#  Optimal Control with Broken Symmetry of
Multi-Agent Systems on Lie Groups

Efstratios Stratoglou, Leonardo Colombo and Tomoki Ohsawa E. Stratoglou
<EMAIL_ADDRESS>is with Universidad Politécnica de Madrid
(UPM), 28006 Madrid, Spain. L. Colombo<EMAIL_ADDRESS>is
with Centre for Automation and Robotics (CSIC-UPM), Ctra. M300 Campo Real, Km
0,200, Arganda del Rey - 28500 Madrid, Spain. T. Ohsawa<EMAIL_ADDRESS>is with Department of Mathematical Sciences, The University of Texas at
Dallas, 800 W Campbell Rd, Richardson, TX 75080-3021. This work was partially
supported by NSF grant CMMI-1824798.

###### Abstract

In this paper we study reduction by symmetry for optimality conditions in
optimal control problems of left-invariant affine multi-agent control systems,
with partial symmetry breaking cost functions. Our approach emphasizes the
role of variational principles. Specifically, we recast the optimal control
problem as a constrained variational problem with a partial symmetry breaking
Hamiltonian and obtain the reduced optimality conditions from a reduced
variational principle via Pontryagin Maximum Principle. We apply the results
to a collision avoidance problem for multiple unicycles in the presence of an
obstacle.

##  I Introduction

Lie groups symmetries appear naturally in many control systems problems [2],
[5], [11]. Examples of invariant control problems on Lie groups include motion
planning for underwater vehicles [19], conflict resolution in differential
games [27], collective motion in biological models [16], controlled
Lagrangians [3, 7], and coordination of multi-agent systems [26].

The goal of this work is to study symmetry reduction in optimal control
problems for left-invariant affine multi-agent control systems on Lie groups
with partially broken symmetries. More specifically, cost functions that break
some of the symmetries but not all. Symmetry breaking is a common phenomena in
several physical contexts [10]. The simplest example is the spinning top
dynamics (the motion of a rigid body with a fixed point under a gravitational
field), where due to the presence of gravity, we get a dynamics that is
$SO(2)$-invariant but not $SO(3)$-invariant (i.e., it is invariant under a
subgroup of the full symmetry group), contrary to what happens for the free
rigid body [12]. In the context of motion planning, symmetry breaking appears
naturally in the form of a barrier function as we shall see in this work.

Symmetries in the condition of Pontryagin’s maximum principle (PMP) were
studied in [5], [6], [8], [18], [23], [24]. As an alternative to the PMP, a
Lagrangian approach can be employed to exploit symmetries in the necessary
conditions for optimal control problems. In the work [4], Lagrangian systems
with broken symmetries have been applied to optimal control problems on Lie
groups for the single agent case. Also, in the single agent case, optimal
control on semidirect products was also studied by using a Clebsch formulation
[9]. Both approaches provide coordinate-free equations describing necessary
conditions for extrema on vector spaces.

Hence, in this work we do not solve differential equations on manifolds which
usually requires embedding the manifold into an Euclidean space, and thus
increasing the dimension of the system with additional constraints
(differentiable manifolds are modeled locally on normed spaces and Lie groups
are examples of differential manifolds). Alternatively, in this work we
provide reduced equations on vector spaces which is useful in practice. In
addition, in this work we build on the results of [4] by considering also
obstacles in the configuration space that agents should avoid while they also
avoid mutual collision.

The structure of the paper is as follows. Section II reviews the basic
concepts of mechanics on manifolds, in particular, mechanics on Lie groups.
Sections III describes left-invariant affine multi-agent control system and
the problem addressed. Section IV presents the main results of the work, where
we develop the reduction for optimality conditions in the optimal control
problem. Section V illustrates the employment of the results with an example
of three unicycles avoiding collisions and obstacles. Conclusions and future
work are discussed in Section VI.

##  II Background

###  II-A Mechanics on Manifolds

Let $Q$ be a differentiable manifold, the configuration space of a mechanical
system, with $\dim Q=n$. By $TQ$ we denote the tangent bundle of $Q$ and by
$T^{*}Q$ its cotangent bundle. In local coordinates $Q$ is described by the
positions $q=(q^{1},\dots,q^{n})$, a tangent vector $v_{q}=(q,\dot{q})\in TQ$
describes positions and the velocities
$(q,\dot{q})=(q^{1},\dots,q^{n},\dot{q}^{1},\dots,\dot{q}^{n})$ and a
cotangent vector $p_{q}=(q,p)=(q^{1},\dots,q^{n},p_{1},\dots,p_{n})\in T^{*}Q$
describes positions and the momenta.

###### Definition 1

Consider a differentiable function $f:Q\to M$ where $Q$ and $M$ are smooth
manifolds. The tangent lift (or tangent map) of $f$ at $q\in Q$ is the map
$T_{q}f:T_{q}Q\to T_{f(q)}M$ and the cotangent lift (or cotangent map) of $f$
is the map $T^{*}_{q}f:T^{*}_{s}M\to T^{*}_{q}Q$ defined by

$\langle
T^{*}_{q}f(\alpha_{s}),v_{q}\rangle=\langle\alpha_{s},T_{q}f(v_{q})\rangle$
(1)

where $\alpha_{s}\in T^{*}_{s}M,\;v_{q}\in T_{q}Q,s=f(q)$ and
$\langle\cdot,\cdot\rangle$ denotes the natural dual pairing between covectors
and tangent vectors.

The dynamics of a mechanical system is described by the equations of motion
determined by a Lagrangian function $L:TQ\to\mathbb{R}$ given by
$L(q,\dot{q})=K(q,\dot{q})-V(q)$, where $K:TQ\to\mathbb{R}$ denotes the
kinetic energy and $V:Q\to\mathbb{R}$ the potential energy of the system. The
equations of motion are given by the Euler-Lagrange equations
$\displaystyle{\frac{d}{dt}\bigg{(}\frac{\partial
L}{\partial\dot{q}^{i}}\bigg{)}-\frac{\partial L}{\partial
q^{i}}=0,\;\;i=1,\dots,n}$, which determine a system of second-order
differential equations. We assume the Lagrangian function is hyperregular and
hence, through the Legendre transform $p=\frac{\partial L}{\partial\dot{q}}$,
the Euler-Lagrange equations are equivalent to Hamilton’s equations of motion
$\displaystyle{\dot{q}=\frac{\partial H}{\partial
p},\;\;\dot{p}=-\frac{\partial H}{\partial q}}$, for the Hamiltonian
$H(q,p)=\langle p,\dot{q}(q,p)\rangle-L(q,\dot{q}(q,p)),$ where $\dot{q}(q,p)$
is the second component of the inverse Legendre transform. Hamilton’s
equations can be defined equivalently in terms of the canonical Poisson
bracket as follows.

###### Definition 2

Let $P$ be a smooth manifold. A Poisson bracket on $P$ is a bilinear, skew-
symmetric operator $\\{\cdot,\cdot\\}:C^{\infty}(P)\times C^{\infty}(P)\to
C^{\infty}(P)$ satisfying the Jacobi identity and the Leibniz rule. The pair
$(P,\\{\cdot,\cdot\\})$ is called Poisson manifold. For any Hamiltonian
function $H:P\to\mathbb{R}$ the corresponding Hamiltonian vector field,
$X_{H}$, which describes the equations of motion, is uniquely determined by
$\dot{F}=\\{F,H\\}$ for all smooth $F:P\to\mathbb{R}$, along solutions to
$X_{H}$ (see Def $4.22$ in [13]).

Consider the cotangent bundle of a manifold $Q$, for $F,G\in
C^{\infty}(T^{*}Q)$ the pair $(T^{*}Q,\\{\cdot,\cdot\\})$ is a Poisson
manifold with Poisson bracket $\\{F,G\\}=\sum_{i=1}^{n}\bigg{(}\frac{\partial
F}{\partial q_{i}}\cdot\frac{\partial G}{\partial p_{i}}-\frac{\partial
F}{\partial p_{i}}\cdot\frac{\partial G}{\partial q_{i}}\bigg{)}$. This
bracket is called the canonical Poisson bracket (for more details see [13],
[20]).

###  II-B Mechanics on Lie groups

If the configuration space is a Lie group $G$, and the Lagrangian possesses
the full $G$-symmetry, Euler-Lagrange equations can be reduced to a first-
order system of differential equations.

