speed wasapproximately25 cm/s (Spray) to35 cm/s (Slocum);this is of the same order as the stronger currents in andaround Monterey Bay. Accordingly, the gliders do notmake progress in certain directions when the currentsare too strong.

The Spray gliders, rated to 1500 meter depth andoperated to 400 meters and sometimes 750 meters duringsummer 2003, were deployed in deep water, traveling asfar as 100 km offshore. The Slocum gliders, operatedto 200 meter depth, were deployed closer to the coast.The gliders surfaced at regular intervals (although notsynchronously) to get GPS fixes for navigation, to senddata collected back to shore and to receive updatedmission commands. The communication to and fromthe shore computers, via Iridium satellite and ethernet,was the only opportunity for communication “between”gliders; the gliders were not equipped with means tocommunicate while they were underwater because ofpower and other constraints.

Fig. 1. Two Slocum gliders in summer 2003. Each is about1.5 meters long. Motion in the vertical plane follows a sawtoothtrajectory. A rudder is used to steer in the horizontal plane. Maximumdepth is 200 meters and average forward speed relative to the flowis approximately 35 cm/s. During the AOSN 2003 experiment, thegliders were configured to surface and communicate as frequently asevery two hours.

On a typical single battery cycle, the Slocum glidersperformed continuously for up to two weeks betweendeployment and recovery while the Spray gliders re-mained in the water for the entire experiment (about sixweeks). Collectively, the gliders delivered a remarkablyplentiful data set. Figures2 and 3 show locations ofthe data collected by all of the gliders over the courseof the month-long field experiment. Each point on theplots refers to the location in the horizontal plane ofa data profile taken, i.e., a series of measurements(including temperature, salinity, chlorophyll fluorescence– for concentration of phytoplankton) as a function ofdepth. Together the points illustrate the paths of thegliders. Figure2 shows the paths of the five Spray gliderstraveling back and forth along lines approximately per-pendicular to the shore. As seen in Figure3, the Slocumgliders, traveled around trapezoidal racetracks closer toshore, other than when used for coordination experimentsas described next.

B. Cooperative Control Sea Trials

In this section we summarize results of sea trials,run as part of the field experiment, with small fleetsof Slocum underwater gliders controlled in formations[13]. The focus was on relatively small scales in theregion (on the order of 3 kilometers) and feature trackingcapabilities of mobile sensor networks. The sea trialswere aimed at demonstrating strategies for cooperativecontrol and gradient estimation of scalar sampled fieldsusing a mobile sensor network comprised of three glidersin a strong flow field with limited communication andfeedback.

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Fig. 2. Sensor measurement locations (Spray). Each pointrepresents a vertical profile of data as a function of depth.

Fig. 3. Sensor measurement locations (Slocum). Each pointrepresents a vertical profile of data as a function of depth.

The control strategy was derived from thevirtual bodyand artificial potential (VBAP) multi-vehicle controlmethodology presented in [29]. VBAP is a general strat-egy for coordinating the translation, rotation and dilationof a group of vehicles and can be used in missionssuch as gradient climbing in a scalar, environmentalfield. A virtual body is a collection of moving referencepoints with dynamics that are computed centrally andbroadcast to vehicles in the group. Artificial potentialsare used to couple the dynamics of vehicles and a virtualbody so that desired formations of vehicles and a virtualbody can be stabilized. Each vehicle uses a control lawthat derives from the gradient of the artificial potentials;therefore, each vehicle must have available the positionof at least the nearest neighboring vehicles and the

nearest reference points on the virtual body. If sampledmeasurements of a scalar field can be communicatedto a central computer, the local gradients of a scalarfield can be estimated. The speed of the virtual bodyis controlled to ensure stability and convergence ofthe vehicle formation. Gradient climbing algorithms canalso prescribe virtual body direction. For example, thevirtual body (and consequently the vehicle group) can bedirected to head for the coldest water when temperaturegradient estimates computed from vehicle measurementsare available.

The control theory and algorithms described in [29]depend upon a number of ideal assumptions on theoperation of the vehicles in the group, including contin-uous communication and feedback. Since this was notthe case in the operational scenario of the field exper-iment, a number of modifications were made. Detailsof the modifications are described in [30]; these includeaccommodation of constant speed of gliders, relativelylarge ocean currents, waypoint tracking routines, com-munication only every two hours when gliders surface(asynchronously) and other latencies.

For the Slocum vehicles, each glider has on-boardlow-level control for heading and pitch which enables itto follow waypoints [31]. A waypoint refers to a verticalcylinder in the ocean with given radius and position.When a sequence of waypoints is prescribed, the gliderfollows the waypoints by passing through each of thecorresponding cylinders in the prescribed sequence usingits heading control. Heading control requires not onlythat the glider know the prescribed waypoint sequence,but also that it can measure (or estimate) its own po-sition and heading. Heading is measured on-board theglider (as is pitch and roll). Depth and vertical speedare estimated from pressure measurements. ¿From thesemeasurements and some further assumptions, the gliderestimates its linear velocity. Position is then computed byintegration, using the most recent GPS fix as the initialcondition. This deduced reckoning approach also makesuse of an estimate of average flow, computed from theerror on the surface between the glider’s GPS and itsdead-reckoned position.

In the cooperative control sea trials of 2003, the glidersused their low-level control to follow waypoints as perusual; however, the waypoint sequences were updatedevery two hours using the VBAP control strategy forcoordination. VBAP was run on a simulation of theglider group using the most recent GPS fixes and averageflow measurements as initial conditions. The trajectoriesgenerated by VBAP were then discretized into waypointlists which were transmitted to the gliders when theysurfaced. The approach is discussed further in [30], [13].

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Fig. 4. Snapshots in time of glider formation starting at 18:03 UTCon August 6, 2003 and moving approximately northwest. The vectorsshow the estimate of minus the temperature gradient at the group’scenter of mass at 10 meters depth. The gray-scale map correspondsto temperature measured in degrees Celsius. The three smaller blackcircles correspond to the initial positions of the gliders.

On August 6, 2003, a sea trial was run in which threeSlocum gliders were commanded to move northwest inan equilateral triangle with inter-glider distance equal tothree kilometers. The desired path of the center of massof the vehicle group was pre-planned. The trial was runfor sixteen hours, with gliders surfacing every two hours(although not at the same time). The orientation of thegroup was unrestricted in the first half of the sea trialand constrained in the second half of the sea trial so thatone edge of the triangle would always be normal to thepath of the center of mass of the group.

Snapshots of glider formations as well as glider groupestimates of temperature gradient are shown in Figure4for the August 6, 2003 sea trial. The group stayed information and moved along the desired track despiterelatively strong currents. Further, the gradient estimate,as seen in the figure, is remarkably smooth over timeand points to the colder water, as verified from indepen-dent temperature measurements. In a second sea trial,described in detail in [13], three gliders again werecontrolled in an equilateral triangle formation. In this seatrial the inter-glider distance was commanded initially tobe six kilometers and then reduced to three kilometersto demonstrate and test the influence of changing theresolution of the mobile sensor array. The glider networkperformed remarkably well despite currents with magni-tude as high as 35 cm/s, which is the effective speed ofthe gliders.

IV. SAMPLING , CONTROL AND NETWORK ISSUES

The knowledge and skills accumulated during the fieldexperiment and the success of the coordinated vehiclesea trials in 2003 provide a great deal of inspiration forfurther possibilities in ocean sampling networks. Indeed,another field experiment is planned for August 2006,again in Monterey Bay, in which a fleet of sensor-equipped, autonomous underwater gliders will be oper-ated continuously for a month as an adaptive samplingnetwork. The fleet will include on the order of tenunderwater gliders and a focus will be on broad-scalecoverage of an area including the upwelling center atPoint Ano Nuevo (just north of Santa Cruz).

The field experiment brought experience with a num-ber of practical challenges associated with sensor net-works in the ocean, including the relatively strong flowfield that pushes the vehicles around and the delays andconstraints on communication.

In SectionIV-A , we reflect on the broad central ob-jective stated in SectionII and list some of the importantand challenging issues in sampling, control and mobilenetworks. In SectionIV-B we clarify which issues weaddress in this paper and we define the boundaries ofthe problem addressed.

A. Catalog of Challenges and Constraints

There are a number of challenges and constraintsto be investigated in order to address our central ob-jective. The interest in optimization of data collected,management of uncertainty and extension of resourcesintroduce conflicting demands which require tradeoffs.Further, it is a goal to make the design methodology assystematic as possible since the ocean observation andprediction system should be autonomous and portable.This motivates simpler and less computationally inten-sive approaches. Major issues involving the performancemetric, optimization of the metric and feedback controldesign for robustness are listed as follows.

• Sampling metric definition. A metric should beselected that defines what is meant by the “best”or “richest” data set. The selected metric should bestudied to evaluate how well it serves the range ofgoals.

• Multiple fields . When there are more than one fieldto be sampled, a choice needs to be made as to howto weight the importance of different fields in thesampling metric.

• Multiple scales. A complete approach to optimalocean sampling needs to address the range of scalescritical to understanding, modeling and predictingocean dynamics. For example in the context we

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study, the spatial scale ranges from 25 kilometersfor the synoptic picture down to 3 to 5 kilometersfor features of the upwelling and even as small ashundreds of meters for some of the biology.

