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Nonlin. Processes Geophys., 20, 657–668, 2013
Advances
in
Annales
www.nonlin-processes-geophys.net/20/657/2013/
doi:10.5194/npg-20-657-2013
Geosciences
Geophysicae
© Author(s) 2013. CC Attribution 3.0 License.
Discussions
Distributed allocation of mobile
sensing swarms in gyre flows
Open Access
Open Access
Atmospheric
Atmospheric 1
K. Mallory , M. A. Hsieh1 , E. Forgoston2 , and I. B. Schwartz3
Chemistry
Chemistry
1 SAS Lab, Drexel University, Philadelphia, PA 19104, USA
2 Department of Mathematical Sciences, Montclair
andState
Physics
and Physics
University, Montclair, NJ 07043, USA
3 Nonlinear
Systems Dynamics Section, Plasma PhysicsDiscussions
Division, Code 6792, US Naval Research Lab, Washington,
DC 20375, USA
Open Access
Open Access
Atmospheric
Atmospheric
Correspondence to: K. Mallory ([email protected])
Measurement
Measurement
Received: 1 March 2013 – Revised: 15 July 2013
– Accepted: 29 July 2013 – Published: 16 September 2013
Techniques
Techniques
Discussions
Open Access
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information about the dynamics and transport of the fluidic
Abstract. We address the synthesis of distributed control
environment. For two-dimensional (2-D) flows, ridges of lopolicies to enable a swarm of homogeneous
mobile sensors
Biogeosciences
ogeosciences
to maintain a desired spatial distribution in a geophysical
Discussionscally maximal finite-time Lyapunov exponent (FTLE) (Shadflow environment, or workspace. In this article, we assume
den et al., 2005) values correspond, to a good approximation
the mobile sensors (or robots) have a “map” of the envi(though see Haller, 2011), to Lagrangian coherent structures.
ronment denoting the locations of the Lagrangian coherent
Details regarding the derivation of the FTLE can be found in
structures or LCS boundaries. Using this information, Climate
we dethe literature (Haller, 2000, 2001, 2002; Shadden et al., 2005;
Climate
sign agent-level hybrid control policies that leverage the surLekien et al., 2007; Branicki and Wiggins, 2010).
of the
Past Recent years have seen the use of autonomous underwater
noise
of therounding
Past fluid dynamics and inherent environmental
to enable the team to maintain a desired distribution Discussions
in the
and surface vehicles (AUVs and ASVs) for persistent monworkspace. We discuss the stability properties of the ensemitoring of the ocean to study the dynamics of various bioble dynamics of the distributed control policies. Since relogical and physical phenomena, such as plankton assemalistic quasi-geostrophic ocean models predict
double-gyre
blages (Caron et al., 2008), temperature and salinity profiles
Earth
System
Earth System
flow solutions, we use a wind-driven multi-gyre flow model
et al., 2008; Wu and Zhang, 2011; Sydney and PaDynamics(Lynch
to verify the feasibility of the proposed distributed control
ley, 2011), and the onset of harmful algae blooms (Zhang
Dynamics
Discussions
strategy and compare the proposed control strategy with
a
et al., 2007; Chen et al., 2008; Das et al., 2010). These studies have mostly focused on the deployment of single, or small
baseline deterministic allocation strategy. Lastly, we validate
the control strategy using actual flow data obtained by our
numbers of AUVs working in conjunction with a few staGeoscientific
Geoscientifictionary sensors and ASVs. While data collection strategies
coherent structure experimental testbed.
nstrumentation
Instrumentationin these studies are driven by the dynamics of the processes
Methods and
Methods andthey study, most existing works treat the effect of the surData Systems
Data Systemsrounding fluid as solely external disturbances (Das et al.,
1 Introduction
2010; Williams and Sukhatme, 2012), largely because of our
Discussions
limited understanding of the complexities of ocean dynamGeophysical flows are naturally stochastic and aperiodic, yet
ics. Recently, LCS have been shown to coincide with opexhibit coherent structure. Coherent structuresGeoscientific
are of signifiGeoscientific
timal trajectories in the ocean which minimize the energy
cant importance since knowledge of
them enables
the predicModel
Development
and the time needed to traverse from one point to another
Development
tion and estimation of the underlying geophysical fluid
dyDiscussions
(Inanc et al., 2005; Senatore and Ross, 2008). And while renamics. In realistic ocean flows, these time-dependent cohercent works have begun to consider the dynamics of the surent structures, or Lagrangian coherent structures (LCS), are
rounding fluid in the development of fuel efficient navigation
into dynamicallyandstrategies (Lolla et al., 2012; DeVries and Paley, 2011), they
Hydrologysimilar
andto separatrices that divide the flow Hydrology
distinct regions, and are essentially extensions of stable and
mostly on historical ocean flow data and do not employ
Earth
Systemrely
Earth System
unstable manifolds to general time-dependent
flows (Haller
knowledge of LCS boundaries.
