Singular Instantons and Painlevé VI

Singular Instantons and Painlevé VI

Richard MUÑIZ MANASLISKI

R. Muñiz Manasliski

Centro de Matemática, Facultad de Ciencias,
Iguá 4225 esq. Mataojo C.P. 11400, Montevideo, Uruguay<EMAIL_ADDRESS>
Received February 26, 2016, in final form June 09, 2016; Published online June
15, 2016

We consider a two parameter family of instantons, which is studied in [Sadun
L., Comm. Math. Phys. 163 (1994), 257–291], invariant under the irreducible
action of $\operatorname{SU}_{2}$ on $S^{4}$, but which are not globally
defined. We will see that these instantons produce solutions to a one
parameter family of Painlevé VI equations ($\text{P}_{\text{VI}}$) and we will
give an explicit expression of the map between instantons and solutions to
$\text{P}_{\text{VI}}$. The solutions are algebraic only for that values of
the parameters which correspond to the instantons that can be extended to all
of $S^{4}$. This work is a generalization of [Muñiz Manasliski R., Contemp.
Math., Vol. 434, Amer. Math. Soc., Providence, RI, 2007, 215–222] and [Muñiz
Manasliski R., J. Geom. Phys. 59 (2009), 1036–1047, arXiv:1602.07221], where
instantons without singularities are studied.

twistor theory; Yang–Mills instantons; isomonodromic deformations

34M55; 53C07; 53C28

##  1 Introduction

Yang–Mills instantons are anti-self-dual (or self-dual) connections over four-
dimensional, orientable, Riemannian manifolds. They are absolute minima of the
Yang–Mills functional

$\displaystyle\operatorname{YM}(M)=-\int_{M}\operatorname{Tr}(F^{\nabla}\wedge\ast
F^{\nabla})$

restricted to a fixed Chern number, where $\ast$ is the Hodge operator.
Instantons appear in many instances in mathematics and physics and since the
work of Simon Donaldson we know that they are a fundamental tool in the study
of the topology of four-dimensional manifolds. In [13] Kronheimer and Mrowka
introduced a moduli space of instantons on a four manifold having certain type
of singularity along an embedded surface. They are called instantons with
holonomic singularity because they have non trivial asymptotic holonomy when
we consider small circles around the surface. The goal in [13] was to study
topological obstructions to the embedding of a surface into a four-dimensional
manifold. This kind of instantons was first introduced by physicists [6] and
they are also known as fractionally charged instantons since their second
Chern number is not necessarily an integer. We consider instantons with
holonomic singularities on $S^{4}$ which are invariant under an action of
$\operatorname{SU}_{2}$ and explore their relation with certain solutions to
the famous Painlevé VI equation $\text{P}_{\text{VI}}$. The relation between
instantons and $\text{P}_{\text{VI}}$ has been extensively studied, see for
example [16, 17, 24].

Painlevé VI equation is an ODE on the complex domain, depending on four
complex parameters, and it is the most important second order equation having
what is called the Painlevé property (that is, absence of movable critical
singularities). Critical singularities of $\text{P}_{\text{VI}}$ can only be
located at 0, 1 or $\infty$. This equation appears when isomonodromic
deformations of certain connections are considered [8, 12]. Recently
$\text{P}_{\text{VI}}$ has been interpreted as the simplest non abelian
Gauss–Manin connection [2]. Typically solutions are “new” transcendental
functions and to find them is a highly non trivial activity (general
transcendental solutions have been obtained in [9]). Nevertheless, for certain
values of the parameters one can find “classical” [18, 23] or even algebraic
solutions. One main problem was to find a list of the algebraic solutions to
$\text{P}_{\text{VI}}$ analogous to the Schwartz’s list for the hypergeometric
equation [1, 7], this is archived in [14].

In [20] we studied symmetric instantons defined on all of $S^{4}$. They form a
countable family and the Painlevé equations related to them are all
equivalent. By equivalent we mean that they are in the same orbit of the
Okamoto affine Weyl group of type $F_{4}$ [2, 21]. Allowing instantons to have
singularities we have a continuous family of non equivalent
$\text{P}_{\text{VI}}$ equations. More precisely, for each real number
$\theta$ we have the equation

$\displaystyle\frac{d^{2}y}{dx^{2}}=\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-x}\right)\left(\frac{dy}{dx}\right)^{2}-\left(\frac{1}{x}+\frac{1}{x-1}+\frac{1}{y-x}\right)\left(\frac{dy}{dx}\right)$
$\displaystyle\hphantom{\frac{d^{2}y}{dx^{2}}=}{}+\left(\frac{1}{8}(\theta\pm
2)^{2}+\frac{1}{8}\theta^{2}\frac{x}{y^{2}}+\frac{1}{8}\theta^{2}\frac{x-1}{(y-1)^{2}}+\frac{1}{8}(\theta^{2}-4)\frac{x(x-1)}{(y-x)^{2}}\right)\frac{y(y-1)(y-x)}{x^{2}(x-1)^{2}}.$

For $\theta=1$ the solution corresponding to the non singular instanton is one
of Hitchin’s octahedral solutions [10, 11].

The above set of parameters lies on the intersection of three reflecting
hyperplanes of the Weyl group. Being more precise, the Painlevé equation
depends on four parameters $(\theta_{1},\theta_{2},\theta_{3},\theta_{4})$ and
the reflections with respect to the planes $\theta_{1}=0$, $\theta_{2}=0$,
$\theta_{3}=0$, $\theta_{4}=1$, $\sum\theta_{i}=0$ generate an affine Weyl
group of type $D_{4}$ of symmetries. In our case
$\theta_{1}=\theta_{2}=\theta_{3}=\theta_{4}=\theta/2$, hence the one
parameter family lies in the intersection of the three reflecting hyperplanes:
$\theta_{1}-\theta_{2}+\theta_{3}-\theta_{4}=0$,
$\theta_{1}+\theta_{2}-\theta_{3}-\theta_{4}=0$,
$\theta_{1}-\theta_{2}-\theta_{3}+\theta_{4}=0$. This family of parameters is
equivalent to that considered by Dubrovin and Mazzoco in [7]; in fact,
parameters of the form $(\theta/2,\theta/2,\theta/2,\theta/2)$ are in the same
orbit (under the Weyl group) as $(0,0,0,\theta)$, and the parameter $\mu$ of
[7] in terms of $\theta$ is $\mu=\theta/2$. The transformation relating both
families of parameters is the so called Okamoto transformation and it is given
by

$\displaystyle(x,y,\boldsymbol{\theta})\mapsto\left(x,y+\frac{\delta}{q},\boldsymbol{\theta}-\boldsymbol{\delta}\right),$

where

$\displaystyle\boldsymbol{\theta}=(\theta_{1},\theta_{2},\theta_{3},\theta_{4}),\qquad\boldsymbol{\delta}=(\delta,\delta,\delta,\delta),\qquad\delta=\frac{1}{2}\sum_{i}\theta_{i},$
$\displaystyle
2q=\frac{(x-1)y^{\prime}-\theta_{1}}{y}+\frac{y^{\prime}-1-\theta_{2}}{y-x}-\frac{xy^{\prime}+\theta_{3}}{y-1}.$

Dubrovin and Mazzoco find all algebraic solutions considering a special class
of solutions having a specific asymptotic behaviour around the critical
points. It is said that a branch of a solution to
$\text{P}_{\text{VI}}(\theta)$ has critical behaviour of algebraic type in 0
if there exist $\ell_{0}\in\mathbb{R}$, $a_{0}\in\mathbb{C}$ and $\epsilon>0$
such that

$\displaystyle
y(x)=a_{0}x^{\ell_{0}}\big{(}1+\mathcal{O}\big{(}x^{\epsilon}\big{)}\big{)}\qquad\text{as}\quad
x\to 0.$ (1.1)

Obviously, any algebraic function verifies this property with $\ell_{0}$
rational. It is easy to see that the Okamoto transformation preserves this
type of solutions, without changing $\ell_{0}$. As it is proved in [7, Theorem
2.1], for each nonresonant value of $\theta$ (i.e., $\theta\notin\mathbb{Z}$)
there exists a solution to $\text{P}_{\text{VI}}(\theta)$ with asymptotic
behaviour prescribed by (1.1). Such a solution will also have critical
behaviour of algebraic type at 1 and $\infty$, i.e., there exist
$(a_{1},\ell_{1})$ and $(a_{\infty},\ell_{\infty})$ such that

$\displaystyle\begin{split}&y(x)=1-a_{1}(1-x)^{\ell_{1}}\big{(}1+\mathcal{O}\big{(}(1-x)^{\epsilon}\big{)}\big{)}\qquad\text{as}\quad
x\to 1,\\\
&y(x)=a_{\infty}x^{1-\ell_{\infty}}\big{(}1+\mathcal{O}\big{(}x^{-\epsilon}\big{)}\big{)}\qquad\text{as}\quad
x\to\infty.\end{split}$

For the solution to be algebraic the parameters $\ell_{i}$ must be rational
and satisfy $0<\ell_{i}\leq 1$. In Section 3 we find an explicit expression
for the solution to Painlevé’s equation in terms of the invariant instanton,
from which, using the previous facts, we show that solutions from Sadun’s
instantons [22] are not algebraic.