###### Definition 3

Let $G$ be a Lie group and $Q$ a smooth manifold. A left-action of $G$ on $Q$
is a smooth map $\Phi:G\times Q\to Q$ such that $\Phi(e,g)=g$ and
$\Phi(h,\Phi(g,q))=\Phi(hg,q)$ for all $g,h\in G$ and $q\in Q$, where $e$ is
the identity of the group $G$ and the map $\Phi_{g}:Q\to Q$ given by
$\Phi_{g}(q)=\Phi(g,q)$ is a diffeomorphism for all $g\in G$.

###### Definition 4

A function $f:Q\to\mathbb{R}$ is called invariant under an action $\Phi_{g}$
if $f\circ\Phi_{g}=f$ for any $g\in G$.

For a finite dimensional Lie group $G$, its Lie algebra $\mathfrak{g}$ is
defined as the tangent space to $G$ at the identity, $\mathfrak{g}:=T_{e}G$.
Let $L_{g}:G\to G$ be the left translation of the element $g\in G$ given by
$L_{g}(h)=gh$ where $h\in G$. It is a left-action and a diffeomorphism on $G$.
The tangent map of $L_{g}$ at $h\in G$ is $T_{h}L_{g}:T_{h}G\to T_{gh}G$ and
the cotangent map is $T^{*}_{h}L_{g}:T^{*}_{gh}G\to T^{*}_{h}G$. By
$\text{ad}:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g},$
$(\xi,\eta)\mapsto\text{ad}_{\xi}\eta$ we denote the adjoint operator which is
defined by the Lie bracket on $T_{e}G$, i.e. $\text{ad}_{\xi}\eta=[\xi,\eta]$,
and by $\text{ad}^{*}:\mathfrak{g}\times\mathfrak{g}^{*}\to\mathfrak{g}^{*}$,
$(\xi,\mu)\mapsto\text{ad}^{*}_{\xi}\mu$ the coadjoint operator given by
$\langle\text{ad}^{*}_{\xi},\mu,\eta\rangle=\langle\mu,\text{ad}_{\xi}\eta\rangle$
where $\eta\in\mathfrak{g}$ and
$\langle\cdot,\cdot\rangle:\mathfrak{g}^{*}\times\mathfrak{g}\to\mathbb{R}$ is
the pairing between vectors and covectors (see [28] and [20] for instance).

###### Definition 5

Denote by $\mathfrak{X}(G)$ the set of vector fields on $G$. A vector field
$X:G\to TG,$ $h\mapsto X(h),$ is called left-invariant if
$T_{h}L_{g}(X(h))=X(L_{g}(h))$, for all $g,h\in G.$

###### Definition 6

Consider a Lie group $G$, a vector space $W$ and the representation of $G$ on
$W$ given by $\rho:G\times W\to W$, $(g,v)\mapsto\rho_{g}(v)$, which is a left
action, and it is defined by the relation
$\rho_{g_{1}}(\rho_{g_{2}}(v))=\rho_{g_{1}g_{2}}(v)$, $g_{1},g_{2}\in G$. Its
dual is given by $\rho^{*}:G\times W^{*}\to W^{*},$
$(g,\alpha)\mapsto\rho^{*}_{g}(\alpha),$ satisfying
$\langle\rho^{*}_{g^{-1}}(\alpha),v\rangle=\langle\alpha,\rho_{g^{-1}}(v)\rangle$.

The infinitesimal generator of the left action of $G$ on $W$ is
$\rho^{\prime}:\mathfrak{g}\times W\to W,$
$(\xi,v)\mapsto\rho^{\prime}(\xi,v)=\frac{d}{dt}|_{t=0}\rho_{e^{t\xi}}(v).$
For every $v\in W$ consider the linear tranformation
$\rho^{\prime}_{v}:\mathfrak{g}\to W,$
$\xi\mapsto\rho^{\prime}_{v}(\xi)=\rho^{\prime}(\xi,v)$ and its dual
$\rho^{\prime*}_{v}:W^{*}\to\mathfrak{g}^{*},$
$\alpha\mapsto\rho^{\prime*}_{v}(\alpha).$ The last transformation defines the
momentum map $\textbf{J}_{W}:W\times W^{*}\to\mathfrak{g}^{*},$
$(v,\alpha)\mapsto\textbf{J}_{W}(v,\alpha):=\rho^{\prime*}_{v}(\alpha)$ such
that for every $\xi\in\mathfrak{g}$,
$\displaystyle{\langle\textbf{J}_{W}(v,\alpha),\xi\rangle=\langle\rho^{\prime*}_{v}(\alpha),\xi\rangle=\langle\alpha,\rho^{\prime}_{v}(\xi)\rangle=\langle\alpha,\rho^{\prime}(\xi,v)\rangle}$.

For $\xi\in\mathfrak{g}$, consider the map $\rho^{\prime}_{\xi}:W\to W$,
$v\mapsto\rho^{\prime}_{\xi}(v)=\rho^{\prime}(\xi,v)$ and its dual
$\rho^{\prime*}_{\xi}:W^{*}\to W^{*}$,
$\alpha\mapsto\rho^{\prime*}_{\xi}(\alpha)$ such that
$\langle\rho^{\prime*}_{\xi}(\alpha),v\rangle=\langle\alpha,\rho^{\prime}_{\xi}(v)\rangle$,
and hence it satisfies
$\displaystyle{\langle\mathbf{J}_{W}(v,\alpha),\xi\rangle=\langle\alpha,\rho^{\prime}(\xi,v)\rangle=\langle\alpha,\rho^{\prime}_{\xi}(v)\rangle=\langle\rho^{\prime*}_{\xi}(\alpha),v\rangle}$.
See [13] and [20] for more details on the momentum map.

##  III Left-invariant multi-agent control systems

Denote by $\mathcal{N}$ a set consisting of $s$ free agents, and by
$\mathcal{E}\subset\mathcal{N}\times\mathcal{N}$ the set describing the
interaction between agents. The neighbor relationships are described by an
undirected graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$, where $\mathcal{N}$
is the set of vertices and $\mathcal{E}$ the set of edges for $\mathcal{G}$.
We further assume $\mathcal{G}$ is static and connected. We assume that each
agent occupies a circular area of radius $\overline{r}$. For every agent
$i\in\mathcal{N}$ the set
$\mathcal{N}_{i}=\\{j\in\mathcal{N}:(i,j)\in\mathcal{E}\\}$ denotes the
neighbors of that agent. The agent $i\in\mathcal{N}$ evolves on an
$n$-dimensional Lie group $G$ and its configuration is denoted by $g_{i}\in
G$. We denote by $\bf{G}$ and by $T_{\bf{e}}\bf{G}=:\bar{\mathfrak{g}}$ the
cartesian products of $s$ copies of $G$ and $\mathfrak{g}$, respectively,
where $\textbf{e}=\underbrace{(e,e,\dots,e)}_{s-copies}$ is the identity of
$\bf{G}$ with $e$ the identity element of the of $G$.

For each agent $i\in\mathcal{N}$ there is an associated left-invariant control
system described by the kinematic equations

$\dot{g}_{i}=T_{e}L_{g_{i}}(u_{i}),\;\;g_{i}(0)=g^{0}_{i},$ (2)

where $g_{i}(t)\in C^{1}([0,T],G),\;T\in\mathbb{R}$ fixed,
$u_{i}\colon[0,T]\to\mathfrak{g}.$ is the control input and $g^{i}_{0}\in G$
is considered as the initial state condition. Note that while for each
$i\in\mathcal{N}$, $\dim\mathfrak{g}=n$ with
$\mathfrak{g}=\hbox{span}\\{e_{1},e_{2},\ldots,e_{n}\\}$, then the control
inputs may be described by $u_{i}=[u_{i}^{1},u_{i}^{2},\dots,u_{i}^{m}]^{T}$,
where $u_{i}(t)\in\mathbb{R}^{m}$, with $m\leq n$. Hence, the control input
for each agent is given by
$\displaystyle{u_{i}(t)=e_{0}+\sum_{k=1}^{m}u^{k}_{i}(t)e_{k}}$, where
$e_{0}\in\mathfrak{g}.$ Thus, the left-invariant control systems (2) for each
agent $i\in\mathcal{N}$ can be written as

$\dot{g}_{i}(t)=g_{i}(t)\bigg{(}e_{0}+\sum_{k=1}^{m}u^{k}_{i}(t)e_{k}\bigg{)}.$
(3)

The problem under study consists on finding reduced necessary conditions for
optimality in an optimal control problem for left-invariant multi-agent
control system. These solution curves should minimize some cost function,
prevent agent collision while also agents avoid static obstacles in the
configuration space.

