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

Underwater Glider Model Parameter Identification

Joshua G. Graver and Ralf Bachmayer and Naomi Ehrich Leonard∗

Mechanical and Aerospace EngineeringPrinceton UniversityPrinceton, NJ 08544

jggraver@princeton.edu, ralf@princeton.edu, naomi@princeton.edu

David M. FratantoniPhysical Oceanography, Mail Stop: 21Woods Hole Oceanographic Institution

Woods Hole, MA 02543dfratantoni@whoi.edu

Abstract

An underwater glider is a buoyancy-propelled, fixed-wing vehicle with attitude controlled completely, or inpart, by means of internal mass redistribution. Wehave developed a physics-based nonlinear model ofthe dynamics of an underwater glider and adaptedit to model the SLOCUM glider’s geometry, rudder,ballast pump and internal movable mass. In thispaper we identify the model parameters to matchthe steady glides in new flight test data from theSLOCUM glider. In the process we also estimate thebuoyancy trim offset of the glider used in the flighttests.

1 Introduction

An underwater glider is a buoyancy-propelled, fixed-wing autonomous underwater vehicle. Attitude iscontrolled by means of internal mass redistributionand in some cases with external control surfaces. Ini-tially conceived by Henry Stommel [12], autonomousunderwater gliders offer many advantages in oceansensing: long duration missions, greater operationalflexibility and low-cost operations. Gliders are moremobile and flexible than fixed moorings, are moremaneuverable than drifters, have greater range thanother AUV’s, and do not need expensive support ves-sels.

∗Research partially supported by the Office of Naval Re-search under grants N00014–02–1–0826 and N00014–02–1–0861.

Several oceangoing gliders are operational or underdevelopment, including the SLOCUM glider [15], theSpray glider [11] and Seaglider [1]. These three glid-ers are designed for long-duration, ocean sensing mis-sions. They collect oceanographic data such as wa-ter temperature, conductivity, depth, and currents.They can also carry other scientific sensors, such asfluorometers, optical backscatter or bioluminescencesensors. The three gliders are similar in size and ge-ometry, each measuring approximately two meters inlength and weighing around 50kg. Each has a cylin-drical hull, two fixed wings and a tail. All are de-signed to be statically stable in a glide. The gliderscontrol pitch by moving an internal mass or battery.In the Spray, Seaglider and the thermally poweredSLOCUM, roll is also controlled by moving an inter-nal mass or battery. Yaw and heading are controlledthrough the hydrodynamic yawing moment due tothe roll. Some of these gliders are capable of divesto depths of 1,500 meters. In the electric SLOCUM,designed for shallower dives from five to 200 metersand thus more inflections (transition between down-wards and upwards glides), roll is set by the glider’sstatic CG position and pitch is controlled by movinginternal mass. Yaw and heading are controlled usingthe rudder mounted on the vertical tail of the glider.

Our research on glider dynamics aims to developa widely applicable, model-based approach to designand control of gliders. This approach, described in [7]and [3] considers a three-dimensional nonlinear dy-namic model of a glider, with hydrodynamic forces,ballast control, internal moving mass control, andnonlinear coupling between the vehicle and movable

1

Figure 1: A SLOCUM Glider.

internal mass. This model is applicable to glider de-sign, to the study of stability and controllability ofglide paths and to the derivation of feedback controllaws. It is emphasized that this approach is intendedto be general rather than vehicle specific and is meantto complement other efforts towards analysis and de-sign of gliders including the SLOCUM, Spray andSeaglider.

In this paper we describe model parameter iden-tification for the SLOCUM using experimental flighttest data, focusing in particular on data from steadystraight glides. We have adapted our model to theSLOCUM electric glider, modelling the location ofthe ballast system, the properties of the moving pitchmass, and the rudder. The resulting equilibriumequations appear in Section 2.3. In Section 3 we de-termine parameter values such that the model willmatch the data set of equilibria for the glider. Thisdetermines the coefficients for our quasi-steady hy-drodynamic model and parameters representing thetrim and buoyancy of the glider. We discuss nextsteps and final remarks in Section 4.

2 SLOCUM Glider Model

2.1 SLOCUM Glider

The SLOCUM glider is manufactured by Webb Re-search Inc., Falmouth, MA, is a buoyancy-driven, au-tonomous underwater vehicle [13, 15]. The opera-tional envelop of the glider includes a 200 m depthcapability and a projected 30 day endurance, whichtranslates into approximately 1000 km operationalrange with a 0.4 m/s fixed horizontal and 0.2 m/svertical speed. The glider has an overall length of1.5 m and a mass of 50 kg. The buoyancy engine isan electrically powered piston drive, located in the

Figure 2: SLOCUM Electric Glider Layout [14].

nose section of the glider, Figure 1. The drive allowsthe glider to take in and expel water, thereby chang-ing its overall buoyancy. The mechanism allows aclose to neutrally buoyant trimmed glider to changeits displacement in water by ±250 ccm, which corre-sponds to approximately ±0.5% of the total volumedisplaced. This change in buoyancy generates a verti-cal force which is translated through two swept wingsinto a combined forward and up/downward motion.Due to the location of the piston drive, also calledbuoyancy engine, the change in direction of the buoy-ant force also creates the main pitching moment forthe glider. Besides the buoyancy engine the gliderpossesses two more control actuators, a 9.1 kg bat-tery pack, referred to as sliding mass, that can belinearly translated along the main axis of the gliderand a rudder attached to the vehicle tail fin struc-ture. The sliding mass is used for fine tuning thepitch angle.