A generalization of the Dubrovin–Mazzoco strategy can be found in [1], where
new explicit algebraic solutions were found the parameters of which lie in the
interior of a fundamental domain. The same circle of ideas leads finally to
the classification of all algebraic solutions in [14].

In Section 2 we briefly describe the action of $\operatorname{SU}_{2}$ on
$S^{4}$ which is considered and we remember the explicit form of the reduced
ASD equations for invariant instantons. In Section 3, which is the main part
of the paper, using the relation between symmetric instantons and
isomonodromic deformations we make the calculations to find an explicit
expression for the solution of $\text{P}_{\text{VI}}$ in terms of the
instanton. That is, we find the explicit form of the map from symmetric
instantons to solutions of $\text{P}_{\text{VI}}$. Section 4 is devoted to
study the special case of instantons with holonomic singularity that were
defined by Kronheimer and Mrowka and we compute the parameters of the
corresponding $\text{P}_{\text{VI}}$ equations. Finally, in Section 5 we state
the result of Sadun showing the existence of instantons of the kind studied in
Section 4, and show that the corresponding solutions to $\text{P}_{\text{VI}}$
are not algebraic unless the instanton can be smoothly extended to all of the
4-sphere.

##  2 The action and some notations

Let us identify $\mathbb{C}^{4}$ with the space of homogeneous polynomials of
degree three, in two variables, and with complex coefficients

$\displaystyle\mathbb{C}^{4}\cong\big{\\{}\mathbf{p}(\mathbf{x},\mathbf{y})=z_{1}\mathbf{x}^{3}+z_{4}\mathbf{x}^{2}\mathbf{y}+z_{3}\mathbf{x}\mathbf{y}^{2}+z_{2}\mathbf{y}^{3}\colon
z_{i}\in\mathbb{C},\,i=1,2,3,4\big{\\}}.$

$\operatorname{SU}_{2}$ acts as usual on the space of polynomials
$g\cdot\mathbf{p}(\mathbf{x},\mathbf{y})=\mathbf{p}((\mathbf{x},\mathbf{y})\bar{g})$,
and this representation is quaternionic if we identify $\mathbb{C}^{4}$ with
$\mathbb{H}^{2}$ through the map

$\displaystyle(z_{1},z_{2},z_{3},z_{4})\longmapsto(z_{1}+z_{2}j,z_{3}+z_{4}j),$

viewing $\mathbb{H}^{2}$ as a left $\mathbb{H}$-module. Using the
identification $\mathbb{H}{\rm P}^{1}\cong S^{4}$ we obtain an action by
isometries of $\operatorname{SU}_{2}$ on $S^{4}$ such that
$S^{4}/\operatorname{SU}_{2}\cong[0,1]$. The curve $c(t)$ given by
$t\mapsto(1,0,t,0)$ composed with the quotient map
$\mathbb{C}^{4}\rightarrow\mathbb{H}{\rm P}^{1}$ parametrizes a great circle
on $S^{4}$ and it is such that for $0\leq t\leq 1$ it intersects each orbit
exactly once (the parametrization here is not geodesic unlike the usual
parametrization in the literature). When $t\in(0,1)$ we have three-dimensional
orbits and for $t=0,1$ the orbits are of dimension two ($c(0)$ obviously has
one-dimensional stabilizer, and it is not difficult to see that the real
twistor line above $c(1)$ has a point with one-dimensional stabilizer). The
exceptional orbits are diffeomorphic to $\mathbb{R}{\rm P}^{2}$ and we denote
them by $\mathbb{R}{\rm P}^{+}$ ($t=0$) and $\mathbb{R}{\rm P}^{-}$ ($t=1$).
For more details and other descriptions of this action we refer to [3, 19, 20,
22]. Any invariant object on $S^{4}$ is determined by its restriction to the
curve $c(t)$.

By one-dimensional reduction, an invariant connection over the trivial complex
vector bundle of rank 2, on an open set of three-dimensional orbits, is given
by a function

$\displaystyle t\mapsto
a_{1}(t)X_{1}\otimes\sigma_{1}+a_{2}(t)X_{2}\otimes\sigma_{2}+a_{3}(t)X_{3}\otimes\sigma_{3}\in{\mathfrak{su}}_{2}\times{\mathfrak{su}}_{2}^{\ast},$

where $\\{X_{1},X_{2},X_{3}\\}$ is the standard basis of ${\mathfrak{su}}_{2}$
and $\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$ is the corresponding dual basis.
For the connection to be anti-self-dual (ASD) the triplet of functions
$a=(a_{1},a_{2},a_{3})$ must satisfy the differential equations (see [4, p.
196], [20, p. 1045, equations (4.1)]):

$\displaystyle\frac{1}{2}K_{1}(t)\dot{a}_{1}=a_{1}-a_{2}a_{3},\qquad\text{and
cyclic permutations},$ (2.1)

where

$\displaystyle K_{1}(t)=\frac{(t^{2}-1)(t^{2}-9)}{4t},\qquad
K_{2}(t)=4t\frac{(t-3)(t+1)}{(t+3)(t-1)},\qquad
K_{3}(t)=4t\frac{(t+3)(t-1)}{(t-3)(t+1)},$

come from the fact that the basis is not orthonormal, e.g.,
$K_{1}=\frac{||X_{2}||||X_{3}||}{||X_{1}||||\dot{c}||}$. The differences
between expressions in [4] and [20] arise from the parametrization of the
curve $c(t)$; in [4] they use a geodesic parametrization, in [20] as here the
parametrization is not geodesic.

Given an initial condition $a(t_{0})$ there exists a unique solution defined
on an open interval containing $t_{0}$. We are interested here mainly in
solutions which are defined on all the open interval $(0,1)$.

##  3 Isomonodromic deformation

Twistor theory provides a way to see one-dimensional reductions of symmetric
instantons as solutions to the Painlevé equation. Remember that the twistor
space of a real antiself-dual (ASD) 4-manifold $M$ is a complex 3-manifold
$Z$, which is a $\mathbb{C}{\rm P}^{1}$-fiber bundle over $M$ (the fibers are
called “real twistor lines”). The pull-back to $Z$ of any instanton over $M$
determines a holomorphic vector bundle on $Z$ (this is called the “twistor
transform” of the instanton).

