AN ANALYTICAL FRAMEWORK FOR PREDICTING THE PERFORMANCE OFAUTONOMOUS UNDERWATER VEHICLE POSITIONING

Brian BinghamUniversity of Hawaii at Manoa

Mechanical Engineering2540 Dole St., Holmes Hall 302

Honolulu, HI 96822(808)956-7645

bsb@hawaii.edu

Abstract— In this paper we present a set of simplemodels to predict the performance of underwater position-ing solutions for autonomous underwater vehicles (AUVs).Improvements in underwater navigation continue to con-tribute to the expansion of our capability for exploration,scientific discovery and environmental monitoring. Despitethe diversity instruments and algorithms employed toimprove performance, the fundamental trade-offs involvedin estimating position can be understood with a fewgeneral stochastic observation models and information-centric analysis.

We present models for the most common oceanographicnavigation observations: absolute reference solutions suchas long-baseline (LBL) acoustic positioning, relative po-sitioning solutions such as Doppler velocity log (DVL)dead-reckoning and accurate heading such as a fiberoptic gyro (FOG) reference. We combine these simplemodels using the Cramer Rao lower bound to predict boththe precision and accuracy of common AUV positioningsolutions. To demonstrate the utility of such predictions, wemodel the uncertainty of current solutions and proposedfuture implementations. Many current solutions use acombination of LBL positioning and DVL odometry. Weshow how this approach can predict the performance ofsuch a configuration.

Finally, we substantiate the use of this frameworkby comparing model-based performance predictions toexperimental evidence. The field experiment we presentis a controlled evaluation of DVL dead-reckoning, usingthe Jason remotely operated vehicle (ROV). We observe atotal error growth that can be approximated as 0.04% ofdistance traveled.

I. INTRODUCTION

Autonomous Underwater Vehicles (AUVs) are power-ful tools for exploring, investigating and managing ourocean resources. As the capabilities of these platformscontinue to expand and AUVs continue to mature asoperational assets, navigation remains a fundamentaltechnological component. Determining position, in real-

time or via post-processing, is a foundational capabil-ity; improved navigational techniques make possible theapplication of AUV technology to a wider variety ofscientific, military and industry needs.

The two basic building blocks of an underwater nav-igation solution are absolute and relative positioning.There are many methods and a variety of sensors for eachtype of positioning (see [1]–[3] for review articles), butthe fundamental trade-off is captured by the dichotomyof absolute versus relative positioning.

The goal of the research presented here is to predictthe performance of state-of-the-art AUV navigation so-lutions. Predicting the resulting navigation uncertaintyrequires a fundamental understanding of how the systemaccumulates information in order to estimate position.

A. Approach

We pose the underwater positioning as an estimationproblem, allowing us to predict the performance of avariety of AUV navigation solutions. To do this we usesimple stochastic models of both absolute and relativeposition references. These models quantify the salientcontributors to system-level positioning performance:uncertainty in range observations, geometry of fixedreference locations, uncertainty in velocity observations,uncertainty in heading observations, update rate of ob-servations and vehicle speed. The stochastic modelscombine each of these parameters, and, by using theCramer Rao lower bound (CRLB), we can predict howeach component of the system affects the overall nav-igation performance. To illustrate the efficacy of thesesimple models we use data from field operations toqualify the agreement between the model and real-worldperformance

II. BACKGROUND AND CLOSELY RELATED WORK

A. Underwater Positioning and Navigation

Absolute position is estimated using an external ref-erence at a known location. At the surface the globalpositioning system (GPS) [4] can provide such a refer-ence using orbiting satellites. Because seawater is opaqueto electromagnetic radiation, underwater applicationsuse acoustic positioning solutions such as long base-line (LBL), short baseline (SBL) or ultra-short baseline(USBL) to provide an absolute position reference [5].For AUV applications the most common of these is LBLpositioning where transponders moored to the seafloorprovide a fixed reference. In either case, using GPSduring a surface interval or acoustic positioning whilesubmerged, the result is a position estimate in threedimensions with respect to an absolute Earth-centered,Earth-fixed reference (ECEF) frame with bounded un-certainty.