The glider has two onboard computers, a control com-puter and a science computer. Navigation sensors onthe glider measure heading, pitch, roll, depth, slid-ing mass position and the piston drive position. Be-sides other internal states and other sensor measure-ments, these readings are recorded and processed bythe control computer. Vehicle position at the surfaceis determined by a GPS receiver, with the antennalocated on the rear fin. Note that, while submerged,the glider velocity and horizontal position are notsensed because of the difficulty in measuring thesestates. While underwater, the glider navigates usinga deduced reckoning algorithm. At present, the pitchangle and depth rate measurements and an assumedangle of attack are used by the onboard computer toestimate the horizontal speed of the glider.

The SLOCUM glider can be programmed to navi-gate in various ways. For a typical mission scenario

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the glider navigates to a set of preprogrammed way-points downloaded prior to execution in a missionspecification file and operates under closed-loop pitchand heading control. A mission is composed of yosand segments. A yo is a single down/up cycle, whilea segment can be composed of several yos and startswith a dive from the surface and ends with a surfac-ing. At all surfacings the glider tries to acquire itsGPS location. On the surface the glider compares itsdesired waypoint to its actual GPS position and de-termines a heading correction for the next waypointbefore it dives again for the next segment of the mis-sion. Other modes of operation such as gliding ata given compass heading, fixed rudder angle or fixedbattery position are easy to implement and were usedin the work presented in this paper.

2.2 Experiments with SLOCUM

We conducted glider in-water flight tests during Jan-uary 2003 near Chub Key, Bahamas, using SLOCUMGlider WE01, owned and operated by Woods HoleOceanographic Institution (WHOI). The principal in-vestigator on this research cruise was Dr. DavidFratantoni from Woods Hole Oceanographic Institu-tion, Woods Hole, MA. Operations were conductedfrom the RV Walton Smith of the University of Mi-ami. Using glider WE01, we conducted a series oftest glides including both steady straight and turn-ing glides and glides with more dynamic behavior.

The glider experiments conducted on the cruisewere designed for model confirmation and parameteridentification. The hydrodynamic properties of theSLOCUM glider were estimated in advance using the-oretical calculations and standard aerodynamic refer-ence data. In order to collect the necessary data, weperformed a set of glides including (1) steady glidesat different pitch angles and (2) glides that exhibitrich dynamic behavior such as unsteady turning andpitching with large actuator excursions.

A typical flight test mission consisted of two glidesto fifty meters depth, enough depth to reach equilib-rium glides. The glider surfaced at the beginning andend of the mission for a GPS position fix and datatransfer. Both fixed control glides and glides usingpitch and heading feedback were conducted. Duringfixed control glides, the rudder and sliding pitch massare held at pre-determined positions for the durationof each downwards and upwards glide. This resultedin the glider reaching a steady glide equilibrium cor-responding to those control settings.

Although there is always some state disturbanceand measurement noise in the data, the steady glidesstand out plainly (see Figures 3 and 4). Choosing the

0 100 200 300 400 500 600 700 800 900 1000−40

−20

0

20

40Data from SLOCUM Glider, flight test Vert21

pitc

h [o ]

0 100 200 300 400 500 600 700 800 900 1000

0

20

40

60

dept

h [m

]

0 100 200 300 400 500 600 700 800 900 1000

−5

−2.5

0

2.5

5

battp

os [c

m]

0 100 200 300 400 500 600 700 800 900 1000−500

−250

0

250

500

balla

st[c

c]

time into mission [s]

Figure 3: SLOCUM Data from Flight Test.

0 100 200 300 400 500 600 700 800 900 1000−0.4

−0.2

0

0.2

0.4

dept

h ra

te [m

/s]

time into mission [s]

0 100 200 300 400 500 600 700 800 900 1000250

275

300

325

350

head

ing

[o ]

0 100 200 300 400 500 600 700 800 900 1000−40

−20

0

20

40

pitc

h [o ]

Data from SLOCUM Glider, flight test Vert21

0 100 200 300 400 500 600 700 800 900 1000−20

0

20

roll

[o ]

Figure 4: SLOCUM Data from Flight Test.

steady glides in the data and computing the averagestate over the interval of the steady glide gives a set ofsteady glides. As an example, average state values forfour steady glides are shown in Table 1, as are valuesfor α, V and CD(αeq) by frontal area computed usingmethods described in Section 3.

