Proc. 13th Int. Symp. on Unmanned Untethered Submersible Technology (UUST), August 2003

Adaptive Sampling Using Feedback Control of an Autonomous

Underwater Glider Fleet

Edward Fiorelli, Pradeep Bhatta, Naomi Ehrich Leonard∗

Mechanical and Aerospace Engineering

Princeton University

Princeton, NJ 08544

eddie@princeton.edu, pradeep@princeton.edu, naomi@princeton.edu

Igor Shulman †

Naval Research Laboratory

Code 7331, Bldg 1009

Stennis Space Center, MS 39529

igor.shulman@nrlssc.navy.mil

Abstract

In this paper we present strategies for adaptive sam-pling using Autonomous Underwater Vehicle (AUV)fleets. The central theme of our strategies is the use offeedback that integrates distributed in-situ measure-ments into a coordinated mission planner. The mea-surements consist of GPS updates and estimated gra-dients of the environmental fields (e.g., temperature)that are used to navigate the AUV fleets enabling ef-fective front tracking and/or feature detection. Tothis effect these fleets are required to translate to col-lect and seek good data, expand/contract to effectchanges in sensor resolution, and rotate and reconfig-ure to maximize sensing coverage, all while retaininga prescribed formation. These strategies play a keyrole in directing a cooperative fleet of autonomous un-derwater gliders in the first experiment of the Office ofNaval Research sponsored Autonomous Ocean Sam-pling Network II (AOSN-II) project in Monterey Bay,during August-September 2003. We present the coor-dination framework and investigate the effectivenessof our sampling strategies in the context of AOSN-IIvia detailed simulations.

∗Research partially supported by the Office of Naval Re-search under grants N00014–02–1–0826 and N00014–02–1–0861, by the National Science Foundation under grant CCR–9980058 and by the Air Force Office of Scientific Research un-der grant F49620-01-1-0382.

†Funded through NRL project under Program Element601153N sponsored by ONR grant N0001403WX20819. NRLcontribution number is AB/7330-03-24.

1 Introduction

We have developed adaptive sampling strategies fora fleet of Autonomous Underwater Vehicles (AUV’s).A central application focus has been the coordinatedand cooperative control of a group of autonomousunderwater gliders as part of the first experiment ofthe Office of Naval Research sponsored AutonomousOcean Sampling Network II (AOSN-II) project inMonterey Bay, during August-September 2003.

AOSN-II is a multi-institutional, collaborative re-search program with the central objective “to quan-tify the gain in predictive skill for principal circu-lation trajectories, transport at critical points andnear-shore bioluminescence potential in MontereyBay as a function of model-guided, remote adaptivesampling using a network of autonomous underwa-ter vehicles” [5]. The long-term goal centers on thedevelopment of a sustainable and portable, adaptive,coupled observation/modeling system. According tothe project charter [1], “the system will use oceano-graphic models to assimilate data from a variety ofplatforms and sensors into synoptic views of oceano-graphic fields and fluxes.” Of central importance,“the system will adapt deployment of mobile sensorsto improve performance and optimize detection andmeasurement of fields and features of particular in-terest.”

A novel feature of AOSN-II and the central themeof this paper is the use of adaptive sampling pathsand sensor network patterns. The objective is to en-

1

able the gliders to collect data that is most useful forunderstanding the science, and for providing accurateand efficient forecasts while making best possible useof the gliders’ unique endurance and maneuverabilitycapabilities ([6], [9]).

The long endurance feature of the glider comes atthe cost of limited onboard processing and communi-cation. The current generation of underwater glidersthat are used for AOSN-II can do basic pre-definedwaypoint tracking and can only communicate whileon the surface. This communication constraint leadsto intermittent feedback, which renders the task ofcoordination challenging since the position and esti-mated gradient information are not available contin-uously.

An important innovation introduced for the Mon-terey Bay 2003 Experiment is multi-scale adaptivesampling (cf. [9]). On a daily basis, the data col-lected by the gliders is assimilated into the oceanforecast models, and the ocean forecasts are used tohelp guide the gliders. On a more frequent basis, e.g.,every two hours, the gliders can feedback their mea-surements and accept updates to their motion plans(waypoint specifications). We incorporate our coop-erative control strategies [14] into the mission plan atthe two-hourly time-scale. In particular, we use themeasurements every two hours to replan the pathsof the gliders for cooperative gradient climbing, forcoordinated sampling patterns and more. The co-operative gradient climbing is intended to allow theglider fleet make use of observational data and there-fore overcome errors in forecast data.

In order to demonstrate and ascertain control pa-rameters, we run detailed simulations of an au-tonomous glider fleet performing cooperative temper-ature front tracking. We describe the results of thesesimulations in this paper. The simulations make useof two sets of temperature data from the MOOSUpper-Water-Column Science Experiment (MUSE)experiment performed in Monterey Bay, Californiain August 2000. Innovative Coastal-Ocean Obser-vation Network (ICON) model data [17] is used asthe forecast model data set and the aircraft-observedSST data is used as the “truth” data set. ICON alsosupplies the flow field truth data set in which thesimulated gliders are advected. We chose to simulateusing data from August 17, 2000. During this time,there is a temperature front near the 36.8-degree Nparallel in the ICON data (forecast set). However theaircraft-observed SST data (truth set) indicates thefront is southwest of the front predicted by the ICONdata. We describe a simulation of a glider fleet thatis directed along the temperature front identified inthe ICON prediction. By utilizing the feedback tem-

perature measurements and generating in-situ gradi-ent estimates, the fleet tries to redirect itself towardswhat it perceives to be the actual temperature front.

The remainder of the paper is organized as follows.In §2 we provide a physical description of the under-water gliders and define their use in coordinated andcooperative adaptive sampling. The methodology bywhich we coordinate fleets of underwater gliders andperform adaptive gradient climbing is presented in§3. In §4 we present example adaptive sampling sce-narios such as adaptive gradient climbing. In §5 wegive details on the simulations conducted to test ourstrategies and ascertain control parameters on truthand model data sets. In §6 we conclude with finalremarks.

2 Autonomous Underwater

Gliders and Adaptive Sam-

pling

2.1 Autonomous Underwater Gliders

Gliders are energy efficient AUV’s designed for con-tinuous, long-term deployment. The underwaterglider concept, which was originally conceived byHenry Stommel [19], offers a highly useful technol-ogy with far reaching implications to ocean science.Currently there are several underwater gliders be-ing developed and employed for scientific research[21],[16],[7].