and Yuan, 2000). As such, they encode a great deal Sciences
of global
Sciences
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Open Access
Open Access
Open Access
Open Access
Open Access
Open Access
Open Access
Discussions
Ocean Science
Discussions
Open Acces
cean Science
Open Acces
Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.
658
A drawback to operating both active and passive sensors in
time-dependent and stochastic environments like the ocean is
that the sensors will escape from their monitoring region of
interest with some finite probability. This is because the escape likelihood of any given sensor is not only a function
of the unstable environmental dynamics and inherent noise,
but also the amount of control effort available to the sensor.
Since the LCS are inherently unstable and denote regions
of the flow where escape events occur with higher probability (Forgoston et al., 2011), knowledge of the LCS are of
paramount importance in maintaining a sensor in a particular
monitoring region.
In order to maintain stable patterns in unstable flows,
the objective of this work is to develop decentralized control policies for a team of autonomous underwater vehicles
(AUVs) and/or mobile sensing resources to maintain a desired spatial distribution in a fluidic environment. Specifically, we devise agent-level control policies which allow individual AUVs to leverage the surrounding fluid dynamics
and inherent environmental noise to navigate from one dynamically distinct region to another in the workspace. While
our agent-level control policies are devised using a priori
knowledge of manifold/coherent structure locations within
the region of interest, execution of these control strategies by
the individual robots is achieved using only information that
can be obtained via local sensing and local communication
with neighboring AUVs. As such, individual robots do not
require information on the global dynamics of the surrounding fluid. The result is a distributed allocation strategy that
minimizes the overall control-effort employed by the team
to maintain the desired spatial formation for environmental
monitoring applications.
While this problem can be formulated as a multi-task
(MT), single-robot (SR), time-extended assignment (TA)
problem (Gerkey and Mataric, 2004), existing approaches do
not take into account the effects of fluid dynamics coupled
with the inherent environmental noise (Gerkey and Mataric,
2002; Dias et al., 2006; Dahl et al., 2006; Hsieh et al., 2008;
Berman et al., 2008). The novelty of this work lies in the
use of nonlinear dynamical-systems tools and recent results
in LCS theory applied to collaborative robot tracking (Hsieh
et al., 2012) to synthesize distributed control policies that enables AUVs to maintain a desired distribution in a fluidic environment.
The paper is structured as follows: we formulate the problem and outline key assumptions in Sect. 2. The development
of the distributed control strategy is presented in Sect. 3 and
its theoretical properties are analyzed in Sect. 4. Section 5
presents our simulation methodology, results, and discussion.
We end with conclusions and directions for future work in
Sect. 6.
Nonlin. Processes Geophys., 20, 657–668, 2013
K. Mallory et al.: Allocation of swarms in gyre flows
2
Problem formulation
Consider the deployment of N mobile sensing resources
(AUVs/ASVs) to monitor M regions in the ocean. The objective is to synthesize agent-level control policies that will
enable the team to autonomously maintain a desired distribution across the M regions in a dynamic and noisy fluidic
environment. We assume the following kinematic model for
each AUV:
f
q̇ k = uk + v q k
k ∈ {1, . . . , n},
(1)
where q k = [xk , yk , zk ]T denotes the vehicle’s position, uk
f
denotes the 3 × 1 control input vector, and v q k denotes the
velocity of the fluid experienced/measured by the kth vehicle.
