Range-Based AUV Navigation Expressed inGeodetic Coordinates

Rami S. Jabari and Daniel J. StilwellBradley Department of Electrical and Computer Engineering

Virginia Techrami, stilwell@vt.edu

Abstract—This work describes a range-based acoustic navi-gation system that utilizes range measurements from a singlemoving transponder with a known location to estimate theposition of an autonomous underwater vehicle (AUV). Addition-ally, the navigation system simultaneously estimates the currentsacting on the AUV. The novelty of our approach is to expressthe navigation algorithm and the position estimate entirely ingeodetic coordinates. Navigation estimates are expressed withrespect to the WGS84 coordinate reference system, althoughother geodetic coordinate systems could be used with very minormodification of our approach. Performance of the navigationsystem is evaluated from field trials conducted using the VirginiaTech 690s AUV.

I. INTRODUCTION

We describe a navigation algorithm for autonomous un-derwater vehicles (AUVs) that uses a sequence of acousticrange measurements between the AUV and a moving beaconto estimate the position of the AUV. The novelty of our workis to express the navigation algorithm entirely using geodeticlatitude, longitude, and altitude rather than local Cartesian co-ordinates. The proposed algorithm can simultaneously estimatecurrents and AUV position.

Due to the rapid attenuation of radio frequency (RF) signalsin water, RF localization, such as GPS, is not available tounderwater vehicles. Instead, underwater vehicles often useinertial or acoustic navigation technologies, or a combinationof both. Acoustic signaling between an AUV and a movingbeacon with a known location is used to acquire range (ordistance) measurements based on the round-trip time-of-flightof an acoustic signal. With additional measurements of thevehicle’s heading and speed (velocity), vehicle’s depth, anda rough estimate of the initial AUV position, the AUV canaccurately estimate its location in geodetic coordinates.

The problem of range-based localization has been studiedby several researchers. An extensive observability analysisin local Cartesian coordinates is presented in [1–4]. Theobservability analysis is extended in [5] to incorporate depth.Advancements in range navigation through the use of moresophisticated sensors are presented in [6]. Most recently, in[7], an observer model is proposed to simultaneously localizean AUV and multiple beacons. All of the previously citedworks have implemented a range navigation algorithm in alocal coordinate frame, such as North-East-Down (NED).

Our contribution is to derive a range navigation algorithmentirely in geodetic coordinates so that no local coordinate

frame is necessary and the bookkeeping that can arise when us-ing a local coordinate system is avoided. We show in Chapter3 of [8] that the results of the observability analysis conductedassuming local Cartesian coordinates are inherited by the rangenavigation problem expressed in geodetic coordinates. Thusthe same trajectories that are unobservable in local Cartesiancoordinates, e.g., a straight-line trajectory passing through thebeacon location, are also unobservable in geodetic coordinates.

An estimate of the AUV position is computed by anextended Kalman filter (EKF) that is derived assuming akinematic model of the AUV that is itself expressed in geodeticcoordinates. The EKF requires an initial estimate of the AUV’sposition, which is acquired via GPS prior to diving. Theestimated states of the EKF are latitude, longitude, and watercurrent speed in the latitudinal and longitudinal directions.The estimated states are computed from measurements of theAUV’s speed, heading, and depth.

We present experimental data acquired using the VirginiaTech 690s AUV shown in Figure 1. The AUV is equippedwith an acoustic modem that periodically computes the rangebetween itself and another acoustic modem attached to aboat. The topside (boat) modem transmits the position ofthe boat, acquired via GPS, to the AUV whenever the AUVacquires an acoustic range measurement. The data was post-processed to analyze the proposed range navigation algo-rithm’s performance. The range navigation algorithm has beenincorporated into the 690s navigation system and tested inreal-time. The proposed geodetic range navigation algorithm’sperformance is compared against the local range navigationalgorithm proposed in [4] and shows similar results. Thegeodetic range navigation solution is also compared to ageodetic dead reckoning solution.