Since gliders are significantly inexpensive com-pared to conventional AUV’s, they can be deployedin large numbers. Their long durability makes themsuitable for autonomous, large-scale ocean surveys.The energy efficiency of the gliders is due in part tothe use of a buoyancy engine. The gliders changetheir net buoyancy to induce motion in the verticaldirection. In addition, they have fixed wings thatprovide lift at non-zero angles of attack, and thus in-duce a motion in the horizontal direction. They alsoredistribute their internal mass to control their atti-tude.

The nominal motion of the glider in the longitudi-nal plane is along a sawtooth trajectory. The electricSLOCUM gliders have a rudder for heading control[21]. Heading is controlled in other gliders using ac-tive roll effected by the internal mass redistributionmechanism.

The AOSN-II project employs twelve SLOCUMgliders, owned and operated by Dr. David Fratan-toni of the Woods Hole Oceanographic Institution.Figure 1 shows a SLOCUM glider.

2

Figure 1: SLOCUM glider.

The SLOCUM has a piston-type ballast tank usedto control the net buoyancy. A battery pack thatcan be moved along the long axis of the glider servesas the controlled internal moving mass. The headingof the glider is servo-controlled using a tailfin ruddermechanism. The glider uses an inclinometer to mea-sure its attitude and uses a pressure measurement toestimate its depth (below sea level). It has an on-board altimeter to measure the distance to sea floor.

In AOSN-II the SLOCUM gliders will use anIridium-based, global communication system. Theglider can communicate only when it is at the sur-face. The Iridium communication currently operateson a limited bandwidth, resulting in slow data ex-change rates.

The glider has an antenna fixed to the tail. In or-der to establish communication it raises the tail bypumping an air bladder (located at the rear of thevehicle) when it is at the surface. The Iridium com-munication is not established immediately. This con-tributes to the amount of time the glider spends onthe surface. After the communication is establishedthe gliders spend several minutes exchanging data,before diving back with updated onboard informa-tion.

The AOSN-II SLOCUM gliders are equipped witha variety of sensors for gathering data useful for oceanscientists. This includes sensors to measure conduc-tivity, temperature, depth, fluorescence and opticalbackscatter. The onboard sensors allow the applica-tion of multi-vehicle control algorithms described in§3 for detecting interesting oceanographic and ecolog-ical features such as upwelling plumes and chlorophyllconcentrations.

In this paper we assume the glider moves at a con-stant horizontal speed of 40 cm/s and a constant ver-tical speed of 20 cm/s, relative to water, whenever itis not at the ocean surface. While at the surface the

glider is assumed to simply drift with the local flow.Adding the local flow velocity to the glider velocityrelative to water yields the absolute glider velocity(with respect to an earth-fixed reference frame).

The glider is simulated to track onboard waypoints.The glider servos its heading such that it alwaystries to move directly towards the next waypoint.The input-output relationship between the rudderposition and heading is simulated using a 3 dimen-sional, black-box model derived from experimentallyobserved data. The data used was taken from gliderexperiments in Buzzard’s Bay and was made availableby Dr. David Fratantoni of the Woods Hole Oceano-graphic Institution. A proportional control servos theheading to the desired heading. The waypoints areupdated every time the glider surfaces.

2.2 Adaptive Sampling

The central objective of adaptive sampling is to im-prove the utility of the sensing array through thefeedback of observations. During AOSN-II, this ob-servational feedback is performed for the gliders attwo distinct time scales: the daily time scale and the2-3 hourly time scale. Glider observations of envi-ronmental variables (together with data from othersensor platforms) are assimilated into ocean modelswhich provide oceanographic forecasts. These fore-casts are interpreted on a daily basis to provide base-line sampling mission plans for the glider fleet for thenext day. Every two to three hours, feedback is usedto provide either coordinated control or cooperativecontrol to direct the glider fleet. Coordinated controluses the GPS fixes of the gliders and local currentestimates measured every two hours to enable thepaths, formations and patterns that were planned atthe start of the day. Cooperative control goes furtherand modifies paths, formations and patterns in re-sponse to the gliders’ science sensor measurements aswell as the GPS fixes and current estimates.

Both our coordinated and cooperative control im-plementations attempt to maintain the glider groupin formation. A central theme of our adaptive sam-pling approach is to use gliders in formation to ac-quire spatially distributed measurements to computegradients in the measured field. In the case of coordi-nated control, the gradient information can be usedfor post processing purposes. In the case of coopera-tive control, the gradient estimate is used in real timeto redirect the group. Two particular formations use-ful for gradient estimation are the ring-type and theline-type. In the plane, the set of ring formationsconsists of inscribed polygons, Figure 2a. These for-mations have been shown to be optimal for estimat-

3

ing gradients in the presence of noise [15]. Line for-mations, Figure 2b, allow computation of directionalderivatives along the direction of the line. With fre-quent enough sampling, the gradient in the directionof motion can be estimated as well, making it possibleto estimate the full gradient if θ 6= 0.

d0

(a)

d0

q

(b)

Figure 2: Formation examples for 3 gliders. Reddotted lines illustrate formations. Blue dashed arrowsdenotes net group direction of travel. Not to scale. (a)3-vehicle triangle formation. Blue circle denotes groupcenter. (b) 3-vehicle line formation. Green line denotestangent to instantaneous group direction of travel. Not

to scale.

Three gliders in formation is the minimum num-ber to compute gradients in the plane given instan-taneous measurements. With four and five glidersin a network, crude second-order derivatives can beestimated, and with six gliders a least-squares es-timate of the second-order derivatives can be com-puted. Second-order derivatives facilitate the com-putation of the thermal front parameter (TFP). Themaxima in thermal front parameters are used to de-fine frontal boundaries. Figure 3 illustrates forma-tions of four and five vehicles of particular utility.

Adaptive sampling is also performed to capturevarying spatial scales. In AOSN-II, the above de-scribed formations are proposed to sample the fol-lowing:

1. Meso-scale features,

2. Finer-scale biological features,

3. Finer-scale internal wave dynamics.

The nature of the sampling strategies to capture thesespatial scales do not differ greatly across the threecategories. One category is distinguished from an-other by the resolution of the glider group (inter-vehicle distance). Illustrations of coordinated and co-operative adaptive sampling scenarios are presentedin §4.