• 2D versus 3D. In the event that sampling in three-dimensional space is desired, any methodologiesderived for two dimensions need to be extended.

• Sampling metric computation and adaptation. Amethodology should be developed for computingthe metric with minimal computational burden andfor computing inputs to the metric that are notdirectly measured and/or that change over time.The tradeoff between optimization of the metricversus computation of the metric may need to beconsidered in the design and real-time control ofoptimal collective motion.

• Optimal, collective motion. An approach to opti-mizing the sampling metric should be developed sothat optimal, collective motion for the mobile sensornetwork can be designed. Low frequency feedbackmeasurements can be used to adapt the optimalcollective motion to the changing fields, changingocean processes, changing operational conditionsand health of the sensors in the network.

• Flow field. Whether or not its components are scalarfields of specific interest, the flow field directlyinfluences sampling performance because it canpush the sensors around and prevent them from car-rying out optimal sampling strategies. Accordingly,the flow field must be considered in the designof optimal, collective motion. A methodology toexploit available estimates or predictions of the flowfield is of significant interest.

• Feedback control of collective motion. Relativelyhigh rate feedback control strategies that stabilizeoptimal collective motion are necessary to ensurerobustness of optimal sampling strategies not onlywith respect to the external flow field but also toother disturbances and uncertainties in the oceanenvironment.

Additionally, there are a number of issues associatedwith the sensor platforms themselves and their networkoperation. A list of these such issues follows.

• Constant speed.Strategies for collective motionmust take into account that gliders effectively op-erate at constant speed (relative to the flow field).Otherwise, patterns may be designed that are notrealizable. Gliders can also be operated as virtualmoorings which may be applicable to the adaptivesampling problem but is not considered here.

• Transit and irregular events. There will be a sig-nificant period of time when mobile sensors are “intransit,” meaning that they are on their way betweenoptimal strategies. For example, when gliders arefirst deployed they should transit to locations wherethey will initiate their optimal strategy. However,gliders are slow and the period of time it will take toget to these locations may be significant. Therefore,their paths should be designed both to optimizesampling during transit and to minimize transit time.Similar strategies should be developed in case amobile sensor encounters a region it must avoid(e.g. due to fishing), is taken out of the water forwhatever reason, experiences a debilitating failure,etc.

• Heterogeneous groups. In case mobile sensors inthe network differ in speed, endurance, sensors, etc.,methodologies should be developed to exploit thediffering strengths and potential roles of the sen-sors in the network. For instance, slow, endurancevehicles might be more useful for larger scaleswhereas faster, shorter-lived vehicles might servebetter collecting data over smaller scales.

• Extending lifetime of sensors. Underwater glidersare designed to be endurance vehicles, a centralobjective being to collect data continuously overweeks or even months at a time. Accordingly,keeping energy use to a minimum is critical. Thisimplies also keeping volume (and therefore mass)to a minimum. There is a direct tradeoff here withimproving sensing, navigation, communication andcontrol. For example, communication on the oceansurface makes possible coordinated control of thesensors. However, surfacings that are too frequentcan be costly in terms of energy expenditure andloss of time collecting data, whereas surfacings thatare too infrequent yield very long feedback sam-pling periods which can diminish the performanceand robustness of the control.

• Communication. Communication between glidersis done above the surface on a central computer.Coordinated control strategies for the network ofsensors that are originally designed assuming con-tinuous control will need to be revisited. Sinceminimizing the frequency of surfacings is desir-able to minimize energy and maximize time spentcollecting undersea data and minimize exposure, itis of interest to determine the maximum tolerablefeedback sampling period that does not degradeoverall sampling performance.

• Asynchronicity. Strategies will need to accommo-date asynchronicity in time of surfacing and com-

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munication. Because the gliders will not surfaceat the same time, information communicated to aglider about any of the other gliders will necessarilybe old.

• Latencies. It may not always be possible to closethe feedback loop on the surface. For example, inthe sea trials of 2003, described in SectionIII-B, data retrieved from a glider at its surfacingcould not be used in the mission update to theglider at that same surfacing. Instead the data wasused to compute new instructions communicatedto the glider at the next surfacing. This introducessignificant delays that need to be accommodated.

• Computing. While low-level control is computedon board the gliders, coordinated control of thenetwork is computed on the central shore computerwhere inter-glider communication occurs. Possibili-ties for further exploiting on-board computation andlocal measurements should be investigated.

B. Problem Definition

In this paper we assume a single, scalar, dynamicfield (e.g., temperature or salinity) is to be sampled. Weconsider a sampling metric, defined in SectionV-A, thatderives from objective analysis and a simple model of theenvironment. This metric provides a measure of modeluncertainty as a function of where and when data iscollected. Since reduced model uncertainty implies bettercoverage, we also refer to this sampling performancemetric as a coverage metric. The choice of a sim-ple model for determining sampling performance keepscomplexity and computational burden to a reasonablelevel. The approach also provides a complement to highresolution ocean forecasting models that run on dataassimilated from the mobile sensor network as part ofthe ocean observation and prediction system [32], [33],[34]. The high resolution ocean forecasting models usequasi-3D dynamic modeling, meaning that the 3D regionof the ocean is treated as layered 2D surfaces. Thetypical operation of the gliders is consistent with thisapproach; their motion in the vertical plane follows aregular (sawtooth) trajectory. To be consistent, we focuson fields defined on a 2D surface, i.e. a single layer.

Although the tools we develop are useful in multi-scale problems, we assume a dominant spatial scaleand temporal scale of interest. We further take the 2Dregion over which the field is defined to be rectangularand homogeneous, i.e., the correlation between any twovalues in the field depends only on their separation inspace and time. We take the corresponding spatial scaleσ and temporal scaleτ as given since they are computed

in the high resolution ocean forecasting models andtherefore, in principle, available in real time. In thispaper, the values we use have been computed from gliderdata collected in the 2003 AOSN field experiment.

We frame the optimal collective motion problem anddefine our approach to design of a (near) optimal mobilesensor network in SectionV. By nearoptimal solutions,we mean that we optimize over a parametrized family ofstructured solutions. For example, we consider a familyof closed curves parametrized by number, location, di-mension and shape as well as the relative phases of thevehicles moving around these curves. This parametriza-tion is discussed in SectionV-D. The relative phasesprovide a low-dimensional parametrization of relativeposition of the vehicles and they make a connectionbetween the optimized trajectories and the coupled phaseoscillator models that we use in our coordinated controllaw. We pay particular attention to gliders moving aroundellipses. In the case of gliders moving with constantspeed around circles, the difference in heading for anypair of gliders can be interpreted as the relative phase ofthat pair of gliders. For example, if for a pair of glidersmoving around the same circle, the difference in headingis 180 degrees, then the relative phase is 180 degreesand the gliders are always at antipodal points on thecircle. For ellipses, the relative phase is not necessarilyequivalent to the relative heading.

In SectionVI we present feedback control laws thatstabilize these kinds of collective motions for glidersmoving at constant (unit) speed on the plane. We focuson the case that there may be multiple ellipses andmultiple vehicles per ellipse. The objective is to ensurethat gliders move around their (optimally located, ori-ented, sized) ellipses with optimal relative phases. InSectionVII we compute and study optimal solutions andwe discuss robustness of the solutions with respect to thecoverage metric. We also investigate the influence of theflow field on the design and control of optimal samplingtrajectories.

In this paper we assume a homogeneous group ofmobile sensors. We do not address the issue of transit andirregular events; preliminary results on minimal time andminimal energy glider paths computed using forecasts ofocean flow fields are presented in [35]. We also do notaddress the problems in communication, asynchronicity,latency and computing described above. In [30], [13] itis discussed how these issues were handled in AOSN2003. In [36] a control law is presented that exploresextended sensing, computing and control on-board aglider. In this paper we let each sensor compute its owncontrol law locally and we assume continuous feedbackcontrol with continuous communication without delay or

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asynchronicity. Because communication is not limitedto neighboring gliders in the operational scenario, weassume an all-to-all interconnection topology.

A number of the issues listed in SectionIV-A remainimportant open problems and a number are the subjectof ongoing work.