In this work, we limit our discussion to 2-D planar flows
and motions and thus we assume zk is constant for all k. As
f
such, v q k is a sample of a 2-D vector field denoted by F (q)
at q k whose z component is equal to zero, i.e., Fz (q) = 0, for
all q. Since realistic quasi-geostrophic ocean models exhibit
multi-gyre flow solutions, we assume F (q) is provided by
the 2-D wind-driven multi-gyre flow model given by
y
f (x, t)
) cos(π ) − µx + η1 (t),
s
s
y df
f (x, t)
) sin(π )
− µy + η2 (t),
ẏ = π A cos(π
s
s dx
ż = 0,
x
f (x, t) = x + ε sin(π ) sin(ωt + ψ).
2s
ẋ = −π A sin(π
(2a)
(2b)
(2c)
(2d)
When ε = 0, the multi-gyre flow is time-independent,
while for ε 6= 0, the gyres undergo a periodic expansion and
contraction in the x direction. In Eq. (2), A approximately
determines the amplitude of the velocity vectors, ω/2π gives
the oscillation frequency, ε determines the amplitude of the
left–right motion of the separatrix between the gyres, ψ is
the phase, µ determines the dissipation, s scales the dimensions of the workspace, and ηi (t) describes a stochastic
√ white
noise with mean zero and standard deviation σ = 2I , for
noise intensity I . Figure 2a and b show the vector field of a
two-gyre model and the corresponding FTLE curves for the
time-dependent case.
Let W denote an obstacle-free workspace with flow dynamics given by Eq. (2). We assume a tessellation of W
such that the boundaries of each cell roughly corresponds
to the stable/unstable manifolds or LCS curves quantified
by maximum FTLE ridges as shown in Fig. 2. In general, it may be unreasonable to expect small resource constrained autonomous vehicles to be able to track the LCS locations in real time. However, LCS boundary locations can
be determined using historical data, ocean model data, e.g.,
data provided by the Navy Coastal Ocean Model (NCOM)
databases, and/or data obtained a priori using LCS tracking
strategies similar to Hsieh et al. (2012). This information can
then be used to obtain an LCS-based cell decomposition of
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65
70
75
80
85
90
95
TEXT:
TEXT
Mallory
et al.: Allocation of swarms in gyre flows
TEXT:K.
TEXT
(a)
659 33
(b)
(a)
(b)
(a)
(b)
Fig. 1. (a) Vector field and (b) FTLE field of the model given by (2)
(a)
(b)
for
two
with
A
10,
= 0.005,
ε model
= 0.1,
ψ =model
0,byI (2)
=given
0.01,
Fig.
1. gyres
(a)field
Vector
and µ(b)
FTLE
of the
Fig. 2. Two examples of LCS-based cell decomposition of the
Fig. 1. (a)
Vector
and field
(b)=FTLE
field
of thefield
given
for
two
gyres
with
A
=
10,
µ
=
0.005,
ε
=
0.1,
ψ
=
0,
by
Eq.
(2)
and
s
=
50.
LCS
are
characterized
by
regions
with
maximum
FTLE
region
interest assuming
a flowcell
fielddecomposition
given by (2).ofThese
for two gyres with A = 10, µ = 0.005, ε = 0.1, ψ = 0, I = 0.01,
2. Twoof examples
of LCS-based
the
I = 0.01, and
s = 50.
by regions
maxi- Fig. cell
(denoted
byLCS
red).areIncharacterized
2D flows, regions
withwith
maximum
decompositions were performed manually. (a) A 4 × 4 timeand s =measures
50. LCS are
characterized
by regions
with maximum
FTLE
region
of
interest
assuming
a
flow
field
given
by
(2).
These
Fig. 2. Two examples of LCS-based cell decomposition of the remum FTLE
measures
(denoted
by curves.
red). In 2-D flows, regions with
FTLE
measures
correspond
to 1D
independent grid of gyres with A = 0.5, µ = 0.005, ε = 0, ψ = 0,
measures (denoted by red). In 2D flows, regions with maximum
cellgion
decompositions
were performed
manually.
4 ×These
4 timeof interest assuming
a flow field
given by(a)
Eq.A (2).
maximum FTLE measures correspond to 1-D curves.
I = 35, and s = 20. The stable and unstable manifolds of each sadFTLE measures correspond to 1D curves.
cell decompositions
werewith
performed
manually.
