# Some inverse problems associated with Hill operator

Alp Arslan Kıraç  Alp Arslan Kıraç
Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University,
20070, Denizli, Turkey<EMAIL_ADDRESS>
###### Abstract.

Let $l_{n}$ be the length of the $n$-th instability interval of the Hill
operator $Ly=-y^{\prime\prime}+q(x)y$. We obtain that if $l_{n}=o(n^{-2})$
then $c_{n}=o(n^{-2})$, where $c_{n}$ are the Fourier coefficients of $q$.
Using this inverse result, we prove: Let $l_{n}=o(n^{-2})$. If
$\\{(n\pi)^{2}:\textrm{$n$ even and $n>n_{0}$}\\}$ is a subset of the periodic
spectrum of Hill operator then $q=0$ a.e., where $n_{0}$ is a positive large
number such that $l_{n}<\varepsilon n^{-2}$ for all $n>n_{0}(\varepsilon)$
with some $\varepsilon>0$. A similar result holds for the anti-periodic case.

###### Key words and phrases:

Hill operator; inverse spectral theory; eigenvalue asymptotics; Fourier
coefficients

###### 2000 Mathematics Subject Classification:

34A55, 34B30, 34L05, 47E05, 34B09

##  1\. Introduction

Consider the Hill operator

$Ly=-y^{\prime\prime}+q(x)y$ (1.1)

generated in $L_{2}(-\infty,\infty)$ , where $q(x)$ is a reel-valued summable
function on $[0,1]$ and $q(x+1)=q(x)$. Let $\lambda_{n}$ and $\mu_{n}$
$(n=0,1,\ldots)$ denote, respectively, the $n$-th periodic and anti-periodic
eigenvalues of the Hill operator (1.1) on $[0,1]$ with the periodic boundary
conditions

$y(0)=y(1),\,y^{\prime}(0)=y^{\prime}(1),$ (1.2)

and the anti-periodic boundary conditions

$y(0)=-y(1),\;y^{\prime}(0)=-y^{\prime}(1).$

It is well-known [5, 7] that

$\lambda_{0}<\mu_{0}\leq\mu_{1}<\lambda_{1}\leq\lambda_{2}<\mu_{2}\leq\mu_{3}<\cdots\rightarrow\infty.$

The intervals $(\mu_{2m},\mu_{2m+1})$ and $(\lambda_{2m+1},\lambda_{2m+2})$
are respectively referred to as the $(2m+1)$-th and $(2m+2)$-th finite
instability intervals of the operator $L$, while $(-\infty,\lambda_{0})$ is
called the zero-th instability interval. The length of the $n$-th instability
interval of (1.1) will be denoted by $l_{n}$ ($n=2m+1,\,2m+2$). For further
background see [14, 15, 13].

Borg [2], Ungar [22] and Hochstadt [13] proved independently of each other the
following statement:

_If $q(x)$ is real and integrable, and if all finite instability intervals
vanish then $q(x)=0$ a.e._

Hochstadt [13] also showed that, when precisely one of the finite instability
intervals does not vanish, $q(x)$ is the elliptic function which satisfies

$q^{\prime\prime}=3q^{2}+Aq+B\quad a.e.,$

where $A$ and $B$ are suitable constants, and, when $n$ finite instability
intervals fail to vanish, $q(x)$ is infinitely differentiable a.e. For more
results concerning the above type and further references, see [8, 9, 10, 11].

Also, by using length of the instability interval, let us consider another
approach to inverse problems. Hochstadt [12] proved that the lengths of the
instability intervals $l_{n}$ vanish faster than any power of $(1/n)$ for an
$L^{2}_{1}$ potential $q$ in $C_{1}^{\infty}$. McKean and Trubowitz [16]
proved the converse: if $q$ is in $L_{1}^{2}$ and the length of the $n$-th
instability interval for $n\geq 1$ is rapidly decreasing, then $q$ is in
$C_{1}^{\infty}$. Later Trubowitz [21] proved the following result: an
$L_{1}^{2}$ potential $q$ is real analytic if and only if the lengths of the
instability intervals are decays exponentially. In [6], Coskun showed that
(see Theorem 6), in our notations,

$\textrm{if}\,\,l_{n}=O(n^{-2})\textrm{ then}\,\,c_{n}=:(q\,,e^{i2n\pi
x})=O(n^{-2})\textrm{ as $n\rightarrow\infty,$}$ (1.3)

where $(.\,,.)$ is the inner product in $L^{2}[0,1].$

At this point we refer to some Ambarzumyan-type theorems in [1, 4, 26, 3]. In
1929, Ambarzumyan [1] obtained the following first theorem in inverse spectral
theory: If $\\{n^{2}:n=0,1,\ldots\\}$ is the spectrum of the Sturm-Liouville
operator (1.1) on $[0,1]$ with Neumann boundary condition, then $q=0$ a.e. In
[4], they extended the classical Ambarzumyan’s theorem for the Sturm-Liouville
equation to the general separated boundary conditions, by imposing an
additional condition on the potential function, and their result supplements
the Pöschel-Trubowitz inverse spectral theory [17]. In [26], based on the
well-known extremal property of the first eigenvalue, they find two analogs of
Ambarzumyan’s theorem to Sturm-Liouville systems of n dimension under periodic
or anti-periodic boundary conditions. In the paper [3], by using Rayleigh-Ritz
inequality and imposing a condition on the second term in the Fourier cosine
series (see (1.4)), they proved the following Ambarzumyan-type theorem:

_(a) If all periodic eigenvalues of Hill’s equation ( 1.1) are nonnegative and
they include $\\{(2m\pi)^{2}:m\in\mathbb{N}\\}$, then $q=0$ a.e._

_(b) If If all anti-periodic eigenvalues of Hill’s equation ( 1.1) are not
less than $\pi^{2}$ and they include $\\{(2m-1)^{2}\pi^{2}:m\in\mathbb{N}\\}$,
and_

$\int_{0}^{1}q(x)\,cos(2\pi x)\,dx\geq 0,$ (1.4)

_then $q=0$ a.e._

More recently, in [18], we obtain the classical Ambarzumyan’s theorem for the
Sturm-Liouville operators with $q\in L^{1}[0,1]$ and quasi-periodic boundary
conditions, when there is not any additional condition on the potential $q$
such as (1.4). See further references in [18].

In this paper, we prove the following results:

######  Theorem 1.1.

If $l_{n}=o(n^{-2})$ then $c_{n}=o(n^{-2})$ as $n\rightarrow\infty.$

######  Theorem 1.2.

Let $l_{n}=o(n^{-2})$ as $n\rightarrow\infty$. Then

(i) if $\\{(n\pi)^{2}:\textrm{$n$ even and $n>n_{0}$}\\}$ is a subset of the
periodic spectrum of Hill operator then $q=0$ a.e.

(ii) if $\\{(n\pi)^{2}:\textrm{$n$ odd and $n>n_{0}$}\\}$ is a subset of the
anti-periodic spectrum of Hill operator then $q=0$ a.e., where $n_{0}$ is a
positive large number such that

$l_{n}<\varepsilon n^{-2}\qquad\textrm{for all $n>n_{0}(\varepsilon)$ with
some $\varepsilon>0$.}$

In Theorem 1.1, we obtain that $O$-terms in (1.3) can be improved to the
$o$-terms $o(n^{-2})$ from which we shall use essentially in the proof of
Theorem 1.2.

Note that the first eigenvalue for Ambarzumyan-type theorems is important to
be given while for some of the other types the multiplicity of some
eigenvalues is important, that is, some of the instability intervals vanish.
Unlike the works in types briefly outlined above, to prove the assertion of
Theorem 1.2 we use not only the length of instability internals $l_{n}$ as
$n\rightarrow\infty$ but also a subset of spectrum of Hill operator as in
Ambarzumyan-type theorems. However, in Theorem 1.2, we assume that, for some
large $n_{0}$, $(n\pi)^{2}$ with $n>n_{0}$ is a periodic eigenvalue for even
$n$ (or anti-periodic for odd $n$) and we do not assume that the given
eigenvalues are of multiplicity 2.

##  2\. Preliminaries and Proof of the results

We shall consider only the periodic (for even $n$) eigenvalues of Hill
operator. The anti-periodic (for odd $n$) problem is completely similar. It is
well known [7, Theorem 4.2.3] that the periodic eigenvalues
$\lambda_{2m+1},\lambda_{2m+2}$ are asymptotically located in pairs such that

$\lambda_{2m+1}=\lambda_{2m+2}+o(1)=(2m+2)^{2}\pi^{2}+o(1)$ (2.1)

for sufficiently large $m$. From this formula, for all $k\neq 0,(2m+2)$ and
$k\in\mathbb{Z}$, the inequality

$|\lambda-(2(m-k)+2)^{2}\pi^{2}|>|k||(2m+2)-k|>C\,m,$ (2.2)

is satisfied by both eigenvalues $\lambda_{2m+1}$ and $\lambda_{2m+2}$ for
large $m$, where, here and in the rest relations, $C$ denotes a positive
constant whose exact value is not essential. Note that when $q=0$, the system
$\\{e^{-i(2m+2)\pi x},e^{i(2m+2)\pi x}\\}$ is a basis of the eigenspace
corresponding to the double eigenvalues $(2m+2)^{2}\pi^{2}$ of the problem
(1.1)-(1.2).