Fig. 1. Virginia Tech 690s AUV

II. PRELIMINARIES

A. Acoustic Ranging

The concept of acoustic ranging is based on the time ofarrival (ToA) of acoustic signals between transducers. Forthe scenario in this paper, an acoustic modem and transducercombination on an AUV pings a remote acoustic modem andtransducer, which can be on a boat, another AUV, a buoy, etc.A return signal is sent and a range measurement is calculatedvia the round-trip time-of-flight of the acoustic signal. Thereturn signal from the remote modem contains encoded in-formation, such as the remote modem’s GPS coordinates anddepth, in its data packet. Figure 2 illustrates an AUV pinging atransducer that is attached to a boat. For this paper, the remotemodem is referred to as the topside modem and the remotetransducer is referred to as the topside transducer. The topsidemodem and transducer are mounted on a boat.

AUV

Transducer

Acoustic Modem

Topside Transducer

Fig. 2. The AUV is shown pinging the topside modem and transducercombination that is fixed to a boat.

For the range measurements to be accurate, it is assumedthat the speed of sound in the operation medium is known.The speed of sound in water can be approximated if the depth,temperature, and salinity are known. One approach to calculatethe speed of sound in ocean water is the Mackenzie equation[9]. The Mackenzie equation is only valid for temperaturesranging from −2 to 30 degrees Celsius, salinity from 30 to40 parts per thousand, and depth from 0 to 8000 meters. Forwater with a lower salinity content, an alternative approach isCoppens’ equation [10]. Coppens’ equation is only valid fortemperatures ranging from 0 to 35 degrees Celsius, salinity

from 0 to 45 parts per thousand, and depth from 0 to 4kilometers.

There are several practical issues that arise with underwateracoustic ranging. Errors in range measurements can arise fromthe assumed speed of sound in water being inaccurate. It isnot uncommon to drop data packets due to multipath or signalfading. Multipath can be extreme in shallow water due toreflections from the water’s surface and the sea floor. Anothercommon source of multipath originates from object reflectionsnear the acoustic modem. Signal fading and attenuation canbe a result of destructive interference due to multipath andsound absorption in water. Researchers in [11] discuss some ofthese issues along with the advancements in acoustic commu-nication. As mentioned in [12], other sources of interferenceinclude bubble plumes and propeller wakes. Small errors arepresent due to vehicle motion that occurs between the time arange measurement is initiated and the time it is received.There are instances when the data in an acoustic message(e.g., coordinates of the reference beacon) is incomplete orcorrupted. Occasionally, a range measurement will be inaccu-rate. If an inaccurate measurement is not rejected, it can resultin poor algorithm performance. This paper uses a probabilisticapproach to reject inaccurate range measurements.

B. Coordinate Frames

1) World Geodetic System 1984: The World Geodetic Sys-tem 1984 (WGS84) [13] is a geodetic coordinate system thatis used in GPS navigation systems. The WGS84 ellipsoid isdefined by the value of the semi-major axis a and flattening f .Other parameters that describe the shape of the ellipsoid arederived from a and f , including the value of the semi-minoraxis b and first eccentricity squared e2. If a geodetic datumother than the WGS84 ellipsoid is used, the constants aredifferent. Table I shows the constants that are used throughoutthis paper.

TABLE IWGS84 PARAMETERS; SIGNIFICANT DIGITS ARE AS SPECIFIED BY

WGS84.

Symbol Description Valuea Semi-major axis 6378137.0 mb Semi-minor axis 6356752.3142 me2 First eccentricity squared 6.69437999014e-31f Inverse flattening 298.257223563

A location relative to the WGS84 ellipsoid is expressed interms of latitude, longitude, and altitude. Latitude ϕ is the an-gular distance from the Earth’s equator, where a measurementnorth of the equator is positive and a measurement south ofthe equator is negative. Longitude λ is the angular distancefrom the Earth’s prime meridian, where a measurement eastof the prime meridian is positive and a measurement west ofthe prime meridian is negative. Lines of latitude range from-90 to 90 while lines of longitude range from -180 to180. Altitude h is the distance along the ellipsoidal normal.Positive altitude points away from the ellipsoid’s interior. The

location of a point relative to the WGS84 ellipsoid is denotedas g = (ϕ, λ, h) ∈ S2 × R where S2 is the unit sphere.

2) Earth-Centered Earth-Fixed (ECEF): The ECEF coor-dinate frame, described in [14], is a Cartesian coordinatesystem and has its origin fixed at the center of the Earth.The ECEF coordinate system is illustrated in Figure 3. TheECEF coordinate frame has the axes XECEF , YECEF , andZECEF . The axis XECEF extends through the intersectionof the prime meridian and the equator (0, 0). The axisYECEF extends through the intersection of the equator and90 longitude (0, 90). The axis ZECEF completes the right-handed coordinate system and extends through the true northpole (90, 0). The location of a point in the ECEF frame isdenoted as (x, y, z).

a

a

b

Prime Meridian

Equator

'

XECEF

YECEF

ZECEF

Fig. 3. Illustration of the ECEF coordinate frame. Variables a and b are thesemi-major and semi-minor axes, respectively.