Sensor coverage and gradient climbing problemswith vehicle networks are gaining interest in the liter-ature. For example, in [4], Voronoi diagrams are usedto design control laws for optimal sensor coverage for

(a) (b)

(c) (d)

Figure 3: Formations with 4 or 5 gliders. Not to

scale. (a) Four-vehicle ring formation (square) providesgood gradient estimation (optimal formation). (b) Four-vehicle complex triangle can be useful for second-orderderivative computation. (c) Five-vehicle ring formation(pentagon) provides good gradient estimation (optimalformation). (d) Five-vehicle cross formation can be usefulfor second-order derivative computation.

a field known a priori . In [2] and [13] gradient climb-ing is explored for vehicles networks wherein vehiclescompute gradients along their line of motion only. In[15] a least-squares approach is presented to estimatethe full gradient using distributed measurements pro-vided by a vehicle network.

3 Coordinated Control of Un-

derwater Gliders

In this section we present the Virtual Body and Artifi-cial Potential (VBAP) multi-vehicle control method-ology used to enable stable group motion while per-forming adaptive sampling. Originally presented in[14] for a general vehicle, here we specialize to under-water gliders. We also extend the method to includea new artificial potential designed to set the orienta-tion of vehicles about virtual leaders.

VBAP relies on artificial potentials and virtualbodies to coordinate a group of point masses (withunit mass) in a provably stable manner. The arti-ficial potentials define interactions between vehiclesthat are realized with control forces. The artificial po-tentials also define interactions between vehicles andreference points on the virtual body that we refer toas virtual leaders.

In §3.1 we present a control law that stably coordi-nates a group of unconstrained fully-actuated vehicles

4

to a desired configuration that is at rest or moving ata constant speed using virtual bodies. We then as-sign dynamics to the virtual bodies to achieve desiredgroup translations, expansions/contractions, defor-mations, and rotations, in §3.2. In §3.3 we modify thecontrol laws to accommodate constant speed equalityconstraints and in §3.4 we discuss how we coordinateour groups in the presence of the external currents,surfacing asynchronicities, and mission-update laten-cies. Finally, in §3.5 we present our method for dis-cretizing the continuous glider trajectories generatedby VBAP into optimal waypoint lists.

Since we will specialize to underwater gliders, weonly consider coordination in the plane. However, theresults of this section are applicable in three dimen-sions as well.

3.1 Stationary Formations

Let the position of the ith vehicle in a group of Nvehicles, with respect to an inertial frame, be givenby a vector xi ∈ R

2, i = 1, . . . , N as shown in Figure4. The position of the kth virtual leader with respectto the inertial frame is bk ∈ R

2, for k = 1, . . . ,M .Assume that the virtual leaders are linked, i.e., letthem form a virtual body. The position vector fromthe origin of the inertial frame to the center of massof the virtual body is denoted r ∈ R

2, as shown inFigure 4. The control force on the ith vehicle is givenby ui ∈ R

2. Since we assume full actuation, the dy-namics can be written for i = 1, . . . , N as

xi = ui.

i

jxi

xjr

bk xij

hik

jkh

X

Y

0

jk

lk

Figure 4: Notation for framework. Blue circlesare virtual leaders.

Let xij = xi − xj ∈ R2 and hik = xi − bk ∈ R

2.Between every pair of vehicles i and j we define anartificial potential VI(xij) which depends on the dis-tance between the ith and jth vehicles. Similarly,

between every vehicle i and every virtual leader k wedefine an artificial potential Vh(hik) which dependson the distance between the ith vehicle and kth vir-tual leader.

In [14] and [15] additional virtual leaders are usedto enforce the orientation of the group as a whole.Here, a third potential, Vr(θik) is introduced for ori-entation control between vehicles within the group.This potential depends on the angle between a refer-ence line attached to the kth virtual leader, lk, andthe vector hik. This potential also provides an in-tuitive means to enable new formations and groupreconfigurations.

The control law for the ith vehicle, ui, is definedas minus the gradient of the sum of these potentials:

ui = upi (1)

= −

N∑

j 6=i

∇xiVI(xij)−

M∑

k=1

(∇xiVh(hik)+∇xi

Vr(θik))

=

N∑

j 6=i

fI(xij)

‖xij‖xij+

M∑

k=1

(

fh(hik)

‖hik‖hik+

fr(θik)

‖hik‖h⊥ik

)

.

h⊥ik is a unit vector perpendicular to hik. At thispoint we have not constrained the vehicle speeds toreflect the constraints of an actual glider. We will doso shortly.

We consider the form of potential VI that yieldsa force that is repelling when a pair of vehicles istoo close, i.e., when ‖xij‖ < d0, attracting when thevehicles are too far, i.e., when ‖xij‖ > d0 and zerowhen the vehicles are very far apart ‖xij‖ ≥ d1 > d0,where d0 and d1 are constant design parameters. Thepotential Vh is designed similarly with possibly dif-ferent design parameters h0 and h1 (among others).Typical forms for fI and fh are shown in Figure 5.

The potential Vr enforces desired orientationsabout virtual leaders. It yields a force perpendicu-lar to the line between vehicles and virtual leaders todrive θik to one of a set of desired angles prescribedby φk. For example, the potential

Vr(θik) =

{

αr

2 (1− cos(N(θik − φk))) ||hik|| ≤ h1

0 ||hik|| > 0

where αr is the amplitude of the potential, has equi-

libria (∇xiVr(θ

∗ik) = 0) at θ∗ik = φk +

∑Nj=1

2π(j−1)N

.Figure 6 illustrates this potential for αr = 1, N = 3and φ1 = 0. The discontinuity in Vr is a result ofthe desire to obtain local interactions between virtualleaders and vehicles while introducing the potentialin summation with Vh and VI . We are currently in-vestigating composite potentials to get both desired

5

radial separation and angular placement without dis-continuities.

Figure 5: Representative control forces fI andfh derived from artificial potentials.

−10−7.5

−5−2.5

02.5

57.5

10

−10−7.5

−5−2.5

02.5

57.5

100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

XY

Vr

Figure 6: Representative rotational artificialpotential, Vr(θik). Virtual leader is located at theorigin.

Note that each vehicle uses exactly the same con-trol law and is influenced only by near neighbor vehi-cles, i.e., those within a ball of radius d1, and nearbyvirtual leaders, i.e., those within a ball of radius h1.The global minimum of the sum of all the artificialpotentials consists of a configuration in which neigh-boring vehicles are spaced a distance d0 from one an-other, a distance h0 from neighboring virtual leaders,while the group is at the desired orientations abouteach virtual leader given by φk. The global minimumwill exist for appropriate choice of d1, h1, and φk butit will not in general be unique. Since a particularvehicle can play any role in the network, any permu-tation of vehicles in a global minimum configurationwill correspond to the global minimum.