To obtain the asymptotic formulas for the periodic eigenvalues
$\lambda_{2m+1},\lambda_{2m+2}$ corresponding respectively to the normalized
eigenfunctions $\Psi_{m,1}(x),\Psi_{m,2}(x)$, let us consider the the well-
known relation, for sufficiently large $m$,

$\Lambda_{m,j,m-k}(\Psi_{m,j},e^{i(2(m-k)+2)\pi
x})=(q\,\Psi_{m,j},e^{i(2(m-k)+2)\pi x}),$ (2.3)

where $\Lambda_{m,j,m-k}=(\lambda_{2m+j}-(2(m-k)+2)^{2}\pi^{2})$, $j=1,2.$ The
relation (2.3) can be obtained from the equation (1.1), first, replacing $y$
by $\Psi_{m,j}(x),$ and secondly, multiplying both sides by $e^{i(2(m-k)+2)\pi
x}$. By using Lemma 1 in [24], to iterate (2.3) for $k=0$, in the right hand-
side of formula (2.3) we use the following relations

$(q\,\Psi_{m,j},e^{i(2m+2)\pi
x})=\sum_{m_{1}=-\infty}^{\infty}c_{m_{1}}(\Psi_{m,j},e^{i(2(m-m_{1})+2)\pi
x}),$ (2.4) $|(q\,\Psi_{m,j},e^{i(2(m-m_{1})+2)\pi x})|<3M$ (2.5)

for all large $m$, where $j=1,2$ and $M=\sup_{m\in\mathbb{Z}}|c_{m}|$.

First, we fix the terms with indices $m_{1}=0,(2m+2)$. Then all the other
terms in the right hand-side of (2.4) are replaced, in view of (2.2) and (2.3)
for $k=m_{1}$, by

$c_{m_{1}}\frac{(q\,\Psi_{m,j},e^{i(2(m-m_{1})+2)\pi
x})}{\Lambda_{m,j,m-m_{1}}}.$

In the same way, by applying the above procedure for the other eigenfunction
$e^{-i(2m+2)\pi x}$ corresponding to the eigenvalue $(2m+2)^{2}\pi^{2}$ of the
problem (1.1)-(1.2) for $q=0$, we obtain the following lemma (see also Section
2 in [20, 19]).

######  Lemma 2.1.

The following relations hold for sufficiently large $m$:

$\textrm{(i)}\quad[\Lambda_{m,j,m}-c_{0}-\sum_{i=1}^{2}a_{i}(\lambda_{2m+j})]u_{m,j}=[c_{2m+2}+\sum_{i=1}^{2}b_{i}(\lambda_{2m+j})]v_{m,j}+R_{2},$
(2.6)

where $j=1,2,$

$u_{m,j}=(\Psi_{m,j},e^{i(2m+2)\pi x}),\quad
v_{m,j}=(\Psi_{m,j},e^{-i(2m+2)\pi x}),$
$a_{1}(\lambda_{2m+j})=\sum_{m_{1}}\frac{c_{m_{1}}c_{-m_{1}}}{\Lambda_{m,j,m-m_{1}}},\,a_{2}(\lambda_{2m+j})=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}}c_{-m_{1}-m_{2}}}{\Lambda_{m,j,m-m_{1}}\,\Lambda_{m,j,m-m_{1}-m_{2}}},$
(2.7)
$b_{1}(\lambda_{2m+j})=\sum_{m_{1}}\frac{c_{m_{1}}c_{2m+2-m_{1}}}{\Lambda_{m,j,m-m_{1}}},\qquad
b_{2}(\lambda_{2m+j})=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}}c_{2m+2-m_{1}-m_{2}}}{\Lambda_{m,j,m-m_{1}}\,\Lambda_{m,j,m-m_{1}-m_{2}}},$
$R_{2}=\sum_{m_{1},m_{2},m_{3}}\frac{c_{m_{1}}c_{m_{2}}c_{m_{3}}(q\,\Psi_{m,j}(x),e^{i(2(m-m_{1}-m_{2}-m_{3})+2)\pi
x})}{\Lambda_{m,j,m-m_{1}}\,\Lambda_{m,j,m-m_{1}-m_{2}}\,\Lambda_{m,j,m-m_{1}-m_{2}-m_{3}}}.$
(2.8)

The sums in these formulas are taken over all integers $m_{1},m_{2},m_{3}$
such that $m_{1},m_{1}+m_{2},m_{1}+m_{2}+m_{3}\neq 0,\,2m+2$.

$\textrm{(ii)}\quad[\Lambda_{m,j,m}-c_{0}-\sum_{i=1}^{2}a^{\prime}_{i}(\lambda_{2m+j})]v_{m,j}=[c_{-2m-2}+\sum_{i=1}^{2}b^{\prime}_{i}(\lambda_{2m+j})]u_{m,j}+R^{\prime}_{2},$
(2.9)

where $j=1,2,$

$a^{\prime}_{1}(\lambda_{2m+j})=\sum_{m_{1}}\frac{c_{m_{1}}c_{-m_{1}}}{\Lambda_{m,j,m+m_{1}}},\qquad
a^{\prime}_{2}(\lambda_{2m+j})=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}}c_{-m_{1}-m_{2}}}{\Lambda_{m,j,m+m_{1}}\,\Lambda_{m,j,m+m_{1}+m_{2}}},$
$b^{\prime}_{1}(\lambda_{2m+j})=\sum_{m_{1}}\frac{c_{m_{1}}c_{-2m-2-m_{1}}}{\Lambda_{m,j,m+m_{1}}},\qquad
b^{\prime}_{2}(\lambda_{2m+j})=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}}c_{-2m-2-m_{1}-m_{2}}}{\Lambda_{m,j,m+m_{1}}\,\Lambda_{m,j,m+m_{1}+m_{2}}},$
$R^{\prime}_{2}=\sum_{m_{1},m_{2},m_{3}}\frac{c_{m_{1}}c_{m_{2}}c_{m_{3}}(q\,\Psi_{m,j}(x),e^{i(2(m+m_{1}+m_{2}+m_{3})+2)\pi
x})}{\Lambda_{m,j,m+m_{1}}\,\Lambda_{m,j,m+m_{1}+m_{2}}\,\Lambda_{m,j,m+m_{1}+m_{2}+m_{3}}}$
(2.10)

and the sums in these formulas are taken over all integers $m_{1},m_{2},m_{3}$
such that $m_{1},m_{1}+m_{2},m_{1}+m_{2}+m_{3}\neq 0,\,-2m-2$.

Note that, by substituting respectively $m_{1}=-k_{1}$ for $i=1$ and
$m_{1}+m_{2}=-k_{1}$, $m_{2}=k_{2}$ for $i=2$ into the relations for
$a^{\prime}_{1}(\lambda_{2m+j})$ and $a^{\prime}_{2}(\lambda_{2m+j})$, we have
the equalities

$a_{i}(\lambda_{2m+j})=a^{\prime}_{i}(\lambda_{2m+j})\quad\textrm{for
$i=1,2$.}$ (2.11)

Here, using the equality

$\frac{1}{m_{1}(2m+2-m_{1})}=\frac{1}{2m+2}\left(\frac{1}{m_{1}}+\frac{1}{2m+2-m_{1}}\right),$

we get the relation

$\sum_{m_{1}\neq
0,(2m+2)}\frac{1}{|m_{1}(2m+2-m_{1})|}=O\left(\frac{ln|m|}{m}\right).$

This with (2.2), (2.3) and (2.5) gives the following estimates (see (2.8),
(2.10))

$R_{2},\,R^{\prime}_{2}=O\left((\frac{ln|m|}{m})^{3}\right).$ (2.12)

Moreover, in view of (2.2), (2.3) and (2.5), we get (see also [24, Theorem 2],
[19])

$\sum_{k\in\mathbb{Z};\,k\neq\pm(m+1)}\Big{|}(\Psi_{m,j},e^{i2k\pi
x})\Big{|}^{2}=O\left(\frac{1}{m^{2}}\right)$ (2.13)

Therefore, the expansion of the normalized eigenfunctions $\Psi_{m,j}(x)$ by
the orthonormal basis $\\{e^{i2k\pi x}:k\in\mathbb{Z}\\}$ on $[0,1]$ has the
following form

$\Psi_{m,j}(x)=u_{m,j}\,e^{i(2m+2)\pi x}+v_{m,j}\,e^{-i(2m+2)\pi x}+h_{m}(x),$
(2.14)

where

$\\!\\!(h_{m},e^{\mp i(2m+2)\pi
x})=0,\,\|h_{m}\|=O(m^{-1}),\,\sup_{x\in[0,1]}|h_{m}(x)|=O\left(\frac{ln|m|}{m}\right)$
$|u_{m,j}|^{2}+|v_{m,j}|^{2}=1+O\left(m^{-2}\right).$ (2.15)

### Proof of Theorem 1.1

First we estimate the terms of (2.6) and (2.9). From (2.1), (2.2) and (2.13),
one can readily see that

$\sum_{m_{1}\neq 0,\pm(2m+2)}\left|\frac{1}{\Lambda_{m,j,m\mp
m_{1}}}-\frac{1}{\Lambda_{m,0,m\mp m_{1}}}\right|$ $\leq
C|\Lambda_{m,j,m}|\sum_{m_{1}\neq 0,\pm(2m+2)}|m_{1}|^{-2}|2m+2\mp
m_{1}|^{-2}=o\left(m^{-2}\right),$ (2.16)

where $\Lambda_{m,0,m\mp m_{1}}=((2m+2)^{2}\pi^{2}-(2(m\mp
m_{1})+2)^{2}\pi^{2})$. Thus, we get

$a_{i}(\lambda_{2m+j})=a_{i}((2m+2)^{2}\pi^{2})+o\left(m^{-2}\right)\quad\textrm{for
$i=1,2.$}$ (2.17)

Here, by virtue of (2.16) we also have, arguing as in [19, Lemma 3](see also
Lemma 6 of [25]),

$b_{1}(\lambda_{2m+j})=\frac{1}{4\pi^{2}}\sum_{m_{1}\neq
0,(2m+2)}\frac{c_{m_{1}}c_{2m+2-m_{1}}}{m_{1}(2m+2-m_{1})}+o\left(m^{-2}\right)$
$=-\int_{0}^{1}(Q(x)-Q_{0})^{2}\,e^{-i2(2m+2)\pi x}dx+o\left(m^{-2}\right)$
$\qquad\;\;=\frac{-1}{i2\pi(2m+2)}\int_{0}^{1}2(Q(x)-Q_{0})\,q(x)\,e^{-i2(2m+2)\pi
x}dx+o\left(m^{-2}\right),$ (2.18)

where

$Q(x)-Q_{0}=\sum_{m_{1}\neq 0}Q_{m_{1}}\,e^{i2m_{1}\pi x}$ (2.19)

and $Q_{m_{1}}=:(Q(x),e^{i2m_{1}\pi x})=\frac{c_{m_{1}}}{i2\pi m_{1}}$ for
$m_{1}\neq 0$ are the Fourier coefficients with respect to the system
$\\{e^{i2m_{1}\pi x}:m_{1}\in\mathbb{Z}\\}$ of the function
$Q(x)=\displaystyle\int_{0}^{x}q(t)\,dt.$ Here only for the proof of Theorem
1.1, we may suppose without loss of generality that $c_{0}=0$, so that
$Q(1)=c_{0}=0.$