There are a number of sources of uncertainty in theflight test data. As noted, the glider velocity and hor-izontal position are not measured. The current condi-tions in the area of operation are unknown. Estimatesof the current may be made using the model of theglider dynamics, but this cannot be used to determinethe model parameters. Glider velocity and currentare important because the hydrodynamic forces onthe glider depend on the glider speed relative to thewater. Other sources of uncertainty in the glider datainclude the trim condition of the glider and the CG

3

Avg. Value Glide 1 Glide 2 Glide 3Pitch θ (deg) -22.77 23.74 -25.78

Depth rate z (m/s) 0.168 -0.224 0.200Battery pos. (cm) -2.4 -1.8 -2.3Ballast mb (cc) 244.4 -237.3 247.7

Rudder δR (deg) 2 4 2Roll φ (deg) 3.34 3.23 3.81

Heading φ (deg) 334 333 333AoA α (deg) 2.7 -2.9 2.3

Speed V (m/s) 0.388 0.499 0.425Drag Coeff. CD(αeq) 0.27 0.31 0.25

Table 1: Example of Steady Glide Data.Flight Vert22 4 on SLOCUM Glider WE01.

position. Some static roll offset appears in the data,i.e. the CG and static trim of the glider induced somestatic roll. Because of operational considerations dur-ing the cruise it was not possible to correct this trimor to obtain completely accurate static mass and trimmeasurements. The wings are made of a thin com-posite material which may deflect during flight andchange the predicted flight performance. When atthe surface to determine GPS position, the glider issubject to wind and current-driven drifting, and thisleads to some uncertainty in the glider’s surfacing po-sition.

Our analysis is designed to minimize the effects ofthese uncertainties, making as much use as possibleof the directly measured states. For example, GPSpositions are not used in the calculation of the glidervelocity.

As mentioned above, the glider used in experimentshad a slight static roll due to miss-trim. Because ofthis, the glider is slightly out of the longitudinal planein flight. This is another possible source of error inthe experimental analysis. The static roll produces asmall yaw moment which is offset by a small rudderangle. This probably results in the glider flying withsome sideslip angle. This could result in additionaldrag on the glider and possibly change the lift andmoment on the glider in comparison with fully lon-gitudinal, zero-sideslip flight. These problems couldbe reduced in future flight tests by correcting the rolltrim of the glider.

2.3 SLOCUM Model Planar Equilib-rium Equations

We have derived a model of glider dynamics, de-scribed in [7] and [3]. Our dynamic glider modeldescribes a glider with simple body and wing shape.Control is applied to two point masses inside the vehi-

cle: we control the mass of a point with fixed positionin the body, representing the ballast tank, and controlthe position of a mass with varying position withinthe body, representing the moving battery pack. Themodel describes the nonlinear coupling between thevehicle and the shifting and changing masses. Themajor forces on a glider are all incorporated into themodel, including buoyancy, the moments and forcesdue to the internal moving mass, and quasi-steady hy-drodynamic forces. Beginning with the glider equa-tions from [7], we add terms to the model to accountfor the SLOCUM ballast system location, the slidingmass range of travel, and the rudder. The aim of themodel is to adequately match the dynamic perfor-mance of the glider while maintaining a level of sim-plicity in the model that allows for analytical workand design insight.

We take the glider hull to be symmetrical withwings and tail attached so that the center of buoy-ancy (CB) is at the center of the hull. We assign acoordinate frame fixed on the vehicle body to have itsorigin at the CB and its axes aligned with the prin-ciple axes of the hull. Let body axis 1 lie along thelong axis of the vehicle (positive in the direction ofthe nose of the glider), let body axis 2 lie in the planeof the wings and body axis 3 point in the directionorthogonal to the wings as shown in Figure 5.

i

j

k

1e

2e

3e

Figure 5: Frame assignment on underwater glider.

The total stationary mass of the glider, ms, (alsoreferred to as body mass) is the sum of three terms:ms = mh + mw + mb. mh is a fixed mass thatis uniformly distributed throughout the body of theglider, mw is a fixed point mass that may be off-set from the CB, and mb is the variable ballastpoint mass, also offset from the CB in the SLOCUM.ms = mh + mw + mb. The vector from the CB tothe point mass mw is rw. The vector from the CBto the variable ballast mass mb is rb. The moving

4

internal point mass is m. The vector rp(t) describesthe position of this mass with respect to the CB attime t. The total mass of the vehicle is then

mv = mh + mw + mb + m = ms + m.

