# Zygmund graphs are thin for doubling measures

Claudio A. DiMarco  1000 E. Henrietta Rd., Mathematics Department, Monroe
Community College, Rochester, NY 14623, USA<EMAIL_ADDRESS>
###### Abstract

The Zygmund functions form an intermediate class between Lipschitz and Hölder
functions; their second order divided differences are uniformly bounded. It is
well known that for $d\geq 1$ the graph of any Lipschitz function
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is thin for doubling measures, and we
extend this result to the Zygmund class.

###### keywords:

Zygmund class, doubling measure, Lipschitz class, Hölder class.

###### MSC:

[2020]Primary 28A35, 26B35; Secondary 28A75

##  1 Introduction

Certain subsets $E$ of a metric space $X$ are “thin” or “fat” for doubling
measures. Wu initiated the discussion of thinness for doubling measures on
$\mathbb{R}$ in [Wu], wherein the relationship between thinness and “porosity”
was also investigated. The results in [Wu] employ a generalized notion of
porosity which easily implies thinness for doubling measures, as Chen and Wen
noted in [Chen, page 1].

In [Ojala2] the authors considered the case when $X$ is uniformly perfect and
$E\subset X$ has the property $\mu(E)=0$ for every doubling measure $\mu$ on
$X$, or the property $\mu(E)>0$ for all doubling measures. Such subsets are
called thin and fat, respectively. Ojala et al provided sufficient conditions
for certain “cut-out sets” being thin or fat, and proved that $E\subset X$ is
thin if and only if $E$ is quasisymmetrically null, i.e.
$\mathcal{H}^{q}(f(E))=0$ whenever $f:X\rightarrow Y$ is quasisymmetric and
$Y$ is Ahlfors $q$-regular, where $\mathcal{H}^{q}$ is $q$-dimensional
Hausdorff measure [Ojala2].

It is interesting to consider the special case when $E\subset\mathbb{R}^{d+1}$
is the graph of a continuous function. This is an area of recent interest in
which little is understood. For a while, an open question in this area was
whether every rectifiable curve in the plane is thin for doubling measures.
Surprisingly, this turned out not to be the case (see [Garnett]). However, if
attention is restricted to so called “isotropic” doubling measures on
$\mathbb{R}^{d+1}$, then indeed the graph of any continuous function
$f:[0,1]^{d}\rightarrow[0,1]$ has measure zero [Chen]. This is easily extended
to include all continuous $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$.

Recall that a bounded function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is
Hölder continuous if there are $0<\alpha\leq 1$ and $C>0$ such that
$|f(x)-f(y)|\leq C|x-y|^{\alpha}$ for all $x,y\in\mathbb{R}^{d},$ and is
called Lipschitz if there is $L>0$ such that $|f(x)-f(y)|\leq L|x-y|$ for all
$x,y\in\mathbb{R}^{d}$. In [Ojala] the authors showed that even a graph,
$y=f(x)$, of a continuous function, need not be thin in general. They remark
that there are few known results in the positive direction, i.e. sufficient
conditions for thinness. In particular, it is known that the graph of a
Lipschitz function is thin, but whether this property extends to Hölder
functions remains unknown [Ojala, Question 1.2].

In 1876 Weierstrass introduced the now familiar family of continuous but
nowhere differentiable functions

$f(x)=\sum_{n=0}^{\infty}a^{n}\cos(b^{n}\pi x),~{}~{}x\in\mathbb{R}$

where $0<a<1$ and $b$ is an odd integer with $ab>1+3\pi/2$. Over time several
improvements were made on the lower bound $ab>1+3\pi/2$, and in 1916 Hardy
proved that any function of the form

$\sum_{n=0}^{\infty}a^{n}\cos(b^{n}\pi
x)~{}~{}\text{or}~{}~{}\sum_{n=0}^{\infty}a^{n}\sin(b^{n}\pi x)$

is nowhere differentiable whenever $ab\geq 1$; see [Hardy]. The critical case
$ab=1$ produces

$f(x)=\sum_{n=0}^{\infty}b^{-n}\cos(b^{n}\pi x),~{}~{}x\in\mathbb{R},$

which can be characterized by the behavior of its second divided differences:

######  Definition 1.1.

The Zygmund class $\Lambda_{*}(\mathbb{R}^{d})$ is the space of bounded
continuous functions $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ for which

$\|f\|_{*}=\sup\left\\{\frac{|f(x+h)+f(x-h)-2f(x)|}{|h|}:x,h\in\mathbb{R}^{d}\right\\}<\infty.$
(1.1)

These spaces were introduced by Zygmund when he noticed that the conjugate
function of a Lipschitz function in the unit circle does not need to be
Lipschitz, but it is in the Zygmund class [Zygmund, Donaire].

For any $0<\alpha\leq 1$, denote the Hölder class by
$\Lambda_{\alpha}(\mathbb{R}^{d})$ and notice that the Lipschitz class is
obtained for $\alpha=1$. It is well known that if $0<\alpha<1$ then
$\Lambda_{1}(\mathbb{R}^{d})\subset\Lambda_{*}(\mathbb{R}^{d})\subset\Lambda_{\alpha}(\mathbb{R}^{d})$,
and because Lipschitz graphs $E\subset\mathbb{R}^{d+1}$ are thin for doubling
measures, it is natural to extend this result to the Zygmund class (which is
the main result):

######  Theorem 2.6.

If $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a Zygmund function, then the
graph of $f$ is thin for doubling measures. That is, if $\mu$ is a doubling
measure $\mu$ on $\mathbb{R}^{d+1}$ and $E$ is the graph of $f$, then
$\mu(E)=0.$

There is a noticeable relationship between Definition 1.1 and that of
differentiability, and recent progress has been made in relating the two
notions. In [Donaire] the authors showed that for any
$f\in\Lambda_{*}(\mathbb{R}^{d})$, the set of points at which the first
divided differences are bounded has Hausdorff dimension at least one. They
also proved that any function in the small Zygmund class (see [Donaire] for
the definition) is differentiable on a set of Hausdorff dimension at least
one. This result contrasts with the large Zygmund class because a function
$f\in\Lambda_{*}(\mathbb{R}^{d})$ may be nowhere differentiable.