Now using the assumption $l_{n}=o(n^{-2})$ of the theorem, it is also
$O(n^{-2})$. In view of (1.3) we get $c_{n}=O(n^{-2})$ as
$n\rightarrow\infty$. Thus, from Lemma 5 of [13], we obtain that $q(x)$ is
absolutely continuous a.e. Hence, for the right hand-side of
$b_{1}(\lambda_{2m+j})$ given by (2.18), integration by parts with $Q(1)=0$
gives

$b_{1}(\lambda_{2m+j})=\frac{1}{2\pi^{2}(2m+2)^{2}}\int_{0}^{1}\left(q^{2}(x)+(Q(x)-Q_{0})q^{\prime}(x)\right)e^{-i2(2m+2)\pi
x}dx+o\left(m^{-2}\right).$

Since $q(x)$ is absolutely continuous a.e., this leads to
$\left(q^{2}(x)+(Q(x)-Q_{0})q^{\prime}(x)\right)\in L^{1}[0,1]$. By the
Riemann-Lebesgue lemma, we find

$b_{1}(\lambda_{2m+j})=o\left(m^{-2}\right).$ (2.20)

Similarly

$b^{\prime}_{1}(\lambda_{2m+j})=o\left(m^{-2}\right).$ (2.21)

Let us prove that

$b_{2}(\lambda_{2m+j}),\,b^{\prime}_{2}(\lambda_{2m+j})=o\left(m^{-2}\right).$
(2.22)

Taking into account that $q(x)$ is absolutely continuous a.e. and periodic, we
get $c_{m_{1}}c_{m_{2}}c_{\pm(2m+2)-m_{1}-m_{2}}=o\left(m^{-1}\right)$ (see p.
665 of [25]). Using this and arguing as in (2.12)

$|b_{2}(\lambda_{2m+j})|=o\left(m^{-1}\right)\sum_{m_{1},m_{2}}\frac{1}{\left|m_{1}(2m+2-m_{1})(m_{1}+m_{2})(2m+2-m_{1}-m_{2})\right|}$
$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=o\left(m^{-1}\right)O\left((\frac{ln|m|}{m})^{2}\right)=o\left(m^{-2}\right).$

Thus, we get the first estimate of (2.22). Similarly
$b^{\prime}_{2}(\lambda_{2m+j})=o\left(m^{-2}\right).$ Substituting the
estimates given by (2.11), (2.12), (2.17) and (2.20)-(2.22) into the relations
(2.6) and (2.9), we find that

$[\Lambda_{m,j,m}-\sum_{i=1}^{2}a_{i}((2m+2)^{2}\pi^{2})]u_{m,j}=c_{2m+2}v_{m,j}+o\left(m^{-2}\right),$
(2.23)
$[\Lambda_{m,j,m}-\sum_{i=1}^{2}a_{i}((2m+2)^{2}\pi^{2})]v_{m,j}=c_{-2m-2}\,u_{m,j}+o\left(m^{-2}\right)$
(2.24)

for $j=1,2$.

Now suppose that, contrary to what we want to prove, there exists an
increasing sequence $\\{m_{k}\\}\,(k=1,2,\ldots)$ such that

$|c_{2m_{k}+2}|>Cm_{k}^{-2}\quad\textrm{for some $C>0$}.$ (2.25)

Further, the formula obtained from (2.15) by replacing $m$ with $m_{k}$ shows
that either $|u_{m_{k},j}|>1/2$ or $|v_{m_{k},j}|>1/2$ for large $m_{k}$.
Without loss of generality we assume that $|u_{m_{k},j}|>1/2$. Then it follows
from both (2.23) and (2.24) for $m=m_{k}$ that

$[\Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^{2}a_{i}((2m_{k}+2)^{2}\pi^{2})]\sim
c_{2m_{k}+2},$ (2.26)

where the notation $a_{m}\sim b_{m}$ means that there exist constants $c_{1}$,
$c_{2}$ such that $0<c_{1}<c_{2}$ and $c_{1}<|a_{m}/b_{m}|<c_{2}$ for all
sufficiently large $m$. This with (2.24) for $m=m_{k}$, (2.25) and the
assumption $|u_{m_{k},j}|>1/2$ implies that

$u_{m_{k},j}\sim v_{m_{k},j}\sim 1.$ (2.27)

Now multiplying (2.24) for $m=m_{k}$ by $c_{2m_{k}+2}$, and then using (2.23)
in (2.24) for $m=m_{k}$, we arrive at the relation

$[\Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^{2}a_{i}((2m_{k}+2)^{2}\pi^{2})]\left([\Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^{2}a_{i}((2m_{k}+2)^{2}\pi^{2})]u_{m_{k},j}+o\left(m_{k}^{-2}\right)\right)$
$=|c_{2m_{k}+2}|^{2}\,u_{m_{k},j}+c_{2m_{k}+2}\,o\left(m_{k}^{-2}\right)$

which, by (2.26) and (2.27), implies the following equations

$\Lambda_{m_{k},j,m_{k}}-\sum_{i=1}^{2}a_{i}((2m_{k}+2)^{2}\pi^{2})=\pm|c_{2m_{k}+2}|+o\left(m_{k}^{-2}\right)$
(2.28)

for $j=1,2$.

Let us prove that the periodic eigenvalues for large $m_{k}$ are simple.
Assume that there exist two orthogonal eigenfunctions $\Psi_{m_{k},1}(x)$ and
$\Psi_{m_{k},2}(x)$ corresponding to $\lambda_{2m_{k}+1}=\lambda_{2m_{k}+2}$.
From the argument of Lemma 4 in [25], using the relation (2.14) with
$\|h_{m_{k}}\|=O(m_{k}^{-1})$ for the eigenfunctions $\Psi_{m_{k},j}(x)$ and
the orthogonality of eigenfunctions, one can choose the eigenfunction
$\Psi_{m_{k},j}(x)$ such that either $u_{m_{k},j}=0$ or $v_{m_{k},j}=0$, which
contradicts (2.27).

Since the eigenfunctions $\Psi_{m_{k},1}$ and $\overline{\Psi_{m_{k},2}}$ of
the self-adjoint problem corresponding to the different eigenvalues
$\lambda_{2m_{k}+1}\neq\lambda_{2m_{k}+2}$ are orthogonal we find, by (2.14),
that

$0=(\Psi_{m_{k},1},\overline{\Psi_{m_{k},2}})=u_{m_{k},2}v_{m_{k},1}+u_{m_{k},1}v_{m_{k},2}+O(m_{k}^{-1}).$
(2.29)

Note that for the simple eigenvalues in (2.28) there are two cases. First
case: The simple eigenvalues $\lambda_{2m_{k}+1}$ and $\lambda_{2m_{k}+2}$ in
(2.28) corresponds respectively to the lower sign $-$ and upper sign $+$. Then

$l_{2{m_{k}}+2}=\lambda_{m_{k},2,m_{k}}-\lambda_{m_{k},1,m_{k}}=2|c_{2m_{k}+2}|+o\left(m_{k}^{-2}\right)$

which implies that (see (2.25)) $l_{2{m_{k}}+2}>Cm_{k}^{-2}$ for some $C$,
which contradicts the hypothesis. Now let us consider the second case: We
assume that both simple eigenvalues correspond to the lower sign $-$ (the
proof for the sign $+$ is similar). Then
$\Lambda_{m_{k},2,m_{k}}-\Lambda_{m_{k},1,m_{k}}=o\left(m_{k}^{-2}\right)$.
Using this, (2.23) and (2.28), we have

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!o\left(m_{k}^{-2}\right)\,u_{m_{k},2}=c_{2m_{k}+2}\,v_{m_{k},2}+|c_{2m_{k}+2}|\,u_{m_{k},2}+o\left(m_{k}^{-2}\right),$
(2.30)
$o\left(m_{k}^{-2}\right)\,u_{m_{k},1}=-c_{2m_{k}+2}\,v_{m_{k},1}-|c_{2m_{k}+2}|\,u_{m_{k},1}+o\left(m_{k}^{-2}\right).$
(2.31)

Therefore, multiplying both sides of (2.30) and (2.31) by $v_{m_{k},1}$ and
$v_{m_{k},2}$, respectively, and adding the two resulting relations, we have,
in view of (2.25),

$u_{m_{k},2}v_{m_{k},1}-u_{m_{k},1}v_{m_{k},2}=o(1).$

This with (2.29) gives $u_{m_{k},2}v_{m_{k},1}=o(1)$ which contradicts (2.27).
Thus the assumption (2.25) is false, that is, $c_{2m+2}=o\left(m^{-2}\right)$.
A similar result holds for the anti-periodic problem, that is,
$c_{2m+1}=o\left(m^{-2}\right)$. The theorem is proved. $\square$

For the proof of Theorem 1.2 we need the sharper estimates of the following
lemma:

######  Lemma 2.2.

Let $q(x)$ be absolutely continuous a.e. and $c_{0}=0$. Then, for all
sufficiently large $m$, we have the following equalities for the series in
(2.7)

$a_{1}(\lambda_{2m+j})=\frac{-1}{(2\pi(2m+2))^{2}}\int_{0}^{1}q^{2}(x)dx+o\left(m^{-2}\right),\quad
a_{2}(\lambda_{2m+j})=o\left(m^{-2}\right).$ (2.32)