Looking at the action described in the previous section and taking twistor
transform, each ASD invariant connection on an $\operatorname{SU}_{2}$-bundle
over an interval of three-dimensional orbits induces an isomonodromic
deformation of connections on $\mathbb{C}{\rm P}^{1}$ having four simple poles
[10, 20]. Each instanton defines a holomorphic vector bundle and the action
can be used to define a holomorphic flat connection there, in such a way that
horizontal sections are essentially given by the orbits of the action. The
holomorphic connection is defined except on certain anticanonical divisor
$Y\subset\mathbb{C}{\rm P}^{3}$ which intersects each real twistor line in
four points, therefore the restriction to each line gives a holomorphic
connection on $\mathbb{C}{\rm P}^{1}$ with four singularities. These
connections on $\mathbb{C}{\rm P}^{1}$ are given by the 1-form ($\lambda$ is
the variable on the line)

$\displaystyle
A(t;\lambda)d\lambda=\sum_{j=0}^{3}\frac{A_{j}(t)}{\lambda-\lambda_{j}}d\lambda=-\sum_{i=1}^{3}a_{i}(t)\alpha_{i}(t,\lambda)X_{i}d\lambda,$

where the $\alpha_{i}(t,\lambda)$ are defined by the inverse of the
complexified infinitesimal action

$\displaystyle\alpha^{-1}(t,\lambda)d\lambda=\sum_{i=1}^{3}\alpha_{i}(t,\lambda)X_{i}d\lambda.$

Remember that the infinitesimal action is the map $\alpha\colon\mathbb{C}{\rm
P}^{3}\times{\mathfrak{su}}_{2}\rightarrow T\mathbb{C}{\rm P}^{3}$ obtained as
the derivative of the $\operatorname{SU}_{2}$-action, and its complexification
(which we denote by the same letter) is the map $\alpha\colon\mathbb{C}{\rm
P}^{3}\times\mathfrak{sl}_{2}(\mathbb{C})\rightarrow
T_{\mathbb{C}}\mathbb{C}{\rm P}^{3}$ which is of rank three, and can therefore
be inverted at each point in $\mathbb{C}{\rm P}^{3}\setminus Y$. As it is
known from general facts about isomonodromic deformations, the square of the
residue of $A(t,\lambda)$ at each pole has constant trace in the deformation.
This gives us a conserved quantity for equations (2.1):

######  Proposition 3.1.

Equations (2.1) have a conserved quantity given by

$\displaystyle\frac{1-t^{2}}{9-t^{2}}a_{1}(t)^{2}+\frac{1+t}{t(3-t)}a_{2}(t)^{2}-\frac{1-t}{t(3+t)}a_{3}(t)^{2}.$

###### Proof.

Choose a pole, say $\lambda_{0}$, and call $\alpha_{i,0}(t)$ the residue of
$\alpha_{i}(t,\lambda)$ at it. Then we have

$\displaystyle\operatorname{tr}\big{(}A_{0}^{2}(t)\big{)}=-2\sum_{i=1}^{3}a_{i}(t)^{2}\alpha_{i,0}(t)^{2}=\operatorname{const}.$

Taking derivative with respect to $t$ we have

$\displaystyle\sum_{i=1}^{3}({\dot{a}_{i}}\alpha_{i,0}+a_{i}{\dot{\alpha}_{i,0}})a_{i}\alpha_{i,0}=0,$

and using the ASD equations (2.1)

$\displaystyle\sum_{i=1}^{3}\left(2\frac{\alpha_{i,0}}{K_{i}}+\dot{\alpha}_{i,0}\right)a_{i}^{2}\alpha_{i,0}-a_{1}a_{2}a_{3}\sum_{i=1}^{3}\frac{\alpha^{2}_{i,0}}{K_{i}}=0.$

As the above equation is true for any solution of (2.1), choosing for example
the one with $a_{2}=a_{3}=0$, we obtain that the residue $\alpha_{1,0}$ has to
satisfy

$\displaystyle\dot{\alpha}_{1,0}(t)=-2\frac{\alpha_{1,0}(t)}{K_{1}(t)},$

and similarly for $i=2,3$. Then, we must also have

$\displaystyle\sum_{i=1}^{3}\frac{\alpha^{2}_{i,0}}{K_{i}}=0.$

Moreover, since for the solution $a_{1}=a_{2}=a_{3}\equiv 1$ the flat
connection is $A=\alpha^{-1}$ we know (see [10]) that
$\operatorname{tr}\big{(}A_{0}^{2}(t)\big{)}=\frac{1}{8}$, hence the residues
satisfy

$\displaystyle\sum_{i=1}^{3}\alpha_{i,0}^{2}=-\frac{1}{16}.$

The last three equations completely determine the $\alpha_{i,0}(t)^{2}$, and
they are given by

$\displaystyle\alpha_{1,0}(t)^{2}=-\frac{t^{2}-1}{16(t^{2}-9)},\qquad\alpha_{2,0}(t)^{2}=-\frac{t+1}{16t(3-t)},\qquad\alpha_{3,0}(t)^{2}=\frac{1-t}{16t(t+3)}.$

As a consequence, the functions $\alpha_{i,0}(t)^{2}$ are independent of the
pole chosen (fact already known), or in other words

$\displaystyle\operatorname{tr}\big{(}A_{0}^{2}(t)\big{)}=\operatorname{tr}\big{(}A_{1}^{2}(t)\big{)}=\operatorname{tr}\big{(}A_{2}^{2}(t)\big{)}=\operatorname{tr}\big{(}A_{3}^{2}(t)\big{)}.$
∎

Let $\frac{1}{8}\theta^{2}=\operatorname{tr}(A_{0}^{2}(t))$ be the above
constant; observe that $\operatorname{tr}(A_{0}^{2}(t))$ is real but not
necessarily positive, and so $\theta$ may be imaginary. Then, the eigenvalues
of $A_{i}$ are $\pm\frac{1}{4}\theta$. This kind of isomonodromic deformation
is a known dress of the VI Painlevé equation [12]. Given a solution to
$\text{P}_{\text{VI}}$ we can construct the matrices $A_{i}$’s and viceversa.
If we start with the matrix $A(t;\lambda)$, then for each value $x$ of the
cross ratio of the four poles there is a unique point $y(x)\in\mathbb{C}{\rm
P}^{1}\setminus\\{0,1,x,\infty\\}$ such that $A(x;y(x))$ has a common
eigenvector with the residue at infinity (corresponding to one of the
eigenvalues). As Jimbo and Miwa showed $y(x)$ is a solution to
$\text{P}_{\text{VI}}$. In terms of $\theta$ the parameters
$(\alpha,\beta,\gamma,\delta)$ of $\text{P}_{\text{VI}}$ are given by

$\displaystyle\alpha=\frac{1}{8}(\theta\pm
2)^{2},\qquad\beta=-\frac{1}{8}\theta^{2},\qquad\gamma=\frac{1}{8}\theta^{2},\qquad\delta=-\frac{1}{8}\big{(}\theta^{2}-4\big{)}.$

Denoting by $P_{t}$ the real twistor line corresponding to the point $c(t)$ we
have

$\displaystyle P_{t}\cap
Y=\left\\{\sqrt{\frac{t^{4}+18t^{2}-27+\sqrt{(t^{2}-1)(t^{2}-9)^{3}}}{8t^{3}}}\right\\}$

(in homogeneous coordinates), see [20]. Then, the cross ratio in our situation
is

$\displaystyle x=\frac{(t+1)(t-3)^{3}}{(t-1)(t+3)^{3}}.$

Since the $A_{i}$’s are determined by the solution to $\text{P}_{\text{VI}}$,
so are the $a_{i}$’s. Following [12] (see also [15]), let $y(x)$ be the
solution to the Painlevé equation. By writing

$\displaystyle z(x)=4A(x;y(x))_{11},$

one has

$\displaystyle\dot{y}=\frac{y(y-1)(y-x)}{2x(x-1)}\left(z+\frac{2}{y-x}\right).$

From this expression, and after some computations, we can see that the
functions $a_{i}$’s are given by the following formulas

$\displaystyle
a_{1}^{2}=\frac{(t^{2}-9)x(x-1)^{2}}{4(t^{2}-1)(y-1)(y-x)}w_{1}w_{2},$ (3.1)
$\displaystyle a_{2}^{2}=\frac{t(3-t)x(x-1)}{4(t+1)y(y-1)}w_{2}w_{3},$ (3.2)
$\displaystyle a_{3}^{2}=\frac{t(t+3)x^{2}(x-1)}{4(1-t)y(y-x)}w_{1}w_{3},$
(3.3)

where

$\displaystyle w_{1}=\left(2\dot{y}+\frac{(\theta-2)y^{2}-2\theta xy+2y+\theta
x}{x(x-1)}\right),$ (3.4) $\displaystyle
w_{2}=\left(2\dot{y}+\frac{(\theta-2)y^{2}+2(1-\theta)y+\theta
x}{x(x-1)}\right),$ (3.5) $\displaystyle
w_{3}=\left(2\dot{y}+\frac{(\theta-2)y^{2}+2y-\theta x}{x(x-1)}\right).$ (3.6)