Since we are considering first-order dynamics, thestate of the vehicle group is x = (x1, . . . , xN ). In[10], local asymptotic stability of x = xeq correspond-ing to the vehicles at rest at the global minimum of

the sum of the artificial potentials is proved with theLyapunov function

V (x)=

N−1∑

i=1

N∑

j=i+1

VI(xij)+

N∑

i=1

M∑

k=1

(Vh(hik) + Vr(θik)) .

(2)This Lyapunov function will be important as a for-mation error function and it will be used in a controllaw that will guarantee bounded formation motion.

3.2 Formation Motion

In §3.1 a method for stabilizing the formation to adesired configuration was presented for a stationaryvirtual body. In this section, we prescribe dynamicsto the virtual body to achieve group translation, ex-pansion/contraction, deformation, and rotation. Weidentify the configuration space of the virtual bodyto include the translations, r and rotations, R, of thevirtual body, i.e. (r,R) ∈ SE(2), and configurationvariables for group size (expansion/contraction) andgroup deformations, i.e. (k, φk) ∈ R

M+1 where Mis the number of virtual leaders. We distinguish be-tween the action of R, which specifies the orientationof the virtual body about its center, and φk whichspecifies desired vehicle orientations about the kthvirtual leader. φk does not alter the location and ori-entation of the virtual leaders which constitute thevirtual body.

Stable group motion is obtained by parameteriz-ing a path in configuration space by a scalar variables and specifying a control law for s in terms of aformation error. This control law defines s, evolv-ing in time, in a manner which guarantees that theformation error remains below a user-specified upperbound for all time or until the destination is reachedat which time s is set to zero.

The particular choice of path in configuration spaceis independent of the stabilizing control law for s,so it can be chosen to satisfy the mission require-ments. Accordingly, there is a decoupling of the for-mation maintenance problem and the mission prob-lem. Paths and mission requirements are the subjectof §4.

As in [15], a trajectory of the virtual bodyin SE(2), parameterized by s, is defined by(R(s), r(s)) ∈ SE(2), ∀s ∈ [ss, sf ] such that

bk(s) = R(s)bk + r(s)

lk(s) = R(s)lk

with R(ss) the 2 × 2 identity matrix. Here, bk =bk(ss) − r(ss), k = 1, . . . ,M , is the initial position

6

of the kth virtual leader. Similarly, lk = lk(ss), k =1, . . . ,M , is the initial reference for the kth virtualleader. Both bk and lk are with respect to a virtualbody frame oriented as the inertial frame but withorigin at the virtual body center of mass.

For expansion and contraction, all distances be-tween the virtual leaders and all distance parameters(di, hi), i = 0, 1, are scaled by a factor k(s) ∈ R.

bk(s) = k(s)R(s)bk + r(s)

hi(s) = k(s)hi

di(s) = k(s)di,

with k(ss) = 1, hi = hi(ss) and di = di(ss). R(s)governs rotation, r(s) translation and k(s) expansionand/or contraction.

Deformation is defined to occur when the angles,φk, in Vr(θik) are changed. The out-of-phase saw-tooth presented in §4 is an example of a formationmaneuver acquired from enforcing virtual body de-formation.

Putting everything together, a parameterizedtrajectory in the configuration space for transla-tion, rotation, expansion/contraction, and defor-mation of the virtual body is thus defined by(R(s), r(s), k(s), φk(s)) ∈ SE(2) × R

M+1, ∀s ∈[ss, sf ].

Given our s parametrization, the total vector fieldsfor the virtual body motion can be expressed as

r =dr

dss, R =

dR

dss, k =

dk

dss, φk =

dφkds

s. (3)

s, the virtual body speed, will be governed by a con-trol law that enforces bounded formation error. How-ever, dr

ds, dRds, dkds, dφk

dsare free to be specified. Virtual

body vector fields that preserve the patterns pre-sented in §2.2 while performing interesting maneuversare presented in §4.

We recognize that s parameterizes an equilibriumtrajectory, xeq(s), in the state space. Furthermore,for every fixed choice of s, the Lyapunov func-tion in (2), extended to include s in the form ofbk(s), h(s), d(s), and φk(s), proves asymptotic sta-bility of the fixed point xeq(s), i.e. V (xeq(s), s) =

0, V (x, s) < 0. The following control law for s pro-vides group motion with bounded formation error,V (x, s) [14, 15]:

s = h(V (x, s)) +−(

∂V∂x

)Tx

δ + |∂V∂s|

(

δ + VUδ + V (x, s)

)

(4)

with initial condition s(t0) = ss. h ∈ C, h : R+ →

R+ has compact support in {V | V ≤ VU/2} and

h(0) > 0. δ ¿ 1 is a small parameter.

This control law ensures the system is stable andasymptotically converges to (x, s) = (xeq(sf ), sf ).Furthermore, if at initial time t0, V (x(t0), s(t0)) ≤VU , then V (x, s) ≤ VU for all t ≥ t0. Thus theformation error remains bounded as long as the set{x : V (x, s) ≤ VU} is bounded. The larger we chooseVU , the larger formation error the control law willpermit. To stop motion, i.e. at the endpoint andbeyond, s ≥ sf , we set s = 0.

Figure 7 illustrates two maneuvers involving threeunderwater gliders in a triangle formation about asingle virtual leader.

d0

d0

(a)

d0

f

d0

(b)

Figure 7: Group expansions. Red triangle denotesinitial configuration, light blue triangle denotes result-ing formation after expansion. Blue-circle denotes vir-tual leader ( group center). d0 denotes initial inter-vehiclespacing, d′0 denotes the expanded spacing. Not to scale.

(a) Group expansion. (b) Group expansion with simulta-neous rotation. Group rotates clockwise at a rate given bydφ

dt= dφ

dss. Note: actual trajectories will be piecewise lin-

ear since paths are discretized into segments representedby waypoints.

3.3 Kinematic constraints

We now describe how we specialize the methodol-ogy presented in §3.1 and §3.2 to underwater gliders.First we impose a constant speed constraint on thevehicle point mass model. To this effect we make thefollowing modification to the control vector for eachglider:

ui = αi (upi − βi u

ci )

where,

βi =

{

ε, ||upi || ≤ ε2

0, ||upi || > ε2 or x = xeq

0 < ε¿ 1,

αi =ug

||upi − βi uc

i ||,

and ug is the nominal glider speed in the lateral plane.The term βiu

ci is necessary to avoid singularities when

7

upi → 0. The vector uc

i is arbitrary but should bechosen based on the particular mission at hand, e.g.for a virtual body in translation uc

i = −drds

or in pureexpansion uc

i = −hik.In the presence of this kinematic constraint, we

present a condition sufficient to guarantee boundedformation error since it will be shown to permit aLyapunov Function for the s-frozen, i.e. s= 0, closed-loop dynamics. By Theorem 4.1 in [15], with s dy-namics as given in (4), bounded formation error re-mains guaranteed.