###### Proof.

First, let us consider $a_{1}(\lambda_{2m+j})$. By virtue of (2.16) we get

$a_{1}(\lambda_{2m+j})=\frac{1}{4\pi^{2}}\sum_{m_{1}\neq
0,(2m+2)}\frac{c_{m_{1}}c_{-m_{1}}}{m_{1}(2m+2-m_{1})}+o\left(m^{-2}\right).$

Arguing as in Lemma 3 in [19] (see also Lemma 2.3(a) of [23]), we obtain, in
our notations,

$a_{1}(\lambda_{2m+j})=\frac{1}{2\pi^{2}}\sum_{m_{1}>0,m_{1}\neq(2m+2)}\frac{c_{m_{1}}c_{-m_{1}}}{(2m+2+m_{1})(2m+2-m_{1})}+o\left(m^{-2}\right)$
$=\int_{0}^{1}(G^{+}(x,m)-G^{+}_{0}(m))^{2}\,e^{i2(4m+4)\pi
x}\,dx+o\left(m^{-2}\right)=$
$\frac{-2}{i2\pi(4m+4)}\int_{0}^{1}(G^{+}(x,m)-G^{+}_{0}(m))(q(x)e^{-i2(2m+2)\pi
x}-c_{2m+2})e^{i2(4m+4)\pi x}dx+o\left(m^{-2}\right)$ (2.33)

where

$G^{\pm}_{m_{1}}(m)=:(G^{\pm}(x,m),e^{i2m_{1}\pi
x})=\frac{c_{m_{1}\pm(2m+2)}}{i2\pi m_{1}}$ (2.34)

for $m_{1}\neq 0$ are the Fourier coefficients with respect to
$\\{e^{i2m_{1}\pi x}:m_{1}\in\mathbb{Z}\\}$ of the functions

$G^{\pm}(x,m)=\int_{0}^{x}q(t)\,e^{\mp i2(2m+2)\pi t}dt-c_{\pm(2m+2)}x$ (2.35)

and

$G^{\pm}(x,m)-G^{\pm}_{0}(m)=\sum_{m_{1}\neq(2m+2)}\frac{c_{m_{1}}}{i2\pi(m_{1}\mp(2m+2))}\,e^{i2(m_{1}\mp(2m+2))\pi
x}.$

Here, taking into account the Lemma 1 of [19] and (2.35), we have the
estimates

$G^{\pm}(x,m)-G^{\pm}_{0}(m)=G^{\pm}(x,m)-\int_{0}^{1}G^{\pm}(x,m)\,dx=o(1)\quad\textrm{as
$m\rightarrow\infty$}$ (2.36)

uniformly in $x$.

From the equalities (see (2.35))

$G^{\pm}(1,m)=G^{\pm}(0,m)=0$ (2.37)

and since $q(x)$ is absolutely continuous a.e., integration by parts gives for
the right hand-side of $a_{1}(\lambda_{2m+j})$ given by (2.33) the value

$a_{1}(\lambda_{2m+j})=\frac{-1}{(2\pi(2m+2))^{2}}\left[\int_{0}^{1}q^{2}+\int_{0}^{1}(G^{+}(x,m)-G^{+}_{0}(m))q^{\prime}(x)e^{i2(2m+2)\pi
x}dx\right]$ $+\frac{|c_{2m+2}|^{2}}{(2\pi(2m+2))^{2}}+o\left(m^{-2}\right)$

for sufficiently large $m$. Thus, by using the Riemann-Lebesgue lemma, this
with $(G^{+}(x,m)-G^{+}_{0}(m))q^{\prime}(x)\in L^{1}[0,1]$ implies the first
equality of (2.32).

Now, it remains to prove that $a_{2}(\lambda_{2m+j})=o\left(m^{-2}\right)$.
Similarly, by (2.17) for $i=2$ we get

$a_{2}(\lambda_{2m+j})=\sum_{m_{1},m_{2}}\frac{(2\pi)^{-4}\,c_{m_{1}}c_{m_{2}}c_{-m_{1}-m_{2}}}{m_{1}(2m+2-m_{1})(m_{1}+m_{2})(2m+2-m_{1}-m_{2})}+o\left(m^{-2}\right).$
(2.38)

As in Lemma 4 of [19], using the summation variable $m_{2}$ to represent the
previous $m_{1}+m_{2}$ in (2.38), we write (2.38) in the form

$a_{2}(\lambda_{2m+j})=\frac{1}{(2\pi)^{4}}\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}-m_{1}}c_{-m_{2}}}{m_{1}(2m+2-m_{1})m_{2}(2m+2-m_{2})},$

where the forbidden indices in the sums take the form of $m_{1},m_{2}\neq
0,\,2m+2$. Here the equality

$\frac{1}{k(2m+2-k)}=\frac{1}{2m+2}\left(\frac{1}{k}+\frac{1}{2m+2-k}\right)$

gives

$a_{2}(\lambda_{2m+j})=\frac{1}{(2\pi)^{4}(2m+2)^{2}}\sum_{j=1}^{4}S_{j},$
(2.39)

where

$S_{1}=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}-m_{1}}c_{-m_{2}}}{m_{1}m_{2}},\;S_{2}=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}-m_{1}}c_{-m_{2}}}{m_{2}(2m+2-m_{1})},$
$S_{3}=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}-m_{1}}c_{-m_{2}}}{m_{1}(2m+2-m_{2})},\;S_{4}=\sum_{m_{1},m_{2}}\frac{c_{m_{1}}c_{m_{2}-m_{1}}c_{-m_{2}}}{(2m+2-m_{1})(2m+2-m_{2})}.$

From (2.19) and the assumption $c_{0}=0$ which implies $Q(1)=0$, we deduce by
means of the substitution $t=(Q(x)-Q_{0})$

$S_{1}=4\pi^{2}\int_{0}^{1}(Q(x)-Q_{0})^{2}q(x)\,dx=0.$ (2.40)

Similarly, in view of (2.19) and (2.34)-(2.37), we get by the Riemann-Lebesgue
lemma

$S_{2}=-4\pi^{2}\int_{0}^{1}(Q(x)-Q_{0})(G^{+}(x,m)-G^{+}_{0}(m))\,q(x)\,e^{i2(2m+2)\pi
x}dx=o\left(1\right),$
$S_{3}=-4\pi^{2}\int_{0}^{1}(Q(x)-Q_{0})(G^{-}(x,m)-G^{-}_{0}(m))\,q(x)\,e^{-i2(2m+2)\pi
x}dx=o\left(1\right)$

and by (2.36)

$S_{4}=4\pi^{2}\int_{0}^{1}(G^{+}(x,m)-G^{+}_{0}(m))(G^{-}(x,m)-G^{-}_{0}(m))\,q(x)\,dx=o\left(1\right).$

Thus, these with (2.39) and (2.40) imply that
$a_{2}(\lambda_{2m+j})=o\left(m^{-2}\right)$. The lemma is proved. ∎

### Proof of Theorem 1.2

(i) First let us prove that $c_{0}=0$. By considering the first step of the
procedure in the Lemma 2.1 and using a similar estimate as in (2.12), we may
rewrite the relations (2.6) and (2.9) as follows:

$\left.\begin{array}[]{ll}\displaystyle[\Lambda_{m,j,m}-c_{0}]u_{m,j}=c_{2m+2}v_{m,j}+O\left(\frac{ln|m|}{m}\right),&\\\
\\\
\displaystyle[\Lambda_{m,j,m}-c_{0}]v_{m,j}=c_{-2m-2}\,u_{m,j}+O\left(\frac{ln|m|}{m}\right)&\\\
\end{array}\\!\\!\\!\\!\\!\\!\\!\right\\}$ (2.41)

for $j=1,2$ and sufficiently large $m$. By using the assumption
$l_{2m+2}=o(m^{-2})$, namely, $l_{n}=o(n^{-2})$ for even $n=2m+2$ and Theorem
1.1 which implies $c_{\mp(2m+2)}=o(m^{-2})$, we obtain the relations (see
(2.41)) in the form

$[\Lambda_{m,j,m}-c_{0}]u_{m,j}=O\left(\frac{ln|m|}{m}\right),$ (2.42)
$[\Lambda_{m,j,m}-c_{0}]v_{m,j}=O\left(\frac{ln|m|}{m}\right).$ (2.43)

Again by (2.15) we have, for large $m$, either $|u_{m,j}|>1/2$ or
$|v_{m,j}|>1/2$. In either case, in view of (2.42) and (2.43), there exists a
positive large number $N_{0}$ such that both the eigenvalues $\lambda_{2m+j}$
(see definition of (2.3)) satisfy the following estimate

$\lambda_{2m+j}=(2m+2)^{2}\pi^{2}+c_{0}+O\left(\frac{ln|m|}{m}\right)$ (2.44)

for all $m>N_{0}$ and $j=1,2$. When $m>\max\\{(n_{0}-2)/2,N_{0}\\}$, from the
assumption of Theorem 1.2 (i) the eigenvalue $(2m+2)^{2}\pi^{2}$ corresponds
to the eigenvalue $\lambda_{2m+1}$ or $\lambda_{2m+2}$. In either case we
obtain $c_{0}=0$ by (2.44).