##  2 Preliminaries

For a metric space $(X,d),$ write $A_{\delta}$ for the open
$\delta$-neighborhood of $A\subset X.$ For
$x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}$ the quantity $|x|=\|x\|$ is the
norm of $x$ and $x[i]$ is the $i$th coordinate of $x$. For $1\leq i\leq n$ we
write $e_{i}\in\mathbb{R}^{n}$ for the standard basis vector defined by
$e_{i}[i]=1$ and $e_{i}[j]=0$ whenever $j\neq i$. As usual $\lfloor p\rfloor$
and $\lceil p\rceil$ are the floor and ceiling of $p\in\mathbb{R}$
respectively.

All cubes $Q\subset\mathbb{R}^{d}$ are assumed to be closed unless otherwise
noted, and we use $B(x,r)$ and $\bar{B}(x,r)$ for open and closed balls of
radius $r$ respectively. Also $B_{d+1}(x,r)$ is used to denote open balls in
$\mathbb{R}^{d+1}$ to avoid confusion in the context of a function
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}.$ For an open ball $B=B(x,r)$, for any
$s>0$ we write $sB=B(x,sr).$ We will also make use of the the function

$\operatorname{sign}(x)=\begin{cases}1,&x\geq 0\\\ -1,&x<0.\\\ \end{cases}$

A metric space is called doubling if there is $C_{1}\geq 1$ so that every set
of diameter $d$ in the space can be covered by $C_{1}$ sets of diameter at
most $d/2$ [Heinonen, page 81]. Every complete doubling metric space supports
a doubling measure [Luukkainen]. The following definition can be found in
[Heinonen, page 3]:

######  Definition 2.1.

A Borel measure $\mu$ on a metric space $(X,d)$ is called doubling if there is
$C>0$ such that $\mu(\bar{B}(x,r))\leq C\mu(\bar{B}(x,r/2))$ for all nonempty
closed balls $\bar{B}(x,r)$.

A doubling metric space that supports a doubling measure $\mu$ is often
denoted $(X,d,\mu)$. Given such a space, the doubling property of $\mu$
ensures that the measure is not disproportionately concentrated around any
particular point $x\in X$.

######  Definition 2.2.

A subset $E$ of a metric space $(X,d)$ is thin if $\mu(E)=0$ for every
doubling measure $\mu$ on $X$.

In [Jonsson] the authors defined so-called Lipschitz spaces $L_{\alpha}(F)$ on
closed subsets $F\subset\mathbb{R}^{d}$ for all $\alpha>0$. An integral part
of this definition involves polynomial approximations $P$ to $f$, where the
degree of $P$ is at most $\lfloor\alpha\rfloor$. The case $\alpha=1$
corresponds to the Zygmund class (i.e. $L_{1}(F)=\Lambda_{*}(F)$), in which
case $\lfloor\alpha\rfloor=1$ furnishes degree one polynomial (i.e. affine)
approximations to $f$ [Jonsson, chapter III, section 2.1]. These polynomial
approximations are valid on cubes in $F=\mathbb{R}^{d}$, and therefore on
balls via restriction. We use the following simplified formulation because it
is easily applied to the problem at hand.

######  Fact 2.3.

For any Zygmund function $f\in\Lambda_{*}(\mathbb{R}^{d})$ and any ball
$B=B(x_{0},r)\subset\mathbb{R}^{d}$, there is a degree-1 polynomial $P$ such
that

$\sup_{x\in B}|f(x)-P(x)|\leq Mr,$ (2.1)

with a constant $M$ independent of both $x_{0}$ and $r$. Moreover, this
property characterizes Zygmund functions ([Jonsson-Wallin, Proposition 3, page
54], [Jonsson, page 155]).

Using the characterization of Zygmund functions in Fact 2.3, Lemma 2.4 shows
that if the gradient of $P$ is not too large, then above each point $z\in
E\subset\mathbb{R}^{d+1}$ there is a ball of suitable size which lies above
$E$. If $\nabla P$ is large, then there is a ball in the direction of $\nabla
P$ where $E$ is above a ball in $\mathbb{R}^{d+1}$ of suitable size.

######  Lemma 2.4.

Suppose $f\in\Lambda_{*}(\mathbb{R}^{d})$, let $E\subset\mathbb{R}^{d+1}$
denote the graph of $f$, and fix $z_{0}=(x_{0},f(x_{0}))\in E$. For any
$\delta>0$ there is an open ball $B=B(z_{0},r)\subset E_{\delta}$ and a
corresponding ball $B(z,r)\subset(E_{\delta}\setminus E)$. Moreover $r>0$ may
be chosen arbitrarily small, and there is $M>1$ such that one of the following
conditions holds for each $B$:

  1. (i)

$z[i]={z_{0}}[i]$ for $i<d+1$ and $|z-z_{0}|=(2M+1)r$ or

  2. (ii)

$|z-z_{0}|\leq(5M+1)r$ where the constant $K$ depends on $M$ and is
independent of $z$ and $r$.