Problem (collision and obstacle avoidance problem): Find reduced optimality
conditions on $g(t)=(g_{1}(t),g_{2}(t),\dots,g_{s}(t))\in\textbf{G}$ and the
controls $u(t)=(u_{1}(t),u_{2}(t),\dots,u_{s}(t))\in\bar{\mathfrak{g}}$ such
that $(g(t),u(t))\in\textbf{G}\times\bar{\mathfrak{g}}$ minimize the cost

$\displaystyle\min_{(g,u)}\sum_{i=1}^{s}\int_{0}^{T}\Big{(}C_{i}(g_{i}(t),u_{i}(t))$
$\displaystyle+U_{i}^{0}(g_{i})$ (4)
$\displaystyle+\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}U_{ij}(g_{i}(t),g_{j}(t))\Big{)}dt$

subject to the kinematics $\dot{g}_{i}(t)=T_{e}L_{g_{i}(t)}(u_{i}(t))$,
boundary values $g(0)=(g_{1}(0),\dots,g_{s}(0))=(g_{1}^{0},\dots,g_{s}^{0})$
and $g(T)=(g_{1}(T),\dots,g_{s}(T))=(g_{1}^{T},\dots,g_{s}^{T})$, under the
following assumptions:

  1. (i)

There is a left representation $\rho$ of $G$ on a vector space $W$, i.e.,
$\rho:G\to\mathrm{GL}(W)$ is a homomorphism.

  2. (ii)

$C_{i}:G\times\mathfrak{g}\to\mathbb{R}$ are $G$-invariant functions for each
$i\in\mathcal{N}$ (under a suitable left action of $G$ on
$G\times\mathfrak{g}$, which will be defined shortly) and is also sufficiently
regular.

  3. (iii)

$U_{ij}:G\times G\to\mathbb{R}$ (collision avoidance potential functions) are
$G$-invariant functions under $\Phi$, defined by

$\displaystyle\Phi:G\times(G\times G)$ $\displaystyle\longrightarrow G\times
G,$ (5) $\displaystyle(g,(g_{1},g_{2}))$
$\displaystyle\longmapsto(L_{g}(g_{1}),L_{g}(g_{2})),$

i.e., $U_{ij}\circ\Phi_{g}=U_{ij}$, for any $g\in G$, that is,
$U_{ij}(L_{g}(g_{i}),L_{g}(g_{j}))=U_{ij}(g_{i},g_{j})$, for any
$(g_{i},g_{j})\in\mathcal{E}$, $j\in\mathcal{N}_{i}$ and they are also
sufficiently regular.

  4. (iv)

$U_{i}^{0}:G\to\mathbb{R}$ (obstacle avoidance potential functions) are not
$G$-invariant functions and they are also sufficiently regular, for
$i\in\mathcal{N}$.

  5. (v)

The obstacle avoidance functions $U_{i}^{0}$ depend on a parameter
$\alpha_{0}\in W^{*}$ for each agent $i\in\mathcal{N}$. Hence, we can define
the extended potential function as $U_{i,0}^{\textnormal{ext}}:G\times
W^{*}\to\mathbb{R}$, with $U_{i,0}^{\text{ext}}(\cdot,\alpha_{0})=U_{i}^{0}$.

  6. (vi)

The extended obstacle avoidance functions are $G$-invariant under
$\tilde{\Phi}$, defined by

$\displaystyle\tilde{\Phi}:G\times(G\times W^{*})$
$\displaystyle\longrightarrow G\times W^{*},$ (6)
$\displaystyle(g,(h,\alpha))$
$\displaystyle\longmapsto(L_{g}(h),\rho_{g^{-1}}^{*}(\alpha)),$

where $\rho_{g^{-1}}^{*}\in\mathrm{GL}(W^{*})$ is the adjoint of
$\rho_{g^{-1}}\in\mathrm{GL}(W)$, i.e.,
$U_{i,0}^{\textnormal{ext}}\circ\tilde{\Phi}_{g}=U_{i,0}^{\textnormal{ext}}$,
for any $g\in G$, or
$U_{i,0}^{\textnormal{ext}}(L_{g}(h),\rho_{g^{-1}}^{*}(\alpha))=U_{i,0}^{\textnormal{ext}}(g,\alpha)$
where $\alpha\in W^{*}$.

  7. (vii)

The obstacle avoidance potential functions are invariant under the left action
of the isotropy group

$G_{\alpha_{0}}=\\{g\in G\mid\rho_{g}^{*}(\alpha_{0})=\alpha_{0}\\}.$ (7)

Note that $G\times\mathfrak{g}$ is a trivial vector bundle over $G$ and define
the left action of $G$ on $G\times\mathfrak{g}$ as follows

$\displaystyle\Psi:G\times(G\times\mathfrak{g})$
$\displaystyle\longrightarrow(G\times\mathfrak{g}),$ $\displaystyle(g,(h,u))$
$\displaystyle\longmapsto(L_{g}(h),u).$ (8)

We further assume that $C_{i}:G\times\mathfrak{g}\to\mathbb{R}$ is
$G$-invariant under (8), i.e., $C_{i}\circ\Psi_{g}=C_{i}$, for any $g\in G$.

##  IV Reduction of necessary conditions

We can solve the proposed problem as a constrained problem using the method of
Lagrange multipliers (see, e.g, [2], [4], [15]) or by using the Pontryagin
Maxiumum principle (PMP) [17], [16], [24]. Our approach is based on the PMP.

Define the augmented Hamiltonian function
$H_{a}:\textbf{G}\times\bar{\mathfrak{g}}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s}\to\mathbb{R}$
as follows

$\begin{split}H_{a}(g,u,\mu,\alpha)=&\sum_{i=1}^{s}\bigg{[}\langle\mu_{i},u_{i}-e_{0}\rangle-
C_{i}(g_{i},u_{i}-e_{0})\\\
&-U_{i,0}^{\text{ext}}(g_{i},\alpha)-\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}U_{ij}(g_{i},g_{j})\bigg{]},\end{split}$

where $\mu(\cdot)\in C^{1}([0,T],\mathfrak{g}^{*})$. By applying all actions
of the previous section component-wise each component of the augmented
Hamiltonian is $G$-invariant thus we define the reduced augmented Hamiltonian
$h_{a}:G^{s-1}\times\bar{\mathfrak{g}}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s}\to\mathbb{R}$,
which is given by

$\begin{split}h_{a}(g,u,\mu,\alpha)=&\sum_{i=1}^{s}\bigg{[}\langle\mu_{i},u_{i}-e_{0}\rangle-
C_{i}(u_{i}-e_{0})\\\
&-U_{i,0}^{\text{ext}}(\alpha_{i})-\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}U_{ij}(g_{i}^{-1}g_{j})\bigg{]},\end{split}$

where $\alpha_{i}=\rho^{*}_{g_{i}}(\alpha)$ and with a slight abuse of
notation, we write $C_{i}(e,u_{i}-e_{0})=C_{i}(u_{i}-e_{0})$,
$U_{i,0}^{\text{ext}}(\alpha_{i})=U_{i,0}^{\text{ext}}(e,\alpha_{i})$ and
$U_{ij}(e,g_{i}^{-1}g_{j})=U_{ij}(g_{i}^{-1}g_{j})$.

Next, we can employ the Pontryagin Maximum Principle for the reduced augmented
Hamiltonian $h_{a}$. Therefore, it follows that necessary conditions for
normal extrema for the OCP can be obtained by the integral curves of the
Hamiltonian vector field for the reduced optimal Hamiltonian
$h:G^{s-1}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s}\to\mathbb{R}$ given by

$h(g,\mu,\alpha)=\max_{u}h_{a}(g,u,\mu,\alpha)=h_{a}(g,u^{*},\mu,\alpha),$

where $u^{*}$ denotes the optimal control. The reduced optimal Hamiltonian
$h:G^{s-1}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s}\to\mathbb{R}$ can now
be obtained and is given by

$\begin{split}h(g,\mu,\alpha)=&\sum_{i=1}^{s}\bigg{[}\langle\mu_{i},u^{*}_{i}-e_{0}\rangle-
C_{i}(u^{*}_{i}-e_{0})-U_{i,0}^{\text{ext}}(\alpha_{i})\\\
&-\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}U_{ij}(g_{i}^{-1}g_{j})\bigg{]},\end{split}$

where $\alpha_{i}=\rho^{*}_{g_{i}}(\alpha)$.