Geodetic coordinates on the WGS84 ellipsoid can be trans-formed from latitude, longitude, and altitude to rectangularECEF coordinates via the closed form expressions

x = (N (ϕ) + h) cosϕ cosλ (1a)y = (N (ϕ) + h) cosϕ sinλ (1b)

z =[(

1− e2)N (ϕ) + h

]sinϕ (1c)

The altitude h is the distance along the ellipsoidal normal,between the ellipsoid’s surface and the point of interest. Theradius of curvature in the prime vertical N(ϕ) is defined

N(ϕ) =a√

1− e2 sin2 ϕ(2)

III. RANGE NAVIGATION SYSTEM

This section documents the formulation of the state transi-tion and measurement models used in the proposed geodeticrange navigation system. The map from geodetic to ECEFcoordinates (1) can be modified to account for the depth δ of

the vehicle

x = (N (ϕ) + h− δ) cosϕ cosλ (3a)y = (N (ϕ) + h− δ) cosϕ sinλ (3b)

z =[(

1− e2)N (ϕ) + h− δ

]sinϕ (3c)

In the sequel, x, y, and z, incorporate a depth measurement.The altitude h is assumed to be constant in the operation area.Prior to conducting missions, the AUV acquires GPS dataon the surface of the water. The altitude is averaged and isassumed constant for that operation area.

The EKF used in this paper is standard and the state vectors[k] = [ϕ[k], λ[k], vϕ[k], vλ[k]]T. The kinematic model is

s[k] = f(s[k − 1], k) + ω[k] (4)

and the measurement model is

z[k] = h(s[k]) + ν[k] (5)

where the noise models, ω and ν, are zero-mean multivariatenormal distributions with covariances Q and R for processand measurement, respectively. The error covariance P isinitialized

P [0|0] = E[(s[0]− E[s[0]])(s[0]− E[s[0]])T] (6)

where E is the expectation operator.

A. Kinematic Model

The kinematic model used for geodetic range navigation isderived in this section. The kinematic model f is based onthe geodetic kinematic model derived in Appendix A of [8].The Euclidean distance between the two points p1 and p2 ∈R3 is denoted as

d(p1,p2) = ||p1 − p2||= [(x1 − x2)2

+ (y1 − y2)2

+ (z1 − z2)2]12 (7)

The AUV used in this work receives propeller rotationrate measurements from the motor controller connected to thepropeller’s motor, depth measurements from a pressure sensor,and heading measurements from an MEMS attitude headingreference system (AHRS) at a rate of 10 Hz. Speed trialshave been conducted to map RPM to AUV speed in metersper second. The distance traveled by the AUV is computedby multiplying the forward speed of the vehicle u with thetimestep ∆T . It is assumed that there is no sideslip and thevehicle travels in the surge direction.

Because ∆T is small, the simplifying assumption that theAUV travels on a sphere with constant radius is made duringeach iteration of the kinematic model. The Earth is not a sphereso the radius is recomputed for every iteration of the kinematicmodel. This assumption is useful because it allows sphericaltrigonometry to be used since a closed-form expression doesnot arise when it is assumed that the AUV moves striclty alongthe surface of an ellipsoid during ∆T .

The sphere’s radius is the Euclidean distance from p =(x, y, z) to its center 0

d(p,0) = ||p− 0||=√x2 + y2 + z2 (8)

which can be rewritten in terms of g = (ϕ, λ, h)

dg(g,0) =

√(µ cosϕ cosλ)

2+ (µ cosϕ sinλ)

2+ (ρ sinϕ)

2

(9)

where µ = (N (ϕ) + h− δ) and ρ = [(1− e2)N(ϕ) +h− δ].The kinematic model f(s[k − 1], k) expressed in geodetic

coordinates is

f(s[k − 1], k) =

ϕ[k]λ[k]vϕ[k]vλ[k]

=

sin−1(Ξ + Σ)

λ[k − 1] + atan2(χ,Ω)vϕ[k − 1]vλ[k − 1]

+ ∆T

vϕ[k − 1]vλ[k − 1]

00

(10)

where Ξ = cos (ξ[u[k]]) sin(ϕ[k − 1]),

Σ = cos(ϕ[k − 1]) cos(ψ[k − 1]) sin (ξ[u[k]]) ,

χ = cos(ϕ[k − 1]) sin(ψ[k − 1]) sin (ξ[u[k]]) ,

and Ω = cos (ξ[u[k]])− sin(ϕ[k]) sin(ϕ[k − 1]).