###### Proof.

Let $\delta>0$, fix $z_{0}=(x_{0},f(x_{0}))\in E$, and let $M>1$ as in Remark
2.3. There are two cases:

  1. (i)

There is $\beta>0$ such that for every open ball
$B=B(x_{0},\varepsilon)\subset E_{\delta}$ with $\varepsilon<\beta$, the
corresponding affine function $P_{B}$ as in Remark 2.3 has $|\nabla P_{B}|\leq
1$. For any such $B$, writing $P_{B}(x)=\sum_{i=1}^{d}a_{i}x_{i}$ with
$x=(x_{1},x_{2},\dots,x_{d})$ shows

$\begin{split}|P_{B}(x)-P_{B}(x_{0})|^{2}&=|P_{B}(x-x_{0})|^{2}\\\
&\leq\left(\sum_{i=1}^{d}|a_{i}|\left|x_{i}-x_{0}[i]\right|\right)^{2}\\\
&\leq\sum_{i=1}^{d}a_{i}^{2}\left|x_{i}-x_{0}[i]\right|^{2}\\\
&\leq\varepsilon^{2}\sum_{i=1}^{d}a_{i}^{2}\\\ &=\varepsilon^{2}|\nabla
P|^{2}.\\\ \end{split}$ (2.2)

Choose $r>0$ small enough that both $B_{d+1}=B(z_{0},r)$ and
$B_{d+1}^{\prime}=B(z_{0}+(2M+2)re_{d+1},r)$ lie in $E_{\delta}$. (Notice that
$M>1$ implies these two balls are disjoint.) Because $|\nabla P_{B}|\leq 1$,
observe that for all $x\in B=B(x_{0},r)$, in light of equation (2.1),

$\begin{split}|f(x)-f(x_{0})|&\leq|f(x)-P_{B}(x)|+|P_{B}(x)-f(x_{0})|\\\ &\leq
Mr+|f(x_{0})-P_{B}(x_{0})|+|P_{B}(x_{0})-P_{B}(x)|\\\ &\leq Mr+Mr+|\nabla
P|r\\\ &\leq(2M+1)r,\end{split}$

from which it follows that $B_{d+1}^{\prime}\cap E=\varnothing$, i.e.
$B_{d+1}$ lies entirely above $E$ because

$\operatorname{dist}(z_{0},B_{d+1}^{\prime})=(2M+1)r.$

  2. (ii)

There is a sequence of open balls $B_{n}=B(x_{0},r_{n})$ such that
$r_{n}\rightarrow 0$ and $|\nabla P_{B_{n}}|>1$.

Choose $r>0$ small enough that both the following conditions are satisfied:

    1. (a)

$B=B(x_{0},r)$ has $P_{B}(x)=\sum_{i=0}^{d},a_{i}x[i]$ with $|\nabla
P_{B}|>1$, and

    2. (b)

$(5M+2)r<\delta$ so that $B(z,(5M+2)r)\subset E_{\delta}$ for every $z\in E$.

Put

$r^{\prime}=\min\left(r/2,\frac{Mr}{|\nabla P_{B}|}\right).$ (2.3)

and let $x_{0}^{\prime}=x_{0}+2r^{\prime}u$ where $u=\frac{\nabla
P_{B}}{|\nabla P_{B}|}$. Observe that for all $x\in
B(x_{0}^{\prime},r^{\prime})$, equations (2.1),(2.2) and (2.3) provide

$\begin{split}|f(x)-f(x_{0}^{\prime})|&\leq|f(x)-P_{B}(x)|+|P_{B}(x)-P_{B}(x_{0}^{\prime})|+|P_{B}(x_{0}^{\prime})-f(x_{0}^{\prime})|\\\
&\leq 3Mr.\end{split}$ (2.4)

Let $z_{0}^{\prime}=(x_{0}^{\prime},f(x_{0}^{\prime}))\in\mathbb{R}^{d+1}$ and
define

$z_{0}^{\prime\prime}=z_{0}+(2r^{\prime}u,\operatorname{sign}(f(x_{0})-f(x_{0}^{\prime}))(3M+1)r),$

so that $B_{d+1}^{\prime}=B(z_{0}^{\prime\prime},r^{\prime})$ resides either
entirely below or entirely above $E$. Indeed,

$\begin{split}|z_{0}-z_{0}^{\prime\prime}|&\leq 2r^{\prime}+(3M+1)r\\\ &\leq
2Mr+(3M+1)r\\\ &=(5M+1)r.\end{split}$ (2.5)

It follows from condition (b) and equation (2.4) that
$B_{d+1}^{\prime}\subset(E_{\delta}\setminus E)$ as required.

∎

######  Remark 2.5.

The constant $2M+1$ in the conclusion of Lemma 2.4 may safely be replaced by
$5M+1$ in case the ball $B_{d+1}^{\prime}$ lies directly above $z_{0}\in E$.
To this end, it suffices to choose $r$ small enough that $(5M+2)r<\delta$.
This observation simplifies the proof of Theorem 2.6 by combining two cases.