V. SAMPLING METRIC AND OPTIMALITY

A. Sampling Metric

In this section, we derive a metric to quantify how wellan array of gliders samples a given region. Recall thatan objective is to assimilate the data in an ocean model.Therefore, the metric should reflect how a particularcollected data set reduces the error in the model. Thisnotion is necessarily dependent on the specific modelor assimilation scheme used. During AOSN 2003, thedata was assimilated in several high resolution oceanmodels [32], [33], [34] and the performance of thesampling array was different (but very similar) for each.Since reliable nowcasts and forecasts of the ocean re-quire concurrent ocean models mutually validating theirresults and the data requirements of these models aresimilar, it is natural to derive the performance metric ona simpler, more general assimilation scheme. This ap-proach also has the advantage of avoiding the complexityand computational effort required to study specific highresolution models [37], [38]. We consider a simple dataassimilation scheme called Objective Analysis2 [39],[40]. In this framework, the scalar field (e.g., temper-ature, salinity) observed at each pointr and at each timet is viewed as a random variableT (r, t) or an ensembleof possible realizations. The algorithm keeps track ofan estimate for the average and second moment of thisdistribution

T (r, t)=E [T (r, t)] ,

B(r, t, r′, t′)=E[[T (r,t)−T (r,t)

][T (r′,t′)−T (r′,t′)

]]whereE [·] represents the expected value of a randomvariable. Notice thatT (r, t) is the best estimate of thestate and the diagonal elementsB(r, t, r, t) represent theuncertainty or error at a given point(r, t). The data col-lected by the gliders is a sequence ofM measurementsTk at discrete points(rk, tk). The objective analysisscheme consists of finding linear incrementsζk such thatthe new estimate of the state,

TA(r, t) = T (r, t) +M∑k=1

ζk(r, t)[Tk − T (rk, tk)

],

2Objective Analysis is also commonly referred to as OptimalInterpolation and is equivalent to 3D-Var. It was originally developedby Eliassen et al [14] in 1954 and independently reproduced andpopularized by Gandin [15] in 1963.

minimizes the least square uncertainty of the new es-timate TA. In other words, theζk must minimize thea-posteriori error∫

dr∫dtA(r, t, r, t)

=∫dr∫dtE

[[T (r,t)−TA(r,t)

][T (r,t)−TA(r,t)

]].(1)

An extensive analysis of the assimilation scheme, equa-tions and generalization (e.g., multivariate, discrete, non-stationary systems) can be found in [39], [40]. Assumingthat the measurement noisen is uniform and uncorre-lated, the solution that minimizes (1) is

ζk(r, t) =M∑l=1

B(r, t, rl, tl)(C−1

)kl, (2)

whereC−1 is the inverse of theM ×M matrix (C)kl =nδkl+B(rk, tk, rl, tl) andδkj is the Dirac delta function.The corresponding a-posteriori error (substitution of (2)in (1)) is given by

A(r, t, r′, t′) = B(r, t, r′, t′)

−M∑

k,l=1

B(r, t, rk, tk)(C−1

)klB(rl, tl, r′, t′) . (3)

The remaining error in the best estimate can be used asa quantitative measure of the impact of the sequence ofmeasurements on the error in the assimilation scheme.Substitution of (3) in (1) gives

φ =∫dr

∫dtA(r, t, r, t)

=∫dr

∫dtB(r, t, r, t)

−∫dr

∫dt

M∑k,l=1

B(r,t,rk,tk)(C−1

)klB(rl,tl,r,t)(4)

and is elected as thesampling performance metrictocompare and optimize sensor paths.

B. Ocean Statistics

The coverage metric defined in (4) contains an un-known term,B(r, t, r′, t′), an estimate of the backgroundstatistics. It represents the estimated statistics of theoceanbefore data assimilation. The diagonal elementsB(r, t, r, t) describe our confidence in the initial state.The non-diagonal elements represent the covariancebetween points at different locations and times. Theyare closely related to the correlation length and thecorrelation time in the domain [16].

The metric in (4) has a broad range of applicationand can be used with any positive-definite covariance

10

Num

ber

ofm

easu

rem

ents

/day

0

300

600

900

Time (days since January 1st 2003)

-log

(Φ/A

)

200 210 220 230 240 2500

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

Fig. 6. Sampling metric (solid curve) in units of entropic informationand number of profiles (shadowed area) for AOSN 2003. Each crosscorrespond to a panel of Fig.5. On August 10th (day 223), thenumber of profiles is still high but the metric indicates relatively poorcoverage. The second panel of Fig.5 explains this loss of performanceby a poor distribution of the gliders in the bay on that day.

function B(r, t, r′, t′). For the purpose of illustratingthe use of the metric, we assume that the backgroundcovariance is given by

B(r, t, r′, t′) = σ0 e−‖r−r′‖2

σ2 −|t−t′|2τ2 . (5)

The parametersσ ≈ 25 km and τ ≈ 2.5 days are thespatial and temperature decorrelation scales of MontereyBay during AOSN 2003, determined empirically usingglider data [28]. Notice that the scaling factorσ0 has noeffect on the sampling paths, provided that the measure-ment noisen is scaled by the same factor. This fact isdiscussed and exploited in SectionVII .

Figure5 shows the a-posteriori error at different timesduring AOSN 2003 using the Gaussian covariance. Thedata used correspond to the Spray gliders [22], [28]and the Slocum gliders [13], [28] that patrolled the bayduring the summer of 2003 (as plotted in Figures2 and3). The metric per unit of time (derivative of (4) withrespect to time) is shown on Fig.6 in units of entropicinformation [41].

C. Optimal and Near-Optimal Collectives

In the context of ocean sampling, not only can (4) beused to quantify the performance of a particular arrayor formation, but it also provides a means to search foroptimal sampling strategies. The glider array is viewedas a set ofN trajectoriesrk(t) satisfying the constraint

rk(t) = v, k = 1, . . . , N , (6)

where v is velocity relative to the flow and speed‖v‖ = v is fixed. Each glider generates a sequence ofmeasurement(rlk, tl) = (rk(l∆t), l∆t), where∆t is thesampling period, i.e. the time between profiles. The setof all measurements at a particular depth gathered bytheN gliders can be substituted in (4) to determine theperformance of the array that we write asφ(~r), where~r = (r1, . . . , rN ). A set of optimal trajectories for thesegliders is a set ofN curves satisfying (6) and such thatφ(~r) is minimum.

Such optimal trajectories are usually complicated andunstructured. In addition, their computation requires aminimization in a large functional space, which is notsuitable for real-time applications. In this work, insteadof optimizing individual trajectories, we consider theoptimization of collectives parametrized by a restrictednumber of parameters. For example, SectionsVI andVIIfocus on arrays of vehicles moving around ellipses.For such trajectories the parameters are the numberof ellipses and the number of vehicles per ellipse, theposition, size and eccentricity each ellipse as well asthe relative position of each pair of vehicles as theymove around their ellipses (formulated below as relativephases). Clearly, the computation of the minimum inparametrized families is a much more tractable prob-lem. However, the interest of optimizing the samplingperformance over parametrized collectives rather thanover individual trajectories extends beyond the numericalconvenience. Parametrized collectives are essential toachieve the following:

• Closed-loop control.For each proposed collective,a feedback control is designed that makes it anexponential attractor of the closed-loop dynamics.Feedback control of the collective motion providesrobustness for the relative motion of the vehicles incontrast to a decentralized tracking control of eachvehicle along its individual reference trajectory.

• Robustness. The robustness of an optimal collec-tive can be studied in terms of the derivatives of themetric with respect to the parameters of the family(see SectionVI andVII ). Small second derivativesindicate flat minima and solutions that are morerobust to perturbations such as uncertainty in GPS

11

-123.5 -123.0 -122.5 -122.0

36.0

36.2

36.4

36.6

36.8

37.0

37.2

37.4

Aug 01, 2003 06:00:00

-123.5 -123.0 -122.5 -122.0

36.0

36.2

36.4

36.6

36.8

37.0

37.2

37.4

Aug 05, 2003 08:00:00

-123.5 -123.0 -122.5 -122.0

36.0

36.2

36.4

36.6

36.8

37.0

37.2

37.4

Aug 10, 2003 18:00:00

-123.5 -123.0 -122.5 -122.0

36.0

36.2

36.4

36.6

36.8

37.0

37.2

37.4

Aug 14, 2003 20:00:00

Fig. 5. Error map at different times during the AOSN 2003 experiment. Blue represents small error (good coverage) and red and whiterepresents high error (poor coverage). The purple line encloses all the points where the error has been reduced from its initial state by atleast 85%. The sampling metric is shown on Fig6. Notice that all the gliders are clustered near the coast on August 10th explaining thedrop in coverage performance visible on Fig.6.

measurements, deviations due to the flow field orcommunication problems.

• Interpretation of the data. By restricting thechoice of collectives to specific geometries, thedata collected along these paths can more easilybe interpreted in terms of curved oceanographicsections [42].

In SectionVI , we present the development of coor-dinated control for gliders on circles and on ellipses.In SectionVII , we investigate a parametrized family ofelliptical collectives in more detail and determine theoptimal collective within this parametrized family.

D. Parametrization of collectives

Parametrized families of collectives over closed curvesinvolving the least number of parameters are circles.If we specialize to circles, the optimal parameters tobe computed are the number of circles, the numberof gliders per circle, the origin and radius of eachcircle and the relative positions of the gliders on theirrespective circles. The relative position of two glidersmoving around the same circle can be represented by the

difference in their headings; this difference is fixed sincethe gliders move at constant speed. The difference inthe headings is equal to the relative phase of the glidersaround the circle. To see this suppose the gliders move atunit speed around a single circle of radiusρ0 = |ω0|−1

and centered at the origin. The position of thekth gliderat time t is rk(t) = (ρ0 cos(ω0t+γk), ρ0 sin(ω0t+γk)),whereγk is the phase of thekth glider. The velocity ofthe glider isrk = (cos θk, sin θk) whereθk is the glider’sheading angle. Since for circular motion about the originrk ⊥ rk, the relative heading of two vehicles is equal totheir relative phase, i.e.,θj − θk = γj − γk. In the topleft panel of Figure7, two vehicles move around circleswith γ2 − γ1 = 0. In the top right panel,γ2 − γ1 = π.