The performance of LBL positioning can vary greatlydepending on the geometric configuration of the fixedtransponders and the uncertainty in the estimation ofrange between the AUV and these fixed locations. Hunt,et al. provided one of the early descriptions of LBLpositioning in 1974. They estimated the accuracy of thisearly deep-water implementation to be between 10 and20 m, depending on signal-to-noise ratio of the transmit-ted pulse [6]. Using similar continuous wave pulse trans-missions, we estimated the range standard deviation tobe 3.97 m during a using Autonomous Benthic Explorer(ABE) AUV mission using Benthos transponders1 [7].The REMUS AUV comes standard with LBL capabilitieswith a reported repeatability of approximately 1.5 m[8]. Using higher frequencies and more complex signalprocessing can yield higher performance positioning.We report have achieved sub-centimeter resolution usinghigh-frequency spread-spectrum signals [9], [10].

Not only are these reported values for performancehighly variable, but the reader will also note that it ischallenging to compare the performance because thereporting of uncertainty is not standard. While someauthors refer to precision, some attempt to quantifythe overall accuracy of the solution. Even the metricsare variable as some report summary statistics such asstandard deviation, CEP or 95% confidence while somesimply report a single value for measured error.

As a complement to absolute positioning, dead-reckoning provides an estimate of relative position, typ-ically provided by either a inertial navigation system(INS) or, more commonly, a DVL integrated with an

1Teledyne Benthos, Inc., TR-6001 acoustic transponders.

accurate heading reference. Dead-reckoning provides anestimate of the distance traveled from an arbitrary refer-ence point. A key advantage of this mode of navigation isthat it is self-contained without the need for an externalreference, making it ideal for rapid deployments. Thedisadvantage is that the uncertainty in dead-reckoningis not bounded, the quality of the position estimatedegrades with time and distance traveled.

Just as with acoustic positioning performance, dead-reckoning performance can vary greatly depending onimplementation. For example, Brokloff reported positionerrors between 0.33 and 0.51% distance traveled [11]and Bluefin Robotics reports 0.5% of distance traveled(three-sigma) for real-time dead-reckoning and 0.3%distance traveled after post-processing [12]. Using amore sophisticated INS, McEwen et al. report a positionerror growth of less than 0.05% distance traveled forusing a DVL/INS combination developed by Kearfottand RD Instruments 2 [13].

Neither absolute or relative positioning is typicallyused in a standalone fashion, nor is there a standard,one-size-fits-all solution. By combining the right dead-reckoning solution with periodic updates from an abso-lute position reference, the advantages of both methodsare combined to deliver high-update rate position updateswith bounded absolute precision [14].

B. Modeling Positioning Uncertainty

This research builds on previous efforts in the mod-eling of navigation uncertainty. The accuracy of multi-lateration solutions as been investigated in the contextof underwater [15], terrestrial (GPS) [4], [16] and space[17] navigation. Specifically for long-distance AUV nav-igation, Eustice, et al., developed a uncertainty model topredict the performance of one-way-travel-time acousticnavigation that utilizes both absolute and relative navi-gation references [18].

To integrate the stochastic models we make use of theCramer Rao lower bound, a standard tool of estimationtheory [19], [20].

III. STOCHASTIC MODEL FOR PERFORMANCE

PREDICTION

The framework we propose uses two simple compo-nent models for quantifying the uncertainty in absolute(range-based) and relative (dead-reckoning) underwaterpositioning solutions. We then use the CRLB to combinethese two observations models allowing us to predict theperformance of today’s integrated navigation solutions.

2The Seaborne Navigation System/Doppler Velocity Log (SEADeViL) from Kearfott Corporation.

Fig. 1. Illustration of spherical positioning by trilateration. In thiscase three ranges are measured with no uncertainty to ascertain thetwo-dimensional position estimate (xh).