The mass of the displaced fluid is denoted m and wedefine the net buoyancy to be m0 = mv −m so thatthe vehicle is negatively (positively) buoyant if m0 ispositive (negative). The different masses and positionvectors are illustrated in Figure 6.

m

CB

pr

mb

mh

mwwr

variableballast mass

fixed mass

, distributed hull mass

movable mass

br

i

j

k

Figure 6: Glider mass definitions.

Here we consider this model specialized to the lon-gitudinal plane (assumed invariant), as in [7], andsolve for the equilibrium steady glides in the equa-tions of motion. The resulting SLOCUM verticalplane equilibrium equations are

x = v1 cos θ + v3 sin θ (1)

z = −v1 sin θ + v3 cos θ (2)

0 = (mf3 −mf1)v1eqv3eq

−mg(rP1eq cos θeq + rP3eq sin θeq)

−mbeqg(rB1 cos θeq + rB3 sin θeq)

−mwg(rW1 cos θeq + rW3 sin θeq)

+MDLeq (3)

0 = Leq sin αeq −Deq cos αeq −m0eqg sin θeq (4)

0 = Leq cos αeq + Deq sin αeq −m0eqg cos θeq(5)

where the subscript eq denotes the state at equilib-rium steady glide. v1 and v3 are the components ofthe glider velocity in the e1 and e3 directions, respec-tively, as shown in Figure 5. Here, θ is pitch angle, αis the angle of attack, D is drag, L is lift and MDL isthe viscous moment as shown in Figure 7. mf3 andmf1 are the added mass terms corresponding to thee1 and e3 directions, as derived by Kirchhoff [6]. Inthese equations, as in [7], we take the added masscross terms to be zero. We note that equilibriumterms corresponding to the offset mass mw and the

a

x

q e1V

i j

k

MDL

L

D

Figure 7: Lift and drag on glider.

location rB of the ballast mass mb do not appear inour earlier model, [7].

As shown in Figure 7, we denote the glide pathangle by ξ where

ξ = θ − α.

At equilibrium, it may be shown that

ξeq = − tan−1(

Deq

Leq

)

We also denote the glider speed by V where

V =√

(v21 + v2

3).

Using Equation (2) and our angle definition in Fig-ure 7, we can write the glider depth rate as

z = −V sin(ξ) = −V sin(θ − α) (6)

The hydrodynamic forces and moment are modelledas

D =12ρCD(α)AV 2 ≈ (KD0 +KDα2)(v2

1 + v23)(7)

L =12ρCL(α)AV 2 ≈ (KL0 +KLα)(v2

1 + v23) (8)

MDL =12ρCM (α)AV 2 ≈ (KM0 +KMα)(v2

1 + v23)(9)

where CD, CL and CM are the standard aerodynamicdrag, lift and moment coefficients by cross sectionalarea, A is the maximum glider cross sectional area,and ρ is the fluid density. For the longitudinal quasi-steady fluid model, CD, CL and CM are functions ofα and the K’s are constant coefficients. This model isa standard one, derived using airfoil theory and po-tential flow calculations and then verified using ex-perimental observations, see for example [2, 8]. Amethod for determination of the coefficients is de-scribed in Section 3.

This quasi-steady hydrodynamic model is expectedto be accurate for equilibrium steady glides. It may

5

be less accurate away from equilibrium glides andwhen the glider experiences high accelerations or an-gular rates. The hydrodynamics of the flow about theglider are much more complex during such motions,requiring a more complex hydrodynamic model. Inthe case of our initial analysis and the standard mis-sion use of the SLOCUM glider, the majority of theoperational time is spent at steady glides. Transi-tions and inflections between steady glide equilibriaare relatively slow and gradual. Because of this, thequasi-steady hydrodynamic model may prove satis-factory for our analysis. Incorporating a more com-plex hydrodynamic model involves adding terms tothe lift, drag and moment model.

An analysis of the equilibrium steady glide equa-tions for a generic glider appears in [7]. One interest-ing property of the equilibrium steady glide equationsis that the glide path angle is independent of the glidespeed. Glide path angle depends only on the equilib-rium angle of attack. When choosing an equilibriumglide, it is possible to specify the glide path angle, de-termine the required angle of attack, and then choosea glide speed V . The glide speed depends on the netbuoyancy of the glider, set by the ballast control andthe glide hydrodynamics.

Determining the steady glides for a glider such asthe SLOCUM requires finding the set of model pa-rameters that describe the glider mass and hydrody-namic characteristics. This is described in Section 3Using one method, the hydrodynamic coefficients ofthe glider are estimated using reference data for ships,submarines and standard shapes. With these esti-mated coefficients, the equilibrium equations may beused to compute the set of steady glide conditions forthe SLOCUM glider. Figure 8 shows the steady glideangles given the estimated lift and drag parameters.Figure 9 shows the steady glide speeds given the sameestimated parameters.