Hamilton’s equations of motion are determined from the equation
$\dot{F}=\\{F,h\\}$ where $\\{\cdot,\cdot\\}$ is the Lie-Poisson bracket on
$G^{s-1}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s}$ and
$F\in\mathcal{C}^{\infty}(G^{s-1}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s})$.
The left hand side of Hamilton’s equations is
$\displaystyle{\dot{F}=\frac{dF}{dt}=DF\cdot(\dot{g},\dot{\mu},\dot{\alpha})=\Big{\langle}(\dot{g},\dot{\mu},\dot{\alpha}),\frac{\partial
F}{\partial(g,\mu,\alpha)}\Big{\rangle}}$. By Prop. 4.4.1 of [1], the Lie-
Poisson bracket on $G^{s-1}\times\bar{\mathfrak{g}}^{*}$ is given by

$\begin{split}\\{F,h\\}(g,\mu)=&\sum_{i=1}^{s}\Bigg{[}\Big{\langle}T_{e}^{*}L_{g_{i}}\Big{(}\frac{\partial
F}{\partial g_{i}}\Big{)},\frac{\partial h}{\partial\mu_{i}}\Big{\rangle}\\\
&+\Big{\langle}T_{e}^{*}L_{g_{i}}\Big{(}\frac{\partial h}{\partial
g_{i}}\Big{)},\frac{\partial
F}{\partial\mu_{i}}\Big{\rangle}-\Big{\langle}\mu_{i},\Big{[}\frac{\partial
F}{\partial\mu_{i}},\frac{\partial
h}{\partial\mu_{i}}\Big{]}\Big{\rangle}\Bigg{]},\end{split}$

where $\frac{\partial F}{\partial g_{i}}$ and $\frac{\partial h}{\partial
g_{i}}$ are seen as covectors and $[\cdot,\cdot]$ is the Lie bracket on the
Lie algebra $\mathfrak{g}$. We apply this Lie-Poisson bracket in our setting
hence, so the Lie-Poisson bracket on
$G^{s-1}\times\bar{\mathfrak{g}}^{*}\times(W^{*})^{s}$ is given by

$\begin{split}\\{F,h\\}(g,\mu,\alpha)=&\sum_{i=1}^{s}\Bigg{[}\Big{\langle}T_{e}^{*}L_{g_{i}}\Big{(}\frac{\partial
F}{\partial g_{i}}\Big{)},\frac{\partial h}{\partial\mu_{i}}\Big{\rangle}\\\
+&\Big{\langle}T_{e}^{*}L_{g_{i}}\Big{(}\frac{\partial h}{\partial
g_{i}}\Big{)},\frac{\partial
F}{\partial\mu_{i}}\Big{\rangle}-\Big{\langle}\mu_{i},\Big{[}\frac{\partial
F}{\partial\mu_{i}},\frac{\partial h}{\partial\mu_{i}}\Big{]}\Big{\rangle}\\\
-&\Big{\langle}\alpha_{i},\frac{\partial F}{\partial\mu_{i}}\frac{\partial
h}{\partial\alpha_{i}}-\frac{\partial h}{\partial\mu_{i}}\frac{\partial
F}{\partial\alpha_{i}}\Big{\rangle}\Bigg{]},\end{split}$ (9)

(for more details one can see Section $9.6$ of [13]).

From the first term of the previous equation, through the property of the
cotangent lift (1), we get:

$\sum_{i=1}^{s}\Big{\langle}T_{e}^{*}L_{g_{i}}\Big{(}\frac{\partial
F}{\partial g_{i}}\Big{)},\frac{\partial
h}{\partial\mu_{i}}\Big{\rangle}=\sum_{i=1}^{s}\Big{\langle}T_{e}L_{g_{i}}\Big{(}\frac{\partial
h}{\partial\mu_{i}}\Big{)},\frac{\partial F}{\partial g_{i}}\Big{\rangle}.$

The Lie bracket in equation (9) is the adjoint map, thus each component of the
sum of the third term yields

$\begin{split}\Big{\langle}\mu_{i},\Big{[}\frac{\partial
F}{\partial\mu_{i}},\frac{\partial
h}{\partial\mu_{i}}\Big{]}\Big{\rangle}&=\Big{\langle}\mu_{i},\text{ad}_{\frac{\partial
F}{\partial\mu_{i}}}\frac{\partial
h}{\partial\mu_{i}}\Big{\rangle}=-\Big{\langle}\mu_{i},\text{ad}_{\frac{\partial
h}{\partial\mu_{i}}}\frac{\partial F}{\partial\mu_{i}}\Big{\rangle}\\\
&=-\Big{\langle}\text{ad}^{*}_{\frac{\partial
h}{\partial\mu_{i}}}\mu_{i},\frac{\partial
F}{\partial\mu_{i}}\Big{\rangle}.\end{split}$

The last term can be written as

$\sum_{i=1}^{s}\Bigg{[}\Big{\langle}\alpha_{i},\frac{\partial
F}{\partial\mu_{i}}\frac{\partial
h}{\partial\alpha_{i}}\Big{\rangle}-\Big{\langle}\alpha_{i},\frac{\partial
h}{\partial\mu_{i}}\frac{\partial
F}{\partial\alpha_{i}}\Big{\rangle}\Bigg{]}=\\\
=\sum_{i=1}^{s}\Bigg{[}\Big{\langle}\mathbf{J}_{W}\Big{(}\frac{\partial
h}{\partial\alpha_{i}},\alpha_{i}\Big{)},\frac{\partial
F}{\partial\mu_{i}}\Big{\rangle}+\Big{\langle}\frac{\partial
h}{\partial\mu_{i}}\alpha_{i},\frac{\partial
F}{\partial\alpha_{i}}\Big{\rangle}\Bigg{]},$

where $\mathbf{J}_{W}$ is the momentum map and the first entry of the last
term is the dual of the infinitesimal generator of the left action of $G$ on
$W$. Thus, for the reduced Hamiltonian function Lie-Poisson equations of
motion, for each agent, are given by:

$\displaystyle\dot{g_{i}}=T_{e}L_{g_{i}}(u_{i}^{*}-e_{0}),$
$\displaystyle\dot{\mu_{i}}=\text{ad}^{*}_{u_{i}}\mu_{i}+\sum_{j\in\mathcal{N}_{i}}T^{*}_{e}L_{g_{i}}\Big{(}\frac{\partial
U_{ij}}{\partial g_{i}}\Big{)}-\mathbf{J}_{W}\Big{(}\frac{\partial
U_{i,0}^{\text{ext}}}{\partial\alpha_{i}},\alpha_{i}\Big{)},$
$\displaystyle\dot{\alpha_{i}}=-\rho^{\prime*}_{u_{i}}(\alpha_{i}).$ (10)

##  V Application to optimal control of unicycles

A unicycle is a homogeneous disk rolling vertically on a horizontal plane.
Consider $s$ such unicycles. The configuration space whose elements determine
the motion of each unicycle is $SE(2)\cong SO(2)\times\mathbb{R}$. An element
$g_{i}\in SE(2)$ is given by
$g_{i}=\begin{pmatrix}\cos\theta_{i}&-\sin\theta_{i}&x_{i}\\\
\sin\theta_{i}&\cos\theta_{i}&y_{i}\\\ 0&0&1\end{pmatrix}$, where
$(x_{i},y_{i})\in\mathbb{R}^{2}$ represents the point of contact of the disk
with the horizontal plane and $\theta_{i}$ represents the angular orientation
of the unicycle. The control inputs, for each agent, are given by
$u_{i}=(u_{i}^{1},u_{i}^{2})$ where $u_{i}^{1}$ denotes the steering speed and
$u_{i}^{2}$ denotes the rolling speed of the unicycle. For more details see
[2].

The kinematic equations for the multi-agent system are:

$\dot{x_{i}}=u_{i}^{2}\cos\theta_{i},\;\;\dot{y_{i}}=u_{i}^{2}\sin\theta_{i},\;\;\dot{\theta_{i}}=u_{i}^{1},\,i=1,\ldots,s.$
(11)

The Lie algebra $\mathfrak{se}(2)$ of $SE(2)$ is described by
$\mathfrak{se}(2)=\Big{\\{}\begin{pmatrix}A&b\\\
0&0\end{pmatrix}:A\in\mathfrak{so}(2)\text{ and }b\in\mathbb{R}^{2}\Big{\\}}$
where $A=-aJ$ with $a\in\mathbb{R}$ and $J=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}$. The Lie algebra $\mathfrak{se}(2)$ is identified with
$\mathbb{R}^{2}$ through the isomorphism $\begin{pmatrix}-aJ&b\\\
0&0\end{pmatrix}\mapsto(a,b).$ The elements of the basis of the Lie algebra
$\mathfrak{se}(2)$ are

$e_{1}=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},e_{2}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
0&0&0\end{pmatrix},e_{3}=\begin{pmatrix}0&0&0\\\ 0&0&1\\\ 0&0&0\end{pmatrix},$

which satisfy $[e_{1},e_{2}]=e_{3},\;[e_{2},e_{3}]=0,\;[e_{3},e_{1}]=e_{2}$.
Thus, the kinematic equations (11) take the form
$\dot{g_{i}}=g_{i}u_{i}=g_{i}(u_{i}^{1}e_{1}+u_{i}^{2}e_{2})$ and give rise to
a left-invariant control system on $SE(2)^{s}\times\mathfrak{se}(2)^{s}$.