A. Model for Range-Based Absolute Positioning

The fundamental concept of estimating positionthrough time-of-arrival measurements is the basis formany underwater, terrestrial and space navigation solu-tions. For the purposes of predicting the performanceof AUV positioning, we use a spherical positioningmodel common for both LBL and GPS applications. Thisconcept is illustrated in two-dimensions in Figure 1 andFigure 2

Spherical positioning is based on observing individualrange values (zri

) between known fixed beacon locations(xbi

) and an unknown mobile host position (xh) wherethe individual range measurements is indexed by i.

zri= ‖xh − xbi

‖+ wri(1)

The vector x is used generically to denote a three-dimensional position, i.e., x = (x, y, z). We considerthe additive noise in each measurement (wri

) as anindependent, zero-mean, Gaussian variable with varianceσ2r .

wri∼ N

(0, σr

)(2)

We do not claim that this normal probability distributioncaptures the full complexity of range observations, butinstead that this Gaussian error model represents thesalient characteristics of the measurements while en-abling a prediction of overall system performance.

Fig. 2. This illustration shows how uncertainty affects the sphericaltrilateration illustrated in Figure 1. In this illustration the concentriccircles illustrate the probabilistic bounds on the actual range, quan-tified by the range standard deviation (σr). The dark shaded areaillustrates the uncertainty in the resulting position estimate.

B. Model for Velocity-Based Relative Positioning

The second stochastic observation model we employconcerns the measurement of vehicle velocity and head-ing to estimate course over ground. As discussed above,this is typically accomplished using the combination ofa DVL with a heading reference. The DVL providesindependent measurements of velocity (vk) in each ofthree dimensions (indexed by k).

zvk= vk + wvk

(3)

We characterize the uncertainty as mutually independentadditive, zero-mean, Gaussian white noise.

wvk∼ N

(0, σ2

vk

)(4)

Transforming these sensor frame measurements into alocal coordinate frame requires knowledge about sensorand vehicle attitude. Heading is the most importantmeasurement for this coordinate rotation [21]. Again weuse a simple additive Gaussian noise model to representthe heading (ψ) measurement.

zψ = ψ + wψ (5)

wψ ∼ N(0, σ2

ψ

)(6)

It is possible to carry forward the complete threedimensional (k = 1, 2, 3) formulation [18], for the claritywe present the two-dimensional case. For the remainderof this development we use the symbol x to signifythe along track dimension and y to signify the acrosstrack dimension. This is a non-limiting simplification

Fig. 3. Illustration of odometry uncertainty dynamics. The ellipsesillustrate the 1-σ uncertainty in the along track (x) and across track(y) directions. Five discrete vehicle positions are shown, indexed byi. The distance between consecutive positions is d.

for two reasons: most underwater vehicles are passivelystable in pitch and roll, furthermore pitch and roll aretypically measured much more accurately than head-ing. We also consider the uncertainty along track tobe independent of the uncertainty across track. Thisconsideration captures the dominant dynamics of errorgrowth and allows us to simplify our two-dimensionalmodel, preserving intuition. This approach leads to atwo-dimensional observation model of odometry whereeach discrete measurement of incremental distance (zoj

,where j is the index for velocity measurements in threedirections) is subject to additive, zero-mean, Gaussiannoise.

zoj=(xhj− xhj−1

)+ wo (7)

The additive noise is characterized by a two-dimensionalcovariance matrix (Σo) in the along track and acrosstrack directions.

wo ∼ N(0,Σo

)(8)

Σoi=[tσ2v 0

0 d2σ2ψ

](9)

The diagonal matrix in Equation (9) is a consequence ofthe independent along track and across track uncertaintygrowth. The along track term (σx =

√tσv) captures

growth of position uncertainty as a function of velocityuncertainty, based on random walk uncertainty growth.The across track term ( σy = dσψ) captures the linearuncertainty growth with distance traveled.

Figure 3 is an illustration of this simple odometrymodel. The aspect ratio of error ellipses increases withtime, illustrating the linear growth of the along trackuncertainty (growing with distance traveled) and thegrowth of the along track position with the square rootof time.

C. The Cramer Rao Lower Bound for Predicting Per-formance

The third and final step in developing our predictionframework is to use the CRLB to quantify the informa-tional contribution of absolute and relative positioningobservations to our ability to estimate the true position.When it exists, the CRLB gives the lower bound onthe variance of any valid unbiased estimator x() ofthe parameters x. The Fisher information, Iz(x), is theinformation about the parameters, x, contained in theobservations, z.