3 Parameter Identification

We wish determine the model parameters matchingthe SLOCUM model equilibria equations (1)-(5) tothe steady glides from data. These parameters rep-resent the physical variables corresponding to theglider’s mass, inertia and hydrodynamic characteris-tics. The parameters that appear in the steady glideequations are the displacement m, the masses mh,mw, and m, the positions rB and rW of the ballastmass and offset mass, and the hydrodynamic param-eters KD0 , KD, KL0 , KL, KM0 , KM , mf3 and mf1.

Parameters corresponding to mass and inertia maybe measured directly. The mass and buoyancy trim of

−10 −8 −6 −4 −2 0 2 4 6 8 10−100

−80

−60

−40

−20

0

20

40

60

80

100

Angle of Attack α [o]

Glid

e P

ath

Ang

le ξ

, Pitc

h A

ngle

θ [o ]

SLOCUM equilibrium glide angle and pitch for estimated L, D parameters

ξθ

Figure 8: Equilibrium glides using lift, drag, esti-mated from reference data.

the glider can be measured by weighing the glider inair and in water. The position of the glider CG maybe determined through direct experimental measure-ment. The position of the CB is the centroid of thedisplaced volume of water, and can be computed fromthe glider geometry. Other mass parameters can bedetermined using similar methods. The inertia char-acteristics of the glider can be measured several ways;one way is the bifilar pendulum method, which usesthe glider’s frequency of oscillation in a pendulum ap-paratus. Note that the moment of inertia does notappear in the equilibrium equations.

If direct measurement is not possible, for examplewhen a glider is already at sea, it may be possible todetermine some of these parameters through analysisof glider data and by comparison of several equilibria.For example, in Section 3.2 we describe a method toidentify a glider’s buoyancy trim offset from flight testdata and in Section 3.3 we use an analogous methodto identify the glider static pitch trim.

A variety of methods were used to determine modelhydrodynamic parameters, including reference hydro-dynamic data for generic shapes, aircraft, ships andsubmarines, computational fluid dynamics (CFD)analysis, wind tunnel data, and flight test data. Anextensive selection of references is available, includ-ing [4], [6] and [10]. Because the hydrodynamic pa-rameters are sensitive to small changes in the vehiclegeometry, the challenge in determining these param-eters is to accurately match actual flight data.

We first estimated hydrodynamic parameters forlift, drag and moment for the SLOCUM geometryusing reference data. This involved calculating thehydrodynamic forces on each of the glider compo-

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−90 −80 −70 −60 −50 −40 −30 −20 −10 00

0.2

0.4

0.6

0.8

1

1.2

1.4V

[m/s

]

Glide path angle ξ [o]

SLOCUM equilibrium glide speeds for estimated parameters, m0 = 250 g

Vx horizontal speedVz depth rateV glide speed

Figure 9: Equilibrium speed using lift, drag, esti-mated from reference data.

nents using theoretically and experimentally deter-mined reference data. These parameters were com-pared to the results of our preliminary wind tunneltests conducted at Princeton. More accurate windtunnel tests are in progress using the methods of [9].In addition, calculations of glider hydrodynamic char-acteristics using CFD analysis appear in [5].

Solving Equation (6) for V gives

V =∣

∣

∣

∣

zsin(θ − α)

∣

∣

∣

∣

(10)

Substituting Equation (10) and the hydrodynamiccoefficients (7), (8), and (9) into Equations (4) and(5) gives us

0 =12ρCL(αeq)A

(

zeq

sin(θeq − αeq)

)2

sin αeq

−12ρCD(αeq)A

(

zeq

sin(θeq − αeq)

)2

cos αeq

−m0eqg sin θeq (11)

0 =12ρCL(αeq)A

(

zeq

sin(θeq − αeq)

)2

cos αeq

+12ρCD(αeq)A

(

zeq

sin(θeq − αeq)

)2

sin αeq

−m0eqg cos θeq (12)

These equations include measured quantities z, θand m0. Angle of attack α is a function of v1 and v3and is not sensed. Hydrodynamic coefficients CL(α)and CD(α) have been estimated but are not knownexactly. These estimates, however, do yield forces inthe form (7) and (8). Substituting (7) and (8) into

(11) and (12) gives two equations with four param-eters KD0 , KD, KL0 , KL and unknown α. We usean existing estimate the value of the lift parametersand then determine drag parameter values consistentwith the flight test data. This is necessary becauseof the limited number of states available from theglider. Angle of attack or velocity data would allowus to determine more parameters from experimentaldata.

3.1 Lift

The SLOCUM glider body is symmetric from top tobottom and the wings are symmetrical flat plates.From this, the reference methods show that liftshould be zero at angle of attack α = 0 and shouldbe antisymmetric about α = 0. We compared esti-mates of the lift coefficient of the glider from threesources: aerodynamic reference data, CFD analysisfrom [5], and preliminary wind tunnel data. Theseestimates are reasonably close to one another. Thelift coefficient from [5] was computed using the mostadvanced methods, so we use this estimate for CL(α)by frontal area:

CL(α) = 11.76 α + 4.6 α|α| (13)

where α is in radians. Note that this is close to, butnot exactly, linear in α as modelled in (8).

