Experiments in Underwater Feature Tracking with Performance Guarantees Using a Small AUV Benjamin Biggs

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Experiments in Underwater Feature Tracking with Performance

Guarantees Using a Small AUV

Benjamin Biggs

Virginia Tech

babiggs@vt.edu

Hans He

Virginia Tech

hjh2bs@vt.edu

James McMahon

US Naval Research Laboratory

Acoustics Division, Code 7130

james.mcmahon@nrl.navy.mil

Daniel J. Stilwell

Virginia Tech

stilwell@vt.edu

Abstract—We present the results of experiments performed

using a small autonomous underwater vehicle to determine the

location of an isobath within a bounded area. The primary

contribution of this work is to implement and integrate several

recent developments real-time planning for environmental map-

ping, and to demonstrate their utility in a challenging practical

example. We model the bathymetry within the operational area

using a Gaussian process and propose a reward function that

represents the task of mapping a desired isobath. As is common

in applications where plans must be continually updated based

on real-time sensor measurements, we adopt a receding horizon

framework where the vehicle continually computes near-optimal

paths. The sequence of paths does not, in general, inherit the

optimality properties of each individual path. Our real-time

planning implementation incorporates recent results that lead

to performance guarantees for receding-horizon planning.

I. INTRODUCTION

Determining the location of a depth contour or isobath is

an important element of quickly ﬁnding navigable paths in

waterways. The conventional approach to underwater feature

mapping relies on performing an exhaustive search over an

area where a human operator believes features of interest

are located. Path planning methods that seek to maximize

information gain by planning sequences of locations that

maximize a speciﬁc reward function are available to attempt

to track underwater features more quickly than exhaustive

search methods. However, many of these methods are suscep-

tible to poor performance caused by myopic planning hori-

zons when using receding horizon methods or local maxima

in the reward function when using gradient-based optimiza-

tion approaches. The primary contribution of this paper is the

presentation of the results of experiments in depth contour

estimation using path planning techniques with performance

guarantees. We integrate several recent developments for

real-time isobath localization, all of which can be easily

modiﬁed for similar real-time mapping applications: (1) use

of a sparse Gaussian process for representing the bathymetry,

which leads to a continuous representation of the environment

*This work was supported by the Ofﬁce of Naval Research via grants

N00014-18-1-2627, and N00014-19-1-2194. The work of J. McMahon is

supported by the Ofﬁce of Naval Research through the NRL Base Program.

James McMahon is with the US Naval Research Laboratory, Code 7130,

Washington D.C., USA

Benjamin Biggs, Hans He, and Daniel Stilwell are with the Bradley De-

partment of Electrical and Computer Engineering, Virginia Tech, Blacksburg,

VA, USA

Fig. 1: 690 vehicles being loaded for transportation to the

operational area.

along with rigorous assessment of uncertainty, and (2) use

of a so-called terminal reward in the optimization problem

that generates each near-optimal path, which ensures that the

cumulative reward attained by the AUV from a sequence of

near-optimal paths is no worse than a desired lower bound.

An important contribution of this work is to describe speciﬁc

implementation solutions needed to integrate and implement

these ideas for real-time operation on an AUV.

Isobath localization, like front tracking, can be formulated

as an informative path planning (IPP) problem [1]. The goal

of the IPP problem is to ﬁnd a path that maximizes an

information utility function while not exceeding a budget

constraint on the cost of the path. In general, the computa-

tional complexity associated with informative path planning

grows exponentially with the length of the path. Heuristic

planning methods may be used to produce reasonable results

[2]–[4]. However, it is desirable to plan paths that are as

nearly optimal as possible. Despite a host of techniques

developed to provide approximate or suboptimal solutions

[5]–[8], receding horizon methods are often used regardless

of the planning techniques used within the shorter planning

horizon [9]–[11].

A receding horizon path is constructed from segments of

near-optimal short paths that are greedily selected. How-

ever, the sequence of path segments in a receding horizon

implementation do not inherit the near-optimality of each

arXiv:2210.02524v1 [cs.RO] 5 Oct 2022

individual segment. In prior work [12], [13] the authors

showed how to append a speciﬁc terminal reward to the short

horizon optimization problem used for each path segment

such that the reward of the overall receding horizon path is

guaranteed to exceed a desirable lower bound. This result was

demonstrated for the explicit reward function associated with

a robotic search problem in a discrete environment through

numerical experiments. In this work, we demonstrate the

utility of adding a terminal reward to a different class of

reward function based on level set estimation using Gaussian

process models. We report on the results of challenging real-

world experiments accomplished using a real autonomous

underwater vehicle (AUV) performing on-line planning using

live data.

Throughout this work, we seek to rapidly map the 15 meter

isobath within a bounded area of Claytor lake in Dublin,

VA, USA. The speciﬁc region of Claytor lake where the

experiments of this work are performed is shown in Figure

4 with the solid red lines indicating the boundaries of the

operational area. The AUV used for these experiments is the

Virginia Tech 690 vehicle shown in Figure 1.

The remainder of this paper is organized as follows.

Gaussian process regression and the speciﬁc considerations

and methods used in this work are presented in Section II.