Iz(x) = E[(

∂∂x ln pz(z;x)

)2](10)

Where pz(z;x) is the known probability density of theobservations z which are dependent upon the parametersx and E [] is the expectation operator. The CRLB,Λx(x), is the inverse of the Fisher information.

Λx(x) = [Iz(x)]−1 (11)

The CRLB is the minimum uncertainty achievable byan unknown optimal estimator. An estimator that ap-proaches this bound is efficient, but the bound does notguarantee that an efficient estimator exists or that onecan be found.

To apply this tool we must express the completeobservation model for integrated absolute and relativepositioning measurements. We present this in two dimen-sions, recalling that x is the along track dimension and yis the across track dimension. The result is a combinedmeasurement model including n measurements of the ab-solute x and y positions along with n−1 measurementsof the relative distance traveled in each direction.

Zc =

x1...xny1...ynδx2

...δxnδy2

...δyn

= [A]

x1...xny1...yn

+ Σc (12)

where

[A] =

In×n 0n×n0n×n In×n

∆n−1×n 0n−1×n0n−1×n ∆n−1×n

(13)

∆n−1×n =

0...0

In−1

− In−1

0...0

(14)

Σc =

σ2pI2n 0 00 σ2

v(dt)In−1 00 0 σ2

ψ(dx)2In−1

(15)

where In is an n × n identity matrix, dt is the elapsedtime between absolute position updates and dx is thecorresponding distance traveled along track betweensuccessive absolute position updates.

The measurement model in Equation (12) relates themeasurements of absolute and relative position to theparameters we wish to estimate, the actual position.Because the integrated model is linear and the error ismodeled as having a Gaussian distribution, the CRLB issimply

Λx(x) =[ATΣ−1

c A]

(16)

The resulting n×n matrix bounds the position covariancematrix.

Λx(x) ≤ E[[xh − xh][xh − xh]T

](17)

where xh is the unknown, true n× 2 position matrix ofthe vehicle and xh is the corresponding estimated 2Dposition matrix at each of the n positions.

D. Evaluating Integrated LBL/DVL Solutions

Using the framework developed in the previous sectionwe develop a generalizable prediction of navigationperformance for integrated solutions that employ bothabsolute and relative positioning techniques. Figure 4illustrates this prediction for a large range of systemparameters. To succinctly capture the summative posi-tioning performance we use the notion of horizontaldilution of precision (HDOP). This scalar figure of meritsummarizes the two-dimensional position uncertainty.

σHDOP =

√σ2x + σ2

y

σp(18)

We normalize the uncertainty in the x and y directionswith the overall absolute positioning uncertainty (σp). Toevaluate the HDOP for a particular integrated navigationsolution requires quantifying the following parameters:

Fig. 4. This figure quantifies the trade-offs in designing an integratedpositioning solution using absolute (LBL) and relative (DVL/Head-ing) positioning. The vertical axis shows predictions of the overallHDOP. The data points labeled “Ex1” and “Ex2” correspond to theillustrative examples described below.

• Position uncertainty (σp) which is the standard devi-ation of the overall absolute position reference com-bining both contributions from range uncertaintyand the geometric effects of the multilaterationgeometry.

• Along track uncertainty (σalong) which is a functionof the velocity uncertainty (σv) and the acousticupdate rate (dt) which specifies the elapsed timebetween absolute position updates.

• Across track uncertainty (σacross) which is a func-tion of the heading uncertainty (σφ) and the dis-tance traveled between acoustic updates (dx). Thedistance traveled is simply the product of the vehiclevelocity and the acoustic update rate.

To interpret the results in Figure 4 we can examine thethree regions labeled A, B and C. On the left, in regionA, the along track uncertainty is much smaller thanthe absolute position uncertainty. This scenario indicatesthat the velocity error is small and that the update rateis relatively fast, preventing large drift in the relativepositioning between updates. The result, indicated by alow HDOP, is that the overall positioning uncertaintyis 10% of the standalone absolute positioning. Becauseof the accuracy of the relative measurements betweenindividual absolute updates, the information from eachabsolute reference cycle accumulates, lowering the over-all uncertainty. In fact, this asymptote achieves the lowerbound for the overall uncertainty

lim(σalong/σp)→0

[σHDOP ] = 1/√n (19)

TABLE ITWO ILLUSTRATIVE EXAMPLES OF PERFORMANCE PREDICTIONS

BASED ON THE STOCHASTIC MODEL OF POSITIONING

UNCERTAINTY.