The angular distance traveled is ξ[u[k]] = ∆Tu[k−1]d(g[k−1],0) and

dg(g[k − 1],0) is the distance from the center of the sphereto g[k−1] = (ϕ[k−1], λ[k−1]), which is recalculated everyiteration. The states are latitude ϕ, longitude λ, water currentspeed in the latitudinal direction vϕ, and water current speed inthe longitudinal direction vλ. The water currents vϕ and vλ areinitially set as 0, but are updated when a range measurementis available.

The model does not account for the AUV diving. This isnot a problem for short dives but is an issue if a large changein depth takes a while to achieve. Pitch should be incorporatedinto the model if long dives are needed.

B. Observation Model

The observation (measurement) model h represents themeasurement of the range between the AUV and the topsidetransducer that is defined in (3). The range measurement isthe Euclidean distance d(p1,p2) between locations p1 and p2.The Euclidean distance between locations g1 and g2 expressedin geodetic coordinates is

dg(g1, g2) = [µ21 cos2 ϕ1 + µ2

2 cos2 ϕ2

− 2µ1µ2[cosϕ1 cosϕ2 cos(λ1 − λ2)]

+ [ρ1 sinϕ1 − ρ2 sinϕ2]2]1/2 (11)

where µ1 = (N(ϕ1)+h1−δ1), µ2 = (N(ϕ2)+h2−δ2), ρ1 =[(1−e2)N(ϕ1)+h1−δ1], and ρ2 = [(1−e2)N(ϕ2)+h2−δ2].

The distance from the AUV to the topside transducer isdenoted dg(gAUV , gT ). The measurement model is

h(s[k]) = dg(gAUV , gT )

= [µ2AUV cos2 ϕAUV + µ2

T cos2 ϕT

− 2µAUV µT [cosϕAUV cosϕT cos(λAUV − λT )]

+ [ρAUV sinϕAUV − ρT sinϕT ]2]1/2 (12)

where explicit dependence on the time index [k] has beendropped for notational convenience.

C. Rejection of Inaccurate Range Measurements

A method to reject an inaccurate range measurement is de-rived in this subsection. Range measurements will occasionallybe inaccurate. To reject an inaccurate range measurement, aradius of error is calculated using variances from the EKFstate estimate error covariance P and measurement noisecovariance R. Measurements outside the radius of error arerejected. The acquired range measurement is dT . The radiusof error re is one standard deviation away from the expectedrange measurement since inaccurate range measurements aremuch larger than the true range. The radius of error is

re[k] =√σ2ϕϕ,m[k] + σ2

λλ,m[k] + σ2dT

[k] (13)

where σ2ϕϕ,m and σ2

λλ,m are the P11 and P22 entries in thecovariance P , respectively. They are expressed in meters andhave the subscript m. The measurement covariance R = σ2

dTis a scalar and is the variance of the measurement noise inmeters.

The variances σ2ϕϕ and σ2

λλ are in units of radians butare approximated in meters since range measurements are inmeters. Denote the length in meters per degree latitude asmϕ and the length in meters per degree longitude as mλ.The length of a degree, in meters, of latitude and longitudeapproximated using the expressions in [15] are

mϕ = 111132.92− 559.82 cos(2ϕ) + 1.175 cos(4ϕ)

− 0.0023 cos(6ϕ) (14a)mλ = 111412.84 cos(ϕ)− 93.5 cos(3ϕ) + 0.118 cos(5ϕ)

(14b)

The approximated variances of the latitude and longitudepositions in meters are

σ2ϕϕ,m = σ2

ϕϕ

(180

πmϕ

)2

(15a)

σ2λλ,m = σ2

λλ

(180

πmλ

)2

(15b)

where the subscript m corresponds to meters.It is assumed that the noise models are independent normal

distributions. Hence, the sum of the distributions is

N (0, σ2ϕϕ,m + σ2

λλ,m + σ2dT ) =N (0, σ2

ϕϕ,m)

+N (0, σ2λλ,m)

+N (0, σ2dT ,m) (16)

and one standard deviation of (16) is re defined in (13).To accept or reject the range measurement, the expected

measurement calculated via the Euclidean distance, is com-pared to the actual measurement dT . If the absolute valueof the difference is less than re, then the measurement isaccepted; if not, it is rejected.