Lemma 3.1 (Control Req. for Boundedness)Given

xi = αi (upi − βi u

ci ) ,

suppose||up

i || > ε2, i = 1, . . . , p ≤ N||up

j || ≤ ε2 j = p+ 1, . . . , N.

Then

V (x) =

N−1∑

i=1

N∑

j=i+1

VI(xij)+

N∑

i=1

M∑

k=1

(Vh(hik) + Vr(θik))

remains a suitable Lyapunov function for the s-frozensystem, and thus the formation error for the completesystem remains bounded, if and only if the inequalitygiven by

p∑

i=1

||upi || >

N∑

j=p+1

−‖upj ‖

2 + 〈upj , ε u

cj〉

||upj − εuc

j ||(5)

holds when V ≥ VU .Further, V (x) remains a suitable Lyapunov func-

tion for the s-frozen system if

1

N − p

p∑

i=1

||upi || >

ε2

ε− ε2(6)

holds when V ≥ VU .

Proof: Taking the time derivative of (2) along trajec-tories of x, we get:

V (x) =N∑

i=1

−upi · ui

=

N∑

i=1

−upi · αi (u

pi − βi u

ci )

=N∑

i=1

−αi||upi ||

2 + αi〈upi , βi u

ci 〉.

Since ||upi || > ε2, βi = 0 for i = 1, . . . , p ≤ N.

Similarly since ||upj || ≤ ε2, βj = ε for j = p +

1, . . . , N .

Therefore, using the hypotheses

V (x) = −

p∑

i=1

αi||upi ||

2 +

N∑

j=p+1

αj(−||upj ||

2 + 〈upj , ε u

cj〉)

= ug

−

p∑

i=1

||upi ||+

N∑

j=p+1

−||upj ||2 + 〈up

j , ε ucj〉

||upj − εuc

j ||

.

Thus, V < 0 if and only if the inequality in (5) holds,and V = 0 if and only if x = xeq.

Furthermore,

N∑

j=p+1

−‖upj ‖

2 + 〈upj , ε u

cj〉

||upj − εuc

j ||≤

N∑

j=p+1

〈upj , ε u

cj〉

ε− ε2

< (N − p)ε2

ε− ε2.

Thus, if the inequality in (6) holds,

V ≤ ug

(

−

p∑

i=1

||upi ||+ (N − p)

ε2

ε− ε2

)

< 0,

and V = 0 if and only if x = xeq.

Remark 3.1 The sufficient condition (6) is satisfiedif O(up

i ) > O(ε) for at least one i ∈ 1, . . . , p. Thisprovides a (conservative) lower bound on the amountof control effort derived from the artificial potentialssufficient to maintain bounded formation error.

The constant-speed equality constraint will ulti-mately restrict what formations are feasible using ourpotential function methods. Numerical simulationshave shown that formations that are not kinemat-ically consistent with the speed constraint will notconverge properly. For example, a “rolling” forma-tion defined by a virtual body that is simultaneouslytranslating and rotating is not kinematically consis-tent with the constant speed constraint. This is be-cause each vehicle must slow down at some pointto be “overtaken” by its neighbor. This formationconstraint may seem obvious, but it is important tonote that our potential function methodology doesnot guarantee exact vehicle trajectories, it only guar-antees that they will be within some tube around thedesired trajectories. For this reason the probabilityof numerical convergence increases with increasing er-ror tolerance VU , since the trajectories will have moreroom within state space to evolve. However, in most

8

simulations of infeasible trajectories, numerical con-vergence could not be obtained for reasonable, i.e.small, values of VU . Feasible formations include puretranslation, rotation or expansion/contraction, com-bined rotation and expansion/contraction, and otherswhich will be discussed in §4.

3.4 External Currents, Surfacing

Asynchronicity, and Latency

In this section we consider the effects of external cur-rents, surfacing asynchronicity, and latency, whichare all relevant for fleets of underwater gliders.

3.4.1 External Currents

Gliders are inherently sensitive to ocean currents andit is important to include estimated and forecast cur-rents in the motion planning. Analysis of depth-averaged, lateral plane ocean current model data(ICON, see §5.1.2) suggests that on the two-hourlytime scales, it is quite reasonable to assume that thelocal flow field is uniform. In this case a simple av-erage of the glider estimated currents, ufavg, could becomputed as

ufavg =1

N

N∑

i=1

ufi

where ufi is the ith vehicle’s estimate of the localadvecting current. For use in simulation and controldesign, the virtual body and vehicle velocity are thencomputed to be

r =dr

dss+ ufavg

xi = ui + ufavg.

This approximation is reasonable since the flow av-erage and ith glider’s flow estimate will not differsignificantly. Furthermore, by applying identical ad-vecting flow to each vehicle and the virtual body, thebounded formation error results of §3.2 and §3.3 re-main valid.

As discussed in §3.2 the motion of the virtual bodyis responsible for directing the motion of the gliderfleet in addition to maintaining formation. For ex-ample, to compensate for the imposed advecting flowduring translation, we direct the virtual body ( dr

ds)

in a direction to cancel the current perpendicular tothe desired direction of travel. The desired direc-tion of travel is denoted by the unit-vector rd (seeFigure 8). The weight wf is chosen such that when||ufavg|| < ug, the current perpendicular to rd is com-pletely cancelled as s converges to ug.

Test

uavgf

rdwf

drds

rd rdwf

=

rd

uavgf

rd rdwf

Figure 8: Virtual body direction in response tocurrent. Blue circle denotes virtual leader.

In some instances, the structure of the flow fieldmay influence the choice of group orientation aboutthe virtual body. For example, if a triangle formation,as shown in Figure 2, is near the center of a rotatinggyre (as detected by comparing local glider currentsor model predictions) it may be best not to fix thegroup’s orientation.

3.4.2 Surfacing Asynchronicity

Unforced synchronous surfacing of a fleet of under-water gliders is improbable and is impractical toenforce. Variabilities across the glider fleet suchas w-component (vertical) currents and the localbathymetry, decrease the probability of synchronoussurfacings. Substantial winds and surface traffic (likefishing boats, etc.) render waiting on the surface toimpose synchronicity impractical.

In simulations we have experimented with twostrategies that differ in the number of VBAP plansgenerated per formation cycle. A formation cycle isdefined as the interval between any particular glider’ssurfacings. Here we assume that the time interval be-tween each glider’s consecutive surfacings is uniformacross the group.