(a) A ε4=
× 0,
4 timeindependent
grid of gyres
A = 0.5,
µ = 0.005,
ψ = 0,
dle point in the system is shown by the black arrows. (b) An FTLE
grid
ofThe
gyres
withand
A =unstable
0.5, µ =manifolds
0.005, ε =of
0, each
ψ = sad0,
I =independent
35,
and
s
=
20.
stable
based cell decomposition for a time-dependent double-gyre system
= 35,in
and
s=
20. The
stable
and
unstable
manifolds
of
each
sadreal
time. However,
LCSmanual
boundary
can beofdeterthe
system
is
shown
by
the
black
arrows.
(b)
An
FTLE
W. Figure
2 shows two
cell locations
decompositions
the dleIpoint
with the same parameters as Fig. 1(b).
dle point
in the system is for
shown
by the black arrows.
(b) An FTLE
mined
usingwhere
historical
data,
ocean model
data,correspond
e.g., data procell decomposition
a time-dependent
double-gyre
system
workspace
the cell
boundaries
roughly
to based
real time.
However,
LCS
boundary
locations
can
be
deterbased
cell
decomposition
for
a
time-dependent
double-gyre
system
vided
by theFLTE
Navyridges.
Coastal
Model
databases,
maximum
InOcean
this work,
we(NCOM)
assume the
tessel- with the same parameters as Fig. 1(b).
with the same parameters as Fig. 2.
mined using historical data, ocean model data, e.g., data pro-
and/or
data
a priori
LCS
lation of
W obtained
is given and
do notusing
address
thetracking
problem strategies
of autosynthesize agent-level control policies, or uk , to achieve and
vided
by
the Navy
Coastal
Ocean
Model
165
similar
to (Hsieh
et
This(NCOM)
can then be 200 maintain a desired distribution of the N agents across the M
matic
tessellation
of al.,
the 2012).
workspace
toinformation
achieve databases,
a decomposiand/or used
data
obtained
aboundaries
priori
using
LCS
tracking
strategies
tion where
cell an
correspond
to LCS curves.
to
obtain
LCS-based
cell
decomposition
of W. Fig. synthesize
agent-level
control
or]Tu, kin, to
achieve and
regions,
denoted
N̄ =are
[N̄policies,
, . . . , N̄
1given
We assume
that by
robots
a Mmap
of an
theenvironment
environsimilar 2toshows
(Hsieh
et
al.,
2012).
This
information
can
be 200 maintain
A tessellation
of thecell
workspace
along boundaries
charactwo manual
decompositions
of thethen
workspace
adynamics
desired
distribution
the N agents
across
the
whose
are given
byof(2).
ment,
G,
and
N̄.
Since
the
tessellation
of
W
is
given,
theM
used towhere
obtainthe
ancell
LCS-based
cellroughly
decomposition
of to
W.
Fig.they regions, denoted by N̄ = [N̄ , . . . , N̄ ]T , in an environment
terized
by
maximum
FTLE
ridges correspond
makes sense
since
boundaries
maximum
1
M
We
assume
that
robots
are
given
a
map
of
the
environLCS locations corresponding to the boundaries of each Vi
2 showsFLTE
two ridges.
manual
ofthe
the
workspace
separate
regions
within
the flow
field that
exhibit
distinctofdyIncell
thisdecompositions
work,
we assume
tessellation
W whose
dynamics
given
by
G, andaare
N̄.
Since
the(2).
tessellation
of W robots
is given,
isment,
also
known
priori.
Additionally,
we assume
co-the
namic
andaddress
denote
regions
in the
flow
fieldtessellawhere
where
the
cellbehavior
boundaries
roughly
correspond
to
maximum
170
is
given
and
do not
the
problem
of automatic
We
assume
that
robots
are
given
a
map
of
the
environ205 located
LCS
locations
corresponding
to
the
boundaries
of
each
within the same Vi have the ability to communicate Vi
moreofescape
events
occur probabilistically
tion
thethis
workspace
to achieve
athe
decomposition
where
FLTE ridges.
In
work,may
we
assume
tessellation(Forgoston
of W cell ment,
is
also
known
priori.
robotsthe
coG,
and
N̄. aThis
Since
theAdditionally,
tessellation
of assume
W isstructures
given,
with each
other.
makes
sense since we
coherent
etand
al.,do
2011).