Finally, for sufficiently large $m$, substituting the estimates of

$a_{i}(\lambda_{2m+j}),\,a^{\prime}_{i}(\lambda_{2m+j}),\,b_{i}(\lambda_{2m+j}),\,b^{\prime}_{i}(\lambda_{2m+j}),\,R_{2},\,R^{\prime}_{2}$

for $i=1,2$, respectively, given by Lemma 2.2 with the equalities
$a_{i}(\lambda_{2m+j})=a^{\prime}_{i}(\lambda_{2m+j})$ (see (2.11)),
(2.20)-(2.22) and (2.12) in the relations (2.6) and (2.9) and using $c_{0}=0$,
we find the relations in the following form

$\left.\begin{array}[]{ll}\displaystyle\left[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^{2}}\int_{0}^{1}q^{2}\right]u_{m,j}=c_{2m+2}v_{m,j}+o\left(m^{-2}\right),&\\\
\\\
\displaystyle\left[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^{2}}\int_{0}^{1}q^{2}\right]v_{m,j}=c_{-2m-2}\,u_{m,j}+o\left(m^{-2}\right)&\\\
\end{array}\\!\\!\\!\\!\\!\\!\\!\right\\}$ (2.45)

for $j=1,2$. In the same way, by using the assumption $l_{2m+2}=o(m^{-2})$ and
Theorem 1.1, we write (2.45) in the form

$\left[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^{2}}\int_{0}^{1}q^{2}\right]u_{m,j}=o\left(m^{-2}\right),$
$\left[\Lambda_{m,j,m}+\frac{1}{(2\pi(2m+2))^{2}}\int_{0}^{1}q^{2}\right]v_{m,j}=o\left(m^{-2}\right).$

Thus, arguing as in the proof of (2.44), there exists a positive large number
$N_{1}$ such that the eigenvalues $\lambda_{2m+j}$ satisfy the following
estimate

$\lambda_{2m+j}=(2m+2)^{2}\pi^{2}-\frac{1}{(2\pi(2m+2))^{2}}\int_{0}^{1}q^{2}+o\left(m^{-2}\right)$
(2.46)

for all $m>N_{1}$ and $j=1,2$. Let $m>\max\\{(n_{0}-2)/2,N_{1}\\}$. Using the
same argument as above, by (2.46), we get $\int_{0}^{1}q^{2}=0$ which implies
that $q=0$ a.e.

(ii) The same argument in Section 2 works for the anti-periodic boundary
conditions

$\qquad\qquad y(0)=-y(a),\qquad y^{\prime}(0)=-y^{\prime}(a)$

and one can readily see the corresponding results for the anti-periodic
eigenvalues $\mu_{2m}$, $\mu_{2m+1}$ from (2.1), (2.2) and (2.3) by replacing
$2m+2$ with $2m+1$. Then, arguing as in the proof of Theorem 1.2 (i), we get
the assertion of Theorem 1.2 (ii). $\square$

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# Adaptive Sampling of Latent Phenomena using Heterogeneous Robot Teams
(ASLaP-HR)

Matthew Malencia, Sandeep Manjanna, M. Ani Hsieh, George Pappas, Vijay Kumar
We gratefully acknowledge the support from ARL Grant DCIST CRA
W911NF-17-2-0181, NSF Grant CNS-1521617, ARO Grant W911NF-13-1-0350, ONR
Grants N00014-20-1-2822 and ONR grant N00014-20-S-B001, and Qualcomm Research.
The first author acknowledges support from the National Science Foundation
Graduate Research Fellowship under Grant No. DGE-1845298.The authors are with
GRASP Laboratory, Levine Hall 4th floor, University of Pennsylvania, 3330
Walnut Street, Philadelphia, PA 19104-6228. corresponding authors
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present an online adaptive planning strategy for a team of
robots with heterogeneous sensors to sample from a latent spatial field using
a learned model for decision making. Current robotic sampling methods seek to
gather information about an observable spatial field. However, many
applications, such as environmental monitoring and precision agriculture,
involve phenomena that are not directly observable or are costly to measure,
called latent phenomena. In our approach, we seek to reason about the latent
phenomenon in real-time by effectively sampling the observable spatial fields
using a team of robots with heterogeneous sensors, where each robot has a
distinct sensor to measure a different observable field. The information gain
is estimated using a learned model that maps from the observable spatial
fields to the latent phenomenon. This model captures aleatoric uncertainty in
the relationship to allow for information theoretic measures. Additionally, we
explicitly consider the correlations among the observable spatial fields,
capturing the relationship between sensor types whose observations are not
independent. We show it is possible to learn these correlations, and
investigate the impact of the learned correlation models on the performance of
our sampling approach. Through our qualitative and quantitative results, we
illustrate that empirically learned correlations improve the overall sampling
efficiency of the team. We simulate our approach using a data set of sensor
measurements collected on Lac Hertel, in Quebec, which we make publicly
available.

###### Index Terms:

Informative Path Planning, Heterogeneity, Multi-Robot Systems

##  I Introduction

Robot teams have proven to be powerful tools for environmental monitoring,
drastically improving efficiency over fixed sensor networks or manual sampling
[1]. This effectiveness is driven in large part due to their ability to move
and collect samples from multiple locations, and due to the sampling
strategies that decide where measurements should be taken [2]. However, many
applications involve phenomenon that either cannot be directly measured by
sensors or require slow and costly sensors to measure, called latent phenomena
or latent spatial fields. For example, in agricultural applications,
chlorophyll is measured as a proxy for health or as an indicator of an
infestation [3]. And in marine settings, scientists infer aquatic health by
measuring various chemical concentration indicators [1]. In this paper, we
answer the question: how can robot teams choose sensing locations that best
inform a latent phenomenon? We formulate this question as an Adaptive Sampling
of Latent Phenomena (ASLaP) problem and propose an informed path planning
technique for a heterogeneous team of robots (ASLaP-HR) that seeks to estimate
a quasi-static latent spatial field. We assume that the observable and latent
spatial fields change slowly or remain static at least for the time span of a
robotic survey and that robot team’s communication is not constrained with the
region.

Consider the example scenario presented in Fig. 1: A farmer monitors plant
nutrition by sending leaf samples to a lab for analysis. However, cheap and
fast sensors, such as chlorophyll and soil conductivity sensors, can provide
in-situ proxy measurements that estimate plant nutrition. Our approach to this
scenario uses historical data of co-located chlorophyll, soil conductivity,
and analyzed leaf samples to learn the relationship between the sensor
measurements and nutrient levels. Then, a team of robots (equipped with soil
conductivity and chlorophyll sensors) collaboratively estimates the nutrition
levels of the whole farm (or plot) by using the learned model to inform where
and which sensor measurements will provide the best estimate of nutrition.

((a)) Spatial fields.

((b)) Historical Data

((c)) Multi-robot information gathering.

Figure 1: An example scenario: (a) Farms can be described by various spatial
fields. A spatial map of nutrient levels is most informative to a farmer, but
it requires costly and time-consuming sampling and analysis of leaves.
Chlorophyll and soil conductivity spatial fields can be measured quickly and
cheaply by in-situ sensors. (b) Historical data is used to train a model to
predict nutrient levels using chlorophyll and soil conductivity measurements.
(c) A team of robots can measure chlorophyll and soil conductivity relatively
efficiently and use the learned model from (b) to make real-time decisions
about where and what to sample next to get best estimates of nutrition.

Current work on sampling approaches for robot teams either optimize objectives
on observable spatial fields [4, 5] or assume an exact proxy for a latent
spatial field [6]. However, this does not necessarily align with optimizing
relevant information of the latent phenomenon of interest [7]. After
collecting data of their observable spatial fields, these teams rely on human
experts for higher level decision making about the phenomenon of interest.

Heterogeneity in robot teams has been considered in robotic sampling [8], with
a focus on diverse robot types, e.g., aerial vs ground robots [9]. While some
papers consider sensor heterogeneity, this area is limited to sensors that
align with prescribed roles [6] and sensors that are homogeneous in type but
heterogeneous in cost, range, and accuracy [10]. In this work, we investigate
heterogeneity in sensor types, where each type measures a different spatial
field. These diverse observations can combat uncertainty of the latent
phenomenon because real world sensor spaces are not independent and map
nonuniformly to the latent spatial field.

In order to leverage the strength of heterogeneous sensors, we use data-driven
methods to model the relationship between the observable spatial fields and
the latent phenomenon. Most machine learning methods are limited by the
resolution of the input data, e.g., using satellite data to make predictions
about entire plots of land rather than individual locations or crops [11], or
they postprocess spatial data to gain meaningful insights. Our work seeks to
enable predictions that reason directly about the latent spatial field in real
time while also providing the precision of studying localized phenomena.

###  I-A Related Work.

There are myriad problem spaces that involve path planning for mobile robots.
Bayesian optimization approaches seek to find the maximum of a spatial field
[12, 13, 14]; exploration and active mapping optimize the observation of an
unknown area [15, 16, 17]; and active information gathering or target tracking
focus on estimating the state of dynamic hidden states [18, 19, 20]. The
objective functions studied among these applications can be categorized as
information-theoretic driven [21, 22], such as mutual information [23, 24],
and sample driven, as is the case of hotspot sampling [25, 26]. This work
seeks to maximize the expected information gained by a team of heterogeneous
robots.

Heterogeneity of multi-robot teams can vary drastically. Manjanna et al. use
two autonomous surface vehicles, one with a water quality sensor and the other
with a water sampler, where the sensor vehicle explores to inform the
sampler’s decisions [6]. Similarly, Manderson et al. use an aerial vehicle to
inform the decisions of a ground vehicle [9]. Chand and Carnegie investigated
heterogeneous processing and sensing capabilities to assign tasks to different
robot types, where robots with low-value sensing do computation rather than
exploration tasks [27]. Shi et al. use a Voronoi approach where the space is
partitioned according to heterogeneous capabilities, like mobility constraints
in ground versus aerial vehicles [10]. Similarly, Cai et al. consider
heterogeneous capabilities such as field of view and traversability of
different environment types [28]. Others still consider resource constraints
such as energy budgets and fuel limitations [29, 30, 31]. Whereas Chand and
Carnegie model different levels of sensing ability (e.g., low vs high), we
model qualitatively different sensor types, which all take high quality sensor
measurements in their respective observable space. Manjanna et al. and
Manderson et al. investigate different robot types (sensor vs sampler, and
aerial vs ground sensing, respectively), but their investigations focus on two
prescribed roles. In this work, we do not consider heterogeneity in robot
resources (e.g., computation power, or mobility type) but rather heterogeneity
in sensor types; we model an arbitrary number of sensor types where tasks and
roles are not pre-designed.