A. Single plan per formation cycle. In this strategy,waypoint plans are generated once per formation cy-cle. The waypoint plans remain fixed during the for-mation cycle even if new information becomes avail-able. The first glider in need of waypoints will startthe formation cycle, and is designated as the leadglider. VBAP is initialized with estimated glider lo-cations and local currents. Glider locations are es-timated using their active waypoint plans and lastreported GPS fixes and local current estimates. Way-point lists for each glider are generated all at once andpassed to each glider as they surface.

The lead glider’s estimated location for planning(the location which initializes VBAP) and its esti-mated surfacing location (where it is expected to ac-quire these waypoints) will coincide. However, for anasynchronous group, the other gliders’ estimated lo-

9

cations for planning may be behind their estimatedsurfacing locations. To avoid backtracking, the way-point plans generated for these gliders are edited toremove waypoints between the estimated planning lo-cation and the estimated surfacing location.

B. Multiple plans per formation cycle. In this sce-nario new waypoint lists are generated during aformation cycle whenever new information is avail-able. New information becomes available every time aglider surfaces. The actual surfacing location (last re-ported GPS) and sensor measurements of the last sur-faced glider and all currently active waypoint planswill be considered in computing waypoint plans forthe glider that is expected to surface next. Every timea new waypoint plan is computed the glider locationsare estimated using their active waypoint plans andlast reported GPS fixes and local current estimates.

In this setting, waypoint plans among gliders maynot be consistent because they will generally be com-puted with different information sets. Note that theinformation set may change any time a glider surfacesdue to the new information it supplies. Simulationsindicate however, that with frequent enough surfac-ings the formation as a whole remains together.

3.4.3 Latency

At each glider surfacing where two-way communica-tion is established, new waypoint lists will be up-loaded to the glider after data obtained during thelast mission has been downloaded. During AOSN-IIthe data from the glider will not be available in atimely enough manner to be used in the generationof the next mission plan. The subsequent missionwill therefore not use the latest GPS fix, local cur-rent estimate and sensor measurements, but will useinformation gathered during the previous surfacingof the glider. This introduces a degree of latencyin the VBAP implementation. Simulations indicatethat with two-hourly feedback the formation is robustto these latencies.

3.5 Waypoint Generation

Trajectories generated by VBAP are continuouscurves that must be discretized into waypoints. Wepropose discretizing via the constrained minimiza-tion of an appropriate cost function. We assume thenumber of waypoints, q, has been preselected. De-note a continuous VBAP trajectory for the ith glidergiven by sVi (t), t ∈ [ts, tf ] and define the waypoint

set as wi = {w1i , w

2i , . . . , w

qi } where wji = sVi (tj) for

j = 1, . . . , q such that ts < t1 < t2 < . . . < tq = tf .Denote the trajectory composed of connecting these

waypoints with straight line segments, with endpointsat the glider’s starting location and last waypoint, asswi . Having a time parameterization of swi implicitlydefined, a constrained minimization problem is spec-ified by,

minwiJ(wi) =

tf∫

ts

||sVi (t)− swi (t)||2 dt

subject to ||wm+1i −wmi || > dmin for m = 0, 1, . . . , q−

1, and w0i = sVi (ts). dmin specifies the minimum

acceptable spacing between two adjacent waypoints.At present we choose q = 2 waypoints

hr tf and dividesVi into q equal segments to specify the initial guessfor wi when numerically optimizing. Numerical op-timizations suggest that the cost function may benon-convex resulting in only locally optimal waypointlists. Thus, there is some degree of sensitivity on ini-tial guesses. We are currently investigating whichinitialization schemes work best.

4 Adaptive Sampling Strate-

gies

In this section we present some example adaptivesampling strategies provided by the VBAP method-ology presented in §3. We present strategies for threegliders but they are extendable to larger groups.

In Figures 9 and 10 we illustrate coordinated sam-pling strategies to acquire data across a fixed pathof interest. In Figure 9, the triangle formation criss-crosses a desired mean path providing measurementssuitable for gradient estimates on each side. In Fig-ure 10, line formations are shown surveying arounda path of interest. In Figure 10a the formation isoriented in a way which provides good coverage onalternating sides of the mean path, while providingdirectional derivatives perpendicular to the directionof travel. The pattern in 10b provides simultaneousmeasurements on one side of the mean path whileproviding directional derivatives in the direction oftravel. In Figure 10c the virtual body deforms froma line configuration to triangle formations about themean path. This pattern provides simultaneous mea-surements on both sides of the mean path while alter-nately providing directional derivative and full gradi-ent estimates.

Given a set of distributed measurements we use aleast squares method for estimating gradients at thecenter of the virtual body as presented in [15]. Inthe cooperative control setting, the virtual body di-rection prescribed on a daily basis is complemented

10

d0

a

l

f =l

ust

ust

Figure 9: Triangle formation with zig-zag. Bluedashed line denotes group mean path. λ denotes zig-zagwavelength along group mean path. ust denotes groupspeed along mean group path. Zig-zag frequency givenby f = ust

λ. Blue circle denotes group center. Not to

scale.

with gradient estimate computed on a two-hourly ba-sis. In the context of AOSN-II, the adaptive path ofthe glider group will be computed as a modificationof the path that one might otherwise select basedon the model forecasts, satellite imagery, aircraft orother data. For example, shown in Figure 11 is aprojected gradient approach in which to induce gra-dient descent; the negative of the estimated gradientis projected perpendicular to the vector from the vir-tual body center to a chosen final destination. Thevirtual body is directed by:

rd = rw − w⊥ˆ∇T⊥est. (7)

The scalar w⊥ weights the influence of the two-hourlygradient estimate on the virtual body path. Project-ing the gradient perpendicular to the vector towardsthe desired destination ensures that rd has a com-ponent heading towards the destination at all times.Gradient ascent is achieved by adding rather thansubtracting the weighted sum of the projected gradi-ent in (7).

Figure 12 illustrates the use of the projected gra-dient to direct a glider formation towards the max-imum of a field, e.g., a high concentration in thefluorescence field indicating biological accumulation.The estimated gradient of the field (at the center ofthe glider group) is used to direct the group towardsthis accumulation. Once the accumulation exceedsa prescribed threshold the group may stop and ro-tate while expanding or contracting to focus on thatregion of interest.

In response to measurements, the group may re-configure as illustrated in Figure 13. In this exam-

d0

a

q

l

ust

(a)

d0

l

a q = 0

ust

(b)

l

ust

(c)

a

dma x

0

f

Figure 10: Line formation with zig-zag. Bluedashed line denotes group mean path. Blue dots representrepresent virtual leaders. λ denotes zig-zag wavelengthalong group mean path. ust denotes group speed alongmean group path. Zig-zag frequency given by f = ust

λ.