In the time-independent
case, thesetessellaboundaries
LCS curves.
is givenboundaries
notcorrespond
address
thetoproblem
of automatic
located
thebarriers
same Vand
ability
to communicate
can
act aswithin
transport
underwater
205
LCS
locations
corresponding
toprevent
thetheboundaries
ofacoustic
each Vi
i have
correspond
to stable
andworkspace
unstable
manifolds
ofwhere
saddlecell
points
Aworkspace
tessellation
the
along boundaries
charaction of the
toofachieve
a decomposition
withknown
each other.
This Additionally,
makes
coherent
structures
wave
propagation
(Wang
et al.,sense
2009;since
Rypina
et al.,
2011).cois
also
a
priori.
we
assume
robots
in the
system.
The
manifolds
can also
be characterized
by
terized
by maximum
FTLE
ridges
makes
sense since they
boundaries
correspond
to LCS
curves.
Finally,
we
individual
robots
have
the
ability
to localcan act
asassume
transport
barriers
andthe
prevent
underwater
acoustic
located
within
the same
Vi have
ability
to
communicate
maximum
FTLE
ridges
where
the
FTLE
is
computed
based
175
separate regions
the flow
fieldboundaries
that exhibitcharacdistinct dy- 210 ize
A
tessellation
of thewithin
workspace
along
within
the workspace,
i.e.,
determine
their
ownetpositions
wave
propagation
(Wang
et
al.,
2009;
Rypina
al.,
2011).
with
each
other.
This
makes
sense
since
coherent
structures
on a backward
structures)
forward
(repelling
behavior (attracting
and denote
regions
inor
the
flow
field
where inFinally,
terized namic
by
maximum
FTLE
ridges
makes
sense
since
they
the
workspace.
These
assumptions
are
necessary
to
enwe
assume
individual
robots
have
the
ability
to
localstructures)
integration
in time.
Since
the manifolds
demar- can act as transport barriers and prevent underwater acoustic
escape
events
may
occur
probabilistically
(Forgoston
able
the
development
of a prioritization
scheme
within
each
separatemore
regions
within
the
flow
field
that
exhibit
distinct
dyize
within
the
workspace,
i.e.,
determine
their
own
positions
cate
basinInboundaries
separating the
distinct dynamical
210
wave propagation (Wang et al., 2009; Rypina et al., 2011).
et
al.,the
2011).
the time-independent
case,
boundaries
Vin
on an individual
robot’s
escape likelihoods
in order
i based
the
These
assumptions
to ennamic behavior
andare
denote
regionsthat
in are
theuncertain
flow these
field
where
regions,
they
also
regions
with
respect
weworkspace.
assume individual
robots have are
the necessary
ability to localcorrespond to stable and unstable manifolds of saddle points Finally,
toable
achieve
the desired allocation.
The prioritization
scheme
the
development
of
a
prioritization
scheme
within
each
more escape
events
may
occur
probabilistically
(Forgoston
to velocity vectors within a neighborhood of the manifold. ize within the workspace, i.e., determine their own positions
180
in
the system. The manifolds can also be characterized by 215 will
allow
robots
to
minimize
the
control-effort
expenditure
V
based
on
an
individual
robot’s
escape
likelihoods
in
order
i
et al., 2011).
In theswitching
time-independent
case, these
boundaries of
Therefore,
between regions
in neighborhoods
thethey
workspace.
These
assumptions
are the
necessary
to enmaximum FTLE ridges where the FTLE is computed based in as
movethe
within
the set
V. We describe
methodology
to achieve
desired
allocation.
The prioritization
scheme
correspond
to stableisand
unstableboth
manifolds
of saddleuncertainty
points
the manifold
influenced
by deterministic
the
development
of
a
prioritization
scheme
within
each
on a backward (attracting structures) or forward (repelling able
inwill
the following
section.
allow robots
to minimize the control effort expenditure
as well asThe
stochasticity
to external
in the system.
manifoldsdue
can
also be noise.
characterized by 215 V based
onmove
an individual
robot’s
escape
likelihoods
in
order
structures) integration in time. Since the manifolds demari as they
within the set V. We describe the methodology
Given anridges
FTLE-based
cell FTLE
decomposition
of W,based
let G =
maximum
where the
computed
to achieve
the desired
allocation. The prioritization scheme
cateFTLE
the basin boundaries
separatingisthe
distinct dynamical
in
the
following
section.