######  Theorem 2.6.

If $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a Zygmund function, then the
graph of $f$ is thin for doubling measures. That is, if $\mu$ is a doubling
measure $\mu$ on $\mathbb{R}^{d+1}$ and $E$ is the graph of $f$, then
$\mu(E)=0.$

###### Proof.

Let $\mu$ be a doubling measure on $\mathbb{R}^{d+1}$ with doubling constant
$C,$ define $M$ as in Fact 2.3, and let $\varepsilon>0.$ Put

$K=\lceil
5M+1\rceil+1~{}~{}\text{and}~{}~{}p=\left\lceil\frac{\ln(5K)}{\ln(2)}\right\rceil.$
(2.6)

Because $\bigcap_{\delta>0}(E_{\delta}\setminus E)=\varnothing,$ there is
$\delta>0$ such that $\mu(E_{\delta}\setminus E)<\varepsilon/(C^{p})$. Choose
a covering $\mathcal{B}$ of $E$ by open balls as in Lemma 2.4.

Consider the collection of open balls $\hat{\mathcal{B}}$ obtained by
translating each $B=B(z,r)\in\mathcal{B}$ to one of its offset balls
$\hat{B}=B(w,r)\subset(E_{\delta}\setminus E)$, where $|z-w|=(5M+1)r$ as in
Lemma 2.4 and Remark 2.5. By construction, each $z\in E$ is the center of some
$B\in\mathcal{B}$, so $z\in K\hat{B}$ by (2.6) and hence

$E\subset\bigcup_{\hat{B}\in\hat{\mathcal{B}}}K\hat{B}.$ (2.7)

The Vitali covering lemma (see e.g. [Heinonen, page 3]) provides a countable
disjoint subcollection $\hat{\mathcal{C}}\subset\hat{\mathcal{B}}$ such that

$\bigcup_{\hat{B}\in\hat{\mathcal{B}}}K\hat{B}\subset\bigcup_{\hat{B}\in\hat{\mathcal{C}}}(5K)\hat{B}.$
(2.8)

The fact that $\hat{B}\subset(E_{\delta}\setminus E)$ for every $\hat{B}$,
combined with the doubling property of $\mu$, equations (2.6),(2.7),(2.8), and
the fact that $\hat{C}$ is disjoint, yield

$\begin{split}\mu(E)&\leq\mu\left(\bigcup_{\hat{B}\in\hat{\mathcal{C}}}(5K)\hat{B}\right)\\\
&\leq\sum_{\hat{B}\in\hat{\mathcal{C}}}\mu((5K)\hat{B})\\\
&\leq\sum_{\hat{B}\in\hat{\mathcal{C}}}C^{p}\mu(\hat{B})\\\
&=C^{p}\mu\left(\bigcup_{\hat{B}\in\hat{\mathcal{C}}}\hat{B}\right)\\\ &\leq
C^{p}\mu(E_{\delta}\setminus E)\\\ &<C^{p}(\varepsilon/(C^{p}))\\\
&=\varepsilon.\qed\end{split}$ (2.9)

The relationship between the graph $E$ and the collection of offset open balls
$\hat{B}\subset(E_{\delta}\setminus E)$ used in the proof of Theorem 2.6
raises the question of porosity. The following definition can be found in
[Chen].

######  Definition 2.7.

A subset $E\subset\mathbb{R}^{d}$ is called porous if there exists $a\in(0,1)$
such that for any ball $B(x,r)$, there is a ball $B(y,ar)\subset B(x,r)$
satisfies

$B(y,ar)\cap E=\varnothing.$

######  Remark 2.8.

The value $r>0$ employed in the proof of Lemma 2.4 is independent of the
chosen reference point $z_{0}\in E$. However, the smaller radius
$0<r^{\prime}<r$, which produces the ball
$B(y,r^{\prime})\subset(B(z_{0},r)\setminus E),$ depends on $z_{0}$. Thus the
question of porosity remains open.

######  Question 2.9.

Given any Zygmund function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$, is the
graph porous in $\mathbb{R}^{d+1}$?

######  Remark 2.10.

Because $\mathbb{R}^{n}$ is uniformly perfect, it follows from Theorem 2.6
above and Proposition 1.2 in [Ojala2] that the graph $E$ of any Zygmund
function is quasisymmetrically null. On the other hand, Theorem 1.1 in [Ojala]
guarantees that $E\subset X$ is thin if there are closed balls

$\\{\operatorname{diam}\bar{B}_{i}\\}_{i=1}^{\infty}\in\ell^{0}:=\bigcap_{p>0}\ell^{p}$

such that $E=X\setminus\bigcup\bar{B}_{i}$. This leads to the following
question of the converse.

######  Question 2.11.

Suppose $f\in\Lambda_{*}([a,b]^{d})$, denote its graph
$E\subset\mathbb{R}^{d+1}$. Are there $M>0$ and a sequence of closed balls
$\bar{B}_{i}\subset\overline{E_{M}}$ such that
$\\{\operatorname{diam}\bar{B}_{i}\\}_{1}^{\infty}\in\ell^{0}$ and
$E=\overline{E_{M}}\setminus\bigcup\bar{B}_{i}$?

Considering the thinness of Zygmund graphs in euclidean space, hope remains
for an affirmative answer to the following question posed in [Ojala].

######  Question 2.12.

Is the graph of every Hölder function $f\in\Lambda_{\alpha}(\mathbb{R}^{d})$
thin for doubling measures?

######  Acknowledgement.

The author thanks Leonid Kovalev for suggesting references [Jonsson],
[Jonsson-Wallin] and [Wu].