Not to scale. (a) Line formation with zig-zag, θ = π

2.

(b) Line formation with zig-zag, θ = 0. (c) Out-of-phase zig-zag. Group alternates between triangle andline formations via deformation while translating. Letthe left-most virtual leader be denoted by k = 1 andthe right-most virtual leader be denoted by k = 2. Thispattern results from switching between dφ1

ds= ± π

8h0and

dφ2

ds= ∓ π

8h0. This formation is useful for front crossing;

there is at least one vehicle on either side of the front atany given time.

11

est

rw

Initial location Destination

rd

Test-T-

Figure 11: Projected Gradient Descent. The to-tal virtual body heading vector rd is composed of theunit-vector directed towards the desired destination, rw,less the normalized projected gradient, ∇T⊥

est, weightedby w⊥. The black-dashed line represents a fixed samplingpath and the red-solid line illustrates the path when di-recting the virtual body with the negative projected gra-dient for gradient descent.

ple, the formation initially in a triangle reconfiguresto a line formation to collect data along a front of in-terest. The formation performs an out-of-phase saw-tooth motion which allows the group to compute al-ternating spatial gradients and directional derivativeswhile each member visits both sides of the perceivedfront.

(i)

(ii)

Group direction

Virtual body path

Individual glider path

Figure 12: Adaptive sampling for coverage in aregion of interest. A survey of a region of interestin which gradient climbing (i) and data-driven rotations,expansions and contractions (ii) are used to focus in onsub-regions of greatest scientific interest.

These are just a few examples of possible adaptivesampling strategies and more details may be found in[11]. Furthermore, we are currently investigating howobjective analysis techniques may be implemented to

Group direction

(i)

(ii)

(iii)

Virtual body path

Individual glider path

Figure 13: Sensor reconfiguration. (i) The trian-gle formation provides gradient estimates to find a front.(ii) The formation reconfigures from a triangle formationto a line formation near a front. (iii) At the front, thegroup performs an out-of-phase sawtooth formation tocollect alternate planar gradients and directional deriva-tives. Furthermore, each glider criss-crosses the front.

improve interpretation of the glider data [12]. Wehope this may provide better models for Kalman fil-tering using vehicle networks. Work on Kalman Fil-tering of data obtained using vehicle networks is pre-sented in [15].

5 Simulations

In this section we describe simulations performedto demonstrate and ascertain control parameters forour cooperative and coordinated control methodol-ogy for underwater glider fleets. In §5.1 we presentthe datasets which take the roles of truth and modelenvironmental fields. We describe how we simulatean underwater glider in §5.2. In §5.3 we describe thesimulations and present some results.

5.1 Truth and Model fields

5.1.1 Aircraft SST Data

The aircraft SST data, which provides the truth tem-perature field for our simulation, was generated dur-ing MBARI’s MOOS Upper-Water-Column ScienceExperiment (MUSE) in August, 2000 by the NavalPostgraduate School in collaboration with Navy’sSPAWAR System Center-San Diego and Gibbs Fly-ing Services, Inc., San Diego, CA. The data was col-lected on the afternoon of August 17 , 2000 using atwin engine Navajo aircraft, which was flown along aregular grid over Monterey Bay at an altitude of lessthan 1000 feet. The aircraft measured sea surface

12

temperature immediately below its location. Moredetails about the data collection and the MUSE ex-periment can be obtained from Monterey Bay Aquar-ium Research Institute (MBARI)’s website [20].

Figure 14 presents the aircraft SST data. The redbox illustrates a region of particular interest whichcorresponds to a cold tongue of water entering the bayfrom the north. This is characteristic of an upwellingevent. In §5.3 we present simulation results of a gliderfleet exploring this region.

10 20 30 40 50 60 7010

20

30

40

50

60

70

Longitude UTM Zone 10 (km)

Latti

tude

UTM

Zon

e 10

(km

)

11

11.5

12

12.5

13

13.5

14

14.5

15

degrees C

Figure 14: NPS Aircraft Sea Surface Tempera-ture, Monterey Bay, Aug. 17, 2000

5.1.2 Innovative Coastal-Ocean ObservingNetwork (ICON)

The fine-resolution numerical ocean model of theMonterey Bay Area (ICON model) was developed inthe “Innovative Coastal-Ocean Observing Network”(ICON) project [17]. The ICON model is a three-dimensional, free surface model based on the Prince-ton Ocean Model (POM) [3]. The model has anorthogonal, curvilinear grid, extending 110 km off-shore and 165 km in the alongshore direction. Thehorizontal resolution is 1-4 km with the maximumresolution in the vicinity of Monterey Bay. Verti-cally, the model is characterized by a realistic bot-tom topography with 30 vertical sigma levels. On theopen boundaries the model is coupled to a larger-scaleNRL Pacific West Coast model. The ICON modelwas run with atmospheric forcings from COAMPS(Navy Coupled Ocean and Atmospheric MesoscalePrediction System) model predictions [8]. HF radar(CODAR)-derived surface currents are assimilatedinto the model [18].

In our simulations we use the ICON data from Au-gust 17, 2000 fixed at 16:00 hours local time. TheICON sea surface temperature serves as a represen-tative model field that we will use to motivate a gliderfleet mission. ICON also supplies the flow field truthset in which the simulated gliders are advected.

The ICON SST data is presented in Figure 15. Thered box illustrates the area of the interest as it ap-pears in the ICON data. Not shown are the ICONcurrents. For reference the depth-averaged current inthat region flows predominately in a south-westerlydirection.

Figure 15: ICON Sea Surface Temperature,Monterey Bay, Aug. 17, 2000 16:00 local time

5.2 Glider Simulator Details

We developed a MATLAB simulator to study theimplementation of the VBAP methodologies in apractical scenario involving SLOCUM gliders usedin AOSN-II. Our goal was to incorporate the oper-ational behaviors, constraints and communication la-tencies of the gliders as much as possible.

The glider simulator consists of two modules -

• Module 1 : Glider in Water - to simulate glidermotion during a dive

• Module 2 : Glider at Surface - to simulate glidermotion (drift) when it is at the surface and toupdate the onboard flow estimate.