Recent work has only just begun to develop models and algorithms to make real-
time decisions with respect to a latent phenomenon of interest. Arora et al.
study multi-modal active perception, where measurements must be combined with
domain knowledge to deduce information about variables or phenomena that
cannot be directly measured [7]. Their paper investigates a single robot with
multiple sensors that make discrete observations and maps the observations to
the variables of interest through an expert-designed Bayesian network. In this
work, we consider a multi-robot team where sensor measurements are continuous
and we learn a continuous representation mapping from the observations to the
latent phenomenon of interest.

###  I-B Contributions

We formalize adaptive informative sampling for multiple continuous sensors to
maximize information theoretic objective of a latent phenomenon. We use a
robot team with heterogeneous sensor types to measure observable spatial
fields. A learned model from historical data enables the robot team to infer
about the latent spatial field in real time and use these inferences to choose
sample locations that maximize the information gain of the latent phenomenon.
Additionally, we learn correlations among the multiple observable spatial
fields; these correlations are learned offline and are used to make real-time
sampling location decisions. We investigate the impact of these learned
correlations on the adaptive sampling of the latent field. Our main
contributions are:

  * •

We formulate the ASLaP problem as the maximization of the expected information
of a latent spatial field.

  * •

We develop ASLaP-HR, an adaptive sampling approach for a team of robots with
heterogeneous sensors using a learned mapping from observations to the latent
phenomenon.

  * •

We show that the learned correlations among the observable spatial fields
improve the sampling efficiency of our adaptive sampling approach.

In Section II, we present a novel ASLaP problem formulation that is not
restricted to myopic or non-myopic approaches, nor does it restrict motion
uncertainties or arbitrary sets of actions,111The problem formulation
considers a discrete action set. Using continuous dynamics would require
slight reformulation. though the methods presented in Section III are myopic
and do not consider motion uncertainty. We validate our approach by running
experiments on real sensor data collected on a lake. The sensor data, which we
make publicly available 222This data is available at:
https://github.com/SandeepManjanna/water_quality_dataset, includes co-located
measurements of ten water quality parameters. Our experiments use three of the
ten sensor fields, two as the sensor spaces and one as the latent variable of
interest. The experimental results show that the learned correlations among
the observable spatial fields improve overall sampling. Future work will
investigate the online learning of these correlations.

##  II Problem Definition

Consider a region of locations $x\in\real[2]$ with a spatial field
$\phi:~{}x\mapsto\real$, such as temperature.333We study 2-dimensional spatial
fields in this work, though 3-dimensional space could be considered. Adaptive
informative sampling seeks to estimate this spatial field $\phi$. The estimate
is a belief model, $\hat{\phi}:x\mapsto\mathbb{D}$, where $\mathbb{D}$ is the
set of distributions in $\real[]$; for each $x$, $\hat{\phi}(x)$ is a one-
dimensional distribution that estimates $\phi\left(x\right)$. To model
$\hat{\phi}$, robots make observations at locations $x$ that maximizes
information gain, $\mathcal{I}$. Often, $\mathcal{I}$ is calculated using
Shannon entropy, $H$. The information gain with respect to $\phi$ of a
measurement at $x$ is the change in entropy of the field estimate due to the
new measurement: $\mathcal{I}_{\hat{\phi}}(x)=H(\hat{\phi})-H(\hat{\phi}|x)$.
Additionally, adaptive sampling problems often include resource constraints,
such as energy budgets [32] or sampling budgets [6]. We limit the number of
measurements using a time horizon constraint, $T$.444The constraint $T$ can be
interpreted as an energy budget, assuming each action takes equal energy, or
as a sensing budget of at most $T$ measurements per robot (we assume one
measurement is taken each time step). Other works define a cost per
action/measurement and constrain the sum of costs. Therefore, a robot seeks a
series of measurements $x_{1},\dots,x_{T}$ that maximize
$\sum_{t=1}^{T}\mathcal{I}_{\hat{\phi}}(x_{t})$. This extends to a team of $N$
robots where the vector $\vec{x}_{t}=(x^{(1)}_{t},\dots,x^{(N)}_{T})$. This
type of problem formulation is found in the previous literature on informative
path planning [7, 33, 34, 25]. Conversely, we consider a latent spatial field
where $\phi$ is not directly observeable, yielding the Adaptive Sampling of
Latent Phenomena (ASLaP) problem formulation:

######  Problem 1.

_ASLaP:_ Given a team of $N$ agents, find observation locations
$\mathbf{x}_{t}$ at all time steps in time horizon $T$ that maximize the sum
of information gained relative to the latent spatial field $\phi$. Formally:

$\underset{\vec{x}_{1},\dots,\vec{x}_{T}}{\text{arg\,max}}\sum_{t=1}^{T}\mathcal{I}_{\hat{\phi}}\left[\vec{x}_{t}\right]$
(1)

The novel problem formulation in Problem 1 is challenging because $\phi$ is a
latent phenomenon. This breaks the fundamental assumption that the robots can
directly measure the field $\phi$, thus requiring new methods that enable
inference over this latent spatial field. In addition to this novel problem
formulation, this work is the first to take advantage of learned spatial field
correlation models to improve adaptive sampling.

In the following section, we present an approach using a team of robots with
heterogeneous sensors, where the different sensor types measure observable
spatial fields to collectively inform the estimate of the latent phenomenon by
using a mapping that is learned from historical data. Additionally, we
investigate how learned correlation models among these observable spatial
fields can be leveraged to improve the efficiency of this team’s sampling.

##  III Approach

We consider a heterogeneous team of $N$ robots with $S$ sensor types among the
team, where $S=N$, which means that each robot has a distinct sensor type. It
is possible to extend our approach to an arbitrary number of robots sensor
combinations where $S\neq N$.555For $S\neq N$, there are $N^{(s)}$ robots per
sensor type such that $\sum_{s=1}^{S}N^{(s)}=N$. Each sensor type measures a
spatial field, and, as with real world observeable spatial fields, sensor
measurements from different sensor types are not independent of one another,
as seen in Fig. 2. We use the index $s=1,\dots,S$ for both robots and sensors.
Each robot $r_{s}$ can measure a spatial field ${z}^{(s)}:x\mapsto\real[]$.
Let $x^{(s)}_{t}$ be the location of robot $r_{s}$ at time $t$ and let the
respective observation be ${z}^{(s)}(x^{(s)}_{t})$. For ease of notation, the
observation made by robot $r_{s}$ at time $t$ will be denoted by
${z}^{(s)}_{t}(x)$. Let $Z:x\mapsto\real[S]$ be the $S$-dimensional observable
spatial field, where
$Z(\vec{x})=\left[{z}^{(1)}(x),\dots,{z}^{(S)}(x)\right]$. The spatial field
$Z$ is unknown but observable, therefore, online measurements are used to
maintain a belief estimate $\hat{Z}:x\mapsto\mathbb{D}^{S}$.

Figure 2: The mutual information between different sensor types. Independent
sensor types are independent sources of information. For perfectly correlated
sensor types, co-located measurements from a second sensor type provide zero
additional information. In reality, different sensor types (and the respective
measurable spatial fields), have complex dependencies with mutual information
that is complex, unknown function of the measurements. The middle scatter plot
above shows the relationship of real sensor data that is used in our
experiments in Section IV.

With this team of heterogeneous multisensor robots, we seek to build an
estimate of the latent spatial field. To do this, a mapping $g$ from
observations in $Z$ to the latent spatial field $\phi$ is learned from
historical data. Robots make measurements of $Z$, build an estimate of the
observable spatial field $\hat{Z}$, and then use the learned mapping
$g:\hat{Z}\mapsto\hat{\phi}$ to maintain the estimate of the latent spatial
field $\hat{\phi}$. Learning $g$ is challenging because the mapping from
$\hat{Z}\in\mathbb{D}^{S}$ to $\hat{\phi}\in\mathbb{D}$ has aleatoric
uncertainty, meaning even an exact measurement of an observable spatial field
yields an uncertain latent spatial field value. Multiple machine learning
methods can capture this aleatoric uncertainty [35], such as the Gaussian
Process approach that we use in Section IV. Because the goal of our approach
is to create a good estimate of the latent phenomenon, the observation
locations $x$ are chosen to maximize the information gain with respect to
$\phi$. Using the learned mapping $g$, the information gain is now calculated
by $\mathcal{I}_{g(\hat{Z})}(x)=H(g(\hat{Z}))-H(g(\hat{Z}|x))$, which yields
the Adaptive Sampling of Latent Phenomena using Heterogeneous Robots (ASLaP-
HR) approach:

######  ASLaP-HR.

Given a team of $S$ robots, each equipped with a distinct sensor $s$, each
robot $s$ takes measurements of the field ${z}^{(s)}$ to collectively form an
estimate $\hat{Z}$ of the entire observable spatial field. This in turn allows
estimates of the latent spatial field by using $g$, a mapping from the
observable spatial fields to the latent spatial field that must be learned
from historic data. At each time step $t$, choose the observation location for
each robot, $\vec{x}_{t}=(x^{(1)}_{t},\dots,x^{(N)}_{T})$, that maximize the
information gain $\mathcal{I}_{g(\hat{Z})}$ with respect to this latent
phenomenon. Formally:

$\underset{\vec{x}_{1},\dots,\vec{x}_{T}}{\text{arg\,max}}\sum_{t=1}^{T}\mathcal{I}_{g(\hat{Z})}\left[\vec{x}_{t}\right]$
(2)

While prior works reason about observable spatial fields, ASLaP-HR reasons
about a latent phenomenon through $g$. The following section presents an
ASLaP-HR planning algorithm, which leverages a learned correlation among the
observable spatial fields for improved information gathering. Section IV
presents a specific implementation of this algorithm and shows the improvement
gained by leveraging the learned correlation.