## References


# Field trial on Ocean Estimation for Multi-Vessel Multi-Float-based Active
perception

Giovanni D’urso1, James Ju Heon Lee1, Ki Myung Brian Lee1, Jackson Shields2,
Brenton Leighton2, Oscar Pizarro2, Chanyeol Yoo1, Robert Fitch1 This research
is supported by an Australian Government Research Training Program (RTP)
Scholarship, the Schmidt Ocean Institute, the University of Technology Sydney
and the Australian Centre for Field Robotics.1Authors are with the University
of Technology Sydney, NSW 2007, Australia {Giovanni.Durso,
<EMAIL_ADDRESS>and {Chanyeol.Yoo,
<EMAIL_ADDRESS>are with the Australian Centre for Field
Robotics, University of Sydney, NSW 2006, Australia {j.shields, b.leighton,
<EMAIL_ADDRESS>
###### Abstract

Marine vehicles have been used for various scientific missions where
information over features of interest is collected. In order to maximise
efficiency in collecting information over a large search space, we should be
able to deploy a large number of autonomous vehicles that make a decision
based on the latest understanding of the target feature in the environment. In
our previous work, we have presented a hierarchical framework for the _multi-
vessel multi-float_ (MVMF) problem where surface vessels drop and pick up
underactuated floats in a time-minimal way. In this paper, we present the
field trial results using the framework with a number of drifters and floats.
We discovered a number of important aspects that need to be considered in the
proposed framework, and present the potential approaches to address the
challenges.

##  I Introduction

Active perception is a critical component towards achieving persistent
autonomy in marine scientific missions such as underwater habitat mapping [17,
2, 23], environmental monitoring [21, 6, 8, 26], geological surveying [25, 4,
15], and plume source detection [16, 13, 11]. When the vehicles operate in an
oceanic environment, any control or planning done to address active perception
must account for time-varying, uncertain oceanic flow field; either by
compensating for it [18], or exploiting it [10]. We are interested in using
simple and low-cost marine robots called  _floats_ shown in Fig. 1 that
exploit the ocean current for horizontal movement while the vertical movement
is controlled.

Typical marine missions involve coverage of vast oceanic environment. In this
setting, the under-actuated nature of floats necessitates a _multi-vessel
multi-float_ (MVMF) system, where autonomous surface vessels (ASVs)
judiciously deploy and pick up the floats. We have previously studied such a
system in the context of visiting points of interest (POIs) while minimising
makespan time [5]. In the context of active perception, these POIs can
represent scientifically interesting areas for benthic habitat survey. To
survey a benthic habitat, a typical mission involves a float diving to an
imaging altitude and then “following” the bottom at a fixed altitude. Given an
oceanic model, the deployment and pick-up of the floats are scheduled such
that the horizontal movement induced by the ocean currents maximise coverage
of the POIs.

Because the floats are dependent on the surrounding oceanic current, and its
sensors have limited field of view, a high-resolution ocean model is necessary
to accurately evaluate float trajectories that pass through the POIs. Such
high-resolution ocean forecast data is not readily available in practice. In
addition, solving for MVMF schedule in a time-varying real-world oceanic flow
field is difficult, and we have not yet addressed this scope of the problem
with our current algorithm. To address these issues, we proposed in our
previous work [5] that a local flow field estimator can be used to provide
ocean data and that the flow field environment was quasi-static; i.e., the
flow field can be considered piecewise-constant.

Figure 1:  Ocean float taking images of underwater features in a field trial
for Schmidt Ocean Institute (Lizard Island, Great Barrier Reef, Australia)

In this paper, we recount our field trial where we evaluated our assumptions
and approaches made in our previous work. The trial was held along the coast
of Jervis Bay, NSW, Australia from the 31st of March to the 1st of April 2021.
During the trial, we demonstrated the applicability of the local flow field
estimator alongside our MVMF scheduler, and that the quasi-static assumption
does hold for valid planning in a real-world environment. However, we found
that the scope of planning available to us with such assumption and approach
was limited to short-term missions. We also observed an interesting surface
ocean current structure that is beyond the assumptions made for the local flow
field estimator. Based on what we learnt from this trial, we propose new
challenges that need to be addressed for a more robust and versatile MVMF
scheduler and propose potential approaches to those challenges.

##  II Related work

Multi-robot systems have been previously used to improve the efficiency of
marine science missions. The current state-of-the-art systems predominantly
use fully actuated AUVs and ASVs, or a combination of both [27]. The sensor
data gathered by these vehicles are assimilated using tools such as Gaussian
processes [7, 9, 11], Bayesian neural networks [23] or reduced-order models
[22]. The vehicles are coordinated to maximise a performance measure generated
from the assimilation result, such as information gain or predictive
uncertainty [7, 9, 23].

We are interested in scaling up such missions to be persistent while ensuring
economic viability. A promising approach in this direction is to use minimally
actuated systems such as underwater gliders [21] or floats [19, 24]. In
particular, floats are more inexpensive than underwater gliders and have been
previously deployed in significant quantities (approximately 4000 units) for
marine science [19]. While these systems enable much longer mission duration
(months as opposed to hours [19, 21, 24]), they have limited control authority
because they are under-actuated. Our MVMF system provides similar persistence
while improving on the control authority through selective pick-up and drop-
off with ASVs.

A work closely related to ours is [24], where a single float is steered
deliberately using an accurate, high-resolution model of the ocean currents.
Meanwhile, an open practical challenge is that it is difficult to obtain an
ocean current model of sufficient accuracy and resolution (see, e.g., [12,
3]). To this end, we present a practical framework that estimates ocean
currents from the execution of plans and generates plans from estimates of
ocean currents.

##  III Multi-vessel multi-float problem

We present a brief overview of the hierarchical planner we developed in our
previous work [5]. We then outline the field operations used to test field-
applicability of a flow field estimator that will be used to support the
operation of the planner.