Equations (11) and (12) may be rearranged, givenglider lift coefficient (13) and the steady-glide-testsensor data described in Section 2.2. Solving (12)for drag Deq and substituting into (11) gives

0 = Leq sinαeq −m0eqg sin θeq

−(

Leq cos αeq −m0eqg cos θeq

sinαeq

)

(cos αeq),

where Leq =12ρCL(αeq)A

(

zeq

sin(θeq − αeq)

)2

.

This equation may be solved for the angle of attackfrom flight data for a given steady glide.

3.2 Drag

In this section we determine a drag coefficient for theglider given (13), such that steady glides computedwith the equilibrium equations are consistent withflight test data. We describe first an analysis usingthe buoyancy tank ballast mb as the glider net buoy-ancy m0. This yields a CD(α) that is inconsistentwith our expected drag in both form and magnitude,as discussed below. We then describe a method used

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to identify a static buoyancy trim offset in the testglider. The buoyancy trim offset is then used to com-pute a CD(α) that is more consistent with theoreticaland other predictions.

Drag estimates calculated using aerodynamic refer-ence methods or preliminary wind tunnel tests eachpredict that glider drag coefficient CD(α) will havethe form given in Equation (7). Because of theglider’s symmetrical design, drag should be symmet-rical (an even function) with respect to angle of at-tack, with the minimum (profile) drag at zero angleof attack.

Using Equation (11) or (12), one can solve forCD(αeq) given data for a steady glide and the lift andangle of attack from Section 3.1. Figure 10 shows thedrag coefficient determined for each glide in the setof steady glides from test data. Each point on theplot corresponds to the coefficient of drag calculatedfor one steady glide.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5CD (by frontal area) vs. AoA α from steady glides

alpha [o]

CD

(α)

Figure 10: CD computed from equilibrium glide dataassuming no buoyancy trim offset.

Note that glides with positive angle of attack α,which are glides downwards, have much higher co-efficients of drag than the group of glides upwardsglides with negative angles of attack. This result isnot consistent with any of the estimates for the gliderdrag dependence on α. Our reference calculation ofdrag predicts a parabolic drag dependence on angleof attack. The drag shown is also higher than thepredicted drag.

One possible explanation for the differences be-tween upwards and downwards glide is that the glideractually has an asymmetrical drag curve. Some ele-ments of the glider geometry are asymmetrical fromtop to bottom, including the CTD sensor located be-low one of the wings and the vertical tail. However,

these items are small compared to the glider bodyand wings, both of which are symmetrical, so it is notexpected that these small differences would accountfor such a large difference in the drag. Regardless ofthis asymmetry, drag is still expected to be close tominimum at zero angle of attack.

The simplest and the most obvious explanation forthe difference between the upwards and downwardsglides is an offset in the glider buoyancy trim. Thisoffset can be found using the symmetry of the gliderand lift coefficient to compare upwards and down-wards glides at the same magnitude pitch angles.As noted, because of the symmetrical design of theglider, the lift curve is an odd function with respectto angle of attack and the drag curve is expected tobe an even function, see Figure 8.

Glides conducted at the same magnitude pitch an-gle upwards and downwards should have the samemagnitude glide path angle ξeq and angle of attackαeq. Given the symmetry in lift and drag, and ourapproximation to the longitudinal plane, differencesin velocity between these glides are caused by differ-ences in the driving buoyant force. By comparingsuch glides in the flight test data, we estimate thetrim offset in the glider buoyancy.

First we substitute m0eq = mbeq +4m0 into Equa-tion (14). Using the steady glide data, we estimatethe buoyancy trim offset 4m0 by requiring glideswith the same |θeq| to have the same |αeq|. Thisinvolves solving for αeq for each of the symmetri-cal glides as a function of 4m0 and determining the4m0 for equal |αeq|. Using the available data, weestimate the buoyancy trim offset to be 4m0 = −73grams. This means that, for the water density andthe weight of the glider WE01 during these tests, theglider is 73 grams light (positively buoyant) when theballast tank is set at the half full, mb = 0, “zero buoy-ancy” point. When this buoyancy trim offset is notaccounted for, as shown in Figure 10, it appears thatthere is more drag going down (i.e., it is harder to godown) and less drag going up (i.e., easier).

Substituting m0eq = mbeq + 4m0 into Equa-tions (11) or (12), CD(αeq) may be computed for eachsteady test glide, see Figure 11.

A least-square fit of the data, assuming drag of theform (7), gives drag parameter

CD(α) = 0.214 α + 32.3 α2. (14)

where α is in radians. As shown in Figure 11, thesteady-glide data points are close to a parabolic func-tion of angle of attack α and are symmetrical aboutα = 0. These properties are consistent with the ex-pectations from our reference calculations for drag.