(V, E) denote
an undirected graph
whose vertex
set V =
on 185a backward
(attracting
(repelling
regions, they
are alsostructures)
regions thatorareforward
uncertain
with respect will
robots to minimize the control effort expenditure
3 allow
Methodology
{V1 ,integration
. . . , VM } represents
collection
of FTLE-derived
cells
structures)
in within
time.theSince
the manifolds
demarto
velocity
vectors
a
neighborhood
of
the
manifold.
as
they
move
within the set V. We describe the methodology
inbasin
W. An
edge eij separating
exists in thethe
setdistinct
E if cells
Vi and Vj
cate theTherefore,
boundaries
dynamical
We
propose
to leverage
switching
between
regions
in
neighborhoods
of
in
the
following
section. the environmental dynamics and the
220
3
Methodology
share a physical boundary or are physically adjacent. In other
regions,the
they
are
also
regions
that
are
uncertain
with
respect
inherent
environmental
noise to synthesize energy-efficient
manifold
is influenced
bothforbyW.
deterministic
words,
G serves
as a roadmap
For the caseuncertainty
shown in
to velocity
vectors
within
a
neighborhood
of
the
manifold.
control
policies
for
a
team
sensing resources/robots
as
well
stochasticity
to external
noise. based on four
We propose to leverage of
themobile
environmental
dynamics and the
2a,as
adjacency
of andue
interior
cell is defined
Fig.
to
maintain
the
desired
allocation
in
W at all times.
We asTherefore,
switching
between
regions
in
neighborhoods
of
190
Given
an
FTLE-based
cell
decomposition
of
W,
let
G
=
inherent
environmental
noise
to
synthesize
energy-efficient
220
3
Methodology
neighborhoods. Let Ni denote the number of AUVs or mosume
each
robot
has
a
map
of
the
environment.
In
our
case,
the manifold
isdenote
influenced
both by within
deterministic
uncertainty
(V,
an undirected
graph Vwhose
vertex set V =
control policies for a team of mobile sensing resources/robots
bileE)
sensing
resources/robots
i . The objective is to
this
translates
to
providing
the
robots
the
locations
of
LCS
, . . . , VMagent-level
} represents
the collection
cells We to
maintain
the desired
in W at
all times.and
Wethe
asas well{V
as1stochasticity
due tocontrol
external
noise.of
propose
to leverage
theallocation
environmental
dynamics
synthesize
policies,
or FTLE-derived
uk , to achieve and
boundaries
that
define
each
V
in
G.
Since
LCS
curves
sepi
in
W.
An
edge
e
exists
in
the
set
E
if
cells
V
and
V
share
a
225
sume
each
robot
has
a
map
of
the
environment.
In
our
case,
Given
an
FTLE-based
cell
decomposition
of
W,
let
G
=
inherent
environmental
noise
to
synthesize
energy-efficient
ij
i
j
maintain a desired distribution of the N agents across the M
arate
W
into regions
with
flow
dynamics,
thisofbeboundary
physically
adjacent.
other
this policies
translates
the robots
the resources/robots
locations
LCS
(V, E) physical
denote
undirected
set
V words,
=
control
fortoa providing
team
ofdistinct
mobile
sensing
regions, an
denoted
byorN̄are
=graph
[N̄1 , . . whose
. , N̄M
]T vertex
, in anInenvironment
comes
analogous
to
providing
autonomous
ground
or
aerial
195
G
serves
as
a
roadmap
for
W.
For
the
case
shown
in
Fig.
boundaries
that
define
each
V
in
G.
Since
LCS
curves
i in W at all times. We sepby Eq. of
(2).FTLE-derived cells
{V1 , . . .whose
, VM } dynamics
representsare
thegiven
collection
to maintain the desired allocation
asvehicles
a map
of
the environment
which
often obtained
2(a),
adjacency
of
an
interior
cell
is
defined
based
on
four
arate
W
into
regions
with
distinct
flowisdynamics,
this
bein W. An edge eij exists in the set E if cells Vi and Vj share a 225 sume
each
robot
has
a
map
of
the
environment.
In
our
case,
a comes
priori. analogous
In a fluidictoenvironment,
the map consists
of
the
neighborhoods.