Reciprocally, it is not difficult, from the above equations, to find an
explicit expression for $y$ in terms of $a_{1}$, $a_{2}$, $a_{3}$. By
eliminating $\dot{y}$ from the above equations, and after some elementary
manipulations of the formulas we find:

######  Theorem 3.2.

To each equivariant ASD instanton determined by $(a_{1},a_{2},a_{3})$, with
two of the $a_{i}$’s not identically zero, there corresponds a solution $y(x)$
to the Painlevé VI equation given by

$\displaystyle
y=\frac{(t-3)^{3}(t+1)((t^{2}-4t+3)a_{2}^{2}-(t^{2}+4t+3)a_{3}^{2})a_{1}}{(t+3)((t^{2}+2t-3)^{2}a_{2}^{2}-(t^{2}-2t-3)^{2}a_{3}^{2})a_{1}\pm
16\theta a_{2}a_{3}t^{3}},$ (3.7)

where

$\displaystyle x=\frac{(t+1)(t-3)^{3}}{(t-1)(t+3)^{3}}.$

The corresponding parameters of $\text{\rm P}_{\text{\rm VI}}$ are determined
by Proposition 3.1.

###### Proof.

Substracting (3.4) from (3.5) we obtain

$\displaystyle y=\frac{x}{2\theta}(w_{2}-w_{1}),$

hence

$\displaystyle
y^{2}=\frac{x^{2}}{4\theta^{2}}\big{(}w_{1}^{2}+w_{2}^{2}-w_{1}w_{2}\big{)}.$

Now, using (3.1)–(3.3) we can find $w_{1}^{2}$, $w_{2}^{2}$ and $w_{1}w_{2}$,
and substituting in the last identity we obtain

$\displaystyle y^{2}=\left(\frac{xa_{1}}{\theta
a_{2}a_{3}}\left(\frac{(1-t)(y-x)a_{3}^{2}}{(t+3)x}-\frac{(t+1)(y-1)a_{2}^{2}}{(3-t)}\right)\right)^{2}.$

Therefore

$\displaystyle y=\pm\frac{xa_{1}}{\theta
a_{2}a_{3}}\left(\frac{(1-t)(y-x)a_{3}^{2}}{(t+3)x}-\frac{(t+1)(y-1)a_{2}^{2}}{(3-t)}\right),$

and solving for $y$,

$\displaystyle
y=\frac{(t+1)(t+3)a_{2}^{2}-(1-t)(3-t)a_{3}^{2}}{\frac{(t+1)(t+3)a_{2}^{2}}{x}-(1-t)(3-t)a_{3}^{2}\pm\theta\frac{(t^{2}-9)(x-1)a_{2}a_{3}}{xa_{1}}}.$

Finally, substituting the value of $x$ we arrive at the expression given in
the statement of the proposition. ∎

######  Remark 3.3.

The condition that two of the functions $a_{i}$ are identically zero is
equivalent to saying that one of them vanishes at some point. Suppose that
$a_{3}(t_{0})=0$ at some $t_{0}\in(0,1)$, then equations (3.1)–(3.3) imply
that one of the other two has to vanish at $t_{0}$ too. If
$a_{3}(t_{0})=a_{2}(t_{0})=0$, the ASD equations imply that necessarily
$a_{3}=a_{2}=0$ at all points. In other words, there are two possibilities:
there are two of the $a_{i}$’s that are identically zero, or
$a_{1}(t)a_{2}(t)a_{3}(t)\neq 0$ for all $t$. On the other hand, given that
two of the $a_{i}$’s are null it is easy to find the third of them. For
example, if $a_{2}=a_{3}=0$ then

$\displaystyle a_{1}(t)=\theta\sqrt{\frac{9-t^{2}}{1-t^{2}}}.$

######  Example 3.4.

For the obvious solution $a_{1}=a_{2}=a_{3}=1$ we obtain

$\displaystyle y=-\frac{(t-3)^{2}(t+1)}{(t+3)(t^{2}+3)},$

which is one of Hitchin’s octahedral solutions (Poncelet polygon with $k=3$,
with a different parametrization). In this case $\theta=1$, therefore the
parameters of the Painlevé equation are
$\big{(}\frac{1}{8},-\frac{1}{8},\frac{1}{8},\frac{3}{8}\big{)}$.

######  Example 3.5.

For the Hopf bundle the ASD connection is given by

$\displaystyle a_{1}(t)=3\frac{1-t^{2}}{t^{2}+3},\qquad
a_{2}(t)=-6\frac{t+1}{t^{2}+3},\qquad a_{3}(t)=6\frac{1-t}{t^{2}+3}.$

Substituting in (3.7) we find

$\displaystyle
y(t)=-\frac{(t-3)^{2}(t-1)(t+1)^{2}}{(t+3)\left(7t^{4}+6t^{2}+3\right)}.$

In this example the parameters of the Painlevé equation are
$\big{(}\frac{1}{8},-\frac{9}{8},\frac{9}{8},-\frac{5}{8}\big{)}$.

##  4 Holonomic singularities

Let $M$ be a four-dimensional Riemannian manifold and $S\subset M$ an embedded
surface. For an $\operatorname{SU}_{2}$ vector bundle over $M$, a connection
defined on $M\setminus S$ has holonomic singularity along $S$ if the
connection 1-form restricted to each normal plane to $S$ can be written as

$\displaystyle i\begin{pmatrix}a&0\\\ 0&-a\end{pmatrix}d\theta+\text{lower
order terms},$ (4.1)

where $(r,\theta)$ are polar coordinates in the normal plane to $S$, and
$a\in[0,1/2]$ is the “holonomy parameter”. The limit of the holonomy for
shrinking circles around $S$ is then given by

$\displaystyle\exp 2\pi i\begin{pmatrix}-a&0\\\ 0&a\end{pmatrix}.$

When $a=0$ the asymptotic holonomy is trivial and, for an appropriate
definition of the lower order terms, the connection is defined on all of $M$.
This is the definition given in [13]. When $a=1/2$ the holonomy goes to $-1$
and it is trivial if we look at the associated $\operatorname{SO}_{3}$ bundle.
The limit holonomies for $a=1/2+\epsilon$ and $a=1/2-\epsilon$ are conjugate
to each other, and therefore equivalent.

Equivariant vector bundles on $S^{4}$ for the action considered here are
classified by a pair of integers congruent with 1 mod 4. These integers
correspond to the weights of the stabiliser of each orbit [3, 4, 22]. For the
existence of ASD connections (without any singularity) the weight of
$\mathbb{R}{\rm P}^{+}$ has to be equal to one. Let us denote by $E_{n}$ the
equivariant vector bundle whose weights are $n$ on $\mathbb{R}{\rm P}^{-}$ and
1 on $\mathbb{R}{\rm P}^{+}$ [5].

We will consider ASD connections on $S^{4}$, defined on $E_{n}$, having
holonomic singularities along both special orbits $\mathbb{R}{\rm P}^{\pm}$.
Their existence is established in [22] and will be explained in the next
section. For the moment we assume that they exist and we compute the
parameters of the Painlevé equation related to them in terms of the holonomy
parameter. We denote by $\mathcal{D}$ the ASD connection in order to avoid
confusion with the flat connection $\nabla$ defined from the action.

######  Lemma 4.1.

Let $(a_{1},a_{2},a_{3})$ be a triplet defining an invariant ASD connection on
$E_{n}\rightarrow S^{4}$ having holonomic singularities along $\mathbb{R}{\rm
P}^{\pm}$ with parameter $a$ on $\mathbb{R}{\rm P}^{-}$. Then,
$\lim\limits_{t\to 1}a_{2}(t)=n+4a$.