Parameter Ex 1 Ex 2

Position Uncertainty(σp) 3.0 m 1.0 cmVelocity Uncertainty (σv) 0.003 m/s 0.003 m/sHeading Uncertainty (σψ) 0.1 degrees 1.0 degreesVehicle Speed 1.0 m/s 1.0 m/sAcoustic Update Rate 10 s 10 s

Along Track (σalong) 0.009 m 0.009 mAcross Track (σalong) 0.017 m 0.17 mσalong/σp 0.003 0.9σacross/σalong 1.9 19σHDOP 0.1 0.75System Accuracy 30 cm 0.75 cm

where n is the number of absolute position updates. Inthis case n = 100.

Another interesting aspect of this limiting case is thatthe across track uncertainty does not affect the resultingperformance. This is an important result because forsolutions that employ only relative positioning (dead-reckoning), the heading reference is a key determinateof performance. In contrast, the prediction illustrated inFigure 4 shows that when both absolute and relativepositioning are combined and the velocity measurementshave low uncertainty, the quality of the heading referencehas little impact on the overall performance.

Region C, in the far right of Figure 4, illustratesthe performance prediction for the opposite extremewhen the relative positioning is poor compared to theabsolute reference. In this case the HDOP asymptoticallyapproaches 1.0, indicating that the standalone absoluteposition uncertainty is equivalent to the combined un-certainty. In other words, between each absolute positionupdate the dead-reckoning drift is so large that the in-formation from the last update is lost. Therefore, overallaccuracy is no better than the standalone accuracy of theabsolute reference.

1) Illustrative Examples: To further illustrate the re-sults in Figure 4 we use the framework to analyze twopossible scenarios for AUV navigation, quantified by theparameter values listed in Table I.

The first example (Ex 1) is based on a typical high-performance AUV deployment using instrumentationthat will be familiar to many practitioners. Absoluteposition references are delivered through an array ofBenthos LBL transponders (σp ≈ 3.0 m) which are inter-rogated once every 10 s to allow for acoustic propagation

times. Relative positioning is accomplished using a 1,200kHz RDI DVL (σv = 3 mm/s) in combination with anOctans true north heading reference (σψ = 0.1 degrees).In order to calculate the across track uncertainty wealso specify a typical vehicle speed of 1.0 m/s. Theresulting system-level positioning performance for thisscenario is illustrated by the annotation in Figure 4. Inthis case the relative positioning is sufficiently accurateachieve significant gains in the overall performance dueto the complementary nature of these two positioningmodalities. The HDOP value of 0.1 indicates that theoverall performance is an order of magnitude moreaccurate than the standalone absolute reference.

In the second example (Ex 2) we examine the effect ofusing a high-performance absolute positioning reference.In this case the absolute reference accuracy is 1.0 cm, alower limit of the performance of available solutions.This scenario also employs a heading solution withmuch more uncertainty (1.0 degrees). The results ofthis scenario are also illustrated with an annotation inFigure 4. Here we see that the HDOP is 0.75 indicatingthat the positioning is only 25% more accurate than thestandalone absolute reference.

IV. EXPERIMENTAL RESULTS

To evaluate our dead-reckoning model we performedan at-sea trial using the ROV Jason, part of the NationalDeep Submergence Facility operated by the Woods HoleOceanographic Institution..

A. Experimental Setup

This experiment took place during an engineeringlowering of the ROV. The lowering took place on June21, 2007 as part of a scientific expedition to in the Gulfof Mexico at roughly 2,200m total water depth. Thenavigation system aboard the ROV has been describedby other researchers [14], [22]. For this trial the ROVwas outfitted with an RDI 1,200 kHz Navigator DVL3

to measure velocity relative to the bottom and an OctansFOG compass and attitude sensor4.