Let
N
denote
the
number
of
AUVs
or
moproviding
autonomous
ground
or
aerial
i
physical boundary or are physically
adjacent. In other words,
this translates to providing the robots the locations of LCS
bile
sensing
resources/robots
within
V
.
The
objective
is
to
230
vehicles that
a map
of theeach
environment
often
obtained
i
G serves as a roadmap for W. For the case shown in Fig.
boundaries
define
Vi in G. which
Since is
LCS
curves
sep-a
www.nonlin-processes-geophys.net/20/657/2013/
Geophys.,
657–668,this
2013be2(a), adjacency
of an interior cell is defined based on four
arate W intoNonlin.
regionsProcesses
with distinct
flow 20,
dynamics,
neighborhoods. Let Ni denote the number of AUVs or mocomes analogous to providing autonomous ground or aerial
bile sensing resources/robots within Vi . The objective is to 230 vehicles a map of the environment which is often obtained a
660
K. Mallory et al.: Allocation of swarms in gyre flows
locations of the maximum FTLE ridges computed from data
and refined, potentially in real-time, using a strategy similar to the one found in Hsieh et al. (2012). Thus, we assume
each robot has a map of the environment and has the ability
to determine the direction it is moving in within the global
coordinate frame, i.e., the ability to localize.
Qi , denoted by QiL ⊂ Qi , are tasked to leave Vi . The number
of robots in Vi can be established in a distributed manner in a
similar fashion. The auction can be executed periodically at
some frequency 1/Ta where Ta denotes the length of time between each auction and should be greater than the relaxation
time of the AUV/ASV dynamics.
3.1
3.1.2
Controller synthesis
Consider a team of N robots initially distributed across M
gyres/cells. Since the objective is to achieve a desired allocation of N̄ at all times, the proposed strategy will consist
of two phases: an auction phase to determine which robots
within each Vi should be tasked to leave/stay and an actuation phase where robots execute the appropriate leave/stay
controller.
3.1.1
Auction phase
The purpose of the auction phase is to determine whether
Ni (t) > N̄i and to assign the appropriate actuation strategy
for each robot within Vi . Let Qi denote an ordered set whose
elements provide robot identities that are arranged from highest escape likelihoods to lowest escape likelihoods from Vi .
In general, to first order we assume a geometric measure
whereby the escape likelihood of any particle within Vi increases as it approaches the boundary of Vi , denoted as ∂Vi
(Forgoston et al., 2011). Given W, with dynamics given by
Eq. (2), consider the case when ε = 0 and I 6 = 0, i.e., the case
when the fluid dynamics is time-independent in the presence of noise. The boundaries between each Vi are given
by the stable and unstable manifolds of the saddle points
within W as shown in Fig. 2a. While there exists a stable
attractor in each Vi when I = 0, the presence of noise means
that robots originating in Vi have a non-zero probability of
landing in a neighboring gyre Vj where eij ∈ E. Here, we
assume that robots experience the same escape likelihoods
in each gyre/cell and assume that Pk (¬i|i), the probability
that a robot escapes from region i to an adjacent region, can
be estimated based on a robot’s proximity to a cell boundary with some assumption of the environmental noise profile
(Forgoston et al., 2011).
Let d(q k , ∂Vi ) denote the distance between a robot k
located in Vi and the boundary of Vi . We define the
set Qi = {k1 , . . . , kNi } such that d(q k1 , ∂Vi ) ≤ d(q k2 , ∂Vi ) ≤
. . . ≤ d(q Ni , ∂Vi ). The set Qi provides the prioritization
scheme for tasking robots within Vi to leave if Ni (t) > N̄i .
The assumption is that robots with higher escape likelihoods
are more likely to be “pushed” out of Vi by the environment
dynamics and will not have to exert as much control effort
when moving to another cell, minimizing the overall control
effort required by the team.
In general, a simple auction scheme can be used to determine Qi in a distributed fashion by the robots in Vi (Dias
et al., 2006). If Ni (t) > N̄i , then the first Ni − N̄i elements of
Nonlin. Processes Geophys., 20, 657–668, 2013
Actuation phase
For the actuation phase, individual robots execute their assigned controllers depending on whether they were tasked
to stay or leave during the auction phase. As such, the individual robot control strategy is a hybrid control policy consisting of three discrete states: a leave state, U L , a stay
state, U S , which is further distinguished into U SA and U SP .