###### Proof.

Take a section $g\colon U\rightarrow\operatorname{SU}_{2}$, on an open
neighbourhood $U$ of $x^{-}$ in $\mathbb{R}{\rm
P}^{-}\cong\operatorname{SU}_{2}/\tilde{\operatorname{O}}_{2}^{\prime}$, such
that $g(x^{-})=id$. Since $N_{+}=S^{4}\setminus\mathbb{R}{\rm P}^{+}$ is
isomorphic to the vector bundle over $\mathbb{R}{\rm P}^{-}$ associated to the
slice representation of $\operatorname{O}^{\prime}_{2}$, the above section
gives an isomorphism

$\displaystyle N_{+}\cong U\times D,$

$D$ being a disc perpendicular to $\mathbb{R}{\rm P}^{-}$ at $x^{-}$.

Since $E_{n}|_{\mathbb{R}{\rm P}^{-}}$ is the vector bundle associated to the
representation of $\operatorname{O}^{\prime}_{2}$ on $E_{x^{-}}$ then $E|_{U}$
is trivial. On the other hand $E|_{D}$ is
$\operatorname{O}^{\prime}_{2}$-equivariantly trivial, hence we have that
$E|_{U\times D}$ is trivial. Taking a frame $\\{s_{1},s_{2}\\}$ for $E$ on
$U\times D$ given by the above trivialization (notice that necessarily
$\mathcal{D}_{\dot{c}}s_{i}=0$), the connection 1-form $\Phi$ with respect to
this frame has the behaviour (4.1). Let us take another frame
$\\{s^{\prime}_{1},s^{\prime}_{2}\\}$ for $E$ on
$U\times(D\setminus[c([-1,0])])$ in the following way

$\displaystyle s^{\prime}_{i}(x,y)=g(x)e^{\tfrac{1}{4}\theta j}\cdot
s_{i}(c(t)),$

where $\theta\in(-\pi,\pi)$ is such that $e^{\tfrac{1}{4}\theta j}\cdot
c(t)=y$. With respect to this “equivariant” frame the connection 1-form
restricted to $\Sigma$ is

$\displaystyle\Phi_{t}^{\prime}=\tfrac{1}{4}a_{2}(t)jd\theta+a_{1}(t)i\sigma_{1}+a_{3}(t)k\sigma_{3};$

notice that the 1-forms $\sigma_{1}$, $\sigma_{3}$ extend to the orbit of
$x^{-}$ whereas $d\theta=4\sigma_{2}$ does not. Both connection 1-forms are
related on $U\times(D\setminus[c([-1,0])])$ by

$\displaystyle\Phi=\Lambda\Phi^{\prime}\Lambda^{-1}+\Lambda d\Lambda^{-1},$

$\Lambda$ being the change of frame matrix. It is easy to see that
$\Lambda(x,y)=e^{\tfrac{n}{4}\theta j}\lambda(x)$, where $\lambda(x)$ is the
matrix associated to the action of $g(x)$ (in particular $\lambda(x^{-})=I$).
We then have

$\displaystyle\Phi_{t}=\tfrac{1}{4}(a_{2}-n)jd\theta+\text{lower order
terms},$

therefore $\lim\limits_{t\to 1}a_{2}(t)=4a+n$, since the singularity is
holonomic. Remark moreover that $a_{1}$ and $a_{3}$ remain bounded as $t\to
0$. ∎

Using the conclusion of the preceding lemma we can establish the following
theorem, which gives the family of $\text{P}_{\text{VI}}$ equations related to
the family of instantons with holonomic singularities along $\mathbb{R}{\rm
P}^{+}$ and $\mathbb{R}{\rm P}^{-}$. Notice that the parameters $\alpha$,
$\beta$, $\gamma$, $\delta$ depend only on the holonomy around $\mathbb{R}{\rm
P}^{-}$ and not on the holonomy around $\mathbb{R}{\rm P}^{+}$; if we consider
self-dual instantons the converse is true.

######  Theorem 4.2.

Let $(E_{n},\mathcal{D})$ be an invariant ASD instanton with holonomic
singularity along the surfaces $\mathbb{R}{\rm P}^{\pm}$, having holonomic
parameter $a$ along $\mathbb{R}{\rm P}^{-}$. Then, this instanton is
determined by a solution to the Painlevé VI equation with parameters

$\displaystyle\alpha^{\pm}=\frac{1}{8}(4a+n\pm 2)^{2},\qquad$
$\displaystyle\beta=-\frac{1}{8}(4a+n)^{2},$
$\displaystyle\gamma=\frac{1}{8}(4a+n)^{2},\qquad$
$\displaystyle\delta=-\frac{1}{8}\big{(}(4a+n)^{2}-4\big{)}.$

###### Proof.

Remember that

$\displaystyle\theta^{2}=8\operatorname{tr}A_{0}^{2}=-16\sum_{i=1}^{3}a_{i}^{2}\alpha_{i,0}^{2}.$

Taking limit when $t\to 1$ in the above formula we obtain
$\theta^{2}=\lim\limits_{t\to 0}a_{2}(t)^{2}=(4a+n)^{2}$, since
$\alpha_{1,0}(1)=\alpha_{3,0}(1)=0$ and $\alpha_{2,0}(1)^{2}=-1/16$. ∎

##  5 Sadun’s solutions

The existence of solutions to ASD equations (2.1) defining instantons with
holonomic singularities was proved by Sadun in [22]. By imposing the condition
of finite energy he proved that there are solutions with certain asymptotic
behaviour (roughly speaking they are perturbations of regular instantons). For
an invariant connection $(a_{1},a_{2},a_{3})$ the finite energy condition
implies that the functions $a_{i}$’s are well defined on all the interval
$(0,1)$ and that the limits

$\displaystyle r_{+}=\lim_{t\to 0}a_{1}(t),\qquad\text{and}\qquad
r_{-}=\lim_{t\to 1}a_{2}(t)$

exist. Furthermore if $r_{+}\neq 1$ then $a_{2}(0)=a_{3}(0)=0$, and if
$r_{-}\neq 1$ then $a_{1}(1)=a_{3}(1)=0$. For the connection to be ASD we must
have $|r_{+}|\leq 1$ and if $|r_{+}|=1$ we also have the equality
$a_{2}(0)=a_{3}(0)$ but they are not necessarily zero. When the connection is
ASD, if $r_{+}=1$ and $r_{-}\equiv 1$ mod 4 it can be extended to a smooth
connection on all of $S^{4}$.

Sadun’s result is summarized in the following proposition.

######  Proposition 5.1 (Sadun).

For any pair of real numbers $(c,r_{-})$ such that $r_{-}\geq 1$, $0\leq c\leq
c_{1}$ for some positive constant $c_{1}$, there exists a finite energy
solution $(a_{1},a_{2},a_{3})$ to equation (1.1) defined on $(0,1)$ with the
following asymptotic behavior around $t=1$:

$\displaystyle
a_{1}(t)=-c(1-t)^{(r_{-}-1)/2}+O\big{(}(1-t)^{(r_{-}+1)/2}\big{)},$
$\displaystyle a_{2}(t)=r_{-}+O\big{(}(1-t)^{2}\big{)},$ $\displaystyle
a_{3}(t)=c(1-t)^{(r_{-}-1)/2}+O\big{(}(1-t)^{(r_{-}+1)/2}\big{)}.$

Moreover, the limit $r_{+}=\lim\limits_{t\to 0}a_{1}(t)$ exists and takes any
value exactly once in the interval $[0,1]$ when we vary $c$.

Instantons given by the above proposition have holonomic singularities around
$\mathbb{R}{\rm P}^{\pm}$, and computing the asymptotic holonomy one obtains

$\displaystyle\exp\left(\frac{1-r_{+}}{2}\pi i\right)$

around $\mathbb{R}{\rm P}^{+}$, and

$\displaystyle\exp\left(\frac{1-r_{-}}{2}\pi j\right)$

around $\mathbb{R}{\rm P}^{-}$. The holonomy parameter on $\mathbb{R}{\rm
P}^{-}$ is then given by $a=\frac{r_{-}-1}{4}-[\frac{r_{-}-1}{4}]$ (here we
take $a\in[0,1)$). The holonomy is trivial only when $r_{\pm}\equiv 1$ mod 4,
which is the case when the connection is defined on all of $S^{4}$.