The inner product on $\mathfrak{se}(2)$ is given by
$\langle\xi_{1},\xi_{2}\rangle=\operatorname{tr}(\xi_{1}^{T}\xi_{2})$ for
$\xi_{1},\xi_{2}\in\mathfrak{se}(2)$ and hence, the norm is given by
$\|\xi\|=\sqrt{\operatorname{tr}(\xi^{T}\xi)},$ for any
$\xi\in\mathfrak{se}(2)$.

The dual Lie algebra $\mathfrak{se}(2)^{*}$ of $SE(2)$ is defined through the
dual pairing, $\langle\alpha,\xi\rangle=\operatorname{tr}(\alpha\xi)$, where
$\alpha\in\mathfrak{se}(2)^{*}$ and $\xi\in\mathfrak{se}(2)$ hence, the
elements of the basis of $\mathfrak{se}(2)^{*}$ are

$e^{1}=\begin{pmatrix}0&\frac{1}{2}&0\\\ -\frac{1}{2}&0&0\\\
0&0&0\end{pmatrix},e^{2}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
1&0&0\end{pmatrix},e^{3}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&1&0\end{pmatrix}.$

Consider the cost function $C_{i}(g_{i},u_{i})=\frac{1}{2}\langle
u_{i},u_{i}\rangle$ and the artificial potential function $U_{ij}:SE(2)\times
SE(2)\to\mathbb{R}$ given by
$\displaystyle{U_{ij}(g_{i},g_{j})=\frac{\sigma_{ij}}{2((x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}-4\overline{r}^{2})}}$,
where $\sigma_{ij}\geq 0$ and $\overline{r}$ is the radius of the disk each
agent occupies as defined at the end of Section III. Consider a spherical
obstacle with unit radius and without loss of generality let it be centered at
the origin. Hence, consider the obstacle avoidance potential function
$U_{i}^{0}:SE(2)\to\mathbb{R}$,
$\displaystyle{U_{i}^{0}(g_{i})=\frac{\sigma_{i0}}{2(x^{2}+y^{2}-(\overline{r}+1)^{2})}}$,
where $\sigma_{i0}>0$.

Note that the obstacle avoidance potential functions are not $SE(2)$-invariant
but $SO(2)$-invariant, so they break the symmetry. Using the norm of
$\mathfrak{se}(2)$ and for $\overline{r}=1$, $U_{ij}$ and $U_{i}^{0}$ are
equivalently given by
$\displaystyle{U_{ij}(g_{i},g_{j})=\frac{\sigma_{ij}}{2(\|\text{Ad}_{g_{i}^{-1}g_{j}}e_{1}\|^{2}-6)}}$
and
$\displaystyle{U_{i}^{0}(g_{i})=\frac{\sigma_{i0}}{2(\|\text{Ad}_{g^{-1}}e_{1}\|^{2}-6)}}$.

Let $W=\mathfrak{g}^{*}$, so we define the extended potential functions
$U_{i,0}^{\text{ext}}:SE(2)\times\mathfrak{g}\to\mathbb{R}$ by
$U_{i,0}^{\text{ext}}(g_{i},\alpha)=\frac{\sigma_{i0}}{2(\|\text{Ad}_{g^{-1}}\alpha\|^{2}-6)}$,
which are $SE(2)$-invariant under the action of $\tilde{\Phi}$, i.e.
$U_{i,0}^{\text{ext}}\circ\tilde{\Phi}=U_{i,0}^{\text{ext}},$ for any $g\in
SE(2)$. Since, $W=\mathfrak{g}^{*}$ we have
$\displaystyle{\mathbf{J}_{W}\Big{(}\frac{\partial
U_{i,0}^{\text{ext}}}{\partial\alpha_{i}},\alpha_{i}\Big{)}=\text{ad}^{*}_{\alpha_{i}}\Big{(}\frac{\partial
U_{i,0}^{\text{ext}}}{\partial\alpha_{i}}\Big{)}}$,
$\rho^{\prime*}_{u_{i}}(\alpha_{i})=\text{ad}_{u_{i}}\alpha_{i},$ and
$\rho^{*}_{g_{i}}(\alpha)=\text{Ad}_{g^{-1}}\alpha$ (for more details see
Corollary 7 of [4]), recall from section III that $\rho^{*}$ is the adjoint of
$\rho$. Thus, from PMP we get $u_{i}^{1}=\frac{1}{2}\mu_{i}^{1}$,
$u_{i}^{2}=\mu_{i}^{2}$ and $u_{3}^{i}=\mu_{i}^{3}$ and the reduced augmented
Hamiltonian $h:G^{s-1}\times(\mathfrak{g}^{*})^{s}\times W^{*}\to\mathbb{R}$
is given by

$\begin{split}h(g,\mu,\alpha)=&-\frac{3}{4}\sum_{i=1}^{s}(\mu_{i}^{1})^{2}+\frac{1}{2}\sum_{i=1}^{s}(\mu_{i}^{2})^{2}\\\
&+U_{i,0}^{\text{ext}}(e,\alpha_{i})+\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}U_{ij}(e,g_{i}^{-1}(t)g_{j}(t)).\end{split}$

Denote by $\Gamma_{ij}:=T_{\overline{e}_{i}}^{*}L_{g_{i}}\left(\frac{\partial
U_{ij}}{\partial g_{i}}\right)$ and note that

$\displaystyle\text{ad}^{*}_{u_{i}}\mu_{i}$
$\displaystyle=\begin{pmatrix}0&-\frac{\mu_{i}^{2}\mu_{i}^{3}}{2}&0\\\
\frac{\mu_{i}^{2}\mu_{i}^{3}}{2}&0&0\\\
\frac{\mu_{i}^{1}\mu_{i}^{3}}{2}&-\frac{\mu_{i}^{1}\mu_{i}^{2}}{2}&0\end{pmatrix},$
$\displaystyle\text{ad}^{*}_{\alpha_{i}}\Big{(}\frac{\partial
U_{i,0}^{\text{ext}}}{\partial\alpha_{i}}\Big{)}$
$\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
\Gamma_{i,0}^{31}&\Gamma_{i,0}^{32}&0\end{pmatrix}$
$\displaystyle=\frac{\sigma_{i0}\alpha_{i}^{1}}{(\|\alpha_{i}\|^{2}-6)^{2}}\begin{pmatrix}0&0&0\\\
0&0&0\\\ -\alpha_{i}^{3}&\alpha_{i}^{2}&0\end{pmatrix},$ $\displaystyle
T^{*}_{\overline{e}_{i}}L_{g_{i}}\Big{(}\frac{\partial U_{ij}}{\partial
g_{i}}\Big{)}$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
\Gamma_{ij}^{31}&\Gamma_{ij}^{32}&0\end{pmatrix}$
$\displaystyle=\frac{-\sigma_{ij}}{((x_{ij})^{2}+(y_{ij})^{2}-4)^{2}}\begin{pmatrix}0&0&0\\\
0&0&0\\\ x_{ij}&y_{ij}&0\end{pmatrix},$
$\displaystyle\text{ad}_{u_{i}}\alpha_{i}$
$\displaystyle=\begin{pmatrix}0&0&-u_{i}^{1}\alpha_{i}^{3}\\\
0&0&u_{i}^{1}\alpha_{i}^{2}-u_{i}^{2}\alpha_{i}^{1}\\\ 0&0&0\end{pmatrix},$

where $x_{ij}=x_{i}-x_{j},\;y_{ij}=y_{i}-y_{j}$ and

$\text{Ad}_{g_{i}^{-1}}\alpha_{0}=\begin{pmatrix}0&-1&x_{i}\sin\theta_{i}-y_{i}\cos\theta_{i}\\\
1&0&x_{i}\cos\theta_{i}+y_{i}\sin\theta_{i}\\\ 0&0&0\end{pmatrix}.$