IV. EXPERIMENTAL TESTBED

The Virginia Tech 690s AUV, pictured in Figure 4, was usedfor initial data collection. The 690s is a streamlined AUV withfour independent control surfaces and a propeller at the stern.The AUV displaces 61 pounds, is 61 inches long, and has a6.9 inch diameter. GPS is used for surface navigation whilerange navigation and inertially aided dead reckoning are usedunderwater.

Fig. 4. The Virginia Tech 690s AUV during Claytor Lake field trials.

A. AUV Sensors

The AUV sensors that are relevant to experiments reportedherein are listed in Table II.

TABLE IITHE VIRGINIA TECH 690S SENSOR SUITE

Measurement SensorAttitude Microstrain 3DM-GX3-25 AHRSAbsolute position Linx RXM-GPS-R4 GPS receiverDepth Keller PA-30X pressure sensorRange Teledyne Benthos ATM-903Propellerangular velocity Castle Creations motor controller

The Microstrain 3DM-GX3-25 AHRS reports roll, pitch,yaw, linear accelerations, and angular rates. Only yaw is usedfor the proposed range navigation algorithm. The TeledyneBenthos ATM-903 acoustic modem and transducer that isintegrated into the AUV, shown in Figure 5, requests a rangemeasurement and location from the topside modem. Thetopside transducer is connected to a Universal Deck Box(UDB) that is located on the boat. The Castle Creations motor

controller reports the propeller’s angular velocity. Speed trialswere conducted to map the speed of the propeller in RPM tothe speed of the AUV in m/s, assuming a linear relationshipbetween the two.

Fig. 5. The 690s AUV contains a Teledyne Benthos ATM-903 acousticmodem (left) and transducer (right).

B. Universal Deck Box

The UDB is pictured in Figure 6. The UDB-9400 containsa Benthos acoustic modem and connects to an AT-440-MFtransducer, which operates between 16-21 kHz. The UDBupdates its location (latitude and longitude) at a rate of 1 Hzfrom an external USGlobalSat BU-353 GPS receiver, whichhas a horizontal position accuracy of 10m 2D RMS. The depthof the transducer below the surface of the water is set to aconstant value.

Fig. 6. The Teledyne Benthos Universal Deck Box is pictured above.

The 690s AUV is set to ping the UDB every ten secondsto calculate range and acquire the UDB location (latitude,longitude, and depth). For our experiments, the transmit powerlevel was set to -9 dB of attenuation and the acoustic bit ratewas set to 800 bits/second.

V. EXPERIMENTAL RESULTS

The performance of the geodetic range navigation systemincluding current estimation is assessed in this section. Trials

were conducted in Claytor Lake, VA, and the ChesapeakeBay. This paper only shows the results from one run in theChesapeake Bay. Chapter 4 of [8] shows results from four runsin Claytor Lake, VA, and two runs from the Chesapeake Bay.

Position and current estimation is assessed in this section.The final estimated position of the AUV at surfacing wascompared to the true position determined from GPS. Theaverage error for two trials conducted in the Chesapeake Baywas 21.53m. To assess the performance of the range navigationalgorithm in estimating currents, the EKF current estimate wascompared with the current estimate provided by the NationalOceanic and Atmospheric Administration’s (NOAA) Tides andCurrents at Sandy Point (38.3083, -76.4550) [16]. The fieldtrials were conducted on 14 April 2016 and the operation areawas near Solomons, MD. The data sets reported herein werecollected between 12:00 and 13:00 local time.

It was assumed that the altitude in the operation area isconstant. To determine the altitude in the operation area, theGPS data from the AUV was recorded on the surface of thewater and then averaged. The speed of sound for the acousticmodem was determined by Coppens’ equation.