The experiment provided a visual ground-truth forevaluating the precision (repeatability) of the relativeposition estimated via dead-reckoning. The ground-truthwas provided by deploying a navigation benchmarkas shown in Figure 2. Using the closed-loop controlcapability of the Jason ROV, we were able to positionthe vehicle directly above the benchmark at both thebeginning and end of the experiment.

3Teledyne RD Instruments4IXSEA, Inc.

Fig. 5. The dead-reckoning error was quantified using a stationarybenchmark fixed on the seafloor. This down-looking image illustrateshow the ROV was positioned directly over the benchmark at the startand the end of the experiment. Two parallel, calibrated laser markerswere present in the image to aid in repeatable positioning relative tothe bottom.

Fig. 6. Two-dimensional plot of the dead-reckoning track during thetest. At both the ”Start” and ”End” points the ROV was positioneddirectly over the bottom-mounted benchmark (Figure 2). The differ-ence between the recorded ”Start” and ”End” conditions serve as ameasure of the DR drift.

B. Experiment Results

Figure 6 and Figure 7 show the two-dimensional posi-tion estimates reported from the dead-reckoning solution.At both the “Start” and “End” locations the vehicle waspositioned directly over the navigation benchmark. TheROV was navigated roughly 90 m due south and returnedto the original start position on a reciprocal headingduring an elapsed time of approximately 23 minutes.

The experiment described above allows us to comparethe predicted performance of the relative positioningwith the actual measured error. Table II quantifies this

Fig. 7. This temporal record illustrates the same track shown inFigure 3. Again, the ”Start” and ”End” locations are shown on thetrack.

TABLE IICOMPARISON OF PREDICTED PROBABILITY AND MEASURED

ODOMETRY DRIFT.

PredictedStdDev (cm)

MeasuredError (cm)

Along Track 11.2 4.0Across Track 35.2 7.0Total (2D) 37.0 8.1% of Dist. Traveled 0.2% 0.04%

comparison. The predicted standard deviation is derivedfrom Equation (9) using published values for the sensoruncertainties. The measured error is smaller than pre-dicted, 36% and 20% of the expected standard deviationin the along and across track direction. This measuredtwo dimensional error corresponds to a 18% cumulativeprobability given the predicted values for the standarddeviation. To compare these results to previous reports,we also calculate the total uncertainty as a fraction ofthe distance traveled. The measured value of 0.04%distance traveled is smaller than, but comparable to,those reported in the literature.

V. CONCLUSION

This article presents two results: a modeling frame-work for predicting the accuracy of combined abso-lute/relative positioning and an experimental evaluationof state-of-the-art relative positioning (dead-reckoning).The theoretical results show how the performance of boththe absolute reference (eg., LBL acoustic positioning)and the relative navigation (eg., DVL dead-reckoning)affect the overall system-level accuracy. Figure 4 pro-

vides a succinct summary that captures the salient trade-offs in position uncertainty, velocity uncertainty, headinguncertainty, update rate and vehicle speed. Each of theseparameters is an important determinate of overall posi-tioning accuracy, and this framework provides a tool formaking decisions on how best to configure a particularsolution.

The experimental evaluation of the dead-reckoningnavigation provides an example of one way of directlymeasuring positioning uncertainty. The positioning errorwas less that predicted, indicating that the combinationof our simple model for error growth along with themanufacturers specifications is a conservative estimateof performance.

The goal of this work has been to provide a gener-alizable tool to guide future decisions. The stochasticmodels we choose are simple enough to provide system-level performance accounting, but detailed enough tocapture the important dynamics. This ability to predictthe performance of such integrated navigation solutionsshould provide guidance for both vehicle and instrumentimprovement. By targeting development efforts on themost important aspects of the solution, future efforts willhave the most impact on the overall system performance.

VI. ACKNOWLEDGMENTS

We wish to thank the entire Deep Submergence Labat the Woods Hole Oceanographic Institution for makingroom in the operation for this navigation evaluationduring the RB-07-04 expedition. Similarly, Dr. ChuckFisher, the Chief Scientist during the expedition, waspatient enough to allow for a full engineering lowering.

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