###  III-A Problem statement

An intituive description of the MVMF problem, as introduced in our previous
work [5], is as follows. Given a team of surface vessels, floats, the flow
field, and the desired locations to visit (points of interests or POIs), we
aim to find a sequence of drop-off and pickup actions that allow the
underactuated-floats to cover the desired locations. There are many possible
candidate solutions to this problem, and we are interested in finding
schedules that visit as many POIs as possible without losing floats by leaving
them unattended for long periods after completing their drifts.

For completeness we provide a short summary of our hierarchical planning
method [5]. The hierarchical method consists of solving two subproblems:

  1. 1.

Given a time-invariant oceanic flow field $f_{c}$, a set of POIs $\mathbf{Q}$,
a set of sampled drop-off actions $\mathbf{D}$ and its corresponding set of
pick-up actions $\mathbf{P}$, find a set of drop-off actions
$\mathbf{D}_{s}\subseteq\mathbf{D}$ and its corresponding pick-up actions
$\mathbf{P}_{s}\subseteq\mathbf{P}$ that observes all unique POIs in
$\mathbf{Q}$.

  2. 2.

Given a set of surface vessels $\mathbf{A}$ and floats $\mathbf{B}$, and the
sets $\mathbf{D}_{s}$ and $\mathbf{P}_{s}$ found in the previous sub-problem,
find the joint schedule $\boldsymbol{\Phi}$ over all $\mathbf{D}_{s}$ and
$\mathbf{P}_{s}$ that minimises the makespan of the overall mission.

Our method solves the first sub-problem by generating a large number of
sampled actions and selecting a subset of pick-up and drop-off actions from
these samples using a Monte Carlo Tree Search (MCTS). The second sub-problem
is solved by scheduling the selected subset of actions and allocating the
actions to the surface vessels using Decentralised Monte Carlo Tree Search
(Dec-MCTS). Each surface vessel independently computes its local plan that
asymptotically leads to the joint optimal solution.

The largest and untested assumption in our previous work is assuming a fully-
known time-invariant flow field. We assumed that this is a reasonable
assumption because ocean currents vary slowly and are driven by prevailing
meteorological conditions such as tides and wind. Typically, these processes
occur on the timescale of hours, while missions can last minutes to few tens
of minutes. In other words, it is approximately static. Another practical
limitation of our method is the requirement of a dense flow field model for
planning the operations due to the float dynamics and our desired application.
To plan trajectories for the floats that are capable of observing the
bathymetric features represented as POIs, and due to the complex bathymetry of
reefs and shorelines, we require a spatially dense (vectors at least every
meter) over a small-scale (hundreds of meters) model of the flow field.
Typical flow field models produced by oceanographic sources are spatially
sparse (vectors every hundreds of meters) due to their large scale (hundreds
of kilometres). These flow fields are not suitable for our application because
the under-actuation of the floats necessitates an accurate knowledge of the
flow field for predicting the movement of the system. The unavailability of
such a model means that we need to produce a local estimate of the flow field
rather than rely on existing sources of information in order to apply the
planner in practice.

To gain a deeper understanding of the limitations of the fully-known time-
invariant flow field assumption, we introduce a process for estimating the
flow field locally which is then tested with real-world experiments at Jervis
bay. To produce the local dense flow field we use the Gaussian process
(GP)-based expectation-maximisation (EM) method developed by [12] because
their approach allows estimation of a dense oceanic flow field from sparse
point measurements. The point measurements we use for estimation are the
deploy and retrieval locations of the floats or subsampled GPS readings from
drifters. The GP method works by exploiting the incompressibility of ocean
currents to deal with the underdetermined nature of the problem. The problem
is underdetermined because infinitely many candidate flow fields could have
generated the same measurements. By exploiting incompressibility, the search
space can be reduced so that a feasible solution can be computed. There are
two open questions that make the applicability of this method to our problem
uncertain. First, the method was tested on sparse GPS point measurements of
gliders instead of more dense GPS measurements of drifters. The sparse GP
based method might be not applicable for this much data. Second, the GP method
was shown to work for gliders in deep ocean environments, this field trial
takes place in regions of much shallower water (around 10-25 meters depth),
near shorelines, and near reefs. It is not yet known if the method will
perform as well in this new environment.

###  III-B Sea trial operations

In this section, we discuss the technical approach that was taken during sea
trials to assess the applicability of the flow field GP estimator to real-
world oceanic environments similar to our expected operating areas.

To gather this data the drifters or floats need to be spread over the
workspace. The size of the workspace and thus needed triangle is limited by
both the area that needs to be surveyed by the imaging floats and the
communication range of the drifters. The communication range is important
because the drifters communicate their locations blindly without caching. If
the vessel is outside the communication range, we partially lose the
trajectory information. Once the drifters are deployed they are left to float
along with the current for a fixed period of time then retrieved by the boat.
A timely retrieval is important in order to not lose the hardware in the
ocean. The selection of the drift length is important because it needs to be
long enough to gather sufficient data while not being so long that the data
gathered is no longer relevant due to the changing flow field. For our tests
it was found that around 10-15 minutes of drifting combined with the time
taken to deploy and retrieve was sufficient.

The gathered trajectories from the drifters are then used by the GP based
flow-field estimator to build an estimate of what flow field best fits the
observed trajectories. The GP method is dependant on a set of hyperparameters
which can require retuning. The tuning process during sea trials uses a grid
search over a range of possible sets of parameters and choosing the best set
based on the error between the projected trajectory from the estimated field
and the observed field.