Suppose now that two gliders move at unit speedabout two different circles, each with radiusρ0 butwith noncoincident centers. In this case the relativeheading (and therefore relative phase) of the two glidersremains constant and the relative position of the glidersis periodic. The periodic function can easily be describedby the relative phase and relative position of the circleorigins. Let the distance between the circle origins be

12

(a)

!1 = !2

"1 = "2

(b)

!1

!2 "1"2

(c)

!0!0

d0

d0 ! 2!0

d0 + 2!0 (d)

L = 4

Fig. 7. Cartoons of vehicles moving around closed curves withprescribed relative phases; a) Two vehicles with relative phase equalto zero move around a circle; b) Two vehicles with relative phaseequal toπ move around a circle; c) Two vehicles with relative phaseequal toπ and each vehicle moving around a different circle; d) Aclosed curve with rotational order of symmetryL = 4. Four vehiclesmove around it with fixed relative phase.

d0. Then, if the relative phase is zero, the gliders aresynchronized and their relative distance remains constantand equal tod0. If the relative phase isπ then therelative distance of the vehicles varies from its minimumatd0−2ρ0 to its maximum atd0+2ρ0. This is illustratedin the bottom left panel of Figure7.

Because relative phase is constant for vehicles movingat constant speed around circles of the same radius, weparameterize relative position of a pair of gliders bytheir relative phase. This makes the stabilizing controlproblem one of driving vehicles to circles of given ra-dius with prescribed, fixed, relative phases (equivalently,relative headings). For example, supposeN gliders are tomove around the same circle. An example of an optimalsolution in a homogeneous field is one in which thegliders are uniformly distributed around the circle (calledthe splay state). This is equivalent to phase lockingwith relative phase between neighboring gliders equalto 2π/N , which we study in the next section.

Relative phase can be useful as a prescription ofrelative position even for closed curves of more generalshape. The choices of relative phase that can be keptconstant for constant speed vehicles moving around agiven shape depend on therotational order of symmetryof the shape. The rotational order of symmetry of ashape is equal toL ∈ N0 if the shape looks unchangedafter it is rotated about its center by angle2π/L. Forexample, a hexagon has rotational symmetry of ordersix, a square has symmetry of order four, a rectangleand an ellipse have symmetry of order two. A shapewith rotational order of symmetry equal to one has norotational symmetry.

Consider a shape with rotational order of symmetryequal toL. If we choose the relative phase for a pairof gliders moving at constant speed around the shape tobe an integer multiple of2π/L, the relative phase willremain constant. An example forL = 4 is shown inFigure7. In the case of circles, as discussed above, anyrelative phase can be selected. In the case of ellipses,only two choices of relative phase can be selected; theseare either relative phase equal to zero or equal toπ,when the gliders are synchronized or anti-synchronized,respectively, as they move around a single ellipse or upto N identical ellipses with noncoincident centers.

In SectionVI we describe steering control laws forstabilization of gliders to circles and ellipses with phaselocking.

VI. COORDINATED CONTROL

This section describes feedback control laws that sta-bilize collective motion of a planar model of autonomousvehicles moving at constant speed. Following SectionV,we consider vehicles moving around closed curves withgiven, fixed relative phases. As described in SectionV-D,relative phases determine, in part, the relative positionsof the vehicles. In the case of collective motion aroundcircles of equal radius, the relative phase is identical torelative heading and is also constant. For more generalshapes, prescribed relative phases are chosen as aninteger multiple of2π/L whereL is the rotational orderof symmetry of the shape. For example, in the case ofcoordinated motion of gliders around ellipses,L = 2 andwe design stabilizing controllers that fix relative phasesto 0 or π.

Each glider is modeled as a point mass with unitmass, unit speed and steering control. We first provide afeedback control law that stabilizes circular motion of thegroup of vehicles about its center of mass. This controllaw depends on the relative position of the vehicles.Next, we address the problem of stabilizing the relativephases of the circling vehicles. An additional controlterm, depending only on the relative headings of thevehicles, stabilizes symmetric patterns of the vehiclesin the circular formation.

As long as the feedback control is a function onlyof the relative positions and headings of the vehicles,the system is invariant to rigid rotation and translationin the plane. This corresponds to the symmetry group,SE(2) = R2 ⊗ SO(2) ≡ R2 ⊗ S1, where⊗ is thesemidirect product. We show how breaking this symme-try can lead to useful variations on circular formations.First, we introduce a fixed beacon to break theR2

symmetry. Second, we introduce a reference heading

13

which breaks theS1 symmetry. In addition, we introduceblock all-to-all interconnection topologies for the spacingand orientation coupling in order to stabilize collectivemotion of subgroups of vehicles. This includes the casein which there are multiple circles with a differentsubgroup of vehicles moving around each circle.

Finally, we describe a control law to stabilize collec-tive motion on more general shapes. More specifically,we stabilize a single vehicle on an elliptical trajectoryabout a fixed beacon. Additionally, we couple vehicleson separate ellipses using their relative headings in orderto synchronize the vehicle phases about each ellipse.

A. Circular Control

The vehicle model that we study is composed ofNidentical point-mass vehicles subject to planar steeringcontrol. The vehicle model is

rk = veiθk

θk = uk, k = 1, . . . N (7)

whererk = xk + iyk ∈ C ≡ R2 and θk ∈ S1 are theposition and heading of each vehicle,v is the vehiclespeed relative to the flow, anduk is the steering controlinput to the kth vehicle. In this section, we assumeunit vehicle speed, i.e.v = 1, and ignore the flow. InSectionV, the positionrk of thekth vehicle was a vectorin R2. In this section, we exploit the isometry betweenR2 and C and we viewrk as an element of thereal3

vector spaceC. The real vector spacesC and CN giveus more flexibility in chosing an inner product4.

For the sake of brevity, we often stack identical vari-ables for each vehicle in a common vector. For example,~θ = (θ1, . . . θN ) ∈ TN contains all the headings and~r = (r1, . . . rN ) ∈ CN contains all the positions.

To help understand the model (7), consider the fol-lowing two examples of constant control input. Foruk = ω0 ∈ R0, the vehicles travel on fixed circles ofradius ρ0 = |ω0|−1. The sense of rotation is given bythe sign ofω0. For uk = ω0 = 0, each vehicle follows astraight trajectory in the direction of the initial heading.

Due to the unit speed and unit mass assumptionswe can relate the coherence of vehicle headings to themotion of the group. Let the center of mass of thegroup beR = 1

N

∑Nj=1 rj . Also, let theorder parameter

3By real vector space, we mean a vector space for which the fieldof scalars isR. Complex vector spaces are defined with complexscalars. For example,CN is both a real and a complex vector space.In this paper, we considerCN as a real vector space only.

4〈z1, z2〉 = Re˘z>1 z2

¯is not an inner product for the complex

vector spacesC because it violates sesquilinearity. However, it is avalid inner product for the real vector spacesC andCN .

p~θ ∈ C, denote the centroid of the vehicle headings onthe unit circle in the complex plane. The order parameteris equivalent to the velocity of the center of mass of thegroup, i.e.

p~θ =1N

N∑k=1

eiθk =1N

N∑k=1

rk = R.

Notice that we have∣∣p~θ∣∣ ≤ 1. We define a potential

functionU1 by

U1(~θ) =N

2|p~θ|

2. (8)

Notice that certain distinguished motion of the groupcorrespond to critical points ofU1. For instance,U1(~θ)is maximum for parallel motion of the group (∀ k :θk = θ0) and minimum when the center of mass isfixed (p~θ = R = 0). We refer to solutions for whichp~θ = R = 0 as balancedsolutions since the headingsare distributed around the unit circle in such a (balanced)way that the center of mass of the group is fixed. Letting~1 = (1, · · · , 1) ∈ RN , we have that

⟨∇U1,~1

⟩= 0; this

corresponds to theS1 rotational symmetry of the system.To stabilize circular motion of the group about its

center of mass, we introduce a dissipative control lawthat is a function of the relative positionsrkj = rk− rj .Let the vector from the center of mass to vehiclek berk = rk −R = 1

N

∑Nj=1 rkj . We propose to control the

vehicles using

uk = ω0 (1 + κ 〈rk, rk〉) , k = 1, . . . N (9)

whereκ>0 is a scalar gain. We define the inner productby

〈z1, z2〉 = Re{z>1 z2} , (10)

where zT1 represents the conjugate transpose ofz1 andRe {·} is the real part of a complex number. We viewz1 and z2 as the elements of the real vector spaceCN

(i.e., isomorphic toR2N ), for which (10) is a valid innerproduct.

The stability of the circular motion of the group abouta common point can be studied using standard Lyapunovfunctions. Consider the function

S(~r, ~θ) =12

N∑k=1

|eiθk − iω0rk|2, ω0 6= 0 , (11)

which has minimum zero for circular motion around thecenter of mass with radiusρ0 = |ω0|−1 and directionof rotation determined by the sign ofω0. DifferentiatingS(~r, ~θ) along the solutions of the vehicle model gives

S =N∑k=1

〈ω0rk, rk〉 (ω0 − uk).

14

Therefore, using the circular control (9), we find that

S = −κN∑k=1

〈ω0rk, rk〉2 ≤ 0 ,

and S is an acceptable Lyapunov function for thissystem. Consequently, solutions converge to the largestinvariant set,Λ, for which S = 0. This yields thefollowing result.