A. Assessment of Position Estimation

We describe performance assessment of the trajectoryshown in Figure 7. The dead reckoning trajectory of the AUVis shown in magenta. The trajectory based on the geodeticrange navigation EKF is shown in green. The green x’sindicate the location where a range measurement is accepted.The white and red diamonds are the true starting and endingpositions of the AUV, respectively. GPS locations before andafter diving are shown as red dots. The location of the UDBon the boat is shown in cyan; its initial position starts at thewhite triangle and ends at the red triangle. The operation areais plotted over a satellite image provided by the Google MapsAPI.

The acquired range measurements during the trajectory areshown in Figure 8. Range measurements are shown at thetimes that they are acquired. Measurements that are acceptedare represented by blue x’s while rejected measurements arered x’s. It is clear by visual inspection that the rejected rangemeasurements are the outliers. It can be observed that rangemeasurements are acquired approximately every ten seconds ifthey are not dropped or rejected. When a range measurementis received and accepted, the EKF update step is performedand the AUV corrects its position.

By visual inspection it is clear that the EKF is correctingthe AUV position throughout the trajectory. The distance errorfrom the GPS fix at surfacing and the final estimated positionis 13.97 meters.

Fig. 7. The mission is illustrated above. The satellite image of the operationarea was acquired from Google Maps.

Fig. 8. Range measurements from mission. 32 range measurements wereacquired over a 750 second time period and ten measurements were rejected.

Another method of evaluating the performance of the rangenavigation algorithm in geodetic coordinates is to compareit to the previously developed algorithm reported in [4] thatoperates in a local Cartesian NED coordinate system. Thecovariance matrices for the local range navigation algorithmare initialized using the same variances in meters and me-ters/second. The origin of the local NED coordinate frame isset as the initial location of the boat and UDB. The navigationsolution generated by our approach is nearly indistinguishablefrom the solution generated assuming Cartesian coordinates,which suggests that there is no loss of accuracy due toexpressing the navigation algorithm in geodetic coordinates.

Fig. 9. The local NED range navigation solution (green) is compared tothe solution from the geodetic range navigation algorithm converted to NED(blue). Performance is similar.

B. Assessment of Current Estimation

This subsection assesses the performance of the currentestimation. The current velocity model in the kinematic model(10) is the mechanism by which all velocity errors arerepresented in our navigation solution. Thus we do expectthat current velocity estimate to be an excellent estimate ofactual current velocity. The estimate also attempts to representerrors in heading (magnetometer) bias and water relativespeed (propeller rotation rate), among other sources of error.Nonetheless, we provide in Figure 10 a rough comparisonof the estimated current velocity with respect to an estimateprovided by NOAA at a nearby location. NOAA’s Tides andCurrents at Sandy Point [16] provides an estimate of thetidal current magnitude and direction corresponding to thetime during which the field trial was conducted. The trialwas conducted at 12:31 local time. NOAA does not providecurrent estimates at that exact times, but a plot is providedthat interpolates data between flood and ebb tides. Of course,the accuracy of NOAA’s estimate when applied to the locationwhere the experiments where conducted is questionable.

The geodetic range navigation algorithm estimates latitudi-nal and longitudinal currents in units of rad/s. For plottingpurposes and to compare against NOAA’s values, the EKFestimates are approximated in m/s. The latitude of the positionmeasurement acquired from GPS when surfacing is used forthe conversion. The latitudinal current is denoted as vNorthand the longitudinal current is denoted as vEast in m/s. Theestimated currents are compared to NOAA’s estimate in Figure10.

Fig. 10. The latitudinal and longitudinal current estimates are converted tounits of m/s and compared against the NOAA current estimate.

VI. CONCLUSION

This work proposes a range navigation algorithm in geodeticcoordinates for AUVs. The main contribution of this workis the implementation of a range-based navigation systemin geodetic coordinates for an AUV. The algorithm wasimplemented on the Virginia Tech 690s AUV and tested inreal-time. Additionally, a method to accept and reject rangemeasurements is proposed. The proposed geodetic algorithmis compared to the local NED range navigation algorithm from[4] and the results are similar. The true trajectory underwaterand the true current were not instrumented. The performanceof the geodetic range navigation algorithm is based on thecomparison with the true position when surfacing and currentsestimated by NOAA.

In future work, the proposed geodetic algorithm can beextended to incorporate the case where the sensor noise isdependent on the state of the system [17]. A kinematic modelthat incorporates pitch can be used to achieve more accurateresults for deeper dives.

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