Figure 2:  Estimated surface ocean current data for a 390m x 390m oceanic
environment. Current estimated from three drifter trajectories deployed for
600 seconds in a triangle formation. The colormap represents the trace
covariance. The resulting drift data estimated using GP-based estimator
presented in [12].

##  IV Sea trial result

Twenty sea trials were conducted over five days at two sites (Murray beach and
Nelson beach) inside Jervis Bay, Australia. We present the most representative
and interesting data gathered from these trials.

The trials were conducted aboard the MV _Kimbla_ , an 18-meter long vessel
from which we deployed a heterogeneous team of under-actuated robotic
_floats_ [20, 1, 14] as shown in Fig. 1. These robots were developed by the
University of Sydney. The two types of robots we used are a passive _“surface
drifter”_ type float which has no actuation, and a bottom-following benthic
imaging-float which is capable of diving to a controlled depth. For both
systems, the motion is mainly driven by the ocean currents which allows
interaction with the environment. Both types of robots are equipped with an
IMU, GPS, and communicate their state information over LORA. The GPS position
information is then filtered by a Kalman filter to reduce noise, During these
trials, one imaging float and three drifters were utilised to gather the
current estimation data. A visualisation of the drifter deployment for flow
field estimation is shown in Fig. 2.

(a) Attempt 1, time: 145 sec

(b) Attempt 1, time: 660 sec

(c) Attempt 1, time: 790 sec

Figure 3:  1 vessel 2 drifter MVMF schedule, planned over the estimated flow
field (blue quiver) shown in Fig. 2, executed one hour after the estimation.
Each subfigure shows a snapshot of the _Kimbla_ (triangle) moving from
scheduled action to another. For clarity, only the _Kimbla_ trajectory (black)
associated for that snapshot is shown. The sequence of positions the _Kimbla_
is to visit are determined _a priori_ and are enumerated from 1 to 4; marked
with either a drop-off (x) or a pick-up (start) action. The actual drifter
trajectory (red and blue solid line) is compared against the expected
trajectory (red and blue dashed lined). The drifter locations at each snapshot
are marked with a circle. The _Kimbla_ cut through the red drifter path at
both (a) and (b) before waiting at the middle of the workspace to ensure
communication to all deployed drifters. At time 600 sec, the _Kimbla_ moves to
position 3 to pickup the drifter before detouring to the actual drifter
location then picking up the drifter at position 4. The detour resulted in a
tardiness of 60 sec for position 3 and 45 sec for position 4.

(a) Attempt 2, time: 230 sec

(b) Attempt 2, time: 680 sec

(c) Attempt 2, time: 855 sec

Figure 4:  Same mission as Fig 3, except the schedule was executed two hours
after estimation. To avoid the effect the _Kimbla_ has on the drifters, this
mission was executed with slower vessel velocity and having the _Kimbla_
travel around the drop-off and pick-up locations. Both drifters however
travelled more northerly compared to the expected drifter trajectory, which
implies the ocean current have changed. Again, the _Kimbla_ visited position 3
at time 600 sec before detouring to the actual drifter location then picking
up the drifter at position 4. The detour resulted in a tardiness of 80 sec for
position 3 and 25 sec for position 4.

###  IV-A Trial 1 – Drop-pick evaluation

We evaluate the consequence of the quasi-static assumption on the MVMF
schedule with a simple drop-off/pick-up mission using the _Kimbla_ with two
drifters. The _kimbla_ operation consists of moving from location to location
in the schedule and deploying or retrieving drifters as needed. If a drifter
is not at the retrieval location when expected the new retrieval location is
set to the current GPS position. The boat then moves to retrieve the drifter
at its new location then the mission continues as expected. We execute the
same MVMF schedule twice; one hour after estimation and two hour after
estimation. The evaluation compares the expected drifter trajectory from the
schedule with the observed drifter trajectory from both schedule attempts. If
the quasi-static assumption holds, then the expected and observed trajectories
should be similar. For the evaluation, we arbitrarily choose two drop-off
locations in the workspace, and define the corresponding pick-up locations and
expected drifter trajectory by forward integrating across the estimated flow
field for 600 seconds. We also define the action sequence _a priori_ as
parallel deployment as shown in Fig. 3.

The snapshot of executing the first schedule attempt is shown in Fig. 3. The
_Kimbla_ deploys the drifter at point 1 and 2, before waiting at the middle of
the work environment to maintain communication between the _Kimbla_ and the
drifters. After 600 seconds, the _Kimbla_ attempts to execute the scheduled
pick-up actions while compensating for any discrepancy between the expected
and observed result. The blue drifter appears to closely follow the expected
trajectory. However, the red drifter trajectory is translated and slower than
the expected result. We suspect that the _Kimbla_ may have created wakes near
the red drifter that stalled and displaced it. We see that the _Kimbla_ cuts
across red drifter’s expected trajectory in both Fig. 3a and Fig. 3b. The
delay caused by the wake forces the _Kimbla_ to deviate from its schedule to
retrieve the drifter, which by then the blue drifter passes over the expected
pickup location before it is retrieved. This result brings up an interesting
challenge when planning for MVMF schedule:

######  Remark 1 (Disturbance due to surface vessel).

Recklessly operating the surface vessel near the float robot or its expected
path can disrupt the intended float trajectory, which can negatively impact
the overall quality of the MVMF schedule.