######  Remark 5.2.

On the complement of the singularities the vector bundles $E_{n}$ are all
trivial, then in fact the connections of Proposition 5.1 are defined on the
trivial vector bundle. Considering the connection with $r_{-}=n+4a$ on $E_{1}$
is the same as considering those with $r_{-}=4a$ on $E_{n}$ for $n\equiv 1$
mod 4.

Let us now look at the behaviour of the corresponding $\text{P}_{\text{VI}}$
solutions around the critical points. The critical singularities of
$\text{P}_{\text{VI}}$ are situated at $x=0,1,\infty$ which correspond to
$t=-1,0,1$ respectively (and $3$, $\infty$, $-3$ since there is a two-to-one
correspondence between $x$ and $t$). Remark that the asymptotic behaviour of
$y$ around a critical point is then determined by the asymptotic behaviour of
$a_{1}$, $a_{2}$, $a_{3}$ at $t=-1,0,1$, and viceversa. Remember that the
resonant values of the parameters are those corresponding to
$\theta\in\mathbb{Z}$.

######  Proposition 5.3.

The solutions to the Painelevé VI equation defined from instantons given by
Proposition 5.1 are not algebraic except for the resonant values of the
parameters.

###### Proof.

To be algebraic the solutions must have critical behaviour of algebraic type
since the corresponding Painlevé equations are equivalent, via Okamoto
transformation, to that of Dubrovin and Mazzoco. In particular they must
satisfy $\lim\limits_{x\to\infty}y(x)=\infty$. But if $t\to 1$ then
$x\to\infty$, and using the asymptotic behaviour of the functions $a_{i}$’s
given in Proposition 5.1 and the expression (3.7) we see that
$\lim\limits_{t\to 1}y(t)=0$ for $\theta>1$ and $\lim\limits_{t\to
1}y(t)=-c^{2}$ for $\theta=1$, hence they can not be algebraic. Certainly they
are algebraic for instantons that can be extended through the singular orbits
which correspond to the parameters with $\theta\in\mathbb{Z}$. ∎

### Acknowledgements

The author would like to thank Nigel Hitchin for his suggestion to look at
instantons with holonomic singularities and Gil Bor for many useful
conversations. We also thank the referees for many useful suggestions that
help to improve the paper. This work was partially supported by Grupo CSIC 618
(UdelaR, Uruguay).

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# Path Planning for Optimal Coverage of Areas with Nonuniform Importance

Gregory Snyder111Graduate Student, Department of Mechanical Engineering,
<EMAIL_ADDRESS>University of Hawai‘i at Mānoa, Honolulu, HI 96822  Sachin
Shriwastav222Graduate Student, Department of Mechanical Engineering,
<EMAIL_ADDRESS>(Corresponding author) University of Hawai‘i at Mānoa,
Honolulu, HI 96822  Dylan Morrison-Fogel333PhD Graduate, Department of
Mechanical Engineering. and Zhuoyuan Song444Assistant Professor, Department of
Mechanical Engineering<EMAIL_ADDRESS>AIAA Member.
(Snyder and Shriwastav contributed equally to this work.) University of
Hawai‘i at Mānoa, Honolulu, HI 96822

###### Abstract

Coverage of an inaccessible or challenging region with potential health and
safety hazards, such as in a volcanic region, is difficult yet crucial from
scientific and meteorological perspectives. Areas contained within the region
often provide valuable information of varying importance. We present an
algorithm to optimally cover a volcanic region in Hawai‘i with an unmanned
aerial vehicle (UAV). The target region is assigned with a nonuniform coverage
importance score distribution. For a specified battery capacity of the UAV,
the optimization problem seeks the path that maximizes the total coverage area
and the accumulated importance score while penalizing the revisiting of the
same area. Trajectories are generated offline for the UAV based on the
available power and coverage information map. The optimal trajectory minimizes
the unspent battery power while enforcing that the UAV returns to its starting
location. This multi-objective optimization problem is solved by using
sequential quadratic programming. The details of the competitive optimization
problem are discussed along with the analysis and simulation results to
demonstrate the applicability of the proposed algorithm.

##  1 Introduction

Over the last century, the Hawai‘i Volcanic Observatory (HVO) has developed
volcano-monitoring systems and networks to record and document activities at
Hawaiian volcanoes. The existing infrastructure includes over a hundred field
stations with instruments that record and measure earthquakes, ground
movement, volcanic gases, sound waves, lava advancement, magma volume below
ground, and visual changes in eruptive activities. Satellite data have been
used to detect changes in ground elevation and surface temperature, which can
indicate lava or other eruptive activities, however direct line-of-sight is
often interrupted with cloud cover and volcanic haze. Without satellite
coverage, the large surface area of Hawai‘i island presents issues for remote
sensing, therefore implementation of autonomous unmanned aerial vehicles
(UAVs) may prove invaluable for HVO. Given the size of commercially available
UAVs, power restrictions limit the total surface area that a single platform
may observe. Therefore, an optimization scheme must be implemented to
determine the most effective battery configuration of UAV for a specific
environment and the optimal path for that specific UAV to take based on this
configuration.

Path planning algorithms for UAV systems have been considered for a variety of
applications, though a majority examine a problem of reaching the greatest
number of targets while expending minimal energy or time [1, 2, 3]. Voronoi
diagrams have been traditionally implemented for graphical methods [4, 5, 6],
while spline generation and single-objective genetic algorithms are becoming
commonplace for both online and offline operations [7, 8]. The means by which
the cost of a system is represented varies considerably; for example, Bortoff
[4] considers both a two-step optimization algorithm that first explores
graphical search methods and secondly generates an equilibrium path solution
for a Lagrangian mechanical system composed of virtual forces. Combinations of
graphical and mechanical approaches provide a means of reducing a solution
search area prior to processing higher-order system dynamics.

In general, path planning has been used to achieve UAV interaction with
specific targets or to maximize coverage over a specific area of uniform
importance. A natural succession is to maximize coverage over a given area,
within which some known locations are assigned higher importance than others.
Topological approaches have been investigated by Li et al. [9] that address
energy maps for UAV paths along variable topological terrain. This approach
considers energy-optimal area coverage under the specific dynamics of the UAV,
but does not consider the possibility of a performance index being nonuniform
along identical topological levels, that is, regions of greater importance to
data collection. The problem that arises for nonuniform area preference,
discussed by Mittal and Deb [10], is the normalization of the objective
function under the influence of a weight vector. This consideration implies
that a topological map should be generated for each performance index
considered, as well as the weight vector at every iteration.

This work intends to explore the coverage optimization problem where power
limitation and nonuniform coverage importance distribution are considered. An
optimization problem will be formulated and solved for a UAV deployed over the
area, with an objective to maximize the area covered, energy utilization and
accumulated coverage importance. The major contributions of the proposed
algorithm are: (i) Optimized coverage of an area for given power for a UAV, by
maximizing the coverage and minimizing the return energy, (ii) Trade-off
between coverage of new area and net information gain, by attempting to cover
areas of higher coverage importance while penalizing revisits, (iii) simple
yet effective coverage algorithm readily applicable to real-world
applications.

The remainder of the paper has been organized as follows. Section 2 defines
and sets up the problem, and Section 3 discusses the optimization problem
formulation and the proposed approach. Preliminary results and analysis are
discussed in Section 5, and Section 6 concludes the paper and lists some
future research directions.

Figure 1: An example volcanic survey area (Hawai‘i Island), with the overlaid
coverage importance distribution (Red: High, Yellow: Medium and Green: Low)

##  2 Problem Definition

Considering the problem of monitoring volcanic activities with a UAV, we wish
to maximize the observation productivity by varying its battery capacity as
well as its flight path design. Observation productivity is defined as the
norm of a performance index encoding the area coverage, the number of
“waypoints", and the unspent energy upon return. The problem does not consider
real-time path planning, rather a prescribed flight plan. To best solve our
problem we need the UAV to maximize its surveyed area, while covering more
important areas and best utilize its limited power before returning to the
base station.

We assume that the coverage importance map (See Fig. 1) is available for
trajectory planning. The problem is then set up by defining a grid over the
target area using the area boundary information. The grid is then created at
the spacing $l$ in both directions of the area, and the undesired coverage
locations outside the area are then identified and assigned with the lowest
importance weights. An example where the coverage importance map is overlaid
on top of the area map is shown in Fig. 2. The optimal path planning is
carried out as discussed in the following section.