5.2.1 Glider in Water

While in water the gliders are simulated to move ata horizontal speed of 40 cm/s, and a nominal vertical

13

speed of 20 cm/s relative to water, consistent with theassumptions made in §2.1. The horizontal velocity isadded to the “truth” local horizontal flow velocity,which is provided by the ICON data set, to computethe total absolute horizontal velocity of the glider.Vertical velocity from ICON model predictions wasnot used in this study. Instead a zero-mean whitenoise is added to the vertical velocity to account forthe vertical component of the true local flow field

The gliders are simulated to dive until 25 m abovethe sea floor, up to a maximum depth of 200 m. Thenthey move upward until they are 5 m from the surfacebefore going downward again, unless they are surfac-ing in which case they go all the way to the surface.

The gliders are commanded to start surfacing 2hours after they dove into the water. Since the glidercould be at any depth between 0 and 200 m when itstarts surfacing, the total time of implementation ofthe ‘glider in water’ module will in general be a littleover 2 hours during every cycle.

The glider has a constant estimate of the local flowvelocity, which is updated every time it surfaces andobtains a GPS fix. To calculate its absolute velocitythe glider adds the local flow velocity estimate to itsrelative velocity (i.e., relative to water). The rela-tive velocity is computed using the heading measure-ment, and the assumptions regarding the horizontaland vertical speeds. A white noise with a nonzerobias is added to the actual heading to simulate theheading sensor. The bias itself is randomly chosen atthe start of every dive cycle.

While in the water the glider servos its heading sothat it always tries to towards the next waypoint. Athree-dimensional, black-box model is used to sim-ulate the experimentally observed input-output re-lation between the tailfin position (input) and theheading (output). The proportional servo controlleris turned off whenever the measured heading is withina fixed deadband (±0.02 rad) of the continuouslychanging desired heading.

A waypoint is considered reached if the horizontaldistance from the waypoint is less than 25 m.

5.2.2 Glider at Surface

The glider is set to drift with the flow when it is on thesurface. As described in §2.1, the glider does not geta GPS fix immediately after it surfaces. We assumethat the first GPS fix is obtained t1 = 2 minutes aftersurfacing.

We assume that the glider uses a nominal time oft2 = 17 minutes for exchanging data collected duringthe previous dive. We add a zero mean white noisewith an amplitude of 2 minutes to t2 to account for

possible communication problems. During this timethe glider also gets several GPS fixes. We use thefirst and the last GPS fix, and the glider’s estimateof the local flow to calculate the actual position of itssurfacing.

The glider corrects its flow estimate by comparingthe deadreckoned and actual positions of surfacing.The corrected flow estimate is used in the computa-tion of deadreackoned positions during the next dive.

During the data exchange the glider also gets anupdated set of waypoints. The glider prepares to diveafter the data exchange is completed.

We assume that the glider gets its last GPS fixat the end of data exchange and that it is at thesurface for another minute before diving. Thus thetotal nominal time the glider spends at the surface is2+17+1=20 minutes.

5.3 Preliminary Parameter Studies

and Results

For our simulations we propose the following scenario:a perceived frontal boundary in the model data (seered box in Figure 15) motivates us to drive a fleetof gliders through this region while performing a pro-jected gradient descent to steer the formation towardsthe colder regions. We assume we have a fleet of threegliders and prescribe a triangle formation since themission entails gradient descent.

During the mission, surface temperature measure-ments for each glider are stored along with their cor-responding location. Whenever new waypoints aregenerated with VBAP, the last reported surface tem-perature measurements and corresponding locationsare used to compute a least-squares gradient esti-mate. The gradient estimate is then used to redirectthe group as described in §4.

We begin the simulation with the gliders near thedesired formation as shown in Figure 16. The virtualbody is composed of a single virtual leader. We wishthe group to head predominately south. So we choosethe destination, rw, to be approximately 70 km duesouth from the virtual leader’s initial location. Wedesign our artificial potentials such that the desiredtriangle formation about the virtual leader will beoriented such that a triangle edge spans from eastto west with the inside-directed normal to that edgepointing northward. The weight w⊥ = 1. Simula-tions were run for approximately 12 hours.

Initially we assume the fleet is synchronized andcompute a VBAP plan based on each glider’s initiallocation. Initial local current estimates are taken tobe the depth-averaged current at each glider’s loca-tion. To induce asynchronous surfacings, we artifi-

14

Figure 16: Single mission plan per formation cy-cle. Red circle indicates initial formation center of mass(virtual leader position). Tracklines denote glider trajec-tories. Magenta triangles illustrate glider relative loca-tions at two hour intervals. Color contour shows “truth”sea surface temperature in degrees Celsius.

cially stagger the next surfacing time of each glider by40 minutes (recall each glider surfaces approximatelyevery 2 hours). Thus, the first glider will begin sur-facing after 40 minutes, the second after 80 minutes,and the third after 120 minutes. We simulated bothasynchronous implementations presented in §3.4.2.

Figure 16 presents the glider trajectories when im-plementing a single plan per formation cycle. Asshown, the fleet maintains the desired formation anddetects the cold water tongue in the truth field (air-craft SST data), redirecting the fleet towards it whileheading towards the destination. Figure 17 presentsthe glider trajectories when planning multiple timesper formation cycle. The glider fleet detects thecold water tongue but does not maintain formationas well. The formation exhibits noticeable stretch-ing and backtracking. In both instances the virtualleader is relocated to the glider formation center ofmass at the start of each VBAP implementation. InFigure 18 the projected gradient weight was increasedto 3 while implementing a single plan per formationcycle. As expected the group exhibits a larger devia-tion towards the cold water region.

Figure 17: Multiple mission plans per forma-tion cycle. Red circle indicates initial formation cen-ter of mass (virtual leader position). Tracklines denoteglider trajectories. Magenta triangles illustrate glider rel-ative locations at two hour intervals. Color contour shows“truth” sea surface temperature in degrees Celsius.

Figure 18: Single mission plan per formation cy-cle, larger gradient weighting, w⊥ = 3. Red circleindicates initial formation center of mass (virtual leaderposition). Tracklines denote glider trajectories. Magentatriangles illustrate glider relative locations at two hourintervals. Color contour shows “truth” sea surface tem-perature in degrees Celsius.

15

6 Final Remarks

In this paper we have described a methodology forperforming adaptive sampling using fleets of under-water gliders. We have addressed the constraintsof the underwater glider, such as kinematic con-straints, external currents, fleet asynchronicity, andlatency, providing strategies for coping with them.We present some useful formations and strategies toperform adaptive sampling and provide detailed sim-ulations for the purpose of both demonstration andcontrol design validation.

Acknowledgments

The authors would like to thank Petter Ogren,Clancy Rowley, Ralf Bachmayer, and RodolpheSepulchre for their contributions to this work.

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