Theorem 6.1:Consider the vehicle model (7) withthe circular control (9). All solutions converge to acircular formation of radiusρ0 = |ω0|−1. Moreover,the asymptotic heading arrangement is a critical pointof the potentialU1(~θ). In particular, balanced circularformations form an asymptotically stable set of relativeequilibria.

The technical details of the proof can be found in [43].

Notice that solutions inΛ have the dynamics~θ = ω0~1,i.e. vehicles follow circles of radius|ω0|−1. The setof balanced circular solutions for which all circles arecoincident corresponds to the minimum of the potentialS(~r, ~θ). Simulations suggest that this set of equilibriahas almost global convergence.

B. Control of Relative Headings

If, in addition to the relative positions, we feed backthe relative headings of the vehicles, we can stabilizeparticular phase-locked patterns or arrangements of thevehicles in their circular formation. Let the potentialU(~θ) satisfy

⟨∇U,~1

⟩= 0. This potential is invariant

to rigid rotation of all the vehicle headings. We combinethe circular control (9) with a gradient control term asfollows:

uk = ω0 (1 + κ 〈rk, rk)〉 −∂U

∂θk. (12)

The circular motion of the group in a phase-lockedheading arrangement is a critical point ofU(θ). Thestability of the motion can be proved by showing theexistence of a Lyapunov function. For instance take,

V (~r, ~θ) = κS(~r, ~θ) + U(~θ), (13)

whereS(~r, ~θ) is defined in (11). The time derivative ofV (~r, ~θ) along the solutions of the vehicle dynamics isgiven by

V =N∑k=1

(κ 〈ω0rk, rk〉 −

∂U

∂θk

)(ω0 − uk). (14)

Substitution of the composite control (12) in (14) gives

V = −N∑k=1

(κ 〈ω0rk, rk〉 −

∂U

∂θk

)2

≤ 0.

Therefore, solutions converge to the largest invariant set,Λ, for which V = 0. A detailed proof can be foundin [43] and yields the following theorem

Theorem 6.2:Consider the vehicle model (7) and asmooth heading potentialU(θ) that satisfies

⟨∇U,~1

⟩=

0. The control law (12) enforces convergence of allsolutions to a circular formation of radiusρ0 = |ω0|−1.Moreover, the asymptotic heading arrangement is acritical point of the potentialκU1 + U . In particular,every minimum ofU for which U1 = 0 defines anasymptotically stable set of relative equilibria.

This result enables us to stabilize symmetric pat-terns of the vehicles in circular formations. Symmetric(M,N)-patterns of vehicles are characterized by2 ≤M ≤ N collocated clusters of vehicles with headingsseparated by a multiple of2πM . There is a one-to-onecorrespondence between these symmetric patterns andglobal minima of specifically designed potentials [43].In order to define these potentials, we extend the notionof the order parameter of vehicle headings to includehigher harmonics, i.e.

pm~θ

=1mN

N∑k=1

eimθk .

The objective is to consider potentials of the form

Um(~θ) =N

2|pm~θ|2,

which satisfy⟨∇Um,~1

⟩= 0. These potentials are used

to prove the following [43]:Lemma 6.1:Let 1 ≤M ≤ N be a divisor ofN . Then

~θ ∈ TN is an(M,N)-pattern if and only if it is a globalminimum of the potential

UM,N =bN

2 c∑m=1

KmUm

where⌊N2

⌋is the largest integer less than or equal toN

2andKm are arbitrary coefficients satisfying{

Km < 0 if mM ∈ N ,

Km > 0 otherwise.Theorem6.2 together with Proposition6.1 yield a

prescription for stabilizing symmetric patterns. Of par-ticular interest for mobile sensor networks is stabilizingthe circular formation in which the vehicles are evenlyspaced, i.e. the(N,N)-pattern orsplay stateformation[44]. This formation is characterized byp

m~θ= 0 for

m = 1, . . . N − 1 and |pN~θ| = 1

N . Consequently, wedefine the splay state potential to be

UN,N = K

bN

2 c∑m=1

Um, K > 0. (15)

15

−30 −20 −10 0 10 20

−20

−10

0

10

x

y

Fig. 8. A numerical simulation of the splay state formation usingthe control (16) with ω0 = 0.1 andK = 1 starting from randominitial conditions. Each vehicle and its velocity is illustrated by ablack circle and an arrow. Note that the center of mass of the group,illustrated by a crossed circle, is fixed at steady-state.

The splay state formation control law has the form (12)with U(~θ) given by (15) and can be written

uk = ω0(1+κ 〈rk, rk〉)+K

N

N∑j=1

bN/2c∑m=1

sinmθkjm

. (16)

A simulation of the splay state formation forN = 12vehicles is shown in Figure8. Twelve vehicles start fromrandom initial conditions and the controller (16) enforcesconvergence to a circular orbit with uniform spacing (i.e.,the phase difference between adjacent vehicles is2π

12 ).

C. Planar Symmetry Breaking

The feedback control laws in sectionsVI-A andVI-Brequire only the relative positions and headings of thevehicles and, consequently, they are invariant to rigidtranslation and rotation in the plane. This corresponds tothe symmetry group,SE(2) = R2 ⊗ S1. In this section,we introduce variations of these control laws which breakthe translation and rotation symmetries. First, we breakthe R2 translation symmetry by stabilizing the circularformation about a fixed beacon. Secondly, we break theS1 rotational symmetry by coupling the vehicles to aheading reference.

The position of the fixed beacon is referred to asR0 ∈ C. The relative position from the beacon is definedas rk = rk − R0. A formal proof uses the Lyapunovfunction S(~r, ~θ) defined in (11) with the new definitionof rk. Furthermore, Theorem6.2 continues to hold forcircular motion about the fixed beacon [43]. That is, thecontrol (12) can be used to stabilize circular motion tothe set of heading arrangements that are critical points

of the potentialU(~θ), where⟨∇U,~1

⟩= 0. Clearly, this

applies to the splay state potential (15).Next, we introduce a heading referenceθ0 whereθ0 =

ω0. Let uk, k = 1, . . . , N − 1 be given by (12) whereU(~θ) is a potential that satisfies

⟨∇U,~1

⟩= 0. TheN th

vehicle is coupled to the heading reference using

uN = ω0(1 + κ(rk, rk))−∂U

∂θk+ d sin(θ0−θN ), (17)

whered > 0. Critical points ofU(~θ) that satisfyθN = θ0define an asymptotically stable set [43]. To prove thisresut, we use the composite Lyapunov function

W (~r, ~θ) = V (~r, ~θ) + d(1− cos(θ0 − θN ))

whereV (~r, ~θ) is given by (13). The complete analysiscan be found in [43]. The set of circular formations thatminimizes U(~θ) and satisfyθN = θ0 are the globalminima of W (~r, ~θ). For ω0 = 0, the control (17) canbe used to track piecewise linear trajectories [45].

D. Coordinated Subgroups

In this section, we design control laws to coordinatevehicles in subgroups using block all-to-all interconnec-tion topologies. In other words, the vehicles can be dis-tributed among subgroups, each subgroup correspondingto vehicles moving on a different circle or ellipse. First,we introduce a block all-to-all interconnection topologyfor the circular control term that depends on the relativepositions. This restriction on the coupling yields stabilityof subgroups of vehicles in separate circular formations.Similarly, block all-to-all coupling applied to the gradientcontrol term that depends on relative headings yieldsheading arrangements within subgroups of vehicles. Weillustrate the use of block all-to-all couplings on ascenario of practical interest. The vehicles are divided inthree subgroups that minimize the splay state potentialsuch that each subgroup is in a splay state formation.

We refer to each vehicle subgroup by its block indexb = 1, . . . , B, whereB is the total number of blocks.Let N b be the number of vehicles in blockb. Note thatN b ≥ 2 except in the case of fixed beacons in whichN b ≥ 1.

We assume that each vehicle is assigned to one andonly one block, so that

∑Bb=1N

b = N . Also, let F b ={f b1 , . . . , f bNb} be the set of vehicles indices in blockB.The center of mass of blockb is given by

Rb =1N b

Nb∑k=1

rfbk.

16

Similarly, them-th moment of the heading distributionof block b is

pbm~θ

=1

mN b

Nb∑k=1

eimθfb

k , m = 1, 2, . . . . (18)

Using (18), we can also define block-specific headingpotentials such as

U bm(~θ) =12|pbm~θ|2. (19)

Note that ∂(Ubm)

∂θk= 0 for k /∈ F b and

⟨∇U bm,~1

⟩= 0.

Using this notation, we summarize the followingcorollaries to Theorems6.1 and6.2 [43]. First, considerblock all-to-all coupling for the circular control termonly. In this case, the control law (12) with rk = rk−Rb

and k ∈ F b enforces convergence of all solutions tocircular formations of radiusρ0 = |ω0|−1 in phasearrangements that are critical points of the potentialκU b1 + U as in Theorem6.2, whereU(~θ) is a potential

that satisfies⟨∇U,~1

⟩= 0. In particular, the circular

motion of all the vehicles in a block have coincidentcenters. Alternatively, suppose we use block all-to-allcoupling only in the gradient control term that dependson relative headings. In this case, the control law is (12),where U(~θ) =

∑Bb=1 U

b(~θ) and U b(~θ) is a potentialdepending only on the headings in blockb that satisfies⟨∇U b,~1

⟩= 0. This control enforces convergence of all

solutions to circular formations of radiusρ0 = |ω0|−1

in phase arrangements that are critical points of thepotentialsκU1 + U b.