Figure 2: Coverage importance distribution grid overlaid on the desired
coverage area. An example trajectory to maximize coverage and net area
information gain with a UAV starting from and returning to the base is also
shown.

##  3 Proposed Approach

The proposed approach to maximize the coverage over a volcanic area and net
coverage information gain is outlined in Algorithm 1, and the path planning
logic is shown in Fig. 3. The boundary information of the area is used to
create a grid over it, and the coverage information importance map is
overlaid. The UAV starts at a prescribed base station ($x_{\text{base}},\
y_{\text{base}}$) as shown by a white square in Fig. 2, and it has the
knowledge of the overall grid, its own battery level, and the cost to move in
each possible direction on the grid. For this work, we assume that the cost to
move is same in each direction. The algorithm also maintains the history of
the visited locations to discount the trajectories containing repetitions.
However, we do not prohibit revisits since they may be necessary as part of
the shortest return path to the base due to battery level or physical map
constraints. The coverage of new areas thus competes with the net coverage
information gain and the battery utilization in the optimization problem. At
each time step, the UAV lists the candidate locations to move, all of which
are $l$ units away. During this process, the UAV keeps a record of the cells
covered and keeps track of its battery level relative to its current distance
from the base. The UAV is programmed to initiate the return to the base when
the battery level is critical for safe return from its present position while
still attempting to maximizing its total coverage on its safe return.

When the UAV starts up it will first determine a target area to reach by
weighing the reward for reaching an area based on its importance, as well as
by a temporary reward map. The weight map corresponds to the importance of a
specific cell. The reward map will be repeatedly generated as seen in Fig. 3,
where the value of each cell is determined by the importance weight of the
cell and penalized by its distance relative to the UAV’s current position as
well whether the cell has been previously visited. Once a reward map is
generated, the UAV will calculate the shortest path to the target cell of the
highest reward using Djikstra’s algorithm [11], where the edge length is the
reward of the cell. While this does not necessarily yield the shortest path to
each following target point, it will generate the shortest path that yields a
positive reward and and maximize the accumulated reward along this path,
provided it has unspent energy to do so. Once the path to the target cell is
calculated, it will then calculate the shortest return path to the base using
Djikstra’s algorithm [11] and Bellman-Ford algorithm [12, 13, 14] from the
candidate target position by choosing the shortest path to base, without
possibly revisiting cells to maximize the total reward. The UAV proceeds to
the next exploration step only if the remaining battery level is higher than
the threshold requirement for the calculated shortest path to base. This is
repeated at every new location to fully use the available power.

Figure 3: Flowchart for the proposed optimal path planning algorithm.

Fig. 4 illustrates how the algorithm would navigate a map of pseudo-randomly
distributed cells with weights of high, medium, low and zero values marked in
red, yellow, green, and blue, respectively. The white cell denotes the base
location where the UAV starts its path planning and ultimately must return to.
The white dots show the target waypoint with the highest expected reward
calculated from a dynamic weight map that the algorithm maintains based on the
importance of the cells, their distances from the UAV, and if it has
previously visited. As seen in Fig. 4(b), the next trajectory is more affected
by the distance to the target rather than the energy expense due to the large
difference in the anticipated reward of the cells the path covers. Despite
this the optimal path will take into account the reward of visiting more
importance cells against the cost of moving to an additional cell.
Trajectories in these circumstances can be altered by adjusting the penalties
of revisiting a cell. The black arrows denote the outgoing trajectory from
each cell and the blue arrows denote what the algorithm deems to be the the
shortest-path of return with the maximum reward based on the current remaining
power. These shortest paths with maximum reward are calculated at every
target, as shown in the figure. These return trajectories are determined using
Djikstra’s algorithm depending on whether a possible trajectory passes over a
visited space as the assigned penalties provide a higher penalty. Finally,
Fig. 4(c) shows the complete exploration and return trajectory for the UAV for
the example setup as the algorithm has determined a path that consumes all the
remaining power from that waypoint.

(a)

(b)

(c)

Figure 4: Illustration of the development of the path planning algorithm. The
red, yellow, green and blue cells denote the highest, medium, low and lowest
reward for covering the respective areas respectively. The white cell denotes
the base location and the white dots show the next location of highest
importance that the algorithm targets based on the reward values of the cells
and its current distance from the UAV. The outgoing and the return
trajectories have been shown in black and blue, respectively. The figure shows
(a) randomized weight map, (b) five next highest importance locations, their
outgoing and respective instantaneous return trajectories, and (c) final
trajectory. Algorithm 1 Optimal Coverage of Area with Nonuniform Importance

1:Input: Area grid, cell indices ($M$), coverage importance weight map
($\gamma(M)$), base coordinates ($x_{\text{base}},\ y_{\text{base}}$), cell
size ($l$), starting battery level ($B_{\text{start}}$), critical battery
level ($B_{\text{min}}$)

2:Output: Optimal trajectory $M_{i}\ (i=1,2,\ldots,n)$

3:Notations:

4: $\bullet$ $D$ = Grid matrix listing for visited ($\nu=1$) and not visited
($\nu=0$)

5: $\bullet$ $B_{i}$ = Current battery level

6: $\bullet$ $B_{\text{min}}$ = Critical battery level required to return to
base from current position

7: $\bullet$ $B_{\text{target},j}$ = Energy cost to move to target, $j$

8: $\bullet$ $W_{i}$ = Generated reward map from current location

9:Initialize reward map $W_{1}$

10:Start at ($x_{\text{base}},\ y_{\text{base}}$)

11:$i=1$

12:while $B_{i}$ > 0 do

13: Update $W_{i}$ for available options where
$W_{i}=\gamma(M_{i})-\|M_{k}-M_{i}\|\ \forall\ k\neq i$

14: $M_{i+1}=M^{*}_{i+1}\triangleq\underset{M_{i+1}}{\mathrm{argmin}}\
W_{i}(\gamma(M_{i+1}))$ $\triangleright$ (1), (3)

15: Update $W_{i}(\cdot)$ with $D_{i}$

16: Calculate to path to ($x_{\text{base}},\ y_{\text{base}}$) from
$\gamma(M_{i})$

17: Calculate $B_{\text{min}}$

18: if $B_{i}>B_{\text{target},j}+B_{\text{min},j}$ then

19: Move to $M^{*}_{i}$ via path generated in Line 14

20: $i$++

21:Return to base using path generated in Line 16 from $M_{i-1}$

##  4 Formulating the optimization problem

In this work we formulate an optimization problem comprised of three
objectives. The performance indices for the proposed approach are total
coverage ($J_{1}$), unspent energy ($J_{2}$), and the net coverage importance
gain ($J_{3}$). To maximize the coverage, $J_{1}$, we have the cost term:

$J_{1}=l^{2}\sum_{i=1}^{n}\nu_{i}(M_{i}),$ (1)

where $i\in\\{1,\ldots,n\\}$ is the waypoint index, $l$ is size of the cell,
$M_{i}\triangleq(x_{i},y_{i})$ is the cell occupied by waypoint $i$, and
$\nu_{i}$ is a Boolean function that equals 0 when waypoint $i$ has been
accounted for and $1$ otherwise..

To minimize remaining energy upon return, the second cost term may be stated
as

$J_{2}=B_{\text{start}}-\sum_{j=1}^{m}B_{\text{target},j}-B_{\text{final}}$
(2)

where $B_{\text{start}}$ is the initial battery level at deployment,
$j\in\\{1,\ldots,m\\}$ where $j$ is the current target, $B_{\text{target}}$ is
the energy cost to move to the target from the current cell and
$B_{\text{final}}$ is the cost of returning to base from the final target
waypoint. $J_{2}$ keeps a check on the total energy left during the trajectory
planning, so as to use the available power to the maximum, while trying to
cover as many new cells as possible. It also keeps track of the power left at
a given waypoint to make sure the UAV has enough power left during its return
to visit the maximum possible uncovered waypoints, instead of having to trace
a shorter return path and overlap with the already covered points.