Thus, the Lie-Poisson equations (IV) are

$\begin{array}[]{ll}&\\\
\dot{\mu}_{i}^{1}=-\frac{1}{2}\mu_{i}^{2}\mu_{i}^{3},&\\\ &\\\
\dot{\mu}_{i}^{2}=\frac{1}{2}\mu_{i}^{1}\mu_{i}^{3}-\Gamma_{i,0}^{31}+\sum_{j\in\mathcal{N}_{i}}\Gamma_{ij}^{31},&\\\
&\\\
\dot{\mu}_{i}^{3}=-\frac{1}{2}\mu_{i}^{1}\mu_{i}^{2}-\Gamma_{i,0}^{32}+\sum_{j\in\mathcal{N}_{i}}\Gamma_{ij}^{32},&\\\
&\end{array}$

together with

$\begin{array}[]{ll}\dot{\alpha}_{i}^{1}=0,&\alpha_{i}^{1}=1,\\\
\dot{\alpha}_{i}^{2}=\frac{1}{2}\mu_{i}^{1}\alpha_{i}^{3},&\alpha_{i}^{2}=x_{i}\sin\theta_{i}-y_{i}\cos\theta_{i},\\\
\dot{\alpha}_{i}^{3}=-\frac{1}{2}\mu_{i}^{1}\alpha_{i}^{2}+\mu_{i}^{2}\alpha_{i}^{1},&\alpha_{i}^{3}=x_{i}\cos\theta_{i}+y_{i}\sin\theta_{i}.\end{array}$

###### Remark 1

Note that, for the unicycle model, alternatively it can be used different
potential functions with both properties (collision and obstacle avoidance).
For instance one can consider the kind of potential functions
$\displaystyle{U(g_{i},g_{j})=\frac{\tilde{\sigma}_{i}}{2(x_{i}^{2}+y_{i}^{2}-(\overline{r}+1)^{2})\Upsilon_{ij}}}$,
where
$\Upsilon_{ij}=\prod_{j\in\mathcal{N}_{i}}((x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}-d^{2}_{ij})$
and $d_{ij}$ stands for the desired distance between two agents which were
used for instance in [25]. $\diamond$

Next we present some simulation results for the previous optimal control
problem for three agents on $SE(2)$ avoiding mutual collision as well as a
static obstacle in the configurations space. For simulation results, we
transform the two-point boundary value problem to an initial value problem by
considering the shooting method. The controlled dynamics of the three agents
are simulated using Euler’s method with time step $h=0.0001$ and $N=50000$.
Initial conditions are $g_{1}(0)=\begin{pmatrix}1&0&-1\\\ 0&1&-5\\\
0&0&1\end{pmatrix}$, $g_{2}(0)=\begin{pmatrix}\sqrt{2}/2&\sqrt{2}/2&-1\\\
\sqrt{2}/2&\sqrt{2}/2&-7\\\ 0&0&1\end{pmatrix}$,
$g_{3}(0)=\begin{pmatrix}1&0&-1\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$, with initial
angular velocities $u^{1}_{1}(0)=u^{1}_{2}(0)=1$, $u^{1}_{3}(0)=1.5$ and
linear velocities $u^{2}_{1}(0)=u^{2}_{2}(0)=0.6$, $u^{2}_{3}(0)=2$.

Fig.1 shows the trajectories of the agents on the configuration space. Initial
values were chosen in order that one of the agents avoids the obstacle and the
other two prevent mutual collision. Fig.2 shows the evolution of controls
inputs through the time. Agents $1$, $2$ and $3$ are depicted by the color
red, blue and green, respectively. Note that in Fig. 2 we can also observe
that the agents (blue and red) avoiding mutual collision have similar angular
velocities, while the agent avoiding the obstacle and depicted by green color,
has less sharp fluctuations. From the second graph we can also see that almost
at the same moment an increase of the linear velocities occurs so that all
three agents prevent collision.

Figure 1: The left plot illustrates the trajectories of the three unicycles.
The right plot shows evolution of their attitude through time.  Figure 2: The
two plots show the change of angular and linear velocities with respect to
time for the three agents.

##  VI Conclusions

We studied the reduction by symmetries for necesary optimality conditions for
extrema in an optimal control problem of left-invariant affine multi-agent
control system, by exploiting the physical symmetries of the agents and static
obstacles in the configuration space. Reduced optimality conditions were
obtained by using techniques from Hamiltonian mechanics on Lie groups together
with the PMP. We applied the results to a collision avoidance problem for
multiple unicycles in the presence of an obstacle.

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# Molecular Quantum Wakes for Clearing Fog

Malte C. Schroeder  1 Ilia Larkin  2 Thomas Produit  1 Eric W. Rosenthal  3
Howard Milchberg2 and Jean-Pierre Wolf1  1Dept. of Applied Physics, University
of Geneva, Chemin de Pinchat 22, 1211 Geneva 4, Switzerland
2Institute for Research in Electronics and Applied Physics, University of
Maryland, College Park, MD 20742, USA
3U.S. Naval Research Laboratory, Washington, D.C. 20375, USA
<EMAIL_ADDRESS>
###### Abstract

High intensity laser filamentation in air has recently demonstrated that,
through plasma generation and its associated shockwave, fog can be cleared
around the beam, leaving an optically transparent path to transmit light.
However, for practical applications like free-space optical communication
(FSO), channels of multi-centimeter diameters over kilometer ranges are
required, which is extremely challenging for a plasma based method. Here we
report a radically different approach, based on quantum control. We
demonstrate that fog clearing can also be achieved by producing molecular
quantum wakes in air, and that neither plasma generation nor filamentation are
required. The effect is clearly associated with the rephasing time of the
rotational wave packet in N2. Pump excitation provided in the form of resonant
trains of 8 pulses separated by the revival time are able to transmit optical
data through fog with initial extinction as much as $-6$ dB.

††journal: oe

##  1 Introduction

Light scattering by small particles is a key issue for several optical
applications, such as imaging through turbid media or free space optical
telecommunication (FSO) between ground based stations and satellites. Many
approaches based on adaptive optics and spatio-temporal shaping of laser beams
have been proposed for improving the propagation through scattering media and
for avoiding speckle patterns formation [1, 2, 3, 4, 5, 6]. These methods,
however, are of limited use for dense and extended structures such as
atmospheric fog and clouds. For free space optical networks, this critical
issue is currently mitigated by the multiplication of transmitting/receiving
stations, which increases complexity and cost. Until recently no active method
for opening clear optical transmission channels through clouds or fog was
available. The propagation of ultrashort and high intensity lasers through the
atmosphere provided a radically new concept, based on plasma generation
inherent to laser filamentation [7, 8, 9]. Laser filaments are self-sustained
light structures with a typical core diameter $d_{core}\sim 100\,\mu$m (at
$800$ nm) and up to hundreds of meters in length, orders of magnitude beyond
the linear diffraction limit. They stem from the dynamic balance between self-
focusing (Kerr effect) and defocusing (diffraction, plasma-induced refraction,
etc.) that takes place during non-linear laser propagation in air [10].
Filaments bear high intensities ($10-100$ TW/cm2) and generate a low density
plasma during propagation. The plasma recombination thermalizes quickly,
resulting in a very rapid local increase of pressure that drives a single
cycle cylindrical acoustic wave [11, 12, 13, 14, 15]. Over a time window of
$0.1-1$ ms, this leads to a reduced air density channel significantly wider
than the filament core [13, 14, 15, 16, 17]. When launched in fog (water
droplet diameter centered around $5$ $\mu$m), this filament-induced shockwave
radially ejects the scatterers out of the beam [8, 9]. This fog clearing
mechanism requires modest energy deposition, of the order of less than $1$
mJ/m [18, 19]. For droplets densities $N_{d}\sim 10^{5}\,$cm-3, it was shown
that the filament formation is not significantly altered [20]. This is the
case in the vast majority of scenarios in the field [20, 21, 22], and is the
case for our fog chamber as discussed below. Under these conditions, there is
little additional energy cost due to this droplet clearing mechanism, as the
energy deposition and acoustic wave generation through plasma generation by
the filament occurs in any case as a consequence of its propagation.

However, applying the filament-induced clearing method to real-scale FSO may
be challenging. Opened communication channels require lengths of at least
several hundred meters and diameters of $\sim 10$ cm. Laboratory scale
experiments demonstrated that filament-induced acoustic waves could clear
regions exceeding the filament dimensions, but not larger than $0.5-1$ cm [9].
Engineering the laser pulse properties (adaptive optics and temporal focusing)
in order to achieve uniform filament excitation over several hundred meters
appears particularly challenging.

In the present paper, we generate and coherently control rotational wave
packets in the air molecules (in particular N2) to produce an acoustic wave to
open transmission channels in fog and clouds over short distances without
requiring plasma formation or filamentation. Larger diameter beams and/or mid-
IR lasers could then be used for fog clearing, which when filamenting generate
lower density plasma unsuited for shockwave creation, but for which long
distance propagation is more easily achievable and controllable [23, 24].