To demonstrate the use of the control law (12), wepresent the result of a useful case of block all-to-allspacing coupling with fixed beacons. In this example,the phase coupling is both all-to-all and block all-to-allwith

U = U (N,N) +B∑b=1

U b (Nb,Nb) (20)

whereN b = N/B for b = 1, . . . , B and isU b (Nb,Nb)

is given by (15). This potential is minimized by thephase arrangement in which the entire group, as wellas each block, are in the splay state of vehicle headings.Simulation results forN = 12 andB = 3 are shown inFigure 9. The twelve vehicles start from random initialpositions and organize themselves in the splay statesusing (12).

E. Shape Control: Elliptical Beacon Control Law

In this section, we modify the circular control law andwe stabilize a single vehicle on an elliptical trajectoryabout a fixed beacon. We use a generalization of the

−40 −20 0 20 40 60−40

−20

0

20

40

x

y

Fig. 9. Simulation results forN = 12 andB = 3 starting fromrandom initial conditions with block all-to-all spacing coupling andthree fixed beacons at(R1

0,R20,R

30) = (−30, 0, 30). Phase coupling

is all-to-all and block all-to-all with the potential (20). The simulationparameters areκ = ω0 = 1/10.

potentialS(~r, ~θ) in (11) to prove Lyapunov stability ofthis trajectory. Additionally, we couple several vehiclesvia their relative headings as in SectionVI-B in order tosynchronize the vehicle phases on each ellipse.

Let R0 ∈ C and µ0 ∈ S1 represent the center andorientation of an ellipse with the lengths of the semi-major and -minor axes given bya and b. The positionsof the focii areR0 ± c eiµ0 , wherec =

√a2 − b2. Let

d ∈ C andd′ ∈ C be the relative positions of the vehiclefrom each focus, defined by

d , ρ ei(ψ+µ0) = r−R0 − c eiµ0 (21)

d′ , ρ′ ei(ψ′+µ0) = r−R0 + c eiµ0 (22)

and shown in Panel (a) of Figure10.For a single vehicle whose position and heading

are r and θ, respectively, motion along the ellipse ischaracterized by

ρ+ ρ′

2= a (23)

andψ + ψ′

2= θ − µ0 ±

π

2. (24)

Condition (23) requires that the average distance to thefocii remains constant. Condition (24) requires that theaverage angular position measured from both focii mustbe separated by±π

2 from the angle made by the velocityvector and the major axis. Notice that the term±π

2corresponds to either clockwise or counter-clockwisemotion around the ellipse. For vehicles moving contin-uously in the plane,ψ andψ′ are continuous functions.Therefore, the average(ψ + ψ′)/2 in condition (24)is moving on only one branch (clockwise or counter-clockwise) and can never switch continuously from one

17

(a)

d

d!

R0

c

rµ0

r

x

y

(b)

! ! µ0

"#

$

$!

x

y

Fig. 10. a) The vectorsd and d′ used to identify the position ofthe vehicle (white circles) relative to the focii (black circles) for anellipse centered atR0 and rotated byµ0. b) Depicts the anglesψ,ψ′, α, andφ used in the control design. Note thatφ = 0 for stableelliptical motion with positive rotation.

branch to the other. The physical interpretation of thisproperty is the following: a vehicle moving along anellipse with a constant speed cannot change its sense ofrotation and keep a continous motion. Without loss ofgenerality, we will only consider the positive (counter-clockwise) branch of condition (24). Building on thesegeometrical considerations, we define the shape coordi-nates(ξ, η, α, β, φ) given by

ξ =ρ+ ρ′

2(25)

η =ρ− ρ′

2(26)

α =ψ + ψ′

2(27)

β =ψ − ψ′

2(28)

φ = α+π

2− θ + µ0. (29)

The anglesα andφ are shown in Figure10.b. In thesecoordinates, the conditions for elliptical motion (23) and(24) are equivalent to(ξ, φ) = (a, 0) and ξ = φ = 0. Wechoose the Lyapunov function candidate

S(ξ, η, α, β, φ) =12|ξ − ae−iφ|2 (30)

which has minimum at zero for an elliptical trajectorycentered atR0 and rotated byµ0 with major and minorsemi-axes(a, b).

The time derivative of the Lyapunov function (30)along the trajectories of (48)-(52) is

S = (ξ − ae−iφ, ξ + iae−iφφ)

= (ξ − a cosφ)ξ + ξa sinφ(α− u).

The dynamics of the single vehicle in the shape co-ordinates are derived in the Appendix. Using thesecalculations and choosing the control,u, with scalar gain,κ > 0,

u = α+ κξ

asinφ+

1ξa

(ξ − a cosφ) cosβ (31)

givesS = −κξ2 sin2 φ ≤ 0. (32)

Note that for the circular casea = b = |ωo|−1, thecontrol reduces to the circular beacon control law (9),which can be expressed in the shape coordinates(ρ, φ)as

u = ω0(1 + κρ sinφ).

We obtain the following result:Theorem 6.3:Almost all trajectories of (7) for a sin-

gle vehicle subject to the control (31) converge to anelliptical trajectory centered atR0 and rotated byµ0.The size of the ellipse is parameterized by the length ofits semi-major and -minor axes,a andb.

Proof: By the Lasalle invariance principle, all tra-jectories converge to the largest invariant set for whichS = 0. Using (32), the invariance condition becomessinφ = 0 since ξ > 0. Subject to this condition, thedynamics of the shape variables(ξ, φ) from equations(48) and (52) in the appendix become

ξ = 0 (33)

φ = − 1ξa

(ξ − a cosφ) cosβ. (34)

Setting equations (33) and (34) equal to zero, we obtainthe solutions(ξ, φ) = (a, 0) and sinφ = cosβ = 0. Thelatter corresponds to trajectories on the major axis ofthe ellipse (between the focii) and does not constitute aninvariant set due to the singularities in (31) at the focii.As a result, all trajectories which do not originate at afocus of the ellipse asymptotically converge to the set forwhich (ξ, φ) = (a, 0). This set corresponds to ellipticalmotion with parameters(a, b).

We briefly discuss how to extend this result to coordi-nate groups of vehicles on (separate) ellipses by couplingtheir headings as in SectionVI-B. Let R1

0, . . . ,RN0 and

µ10, . . . , µ

N0 be the location and orientation ofN ellipses

with parameters(ak, bk). Also, let uek

k be the ellipsecontrol (31) corresponding to thekth ellipse. We assumethat the ellipses’ circumferences are all the same. Then,in order to stabilize each vehicle to its ellipse and tosynchronize the phases of all the vehicles, we proposethe control

uk = uek

k +K∂U1

∂θk, k = 1, . . . , N, (35)

18

for K > 0, whereU1 is the potential function (8). Theconvergence analysis of this control law is not pursued inthe present paper but simulations suggest good conver-gence properties. In SectionVII , we compute the optimalsampling ellipses for a group of two gliders. The optimalellipses have the same circumferences and the controllerderived here is applied to this case (see Fig.15).

VII. O PTIMAL COORDINATED SOLUTIONS

In this section, we use the sampling metric definedby (4) to compute near-optimal vehicle trajectories con-strained to ellipses. The objective of this section is todetermine the optimal ellipse parameters as a functionof the size, shape and characteristic scales of the regionof interest and the capabilities of the sensor platforms.We start by introducing a convenient formalization ofthe adaptive sampling problem using non-dimensionalparameters. Next, we present the results of numericaloptimization experiments for a single vehicle on anelliptical trajectory and for a pair of vehicles on separateellipses. Lastly, we consider the influence of a uniformflow field on the sampling performance of the ellipsefeedback control from the previous section. We antici-pate that the insights from these numerical results willextend to larger groups of vehicles.

A. Sampling Numbers

We consider a rectangular domainB of sizeBa × Bbin which we would like optimal sensor coverage duringa finite duration of timeT . The trajectories of theN vehicles, given byrk(t), and the sampling metric,φ(~r) (see (4)), determine the locations and effectivenessof the sensor measurements, respectively. The optimaltrajectories,r∗k(t), and the value of the metric at theoptimum, φ∗ , φ(~r∗), are obtained by minimizingthe metricφ among all acceptable sets of curves,rk,k = 1, . . . , N , satisfying the constant velocity constraintin (6).

We decrease the number of dimensions of the opti-mization problem by applying the Buckinghamπ theo-rem [46] to reduce the number of parameters. LetAv{·}represent the space-time average over the domainB×T .Then the initial uncertainty on the field,σ0, is given by

σ0 = Av{B(r, t, r, t)} . (36)

Recall from SectionV-A, the measurement noise isdenoted byn. In addition, we define the correlation

Name Description Dist. Time Temp.