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−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5CD (by frontal area) vs. AoA α from steady glides

α [o]

CD

(α)

Figure 11: CD computed from equilibrium glide dataassuming buoyancy trim offset of - 73 grams.

The magnitude of the drag least-squares fit (14)is about 75% greater, at α = 0, than the drag calcu-lated from references, and as much as 150% greater atα = −3o. There are several possible explanations forthe difference between the drag found here and thedrag predictions using reference calculations. One ex-planation is that the drag model is based on an idealgeometric model and does not include variations inthe geometry (e.g. surface roughness, wing deforma-tion, ...), protrusions and additions such as the CTDsensor. The model therefore does provide a drag esti-mate that is lower than the measured drag. Anotherpossible explanation is that the steady glides mea-sured in the flight data deviate from the longitudinalplane. This is highly probably because of the gliderstatic roll miss-trim. It can be seen in the flight testdata that, when the glider is set to glide with rud-der fixed at zero, there is some small yaw rate. Thisshows that at least some of the glides have a nonzerosideslip angle. Because the glider has no sideslip andangle of attack sensors, the order of the sideslip anglemust be estimated from other sensor data. By exam-ining the data it may be seen that the yaw rates arevery low in these cases, suggesting the sideslip angleis small and that its effects on the glider yaw rateare small. In other test glides a small rudder anglewas used and was enough to offset the yaw rate dueto roll. It is possible that this sideslip angle due tothe static roll is of the same order of magnitude asthe angle of attack, and may account for the differ-ences between the expected and estimated drag. Thedifference may also be explained by a combination ofthese factors. This is a continuing subject of analysis.

Using the hydrodynamic coefficients determined

from the data, the equilibrium equations may be usedto compute a new set of steady glide conditions, aswas done for Figures 8 and 9 using the estimated pa-rameters. Figure 12 shows the steady glide anglesgiven the parameters identified from the data. Fig-ure 13 shows the steady glide speeds given the sameidentified parameters. For a 25o glide angle, the iden-tified parameters yield a depth rate of 20 cm/s and ahorizontal speed of 42 cm/s, as can be seen from Fig-ure 13. This is consistent with estimates from glideroperations conducted by Webb Research Corporationand WHOI.

−10 −8 −6 −4 −2 0 2 4 6 8 10−100

−80

−60

−40

−20

0

20

40

60

80

100

Angle of Attack α [o]

Glid

e P

ath

Ang

le ξ

, Pitc

h A

ngle

θ [o ]

SLOCUM equilibrium glide angle and pitch for identified L, D parameters

ξθ

Figure 12: Equilibrium glides using Lift, Drag fit todata

−90 −80 −70 −60 −50 −40 −30 −20 −10 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V [m

/s]

Glide path angle ξ [o]

SLOCUM equilibrium glide speeds for identified L, D parameters, m0 = 250 g

Vx horizontal speedVz depth rateV glide speed

Figure 13: Equilibrium speed using Lift, Drag fit todata.

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3.3 Pitch Moment

To compute the hydrodynamic moment on the gliderduring steady flight, we use Equation (3). The mo-ments due to the internal mass, the ballast tank andthe offset mass may be computed from the steady-glide sensor data. Other terms in Equation (3) rep-resent the hydrodynamic moments due to the glider’sadded mass and the rest of the glider hydrodynam-ics. For the steady-state analysis, we will group theseterms together as

12ρAV 2CM (α) = (mf3 −mf1)v1eqv3eq + MDLeq . (15)

Substituting (15) into Equation (3) and rearranginggives

12ρAV 2CM (α) = (mg(rP1eq cos θeq + rP3eq sin θeq)

+ mbeqg(rB1 cos θeq + rB3 sin θeq)

+ mwg(rW1 cos θeq + rW3 sin θeq)) (16)

which may be solved for CM (α) given the steady glideflight data.

During trimming of the glider before a mission,static weights are positioned within the hull and in-struments may be installed or moved. This changesthe mass and trim of the glider. In the model, the uni-formly distributed hull mass mh and the offset massmw represent the distribution of fixed components inthe glider. The position of the offset mass may bedetermined using static measurements during trim-ming of the glider before launch, or calculated fromflight test data. Before launch the SLOCUM glidersare trimmed manually using a static buoyancy tank.The glider ballast tank is set to half-full and weightsare adjusted within the hull to make the glider neu-trally buoyant and level. Using data from the statictrim process, we may determine mw and rw by solv-ing Equation (3) with v1 = 0 and v3 = 0. The massand position of the ballast and sliding mass are de-termined from flight test sensor data.