##  2 Coherent control of rotational wave packets in air molecules

The effects of laser induced molecular alignment and field-free coherent
rotation on laser propagation have been investigated by several authors [25,
26, 27, 28, 29]. Impulsive Raman scattering induces a coherent superposition
of rotational states $|J,m\rangle$ in N2 and O2, i.e. rotational wave packets

$|\Psi_{R}(t)\rangle=\sum_{J,m}c_{J,m}|J,m\rangle e^{-iE_{R}t/\hbar},$ (1)

where $J$ and m are the quantum numbers of the total and z-component of the
rotational angular momentum and $E_{R}=hcBJ(J+1)$ the rotational state
energies of the rigid rotor Hamiltonian (with $B$ the rotational constant, $h$
the Planck constant and $c$ the speed of light). The instantaneous non-
adiabatic laser excitation locks in phase all the Raman-excited rotational
states at $t=0$, and the resulting wave packet evolves as in Eq. (1). For
$t>0$, the states in the sum dephase with respect to each other and the wave
packet spreads. However, rephasing occurs in the form of revivals at specific
times, leading to full re-alignment at periodic intervals of $\Delta
t=T_{R}=1/2Bc$ ($8.36$ ps in N2 and $11.2$ ps in O2). These effects have been
measured using Coulomb explosion imaging in very low density gases [30] and,
importantly for laser propagation applications, the effects persist at
atmospheric pressure [29, 31]. Due to the even integer distribution of the
rotational frequencies, additional alignment or anti-alignment recurrences
also occur at $t=T_{R}/4$, $t=T_{R}/2$, and $t=3T_{R}/4$ [30, 31].

Figure 1: Quantum wake clearing of fog: a series of $8$ pulses of $56$ fs are
generated by a nested Michelson interferometer (pulse stacker [32]) and softly
focused into a fog chamber. The interval between the pulses can be tuned on-
or off-resonance with the rotational revival times $T_{R}$ of the nitrogen
molecules in air. Clearing of the fog is assessed with a counter-propagating
CW laser at telecom wavelength ($1.55$ $\mu$m) and a photodiode that measures
the transmission. In addition a synchronized Nd:YAG laser is used for imaging
the induced shockwave by shadowgraphy and interferometry

The periodic field-free alignment takes place until decoherence occurs, due to
inter-molecular collisions and/or centrifugal distortion (deviation from the
rigid rotor Hamiltonian). Typical rotational coherence times observed in N2 at
standard temperature and pressure are $\sim 80$ ps [33]. Within this coherence
time window, the rotational quantum wake can be coherently controlled by a
train of short pulses precisely separated by the full revival time $T_{R}$,
matching the «quantum echoes» of the rotating molecule, as demonstrated by
Cryan et al [33]. This coherent control method based on repetitive impulsive
Raman excitation was originally proposed by Weiner et al [34] for controlling
phonons in organic solids. By shaping the laser excitation as a sequence of
$4$ pulses separated by $T_{R}$, Zahedpour et al [29] recently demonstrated
that the molecular rotation of N2 was driven to equivalent temperatures as
high as $450$ K ($30$ meV/molecule of additional heating). Conversely,
rotation was coherently hindered by applying series of pulses resonant at
times for which molecules are anti-aligned $(T_{R}/2)$. Collisional
decoherence and thermalization of the rotationally excited molecules after
$\sim 100$ ps [13] generates an on-axis pressure spike that drives a single-
cycle acoustic wave. The amplitude of this wave was shown to exceed that
previously observed in plasma filaments [29]. In the present paper, we
demonstrate that this plasma-free acoustic wave can also efficiently eject
water droplets out of the beam, and that the coherently controlled quantum
wake creates clear channels in fog without the need for filamentation-based
ionization.

##  3 Experiment

For the experimental demonstration of the concept, we used a train of 8
ultrashort pulses, which provides an optimal compromise between the number of
successive Raman excitations and the onset of decoherence due to collisions
and centrifugal distortion.

A sketch of the experimental set-up is shown in Fig. 1. The laser pump
consists of either a single pulse or a train of eight $\lambda=800$ nm, $56$
fs Ti:Sapphire (TiSa) pulses produced with a Michelson interferometer [32]
(«pulse stacker»). The beam is focused with an f/$80$ lens into the fog
chamber. The FWHM of the vacuum beam waist is about $w_{FWHM}=25$ $\mu$m and
the femtosecond laser train is repeated at a frequency of $1$ kHz. The energy
per pulse in the train is $0.48$ mJ. A continuous wave (CW) counter-
propagating laser at $\lambda=1.55$ $\mu$m, spatially overlapping the TiSa
beam, is used for measuring changes in transmission through the fog, and for
investigating the dynamics of the droplets expulsion by the acoustic wave. In
addition, a $10$ Hz frequency doubled Nd:YAG laser ($\lambda=532$ nm) is
synchronized to the TiSa pump beam to probe the 2D gas density at variable
delays by shadowgraphy and interferometry.

In order to simulate realistic fog/cloud conditions, the size distribution of
the droplets in the cloud chamber is centered on a radius of $5$ $\mu$m
(measured by a Grimm 1.109 particle sizer). The droplet number density is
variable from $0$ to $1.5\times 10^{5}$ cm-3 (1000 times more than in a
cumulus cloud) [9]. This variation in Nd corresponds to attenuation ranging
from $0$ dB (no fog) to $-10$ dB. The droplets were generated and dispersed by
a piezo-driven ultrasonic source.

Figure 2: Coherent control of the rotational quantum wake induced shockwave in
fog. Visualized through shadowgraphic measurements. Left: train with pulse
intervals slightly detuned from resonance: $8.66$ ps; Right: train of $8$
pulses tuned in resonance with the full revival time for N2: $8.36$ ps . Delay
between TiSa pump and Nd:YAG probe: $4$ $\mu$s. Total energy of the pump pulse
train: $3.8$ mJ

The shadowgraphic measurements allow for the observation of changes in the
atmospheric pressure caused by the acoustic wave produced by the resonantly
and off-resonantly spaced pulse trains. The shadowgraphic image in figure 2(b)
shows the acoustic wave produced in fog (droplet concentration $1.3\times
10^{5}$ cm-3) when the pump laser consists of a train of $8$ pulses of $56$ fs
duration, separated by $8.36$ ps, i.e. resonant with the full rephasing time
of the rotational wave packet. Each of the $8$ pulses bears an intensity of
the order of $\sim 10^{13}$ W/cm2 at the beam waist. A clear radially
expanding wave is produced, which reduces the air density in the center of the
beam (typ. $500-700$ $\mu$m diameter) during the first hundred $\mu$s after
the pulse train. The reduced density region exhibits a lower refractive index
(Fig. 3(b)), which induces defocusing of the Nd:YAG probe laser, and thus a
loss in the transmitted light. The depleted density region is eventually
refilled by the surrounding gas after a few hundreds microseconds, as shown in
the supplementary videos Visualization 1 and Visualization 2. The acoustic
wave expansion characteristics, in terms of amplitude and speed, appear almost
unaffected by the presence of droplets, similar to previously reported
measurements in clean air [12]. Conversely, if the time separation between the
pulses within the train is slightly detuned away from resonance ($8.66$ ps), a
much weaker shockwave is produced (Fig. 2(a)), revealing a smaller temperature
rise and refractive index shift in the gas (Fig. 3(a)). These results
demonstrate that the dominant mechanism responsible for the acoustic wave
production here is the coherent control of the molecular quantum wake and not
plasma generation (also confirmed by single pulse measurements, Fig. 5).

Figure 3: Change in air refractive index due to rotational gas heating,
measured by interferometry. Left: detuned pulse train: $8.66$ ps; Right:
resonant pulse train at full revival time for N2: $8.36$ ps . Delay between
TiSa pump and Nd:YAG probe: $500$ $\mu$s. The insets show the shift in the
refractive index for time delays of $200$, $400$, $600$ and $800$ $\mu$s after
the pump, in order of deepest to shallowest.