This remark applies for both drifter and shallow diving float implementation.
Though the disturbance on the drifters due to the _Kimbla_ is more obvious for
our small-scale trial, it nonetheless should be avoided for general cases as
the ocean current is a chaotic system. In particular, care needs to be taken
when deploying drifters for flow field estimation as they deploy around the
same time scale as our trial missions.

Snapshot of the second schedule attempt is shown in Fig. 4. After learning
from our previous attempt, we were cautious to operate the _Kimbla_ without
creating wake near drifters or its expected path by travelling around the drop
and pick location. While there was no evidence of the _Kimbla_ affecting the
drifter trajectory, we observed they both drifted more northerly than the
expected trajectory. This implies that the ocean current has changed between
the first and second attempt. However, it remained reasonably static between
the estimation step and the first attempt as described in:

######  Remark 2 (Valid time window for ocean estimation).

The quasi-static flow field assumption holds true for real-world
implementation for about an hour.

(a) Mission1, time: 1347 sec

(b) Mission1, time: 1795 sec

(c) Mission1, time: 2084 sec

Figure 5:  Snapshot of two drifter trajectory (red and blue) crossing.
Expected trajectory and estimated flow field shown for comparison.  Figure 6:
Drifters on the ocean surface in an environment consistent with Langmuir
Circulations

###  IV-B Trial 2 – Langmuir circulation

During our trial, we have also observed a special flow field structure that
contradicts the planar incompressibility assumption made for estimating the
flow field. Two deployed drifters that were expected to move parallel to each
other instead moved towards each other until eventually, their trajectories
crossed. The snapshots of this case are shown in Fig 5.

As both drifters were deployed relatively close together mere minutes apart,
the trajectories crossing implies that the oceanic streamline are crossed,
which is impossible in a planar incompressible flow field. One hypothesis is
that the drifters were sucked into a Langmuir circulation cell. That is, there
is a pair of corkscrew vortex rotating about the direction of the flow that
draws nearby water into a stream and pushes it down underwater. This
oceanographic process occurs in the presence of wind shear and when wind and
waves move in a similar direction. We were also able to observe this current
phenomenon. Figure 5 shows sea debris and slick were collected along a
streamline and the floats were observed to be riding these streams. The
existence of such flow structure invalidates the assumption we use for our
estimator; that the flow field is planar and incompressible.

######  Remark 3 (Violation to assumption on incompressibility).

There exist oceanographic processes that may violate the incompressibility
assumption for explicit 2D flow field case near the ocean surface.

##  V Potential approaches

During our field trial, we have identified various challenges that can be
addressed to improve the overall performance of executing a MVMF schedule in a
difficult real-world environment. In this section, we reflect on these
challenges and discuss potential approaches to address these challenges.

###  V-A Disturbance due to surface vessel

From Remark 1 and from the second schedule attempt shown in Fig. 3d-f, we need
to operate the surface vessels with caution when near a drifter or shallow
depth float robot. There are several operation-based approaches to mitigate
the effect of surface vessels on floats, such as delaying departure to the
next position or reducing surface vessel speed when nearby floats. One
possible algorithmic solution is to plan the surface vessel trajectory such
that, topologically, it does not intersect the drifter trajectories nor come
near the floats during deployment. Such an approach allows the surface vessel
to operate at max capacity while guaranteeing it will not influence the float
trajectory.

###  V-B Valid time window for ocean estimation

As stated in Remark 2, Based on observed data the quasi-static flow field
assumption is reasonable for approximately an hour. However, such a window of
operation limits the scope and size of the mission due to the slow speed of
the floats. In addition, the ocean flow field appears to make sudden changes.
That is, we can not use the drift information from the first schedule attempt
to predict the flow field for the second schedule attempt. This challenge
motivates a time-varying flow field estimator and planning a MVMF schedule
over it, however the time-varying problem is intrinsically difficult. Another
approach could be to integrate the estimation step into the MVMF schedule
planner and have the system be closed-looped. This approach has the added
bonus of being able to autonomously recall floats when the environment becomes
too unstable mid-mission.

###  V-C Violation of incompressibility assumption

From Remark 3, it is clear that we do not yet fully understand the ocean
dynamics enough to make robust flow field estimations. For instance, the GP-
based estimator presented in [12] does not account for wind, tide, or
bathymetry; any of which has an impact on both ocean surface and depth flow
field. This challenge motivates a more comprehensive flow field estimator that
accounts for other oceanic factors.

The flow field estimator assumes a valid 2D incompressible flow field.
External disturbances from wakes or 3D oceanographic phenomena invalidate its
assumptions and will lead to incorrect estimations. So care needs to be taken
to ensure that the system operates under conditions in which these assumptions
hold.

##  VI Conclusion

Our previous work [5] was validated in a series of field trials with a number
of drifters and floats. We have found that there exists several of important
extensions to the hierarchical framework in order to improve the overall
performance. First, the scheduler should consider the disturbance caused by a
surface vessel immediately after deploying drifters and floats. Intuitively,
the planner should enforce surface vessel to manoeuvre in a disturbance-
minimal manner. Secondly, the oceanic flow changes more rapidly than we
anticipated in our previous work. Therefore, there should be a more frequent
online update to ocean currents to allow more accurate result. Lastly, our
incompressibility assumption on ocean currents may be violated near the
surface. Although the assumption still holds true for underwater, such
violation should be considered when using drifters and floats near the
surface.

## Acknowledgement

We would like to thank the crew of the MV _Kimbla_ for helping us perform the
MVMF field trial. Additional thanks go to the Schmidt Ocean Institute for
funding and the University of Technology Sydney and the Australian Centre for
Field Robotics for their support in the project.

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