σ0 Initial Uncertainty, (5), (36) 0 0 1σ Correlation Length, (5), (37) 1 0 0τ Correlation Time, (5), (38) 0 1 0n Measurement Noise 0 0 1v Speed of Sensors 1 -1 0Ba Width of Domain 1 0 0Bb Height of Domain 1 0 0T Duration of Experiment 0 1 0

r∗k(t) Optimal Traj. forkth Veh. 1 0 0φ∗ Minimum Metric 2 1 1

TABLE I

RELEVANT PHYSICAL QUANTITIES AND THEIR DIMENSIONS

length and the correlation time by

σ , Av{∫

R2

dr′B(r, t, r′, t)B(r, t, r, t)

}(37)

τ , Av

+∞∫

−∞

dt′B(r, t, r, t′)B(r, t, r, t)

. (38)

One can easily check that, for a Gaussian correlationfunction given by (5), the equations above an equiva-lent definition of the correlation length and time. Theadvantage of (37) and (38) is that they extend thedefinition to arbitrary correlation functionB. Notice that,strictly speaking, the integrals cannot be taken over aninfinite domain (in space and time). The domain ofinterest is usually finite andB(r, t, r′, t′) is not definedoutside this domain. In practice, the correlation functionbecomes quickly negligable when‖r− r′‖ or |t− t′|grows, therefore, integrating over a finite spatial domainB and finite interval of time[0, T ] gives essentially thesame result as (37) and (38).

We assume that the stochastic component of the fieldis well captured by these variables [47], [48]. Note, inthe case of a Gaussian covariance model in the limitB → R2 and T → ∞, the definitions (37) and (38)evaluate to the1/e ≈ 37% decorrelation scale.

Table I lists the eight relevant variables and theirrespective dimensions. We use temperature as a proxy forthe (arbitrary) units of the sensor measurements. Sincewe are looking for the minimum value of the metric,we add the variableφ∗ to the first eight variables. Therank of the matrix made by the units of this system is3(see TableI). According to the Buckinghamπ theorem[49], the relationship giving theφ∗ can be reduced toa relationship between6 non-dimensional numbers. Forpractical reasons, the following choices of these numberswill be used in this work:

• Φ = φ∗/σ0BaBbT , the normalized metric;and thesampling numbers,

19

• Sz =√BaBb/σ, the size of the domain,

• Sh = Bb/Ba, the shape of the domain,• St = T/τ , the sampling time interval,• Sp = vτ/σ, the normalizedspeed of the vehicle,• Sn = n/σ0, the sensornoise.

A similar development for the optimal trajectoriesleads to the definition of the scaled optimal trajectories

r∗k(s) =1Ba

r∗k(τ t), k = 1, . . . , N.

where s = τt is the normalized time variable. Theconstant speed constraint (relative to the flow) in (6)translates into

‖v‖ =∥∥∥∥dr∗kds

∥∥∥∥ =vτ

Ba=

Sp

Sz

√Sh . (39)

Notice that the use ofSz andSh allows us to study thesystem in terms of its size (the area of the box isσ2Sz2)and its aspect ratio (shape). BothSp andSn can be fixedor limited to a small range for a specific experiment witha homogenous group of vehicles and sensors. During anexperiment, the survey speed of the sensor platforms,v, is typically known and fixed and the characteristicspatial/temporal scales can be estimated. For example,during the AOSN 2003 experiment, the effective gliderspeed,v, (including surface intervals) was between25and 35 cm/s. The glider data was used to approximatethe average correlation length,σ≈25 km, and time,τ≈2.5 days (see [28] for details). Therefore, the samplingnumberSp was between2 and 3 for this experiment.Similarly, Sn only depends on the sensor noise and thea-priori uncertainty of the model.

In the remainder of this paper, we will only considerexperiments that last much longer than the characteristictime scale. In other words, we assume thatT >> τ or,equivalently,St >> 1. For the AOSN experiment, theestimated correlation time was 2.5 days (see [28]). Thegliders sampled the region for about a month, soSt ≈12, which is sufficiently high to validate our analysis.For T >> 1, one expects to get the same normalizedperformance for any interval of timeT . In other words,we assume that the metric per unit of area and time,Φ, isindependent of the sampling time,St. We summarize thefunctional dependence of the normalized performancemetric on the four remaining sampling numbers by

Φ = Φ(Sz,Sh,Sp,Sn). (40)

In the next subsection, we compute the near-optimaltrajectories of a single vehicle among a family of ellipses.These racetracks can be pre-computed or, alternatively,optimized in real-time to maximize the steady-state per-formance of the array. The feedback control presented

in SectionVI is essential to maintain the vehicles onthese optimal tracks in the presence of strong currentsand communication difficulties.

B. Optimal Ellipses in Rectangular Domains

In this subsection, we present optimization resultsfor a single vehicle following a parameterized ellipticaltrajectory in a rectangular domain. The objective is tofind the set of parameters yielding the smallest value ofthe metric, (40), as a function of the sampling numbers.A system with only one sensor moving on an ellipticalpath has six degrees of freedom: the position and orien-tation of the ellipse, the lengths of the semi-major and-minor axesa andb, and the initial phase,γ(0). One caneasily check that these six parameters determine a uniquetrajectory for the vehicle (up to the sense of rotation).

Inspection of (4) directly reveals that the center of theoptimal ellipse necessarily coincides with the middle ofthe boxB. Moreover, the angleγ(0) has no influence onthe metric forSt>> 1 and can be ignored. In addition,we assume that the ellipse orientationµ0 is parallel tothe long side of the box.

For given sizeSz, shapeSh, sensor noiseSn andrelative vehicle speedSp, the problem reduces to a two-dimensional space where the variables are the lengthsof the semi-major and -minor axes of the ellipse,a andb. For example, Figure11 shows the contour levels ofthe metric, i.e. the error map, as a function ofa and bfor the sampling numbersSz = 2, Sh = 1, Sn = 0.1and Sp = 3. There is a unique minimum for a vehiclemoving on a circle of radiusa = b = 0.256. The factthat the optimal ellipse is a circle is consistent with thesquare shape of the domain.

Also notice that the minimum in Figure11 is relatively“flat”. Small deviations from a prescribed optimal plando not have much influence on the metric; this suggestsrobustness to disturbances such as strong currents andintermittent feedback. The error map associated with thisoptimal trajectory is shown in the upper left panel ofFigure 12. Next, we investigate the influence of eachsampling number on the optimal elliptical solution.

1) Independence of the ShapeSh: In Figure 13, weplot the performance of optimal elliptical trajectoriesfor a single vehicle within the rectangular boxB as afunction of the sampling numbersSz andSh. The shapeof the optimal trajectory varies with the shape of thedomain; however, the contour levels ofΦ in the(Sz,Sh)-plane reveal thatΦ does not depend onSh. As a result,the same performance can be achieved on rectangleswith different aspect ratio but with the same area. Inparticular, if a complex domain such as Monterey Bay

20

r

s

0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4

Fig. 11. Non-dimensional metricΦ for one vehicle on an ellipticaltrajectory with semi-major and -minor axis lengthsa and b. Thesampling numbers areSz = 2, Sh = 1, Sn = 0.1, Sp = 3. Theminimum gives the near optimal ellipse (a circle)a = b = 0.256.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

r

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

q

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

s

Fig. 12. Snapshots in time of the error maps associated with the nearoptimal elliptical trajectories for selected values of the parameters.Sn = 0.1 andSp = 3.

0.010.01

0.10.1

0.20.2

0.30.3

0.4

0.50.5

0.6

0.7

0.80.8

0.9

0.9

D

S

0 2 4 6 81

2

3

q

Fig. 13. Optimal metricΦ as a function ofSz and Sh for Sn =0.1 and Sp = 3 with a single vehicle on an elliptical trajectory.Within numerical accuracy,Φ is independent ofSh, the shape ofthe domain. The same performance can be achieved on a rectangleof any aspect ratio (with the appropriate optimal trajectory that willvary with shape).

is divided in several sub-regions patrolled by groups ofgliders, the shape of the sub-regions can be chosen freely.This permits a greater flexibility in designing samplingplans.

2) Role of SpeedSp and NoiseSn: To study theinfluence of the sampling numbersSp, Sn andSz on theoptimal trajectories, the optimal ellipses and the mini-mum value of the metric are computed for several valuesof the sampling numbers. For example, see Figure12 fortypical error maps. We have already determined that theshapeSh and the time numberSt do not influence thesolutions so we present results forSh = 1 andSt � 1.Figure14 gives the optimal non-dimensionalized radius(a = b) and the minimum value of the metric,Φ, asa function of Sz. Each curve corresponds to differentvalues ofSn andSp. Notice that, forSz > Sp, Φ becomesindependent ofSn.

Figure14 also shows thatSp has no influence on theperformance (although it does determine the perimeterof the optimal trajectory). The minimum value ofΦ isdetermined entirely by the noiseSn and the size of thedomain Sz. On the other hand, the optimal trajectory(i.e., the radius of the circle) is a function ofSz andSp,but does not depend on the measurement noiseSn. Thisis an important result that allows us to design optimaltrajectories independently of the precision of the sensors.

C. Multiple Vehicle Results

In this section, we study the optimal elliptical trajec-tories for two vehicles in a square spatial domain. We

21

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

Fig. 15. Optimal ellipse trajectories for two vehicles in a square domain withSz = 1. The left column shows the simulated trajectories usingthe feedback control from SectionVI to stabilize the vehicles to the optimal ellipses with the control gainsκk = 1/ak andK = 0.05, whereak is the semi-major axis of thekth ellipse fork = 1, 2. The right column shows the resulting error maps for the steady-state measurementdistribution. The rows represent simulations #1, #2, #3, #6 (see TableII ).