###  III-A Planning Algorithm Details

We present Algorithm 1 for ASLaP-HR, where a team of $S$ robots take
measurements of $Z$, build an estimate of the observable spatial field
$\hat{Z}$, and then use the learned mapping $g$ to build an estimate of the
latent spatial field $\hat{\phi}$. Algorithm 1 takes the following inputs:

  * •

$\vec{x}_{0}\in\real[2\text{x}S]$ initial positions of all robots

  * •

$T\in\mathbb{Z}^{+}$ time horizon

  * •

$g:\hat{Z}\mapsto\hat{\phi}$ mapping from observations to latent field

  * •

$h_{s}:{z}^{(s)}\mapsto Z$ correlation model for sensor type $s$

Algorithm 1 begins with each robot $r_{s}$ observing the spatial field
${z}^{(s)}$ at their initial position $x^{(s)}_{t}$, yielding measurements
$Z(\vec{x}_{0})$. These measurements are used to update the belief $\hat{Z}$
(line 3) which is then used by $g$ in line 4 to update the belief
$\hat{\phi}$. In Section IV, we model $\hat{Z}$ using Gaussian Processes,
which produce inputs to the learned model $g$, which in turn outputs
$\hat{\phi}$. The team then communicates the sensor measurements.666We assume
robots can communicate with each other within the region, and that time needed
for communication and belief updating is significantly shorter than the time-
step duration. For each time step $t$, and robot $r_{s}$, the next waypoint
$x_{t}^{(s)}$ is selected to maximize the expected information gain.
Calculating the information gain using Shannon Entropy (as shown in Section
II) is computationally intractable, therefore, the variance of $\hat{\phi}$ is
used as an estimate of the expected information gain [36].

Next, $sense(s,x_{t}^{(s)})$ on line 8 returns a new sensor measurement
${z}^{(s)}_{t}$. This new sensor measurement is used to refine the model
$h_{s}$ using any measurements from other sensors that are located together at
$x^{(s)}_{t}$. We assume that GPS localizes the robots; the uncertainty in
this localization does not hinder the estimate of the beliefs. The new sensor
measurements are communicated, and the correlation models $h_{s}$ are used in
line 11 to update the beliefs $\hat{Z}$ at the points $x_{t}^{(s)}$. Line 12
updates the belief $\hat{Z}$, and line 13 updates the belief $\hat{\phi}$.
Algorithm 1 then repeats this sequence to iterate over all $T$ time steps, at
the end returning the most up-to-date belief $\hat{\phi}$ of the latent
phenomenon. The next section elaborates on the line 11 update of $\hat{Z}$.

Algorithm 1 Pseudocode for ASLaP-HR Planner

0: $\vec{x}_{0},T,g,h_{1},\dots,h_{S}$

0: Estimated latent spatial field $\hat{\phi}$

1: Initialize $\hat{\phi},\hat{Z}$

2: Measure $Z(\vec{x}_{0})$

3: $\hat{Z}\xleftarrow{}update\left(Z(\vec{x}_{0})\right)$

4: $\hat{\phi}\xleftarrow{}g\left(\hat{Z}\right)$

5: for $t=1,\dots,T$ do

6: for $s=1,\dots,S$ do

7:
$x^{(s)}_{t}\xleftarrow{}\underset{x}{arg\,max}\quad\mathcal{I}_{g(\hat{Z})}\left[\vec{x}_{t}\right]$

8: ${z}^{(s)}_{t}=sense(s,x^{(s)}_{t})$

9: $h_{s}\xleftarrow{}refine(x^{(s)}_{t},{z}^{(s)}_{t})$

10: end for

11:
$\hat{Z}(x^{(s)}_{t})=h_{s}\left({z}^{(s)}_{t}\right)\;\forall\;s=1,\dots,S$

12: $\hat{Z}\xleftarrow{}update\left(Z(\vec{x}_{t})\right)$

13: $\hat{\phi}\xleftarrow{}g\left(\hat{Z}\right)$

14: end for

15: return

###  III-B Learned Correlations among the Observable Spatial Fields

Through this point in Section III, we have presented the basic structure of
Algorithm 1. The last core piece, presented here and investigated in Section
IV, is the learned correlations among the observable spatial fields $h_{s}$.
Algorithm 1 does not require line 11. However, we hypothesize that
heterogeneous robots can leverage the relationships between sensor-types to
achieve more efficient information gathering.

When $g$ updates the belief of the latent spatial field at a point $x$, it
calculates $\hat{\phi}(x)$ as a function of the entire observable field
$\hat{Z}(x)$. Therefore, the precision of the value and variance of
$\hat{\phi}(x)$ depends on the precision of the entire belief $\hat{Z}$.
However, a robot $r_{s}$ measures only part of $\hat{Z}$, specifically
${z}^{(s)}(x)$.

The precision of the variance of $\hat{\phi}(x)$ is especially important
because it informs the selection sensing locations. If $\hat{\phi}(x)$
underestimates variance, line 7 calculates a lower expected information gain
than is actually present, causing robots to avoid informative measurements. If
$\hat{\phi}(x)$ overestimates variance, robots may take redundant measurements
because they expect more information than is actually available.

In Algorithm 1, we see that the observable spatial field correlation models
$h_{s}$ are used to improve the estimate of $\hat{Z}$, which in turn improves
the estimate of $\hat{\phi}$. Given a measurement ${z}^{(s)}(x)$, the
correlation model $h_{s}$ estimates the value (with variance) of the entire
observable spatial field $\hat{Z}(x)$. These models capture the aleatoric
uncertainty in the correlations among the observable spatial fields. For
example, if there is low aleatoric uncertainty between two sensor types, one
sensor acts almost as a noisy sensor of another spatial field. Conversely,
correlations with high aleatoric uncertainty only marginally diminish the
variance of other spatial fields. The line 11 of Algorithm 1 shows how the
correlation models are used after a robot moves to a new $x$ and takes a new
sample ${z}^{(s)}$. The new sample is the input to model $h_{s}$ that provides
updated estimates of the value and variance of the entire observable space at
that location, $\hat{Z}(x)$.

##  IV Results

Figure 3: Our simulated environment includes two observable spatial fields:
$z^{(0)}$ temperature (left), and $z^{(1)}$ pH (middle). These fields are
unknown but observable:f $z^{(0)}$ can be measured by robot $r_{0}$ and
$z^{(1)}$ can be measured by robot $r_{1}$. The simulated environment also
includes a latent spatial field $\phi$ of ORP values (right) which cannot be
measured by either robot.

We present an instantiation of Algorithm 1 and investigate the impact of the
correlations among the observable spatial fields described in Section III-B.
This section presents simulated experiments where a team of robots estimate a
latent phenomenon. We use real sensor data collected from a lake to create a
simulated environment of two observable spatial fields and a latent phenomenon
of interest. In the following subsections, we present a marine data set, we
overview a simple myopic planner that instantiates Algorithm 1, we describe
the modeling of the correlations between the observable spatial fields, and we
present results from simulated experiments. We show qualitatively that the
learned correlations between the observable spatial fields $h_{s}$ improve the
performance by causing the robot team to spread their samples out. Our
quantitative evaluations show that these correlations improve the accuracy of
the latent field estimate.

###  IV-A Experimental Setup - Data and Simulated Environment

We simulate a team of two robots in a marine setting. One of the robots is
equipped with a temperature sensor and the other one with a pH sensor.
Together, they are seeking information about oxidation reduction potential
(ORP), the latent phenomenon of interest in these simulated experiments.
Historical data temperature, pH, and ORP are needed to learn the correlation
model $g$.

Lac Hertel Data. The historical data used in our simulated experiments was
collected on Lac Hertel in Mont Saint-Hilaire, Quebec. The full data set,
detailed in the Appendix, is made publicly available. For this paper, we use
temperature and pH data as our observable spatial fields, and the oxidation
reduction potential (ORP) data as the latent phenomenon. ORP measures the
ability of a lake or river to cleanse itself or break down waste products; ORP
is aligned with the "knowledge" that might be sought in the environmental
monitoring of a lake because it indicates whether drinking water is sanitary
or if the water is suitability for anaerobic microbial processes.
Additionally, we found that ORP, pH, and temperature had well-formed relations
that could be modeled. Let the historical data
$\mathcal{D}=(\mathcal{X}_{\mathcal{D}},d_{0},d_{1},d_{2})$ be a tuple of the
sets of sample locations, $\mathcal{X}_{\mathcal{D}}$, the sets of
measurements from two observable spatial fields (temperature:$d_{0}$ and
pH:$d_{1}$) and the set of measurements from the latent spatial field
(ORP:$d_{2}$). This historical data is used to learn the mapping from
observations to the latent spatial field $g$ and the correlations $h_{0}$ and
$h_{1}$.

Simulated Environment. To test Algorithm 1, we create a simulated environment
of observable and latent spatial fields. Fig. 3 shows the three spatial fields
of our simulated test environment. All three fields are unknown. However,
$z^{(}0)$ can be measured by robot $r_{0}$ and $z^{(}1)$ can be measured by
robot $r_{1}$. The latent field $\phi$ cannot be measured by the robot team;
this spatial field is used only to evaluate the performance of our algorithm.
We normalize all spatial fields to a scale from zero to one.777Normalizing the
sensor data aids in learning correlation models. To simulate our robots’
motion in this simulated environment, we discretize the region into a grid,
defining the set $\mathcal{X}$ of locations and corresponding ground truth
sensor measurements $\mathcal{GT}=(gt_{0},gt_{1},gt_{2})$.

###  IV-B Myopic Planner

For our simulated experiments, we assume a robot can move to any adjacent cell
during a time step. For a given location $x_{t-1}^{(s)}\in\mathcal{X}$, the
possible next locations $x_{t}^{(s)}$ in line 7 are the eight grid cells
surrounding the current location (excluding cells outside of the boundary,
i.e., $\notin\mathcal{X}$). Lines 3 and 12 create belief models of the
observable spatial field $\hat{Z}$. At each time step $t$, the sensor
measurements and respective sample locations are used to fit the belief model
$\hat{Z}$ of the observable fields using Gaussian Processes. To create these
beliefs, we assume known spatial correlations (covariance kernels), which is a
common assumption because underlying physics informs these correlation (e.g.,
how temperature changes spatially). The belief model of the latent phenomenon,
$\hat{\phi}$, is created/updated using the pre-learned mapping $g$ to predict
the mean and variance of the latent spatial field. The input to $g$ is
$\hat{Z}$ and the output is $\hat{\phi}$. Importantly, we do not assume known
spatial correlations of the latent variable; Mean and variance predictions are
strictly a function of the observable spatial field belief model and the
mapping $g$.