The speciﬁc reward function used for evaluating paths is

presented in Section III. The path planning methods proposed

in [13] and used in this work are presented and discussed

in Section IV. The usage and relevant capabilities of the

690 vehicle are discussed in Section V. The experimental

procedure and selection of parameters is presented in Section

VI. Results and conclusions are presented in Section VII.

II. GAUSSIAN PROCESS REGRESSION

In this section we provide a brief review of Gaussian

process regression. We refer the reader to [14, Chapter 2]

for a more thorough discussion of the topic.

A Gaussian process is a collection of random variables,

any ﬁnite number of which have a joint Gaussian distribution

[14, Deﬁnition 2.1]. A GP is completely speciﬁed by its mean

and covariance functions m(p)and k(p, p0)respectively and

is written

f(p)∼ GP(m(p), k(p, p0)) (1)

where f(p)is a real process. In this work, we assume that the

bathymetry or depth of Claytor lake within the operational

area may be modeled using a GP.

A sample or datum d= (p, z)is an input-measurement

pair with p∈Pand z∈R. Let Dtbe the set of all

samples acquired up to time t. For any input pi, an associated

measurement ziis related to pivia

zi=f(pi) + i,for i= 1,...,|Dt|(2)

where |D| denotes the cardinality of the set Dand i∼

N(0, σ2

n)is drawn from a zero mean Gaussian distribution

with standard deviation σn.

Given the data set Dt, one may wish to predict f(p∗)for

an arbitrary input p∗. From [14, Chapter 2.7], the posterior

predictive distribution of f(p∗)conditioned on Dtis Gaus-

sian and speciﬁed by the mean µp∗|Dtand variance σ2

p∗|Dt

given as

µp∗|Dt=m(p∗) + k(p∗)>(Kt+σ2

nI)−1(Zt−m(Pt)) (3)

σ2

p∗|Dt=k(p∗, p∗)−k(p∗)>(Kt+σ2

nI)−1k(p∗).(4)

Zt= [z1, . . . , z|Dt|]>is the vector of all measurements

contained in Dtand Pt={p1, . . . , p|Dt|}is the set of as-

sociated inputs. The prior mean vector is [m(Pt)]i=m(pi),

[k(p∗)]i=k(p∗, pi),[Kt]i,j =k(pi, pj), and Itis the

|Dt| × |Dt|identity matrix.

When computing the predictive distribution at p∗, the

matrix inversion (Kt+σ2

nI)−1can become intractable as |Dt|

grows large. To address this complexity, we utilize a distance

metric d(p, p0)on Pto reduce the number of samples

retained in Dtand make predictions using local Gaussian

processes. To reduce the number of samples retained in

Dt, a new sample dc= (pc, zc)is only added to Dtif

d(pc, pi)≥δffor all pi∈ Dt. For a predictive point p∗,

the associated data is limited to a locality around p∗giving

the local data set as Dp∗,t ={di∈ Dt|d(pi, p∗)≤δc}. The

local predictive mean and variance are given by equations

(3) and (4) using Dp∗,t in the place of Dt.

Throughout this work, we deﬁne the input space Pto be

two dimensional euclidean space (R2) with distance metric

d(p, p0) = kp−p0kbeing the euclidean distance. We use the

squared exponential kernel function

k(p, p0) = σ2

fexp kp−p0k2

2l2

c(5)

as the prior for the covariance function. The prior mean

function used in this work is presented in Section VI.

III. OBJECTIVE FUNCTION

We wish to determine the level set L(l∗) = {p∈

P|f(p) = l∗}. To direct the selection of locations that

should be measured in order to obtain information regarding

this level set, we require an objective function that encodes

uncertainty regarding whether p∗is an element of L(l∗). To

provide such a function, we use a measure of ambiguity

presented in [15] which is closely related to the straddle

heuristic presented in [16].

Deﬁnition 1 (Ambiguity): The ambiguity associated with a

point p∗with respect to a level lis

ap∗|Dt=βσp∗|Dt− |µp∗|Dt−l|.(6)

Concretely, if β= 2 and ap∗|Dt<0, then the model indicates

that the probability that f(p∗)falls on the opposite side of l

as µp∗|Dtis less than 2.5 percent.

Similar to [15] where batches of points are selected for

sampling, we seek to select a path or a sequence of points to

sample. Considering the ambiguity of sets of points requires

some slight modiﬁcations to account for locations that will

be sampled while no samples have yet been acquired.

Deﬁnition 2 (Anticipated Ambiguity): Let P∗=

{p∗

0, . . . , p∗

n}and P∗

i={p∗

0, . . . , p∗

i−1}with P∗

0=∅. Then

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ExperimentsinUnderwaterFeatureTrackingwithPerformanceGuaranteesUsingaSmallAUVBenjaminBiggsVirginiaTechbabiggs@vt.eduHansHeVirginiaTechhjh2bs@vt.eduJamesMcMahonUSNavalResearchLaboratoryAcousticsDivision,Code7130james.mcmahon@nrl.navy.milDanielJ.StilwellVirginiaTechstilwell@vt.eduAbstractWepresentthe...

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Experiments in Underwater Feature Tracking with Performance Guarantees Using a Small AUV Benjamin Biggs

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