22

z

p

10-4 10-3 10-2 10-1 100 101 102 103

10-5

10-4

10-3

10-2

10-1

100

C

D

A

B

F

z

p

10-1 100 101 102

0.1

0.2

0.3

0.4

0.5

C

D

A

B

Fig. 14. Top Panel: Value of the metric for the optimal circulartrajectory of one vehicle as a function ofSz. Bottom Panel: Radiusof the optimal circle as a function ofSz. Each curve correspond todifferent values of the sensor noiseSn and the vehicle speedSp.Notice thatΦ does not depend onSp. Moreover, the optimal radiusdoes not depend onSn.

also consider the influence of the flow field on the ellipsefeedback control from SectionVI using the performancemetric. We assign the sampling numbersSz = 1 andSh = 1 in order to simplify analysis of the results.We use the feedback control to simulate the vehicletrajectories on the optimal ellipses. The top panels ofFigure 15 show these trajectories and snapshots of theresulting error map.

For these sampling numbers, the coverage metric isminimized for two ellipses that are (nearly) centeredalong the horizontal axis. The optimal relative phasedifference between the vehicles is zero, i.e. they aresynchronized. The vehicles remain synchronized despitethe fact that the optimal ellipses have different eccentric-

Sim Flow direction Heading Coupling Metric (< better)#1 N/A on 0.018#2 0o on 0.020#3 90o on 0.054#4 180o on 0.023#5 270o on 0.101#6 N/A off 0.236

TABLE II

METRIC FOR SIMULATED OPTIMAL TRAJECTORIES.

ities because they have the same perimeter5. Any shiftin the respective position of the vehicles (e.g., delay orcurrent impeding one vehicle) decreases the performanceof the coverage metric [50]. Notice that, in the absence ofinhomogeneities and currents, there are four equivalentsolutions corresponding to the two ellipses of Figure15and the same ellipses rotated by90, 180 and270 degrees.

1) Influence of Flow Field:To study the robustnessof the solution, we used the controller designed inSectionVI to stabilize the vehicles to the optimal ellipsesin the presence of currents. TableII summarizes theseexperiments with the magnitude of the flow speed equalto 2% of the vehicle speed6. The path of the vehiclesconverging toward their optimal ellipses can be seen onthe left panels of Figure15. The corresponding errormaps are shown in the right panels of Figure15.

Comparing simulations #2 and #4 in TableII , weobserve that currents in the longitudinal direction (i.e.,aligned with the major axis of the ellipses) have avery small effect on the performance. On the otherhand, transverse currents have a dramatic effect on thesampling metric. This result contradicts the intuitiveresult that high eccentricity vehicle trajectories shouldnot be aligned with the prevailing currents.

2) Role of Heading Synchronization:Clearly, the abil-ity of the controller to maintain the “synchronization”of the vehicles is, in large part, responsible for theperformance achieved by simulations #1, #2 and #4. Todemonstrate the influence of the synchronization, a sim-ulation was run without the heading coupling describedin Section VI . The performance for such an array is

5We attribute the1.4% difference in the optimal ellipse perimetersto numerical errors in the computation of the metric as well as to thefinite optimization time (i.e., the solution may not have completelyconverged). For the numerical simulation of the ellipse control law,we perturbed the four optimal ellipse parameters(a1, b1, a2, b2) inorder to more precisely match their perimeters without any apprecia-ble degradation of the performance metric.

6We limited the flow speed to2% because larger magnitudeflow velocity significantly distorted the vehicle trajectories due tosingularities in the ellipse control law which occur when the vehiclepasses near a focus of the ellipse. This is a deficiency in the controllerwhich needs to be addressed in future work.

23

dramatically worse than the synchronized case. TableIIshows that, without heading coupling, the network ofvehicles performs even worse in the absence of currentsthan the synchronized array in the presence of currents.

VIII. F INAL REMARKS

We present developments on the design of mobilesensor networks that optimize sampling performance de-fined in terms of uncertainty in an estimate of a sampledfield over a fixed area. The general problem that we pose,and thus, the methodology that we develop, pertains tomobile sensor networks in a number of domains: land,air, space and underwater.

We address a number of general issues as well assome of the particular issues that distinguish mobilesensor networks in the ocean. For example, we make oursolutions robust to strong currents that can push aroundslow moving mobile sensors by determining optimalsolutions in the presence of currents, choosing solutionswith performance robust to small deviations and design-ing feedback control to stably coordinate vehicles.

We determine optimal, coordinated trajectories ofmobile sensors over a parametrized family of trajec-tories. This family consists of multiple closed curves(we specialize to ellipses), each with multiple sensorsmoving at constant speed. The relative positions of thesensors on these curves are parametrized by relativephases. This low-dimensional parametrization simplifiesthe optimization problem and motivates the coordinatedfeedback control laws that include terms modelled aftercoupled phase oscillator dynamics.

We present optimal solutions in several cases. Forexample, two sensors, each moving around a differentellipse, are optimized when their phases are synchro-nized. Sampling performance is significantly enhancedfor the closed-loop system with the coordinating feed-back control enabled. In the presence of a constant flowfield, the solution (with feedback control) with the majoraxes of the two ellipses aligned with the flow provideshigher performance than in the case the flow is alignedwith the minor axes of the ellipses.

In related work we are investigating inhomoge-neous statistics and alternative methods for computingand adapting the sampling metric. We are developingmethodology to further treat and exploit the flow field,to address a range of scales in the sampled field ofinterest and to make use of a heterogeneous sensornetwork. We are also investigating how well the dataset that optimizes the coverage metric presented in thispaper serves the needs of specific high resolution oceanforecasting models.

We describe in the paper a number of practical andcritical challenges of operating mobile sensor networksin the ocean: limitations on communication, computingand control, including inherent asynchronicities and la-tencies. We discuss how we have handled these chal-lenges in previous field work. However, these and otherproblems related to time and energy optimality remainoutstanding open problems of great interest.

Up until recently, our focus has been the optimal de-sign for Eulerian data assimilation. Recent developmentsin data assimilation extend this concept to Lagrangiandata assimilation [51], [52]. In a Lagrangian assimilationscheme, the paths of passive tracers or drifters (asopposed to an estimate of the Eulerian velocity) areassimilated directly into the ocean model. Although itwas developed for float data [53], [52], Lagrangian dataassimilation represents an exciting application for quasi-Lagrangian (i.e., weakly propelled) gliders. In particular,a Lagrangian metric and corresponding optimal trajecto-ries could be substituted into the usual objective analysisscheme.

IX. A CKNOWLEDGMENTS

The authors are grateful to Russ Davis (Scripps In-stitution of Oceanography) for sharing his experimentaldata and providing assistance in using it. This paperhas greatly benefited from many enlightening discussionswith Russ Davis, Pierre Lermusiaux (Harvard Univer-sity), Eddie Fiorelli, Pradeep Bhatta, Fumin Zhang andSpring Berman (Princeton University) and Ralf Bach-mayer (National Research Council of Canada).

Figure 4 was provided by Eddie Fiorelli, PradeepBhatta and Ralf Bachmayer. The authors are grateful toEddie Fiorelli and Fumin Zhang (Princeton University)for sharing their latests advances in formation design andcontrol.

This research was funded in part by Office of NavalResearch grants N0014-02-1-0826 and N00014–02–1–0861 and N00014–04–1–0534 and in part by the Bel-gian Programme on Inter-university Poles of Attraction,initiated by the Belgian State, Prime Minister’s Officefor Science, Technology and Culture. Derek Paley wassupported by a National Science Foundation GraduateResearch Fellowship, the Princeton Gordon Wu GraduateFellowship and the Pew Charitable Trust Grant 2000-002558.

APPENDIX

SHAPE DYNAMICS FOR ELLIPTICAL CONTROL

We first derive the dynamics of a single vehicle inthe coordinates(ρ, ρ′, ψ, ψ′, θ). Differentiating the defi-nitions (21) and (22) using R0 = µ0 = 0 and applying

24

the model (7) for a single vehicle gives

d = ρei(ψ+µ0) + iρei(ψ+µ0)ψ = eiθ (41)

d′ = ρ′ei(ψ′+µ0) + iρ′ei(ψ

′+µ0)ψ′ = eiθ. (42)

Identifying the real and imaginary terms of (41) and (42)produces the system of equations,

ρ = cos(θ − µ0 − ψ) (43)

ρ′ = cos(θ − µ0 − ψ′) (44)

ψ =1ρ

sin(θ − µ0 − ψ) (45)

ψ′ =1ρ′

sin(θ − µ0 − ψ′) (46)

θ = u. (47)

In shape coordinates(ξ, η, α, β, φ), the system of equa-tions (43)-(47) becomes

ξ =12(sin(β+φ)− sin(β−φ)) = cosβ sinφ (48)

η =12(sin(β+φ) + sin(β−φ)) = sinβ cosφ (49)

α =12

(1

ξ+ηcos(β+φ) +

1ξ−η

cos(β−φ))

(50)

β =12

(1

ξ+ηcos(β+φ)− 1

ξ−ηcos(β−φ)

)(51)

φ = α− u. (52)

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