The mapping $g$ can be learned by various machine learning methods which
capture the aleatoric uncertainty between observations and the latent spatial
field [35]. Because we are only considering $S=2$ observable spatial fields,
deep learning methods are not required. We use Gaussian Process Regression
with inputs from $d_{0}$ and $d_{1}$ and targets of $d_{2}$ to train our
mapping $g$. Then, to capture the uncertainty in the latent variable, we use
samples at each location of our belief $\hat{Z}$ to capture the variance in
the belief; we predict the value and variance of the latent spatial field for
all samples; then, we calculate the mean and variance of the latent spatial
field predictions at each point to create our latent belief $\hat{\phi}$.

To calculate the expected information gain in line 7, we use the variance of
the latent phenomenon estimate, $var(\hat{\phi})$. This is an approximation of
Shannon entropy that is often used to reduce computational cost. The
calculated Shannon entropy exactly, $H(\hat{\phi})-H(\hat{\phi}|x)$, is
computationally expensive because it requires that $\hat{\phi}$ be estimated
for each action under consideration. When using a GP belief model, this
requires a GP update for every action, for every robot, at each time step. The
last piece of Algorithm 1 is the learned correlation model, $h_{s}$, which we
detail in the next section.

((a)) Correlation models

((b)) Mapping $g$ from $Z$ to $\phi$

Figure 4: (a) The learned correlation mappings $h_{0}$ from the pH spatial
field to the temperature spatial field and $h_{1}$ from temperature to pH. The
solid line shows the mean prediction and the bars show the variance
prediction. (b) The learned mapping $g$ from observations to the latent
spatial field.

###  IV-C Modeling Correlations among Observable Spatial Fields

The core hypothesis of this work is that modeling correlations among
observable spatial fields can improve information gathering. By capturing the
aleatoric uncertainty in the relationship, robots can sense their own
observable field and use the learned model to reduce the uncertainty of their
teammates’ observable spatial fields.

The goal here is to learn both $h_{0}$, the mapping from a sensor measurement
in ${z}^{(0)}$ to a predicted value and variance in ${z}^{(1)}$, as well as
$h_{1}$, the mapping from ${z}^{(1)}$ to ${z}^{(0)}$. Note that $h_{0}$ is not
equivalent to the inverse of $h_{1}$ because both $h$ map to a variance, not
just a predicted value. Learning such models requires supervised learning with
input values and both target values and variance. However, the data discussed
in Section IV-A does not include the target variance. For example, learning
the mapping from temperature to pH requires the variance of pH for a given
temperature data point, but we only have (temperature, pH) value pairs,
therefore, we transform the data. To learn $h_{0}$, the $d_{0}$ data binned
and the the mean and standard deviation of the respective $d_{1}$ data are
calculated. GP regression takes the $d_{0}$ bins as inputs and the $d_{1}$
means and standard deviations as targets. For learning $h_{1}$, the process is
the same, flipping $d_{0}$ and $d_{1}$.

Fig. 4 shows the learned mappings $h_{0}$ and $h_{1}$ plotted over the
respective historical data $d_{0}$ and $d_{1}$. These models are used in line
11. When the robot equipped with the temperature sensor takes a measurement at
the point $x^{(0)}_{t}$, the sensor observation ${z}^{(0)}_{t}$ is used as the
input to $h_{0}$ to update ${z}^{(1)}_{t}(x)$, the belief of pH at point
$x^{(0)}_{t}$.

###  IV-D Evaluation

We evaluate the performance of Algorithm 1 with and without the correlation
model, i.e., we omit line 11 in the latter. As the goal of Problem ASLaP-HR is
to maximize the information with respect to the latent spatial field, the
evaluation metric we use is the mean squared error between the ground truth of
the latent phenomenon, $gt_{2}$, and the estimate $\hat{\phi}$, calculated
using all points $x\in\mathcal{X}$ in the discretized space:

$MSE=\frac{1}{|\mathcal{X}|}\sum_{x\in\mathcal{X}}\left(gt_{2}(x)-\hat{\phi}(x)\right)^{2}$
(3)

First, we run Algorithm 1 with the robots starting in the center of the
region. The short time horizon mission in Fig. 5 shows a qualitative
comparison of the algorithm’s behavior with and without use of the correlation
model. We see that when the correlation model is used, the robot team
naturally spreads out to explore different parts of the region. Because of the
correlations among the observable spatial fields, the robots are able to
better estimate the expected information gain, finding that there is less
information in the regions where other sensors have sampled because of the
mutual information between the observable spaces.

Figure 5: Left: The ground truth of latent spatial field $\phi$. Middle and
right: The belief model $\hat{\phi}$ of the latent phenomenon with robot
measurements shown in red (square) and blue (circle) for the temperature and
humidity robot, respectively; their sensing locations are indicated by color,
with darker shades indicating later time steps. Middle: The correlation models
$h_{s}$ are not used by the robot team. Right: When using the correlation
models $h_{s}$, the team naturally spreads out their samples to maximize
information gain.

Next, we generate 200 trials by sampling sets of random starting locations
from $\mathcal{X}$. We evaluate all time horizons up to $T=150$, reporting the
MSE in Equation 3 for each time horizon. Fig. 6 shows the median MSE among the
200 trials as well as the 25th to 75th percentile range for Algorithm 1. The
comparison of this algorithm with (dotted line) and without (solid line) the
learned correlations illustrates that the learned correlations among the
observable spatial fields improve the efficiency of the adaptive sampling
algorithm.

In the earlier phase of the sampling survey, the improvement is small because
the mutual information between sensor types can not yet be exploited and the
randomness of starting locations has a large impact on performance. On the
other hand, during the later phase of the sampling survey, the robot samples
begin to cover the entirety of the space. Therefore, both methods converge
close to zero MSE and zero variance. In practice, it is prohibitively
expensive to sample the entire space (and robot sampling capacity or battery
life may preclude this). Therefore, adaptive sampling methods often exist in
the middle ground of time horizons, where enough samples are taken to provide
good information about the region, but not so many samples that it becomes
overly costly. In this realm, our method performs well, using the learned
correlations among the observable spatial fields to maximize the amount of
information gained about the latent phenomenon in a limited number of samples.

Figure 6: The mean squared error (MSE) of the belief model $\hat{\phi}$ shows
that using learned correlations among the observable spatial fields improves
the performance of the adaptive sampling approach. The solid and dashed lines
show the median MSE at each time step for our instantiation of Algorithm
without and with correlation model $h_{s}$, respectively. The shaded regions
show the 25th to 75th percentile range.

##  V Conclusion

Real world robotic sampling methods often collect data to be analyzed offline
by humans or machine learning algorithms to gain information about a latent
phenomenon. However, the strength of robotic sampling is the ability for real-
time adaptation. To bridge this gap, we present the ASLaP problem, where robot
sampling decisions are made with respect to the information gain of a latent
spatial field. We propose an informed path planning approach, ASLaP-HR to this
problem using a team of heterogeneous robots.

Each robot on the team has a distinct sensor type that can measure an
observable spatial field. Collectively, the measurements made by the robot
team inform an estimate of the latent spatial field by using a machine
learning model trained on historical data. Importantly, this learned model is
used online by the robots to calculate the expected information gain of their
actions with respect to the latent phenomenon. Our method is among the firsts
to explore adaptive sampling of a latent phenomenon with a team of robots
using a learned model for online decision making. Additionally, our adaptive
sampling strategy uses learned correlation models among heterogeneous robots
to improve the performance of the team.

We recognize that different sensor types in the real world are not independent
but rather have mutual information that is a complex unknown function of their
measurements. Therefore, we hypothesize that these heterogeneous robots can
improve their information gathering by modeling this mutual information. We
use historical sensor data to learn correlations among the observable spatial
fields, and we use these learned models in our informed path planning
approach. To test the impact of these correlation models, we create a
simulated environment of two observable spatial fields (temp. and pH) and a
latent spatial field (ORP). We run our algorithm in this simulated environment
and calculate the mean squared error between the latent spatial field estimate
returned by our algorithm and the ground truth latent spatial field. We
aggregate our results across hundreds of randomized starting locations,
showing that the learned correlations improve the efficiency of our ASLaP-HR
approach.

##  VI Future Directions

As seen in Section I-A, there are myriad approaches to adaptive sampling and
information gain with robot teams. The many dimensions of these approaches can
be incorporated into our latent sampling framework; this includes complex
motion primitives, long planning horizons, time-varying phenomena, objectives
such as seeking the maximum of the latent spatial field, and uncertainty in
sensors or motion, to name a few. While these open problems are shared among
the adaptive sampling community, latent adaptive sampling brings new open
problems. For example, new learning methods are needed for the mapping $g$ to
better capture aleatoric uncertainty in the relationship between observable
and latent spatial fields; current supervised learning methods focus on
aleatoric uncertainty in classification, with less work on regression. An
important next step that this paper enables is further investigation of the
correlation models. Specifically, these systems would benefit from online
learning or continual learning of these models. Such approaches would require
the robots to navigate the exploration-exploitation trade-off with respect to
the correlation models. In this work, we assume full communication between the
robots. One near future extension of this work will be to consider varying
communication ranges and intermittent communication [2].

We make public the data set used in our second set of experiments. Data
consist of water quality measurements collected at Lac Hertel in Mont Saint-
Hilaire, Quebec. This data is collected using a multiparameter Sonde, Exo1
from YSI. The sensor was mounted on an autonomous surface vehicle to collect
these data measurements. The data includes geotagged and timestamped
measurements of 10 water quality parameters measured using four sensors:
temperature, conductance, salinity, pH, oxidation reduction potential (ORP),
depth, two turbidity measures and two optical dissolved oxygen measures. We
are making this data public through the following Github repository:
https://github.com/SandeepManjanna/water_quality_dataset

((a)) ASV collecting data at Lac Hertel

((b)) Temperature measurements

Figure 7:

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