To maximize the allotted way point values to favor the covered area to
specific targets, we will add the following cost term:

$J_{3}=l^{2}\sum_{i=1}^{n}\gamma_{i}(M_{i})\nu_{i}(M_{i}).$ (3)

Along with other variables explained in $J_{1}$, $\gamma_{i}$ is the function
relating the reward for the cell at $M_{i}$; the higher value of which means
the point is expected to cover some crucial information and it is beneficial
to visit it. This term is unique from $J_{1}$ in that if needed we can
increase the weights of visiting new locations to counter higher weighted
reward cells. $J_{3}$ allows us to minimize the coverage importance gain, and
dictates the likely order of the area coverage. The resulting cost function
for the resulting minimization problem becomes,

$J_{\text{total}}=-\alpha_{1}J_{1}+\alpha_{2}J_{2}-\alpha_{3}J_{3},$ (4)

where $\alpha_{1},\ \alpha_{2}\text{ and }\alpha_{3}$ are the tunable weights
imposed on $J_{1}$, $J_{2}$ and $J_{3}$ respectively.

##  5 Simulation Results

We present the simulation results that show a series of trajectories chosen by
Algorithm 1 in Fig. 5 and 6. The trajectory consists of two segments, the
optimal exploration segment from the base and then to the optimal return
segment to the base. The intermediate targets are chosen by comparing the
number of cells that need to be visited to reach the position against the
values assigned to the color of the cell and the penalty of revisiting a cell.
The simulation uses a $15\times 15$ grid and creates two different types of
importance maps with nonuniform reward as discussed below. The trajectories
were also generated on multiple maps by varying the base locations and battery
capacities for performance analysis.

(a) $[x_{\text{base}},y_{\text{base}}]=[7,6]$, $B_{\text{start}}=50,\ J_{2}=4$

(b) $[x_{\text{base}},y_{\text{base}}]=[7,6]$, $B_{\text{start}}=25,\ J_{2}=3$

(c) $[x_{\text{base}},y_{\text{base}}]=[13,8]$, $B_{\text{start}}=50,\
J_{2}=2$

(d) $[x_{\text{base}},y_{\text{base}}]=[13,8]$, $B_{\text{start}}=25,\
J_{2}=1$

Figure 5: Application of the proposed algorithm on an example multivariate
Gaussian distribution map for different initial locations and battery
capacities. The red, yellow, green and blue cells denote the highest, medium,
low and lowest coverage importance, respectively. The white cell denotes the
base location and the white dots show the next instantaneous location with the
highest importance. The black line denotes the outgoing trajectory, the purple
line denotes the shortest-path return trajectory, and the arrowheads denote
the directions.

Fig. 5 presents optimal trajectory planning on a map with cell coverage
importance values following a multivariate Gaussian distribution. For (a) and
(b), the base is at $[x_{\text{base}},\ y_{\text{base}}]=[7,6]$ with battery
capacities 50 and 25 respectively and for (c) and (d), the base is at $[13,8]$
with the same battery capacities. This map is relatively simple and the
trajectories in all the sub-figures of Fig. 5 use the available battery
capacity to get to the higher reward locations and explore them efficiently
before covering the areas of lower importance, which is shown by the black
lines. The optimal return path (purple lines) is calculated at each of the
intermediate white dots (omitted to avoid cluttering) and the algorithm keeps
exploring until it reaches the critical battery level, and finally returns to
the base through the shortest path of highest reward from that particular cell
as shown in the figures.

(a) $[x_{\text{base}},y_{\text{base}}]=[7,6]$, $B_{\text{start}}=50,\ J_{2}=0$

(b) $[x_{\text{base}},y_{\text{base}}]=[7,6]$, $B_{\text{start}}=25,\ J_{2}=5$

(c) $[x_{\text{base}},y_{\text{base}}]=[13,8]$, $B_{\text{start}}=50,\
J_{2}=6$

(d) $[x_{\text{base}},y_{\text{base}}]=[13,8]$, $B_{\text{start}}=25,\
J_{2}=3$

Figure 6: Application of the proposed algorithm on the Hawai‘i island map in
Fig. 2 for different initial locations and battery capacities. The red,
yellow, green and blue cells denote the highest, medium, lowest and zero
coverage importance, respectively. The white cell denotes the base location
and the white dots show the next instantaneous location with the highest
importance. The black line denotes the outgoing trajectory, the purple line
denotes the shortest-path return trajectory, and the arrowheads denote the
directions.

Similarly, Fig. 6 shows the performance of the proposed algorithm on a
relatively complicated and real-life inspired map of (exaggerated) volcano
flow on the island of Hawai‘i. The coverage importance map is defined and
overlaid on a square grid over the map of the island of Hawai‘i as explained
in Fig. 2. For (a) and (b), the base is at $[x_{\text{base}},\
y_{\text{base}}]=[7,6]$, which corresponds to the peak of Mauna Lua, with
battery capacities 50 and 25, respectively, and for (c) and (d), the base is
at $[13,8]$, which corresponds to the general location of the city of Hilo,
with the same battery capacities. Even for this complicated map, the
trajectories in all the sub-figures of Fig. 6 efficiently perform the
exploration of the higher importance areas before covering the areas of lower
importance (shown by the black trajectory). The proposed algorithm performs
generally well for both the maps as the UAV is always able to return to base
and chooses a route that maximizes reward and minimizes revisiting penalties.
The minimization of unspent energy varies depending on the geometry of cells
of high reward and the path taken thus far.

In Fig. 7 we varied the starting battery capacities for all the maps and
initial location scenarios in Figs. 5 and 6 to analyze the reward performance
of the proposed algorithm. Since the Hawai‘i island map has more higher
importance cells, the total accumulated reward has higher values for both base
locations for. In addition, the overall performance is similar for the
different initial locations and initial battery capacities, which shows that
the trajectories generated are fairly consistent and optimal. For the two
probability maps the differences are most likely due to the clustering of the
high importance cells in the Gaussian map. This is ultimately the result of
only looking one step forward when determining a target and a respective
return path. As we can see in Fig. 5(c), if there was additional energy, the
next immediate step would cause the trajectory to circumnavigate the visited
cells, which is also shown in Gaussian Map (Base-2) plot of Fig. 7, where
there was not enough energy to reach the next target waypoint after the 30
energy run until 50 energy was provided. The algorithm then followed the same
trajectory for the 40 energy trial as it did for the 30 energy and took the
penalty. The trajectory planning and reward performance results are in
agreement with the algorithm expectations and show its applicability for a
range of maps and scenarios.

Figure 7: Comparison of the trajectory performance in terms of the total
reward collected for the maps and base locations seen in Figs 5 and 6 with by
varying the initial battery capacities ($B_{\text{start}}=20,30,40,50,60$).
The cell importance scores are set as $0,1,10\text{ and }100$ (blue, green,
yellow and red), and the penalty for revisiting a trial was $-100$ and an
additional penalty of $-1$ for each unspent energy tile. Base 1 and Base 2 for
both maps correspond to the initial locations $[7,6]\text{ and }[13,8]$,
respectively.

##  6 Conclusion

In this work we presented an algorithm that finds the optimal paths through a
survey area with nonuniform importance to maximize the coverage and net
coverage importance reward for a given initial battery capacity. We follow a
set of dynamically generated trajectories dependent on the surrounding weights
of the area. The UAV navigates through the area while avoiding the visited
locations to maximize the exploration gain while being constrained by the
minimum power to return to base. Further research will include dynamic
targeting system that takes into account the weights of all possible points
that can be visited in between targets as well as linking the trajectories
that the UAV uses to the optimal energy required for the specific
configuration and the total battery power allowance. We are also working on
coordination of UAVs for exploration of a large area, where the sensing
workload will be divided by an optimal mission coordination algorithm.
Inclusion of wind (static and dynamic) over the deployment area and its effect
on trajectory optimization is another important problem under investigation.

##  7 Acknowledgments

Snyder, Shriwastav, and Morrison-Fogel would like to thank Dr. Reza Ghorbani
at the University of Hawai‘i at Mānoa for his insights and extensive
discussions on this work. Song would like to thank the support from the
National Science Foundation under awards CISE/IIS-2024928 and OIA-2032522.

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