In some cases, as discussed in Section 3.2, it maynot be possible to measure the static trim of theglider. In this case it is possible to determine the massoffset rW1, given rW3 and a set of data from symmet-rical steady glides up and down. Using a method ofcomparison analogous to that in Section 3.2, we usethe symmetry of the glider to compare upwards anddownwards glides at the same pitch angle. To esti-mate rW1 we use the sensor state data from theseglides and first compute the moment due to inter-nal masses (as a function of unknown rW1) for eachglide. This is set equal to the hydrodynamic momentaccording to (3) for each glider. We then equate the

magnitude of the moment coefficients for an upwardglide and a downward glide corresponding to the samepitch angle magnitude and solve for rW1. For gliderWE01 as trimmed in these tests, this analysis givesrW1 = −0.093 m.

Once we have determined the internal masses andpositions, we may solve for CM (α) for each glide inour set of steady glides. The result and a least squaresfit are shown in Figure 14.

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

CM

(by frontal area) vs. AoA α from steady glides

α [o]

CM

(α)

Figure 14: CM computed from equilibrium glidedata.

The least-square fit of this data, using the formgiven in Equation 9 with KM0 = 0 because of thesymmetry of the glider, is

CM (α) = 0.63 α.

where α is in radians.The hydrodynamic moment on the glider is small

compared to the moments due to the internal masses.The moment due to the ballast and sliding mass to-gether is around 35 N·m in the nose-down direction.The offset mass, located behind the vehicle CB, pro-vides a countering nose-up moment. At equilibrium,the hydrodynamic moment is the difference betweenthese moments, as shown in Equation (16). We esti-mate the hydrodynamic moment to be of the order 0.1N·m or less. Because this moment is small in magni-tude as compared to the other moments, small uncer-tainties in the positions of the internal masses resultin relatively large uncertainties in the moment coeffi-cient. Note that some variation in CM (α) would yieldplots lying within the error bounds shown in Figure14. Effects due to glider motion out of the longitu-dinal plane in some flight tests will also influence theaccuracy of the analysis, as noted in Section 3.2. In

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order to determine the glider coefficient of momentmore accurately, other methods such as wind tunneltests and CFD analysis may be employed. The rel-atively small hydrodynamic moment means that, forgliders like SLOCUM travelling at low velocities, thepitching effect of the internal mass controls can easilyovercome the vehicle hydrodynamic moment.

4 Final remarks

Once the equilibrium steady glides are matched, theremaining unknown parameters are those that appearonly in the dynamic equations of the vehicle and notat the equilibria. To solve for these parameters wemake use of a simple metric. The metric defines howthe parameters should provide an adequate match ofour dynamic model output to the flight test data.The parameters are determined through an iterativeprocess that involves simulating the glider dynamics,measuring the quality of fit to the experimental datausing the metric, and adjusting the parameters fora better fit. The process is carried out numerically,using a steepest-descent search method to adjust themodel parameters before each iteration and compar-ison of model output with the flight test data. Theresults of this analysis will be reported in a futurework.

The method, in Section 3.2, for determining thestatic buoyancy trim offset of the glider could pos-sibly be adapted to trim the glider at the beginningof a deployment and to detect system changes in theglider during deployment. Possible system faults inthe glider that could occur during a mission include(1) fouling by seaweed, (2) taking on water througha small leak in the hull or (3) a problem with theballast system. These faults could be detected anddistinguished using the methods described here bycomparing upwards and downwards glides.

The results and analysis of the flight data usedin this paper suggest benefits to making use of ad-ditional sensors and methods for future flight tests.Moorings or fixed sensors located off the glider butin the flight test area could be used to measure thecurrent conditions at the operational depths. For thepurpose of flight tests, sensors could be temporarilyinstalled on the glider to measure its velocity and an-gle of attack. Such sensors are standard in aircraftflight test but would require adaptation for use onthe glider, both because it is underwater and travelsat a low velocity. During flight testing, position andvelocity could both be measured by an acoustic rang-ing system. Measuring data with high enough accu-racy would probably require a purpose-built acous-

tic range, the use of an existing naval test range orthe like. Some acoustic systems are already in theprocess of being adapted for use on the SLOCUMglider, and these could provide a useful estimate ofthe glider position and velocity during tests. Thesesensor systems vary in size and expense, with the useof an existing doppler current measuring installationbeing relatively inexpensive. It would not be neces-sary to add these flight test sensors permanently tothe glider, but rather install them temporarily for theduration of flight tests. These types of data wouldallow more accurate measurement of the glider dy-namics and hydrodynamic characteristics.

5 Acknowledgements

The first three authors would like to thank DavidM. Fratantoni from WHOI for the opportunity tojoin the glider test cruise on board the RV WaltonSmith in January 2003 and for providing the glidersand assistance in performing the glider flight tests.Thanks also to the captain and crew of the RV Wal-ton Smith for their help and support. We would alsolike to thank Clayton Jones and Douglas C. Webbfrom Webb Research Corporation and Tom Camp-bell from Dinkum Software for their expertise andhelp in all areas related to the SLOCUM gliders.

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