A folded wavefront interferometer was insterted upstream of the imaging
apparatus to measure the refractive index shift in the reduced density region
surrounding the beam center. Gas density was extracted from retrieved
interferograms using well-established fringe analysis techniques [35]. For
these measurements no fog was added to the chamber, as to prevent turbulence
in the fog from interfering with the acquisition of the interferograms. Figure
3(a) shows the variation of the refractive index for the off-resonant case,
where the pulses in the train are detuned by $300$ fs relative to each other,
at a pump-probe delay of $500$ $\mu$s. The reduced density region is visible
at the prior location of the beam center, gradually expanding outward from
this point. At the center, the relative refractive index shift is $\Delta
n_{\textit{off}}\approx-7.8\times 10^{-6}$. If the pulse train is tuned to the
revival time of molecular nitrogen, the reduced density region increases
significantly in size and depth. At the same time delay of $500$ $\mu$s, the
resulting refractive index shift increases approximately by a factor of $2$ to
$\Delta n_{\textit{on}}\approx-16.2\times 10^{-6}$ (Fig. 3(b)). The asymmetry
visible in the figure stems from convection [28]. In the insets the filling in
of the reduced density region can be observed by the smoothing of the lineouts
over time. The lineouts depict the spatially resolved change in refractive
index along the x-axis, centered on the minimum value at the core.

These observations are consistent with the shadowgraphy measurements. The more
efficient rotational molecular gas heating in the resonant case results in
more heat deposition in the air. Consequently, the higher temperature gradient
causes a stronger acoustic wave, which more efficiently reduces the air
density over a more extended region.

Figure 4: Fog clearing induced by the rotational quantum wake. Transmission
gain in the case of (a) No fog with the $8$ pulse train pump laser, (b) Fog
($1.3\times 10^{5}$ cm-3 droplet concentration) with the $8$ pulse train tuned
on-resonance with the quantum wake rephasing time, and (c) Fog ($0.7\times
10^{5}$ cm-3 droplet concentration) with the $8$ pulse train tuned off-
resonance with the quantum wake rephasing time. Total energy of the pump:
$3.8$ mJ. Transmission is normalized to 1 in each case, corresponding to the
situation where no pump is present ($t<0$).

The gain in transmission provided by the quantum wake clearing process was
assessed by running the probe telecom laser in a CW mode (Fig. 4). Fig. 4(a)
shows the effect of the 8-pulse train pump on the transmission of the
$\lambda=1.55$ $\mu$m CW laser in the absence of fog. The air density
depletion due to rotational heating causes an initial defocusing of the
telecom laser due to the induced refractive index gradient. This leads to the
observed modulation of the signal at $1$ kHz, due to the finite aperture of
the detection system. In this geometry, the envelope of the modulation
requires $2-3$ ms to reach a steady state. In the presence of fog ($N_{d}\sim
1.3\times 10^{5}$ cm-3), corresponding to $9$ dB attenuation, the 8-pulse
train dramatically increases the telecom laser transmission by a factor $3$
(i.e. $4.8$ dB) when the pulse interval in the train is tuned on-resonance
with the full revival time of $8.36$ ps (Fig. 4(b)). Interestingly, the
modulated envelope reaches a steady state after $\sim 7-8$ ms, longer than in
the droplet-free case. This can be interpreted by the fact that the first few
8-pulse trains, delivered at $1$ kHz, are used to clear the initial local
droplet concentration to make its way for the following pulses. Conversely,
when the 8-pulse interval is tuned slightly off-resonance, the gain drops to a
factor of only $\sim 1.5$ ($1.7$ dB, Fig 4(c)). This remaining gain is caused
by partial rotational heating.

Figure 5: Fog clearing induced by the rotational quantum wake as a function of
the initial signal attenuation caused by different fog densities 5(a).
Clearing is efficient for all densities in the case of a pulse train in
resonance with the rephasing time, and the efficiency increases with the
initial density (see text). Clearing is strongly reduced for the off-resonant
case. 5(b) comparison with the plasma induced shockwave - Purple: single pulse
of $3.8$ mJ (entire energy of the train in one pulse), Blue : single pulse of
$0.48$ mJ (one pulse of the train). Transmission is normalized to $1$ in each
case, corresponding to the situation where no pump is present ($t<0$).

The efficiency of rotational heating-induced fog clearing was assessed as a
function of the fog’s optical density, by varying the droplet number density
in the chamber. Although large fluctuations are present due to instabilities
in the cloud chamber (entrance and exits are open to the ambient air), a clear
enhancement in transmission is observed for all the optical density values in
the case of the on-resonance pump pulse trains. As quantitative assessment of
the process efficiency, we consider the maximum initial cloud extinction value
for which transmission is fully recovered ($100\%$, i.e. $0$ dB extinction).
For the on-resonance pulse train, this value corresponds to $-6$ dB initial
extinction, while for the slightly detuned pulse train, it drops to $-2$ dB.
It is now relevant to compare these values to the values achieved with a
single pulse (Fig. 5(b) (with all other experimental conditions kept identical
to the 8 pulses train case). When only one pulse of the train is used (bearing
$0.48$ mJ as in the pulse train), no transmission gain is observed, which
suggests that the plasma generation of individual pulses of the train is
negligible. Conversely if the whole energy of the pulse train, $3.8$ mJ, is
concentrated in a single pulse, plasma generation and filamentation are
initiated, and we observe the associated acoustic wave clearing process seen
previously [8, 9], i.e. $-8$ dB, which exceeds the quantum wake clearing
process with 8 pulses. Also, the slope significantly exceeds the slope in the
case of rotational heating with the pulse train. The linear increase in the
transmission gain was previously attributed [8] to the fact that the number of
droplets $\Delta N$ ejected by the shockwave is proportional to the number of
droplets $N$ initially present in the volume (percentage of droplets cleared
in the channel: on resonance: $57.4\%$; off resonant: $28.0\%$ ; 1 pulse full
energy : $75.1\%$ ; 1 pulse 1/8th of energy: $0\%$). In other words, the
energy of the shockwave is not significantly altered by the quantity of
droplets that have to be moved.

As already mentioned above, the clear enhancement of the transmission at the
rephasing time of the N2 rotational wave packet, as compared to the non-
resonant case, excludes dominant clearing processes like droplet explosion,
shattering or direct evaporation by the laser beam. However, in the present
configuration, the rotational heating of the gas itself ($450$ K) exceeds the
vaporization temperature of water, so that the partial or complete evaporation
of the droplets by heat transfer from the hot air to the droplets may have a
contribution in the fog clearing as well. Although the exact calculation of
the transient evaporated mass rate from a moving droplet is a complex problem
[36], some qualitative arguments can be drawn for assessing this role in the
clearing. A necessary (but not sufficient) condition for the process to occur
is that the amount of energy $Q$ required for vaporizing the ensemble of
droplets in the channel is less than or equal to the energy deposited by the
laser. With $1.5\times 10^{5}$ cm-3 droplets in a channel of $0.5$ mm radius
and $10$ cm length, this leads to $Q=13$ mJ, which means that all the incoming
laser energy should be deposited in the gas and then transferred from the
heated gas to the droplets with an efficiency of $100\%$. These considerations
suggest that the droplet vaporization by the heated gas does not play the
major role in the fog clearing process, in contrast to the displacement of the
drops by the acoustic wave as previously observed for the filament induced
clearing process [8, 9, 37].

##  4 Conclusion

In this article, we demonstrated that neither ionization nor filamentation are
required to produce a fog clearing acoustic wave, as they can be substituted
by quantum controlled rotational heating in air. This discovery opens new
perspectives in the application of the method for real, atmospheric scale
operation like FSO. For instance, intense mid-IR lasers, which provide only
moderate ionization as they propagate in air [38, 39, 40, 41], are now
potentially very attractive candidates for fog clearing applications. It was
recently reported in particular that TW peak powers CO2 lasers can provide
wide diameter self-guiding channels over kilometric distances [24], fulfilling
the requirements for FSO between earth and satellites.

##  5 Funding

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
(SNF) (200021-178926); Horizon 2020 Framework Programme (H2020) (737033-LLR);
Office of Naval Research (ONR) (N00014-17-1-2705, N00014-17-1-2778); Air Force
Office of Scientific Research (AFOSR) (FA9550-16-1-0284, FA9550-16-1-0121).

##  6 Acknowledgments

MCS, TP and JPW gratefully acknowledge technical support from Michel Moret and
financial support from the Schweizerischer Nationalfonds zur Förderung der
Wissenschaftlichen Forschung under grant no 200021-178926. TP and JPW
acknowledge the European Union’s Horizon 2020 research and innovation program
under grant agreement no 737033-LLR.

IL, EWR, and HMM gratefully acknowledge support from the Office of Naval
Research (ONR) (N00014-17-1-2705, N00014-17-1-2778) and the Air Force Office
of Scientific Research (AFOSR) (FA9550-16-1-0284, FA9550-16-1-0121).

##  7 Disclosures

The authors declare no conflicts of interest.

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