Robots who are tasked to leave will execute U L until they
have left Vi or until they have been once again tasked to
stay. Robots who are tasked to stay will execute U SP if
d(q k , ∂Vi ) > dmin and U SA otherwise. In other words, if a
robot’s distance to the cell boundary is below some minimum threshold distance dmin , then the robot will actuate and
move itself away from ∂Vi . If a robot’s distance to ∂Vi is
above dmin , then the robot will execute no control actions.
Robots will execute U SA until they have reached a state
where d(q k , ∂Vi ) > dmin or until they are tasked to leave at
a later assignment round. Similarly, robots will execute U SP
until either d(q k , ∂Vi ) ≤ dmin or they are tasked to leave. The
hybrid robot control policy is given by
F (q k )
,
kF (q k )k
F (q k )
U SA (q k ) = −ωi × c
,
kF (q k )k
U SP (q k ) = 0.
U L (q k ) = ωi × c
(3a)
(3b)
(3c)
Here, ωi = [0, 0, 1]T denotes counterclockwise rotation
with respect to the centroid of Vi , with clockwise rotation
being denoted by the negative and c is a constant that sets the
linear speed of the robots. The hybrid control policy generates a control input perpendicular to the velocity of the fluid
as measured by robot k 1 and pushes the robot towards ∂Vi if
U L is selected, away from ∂Vi if U SA is selected, or results in
no control input if U SP is selected The hybrid control policy
is summarized by Algorithm 1 and Fig. 3.
In general, the auction phase is executed at a frequency
of 1/Ta which means robots also switch between controller
states at a frequency of 1/Ta . To further reduce actuation efforts exerted by each robot, it is possible to limit a robot’s actuation time to a period of time Tc ≤ Ta . Such a scheme may
prolong the amount of time required for the team to achieve
the desired allocation, but may result in significant energyefficiency gains. We further analyze the proposed strategy in
the following sections.
1 The inertial velocity of the fluid can be computed from the
robot’s flow-relative velocity and position.
www.nonlin-processes-geophys.net/20/657/2013/
11:
uk ← US
12:
end if
13: end if
K. Mallory et al.: Allocation of swarms in gyre flows
661
36
Algorithm 1 Auction phase
1: if ElapsedT ime == Ta then
2:
Determine Ni (t) and Qi
3:
∀k ∈ Qi
4:
if Ni (t) > N̄i then
5:
if k ∈ QL then
6:
uk ← U L
7:
else
8:
uk ← U S
9:
end if
10:
else
11:
uk ← U S
12:
end if
13: end if
4
37
37
Analysis
In this section, we discuss the theoretical feasibility of the
proposed distributed allocation strategy. Instead of the tradiFig. 3. Schematic of the single-robot hybrid robot control policy.
tional agent-based analysis, we employ a macroscopic analFig. 3. Schematic of the single-robot hybrid robot control policy.
38
ysis of the proposed distributed control strategy given by
Algorithm 1 and Eq. (3). We first note that while the single robot controller shown in Fig. 3 results in an agent-level
The above expression represents a stochastic transition
4 Analysis
stochastic control policy, the ensemble dynamics330
of a team
rule with aij as the per unit transition rate and Ni (t) and
of N robots each executing the same hybrid control stratNj (t) as discrete random variables. In the robotics setting,
egy can be modeled using a polynomial stochastic hybrid
(4) implies
robots at
Vi will
move to Vjfeasibility
with a rate of the
In this Eq.
section,
we that
discuss
the
theoretical
system (pSHS). The advantage of this approach is that it alof aij Ni . We assume the ensemble dynamics is Markovian
distributed allocation strategy. Instead of the tradi- 38
lows the use of moment closure techniques to model the proposed
time
and note that in general aij 6= aj i and aij encodes the inverse
evolution of the distribution of the team across the various
tional agent-based
analysis,
we employ
of the average time
a robot spends
in Vi . a macroscopic analcells. This, in turn, enables the analysis of the stability of the
Given
Eq.
(4)
and
employing
the
extended
generator
we by Alysis of the proposed distributed control
strategy
given
closed-loop ensemble dynamics. The technique was prevican obtain the following description of the moment dynamics
gorithmof1theand
(3). We first note that while the single robot
ously illustrated in Mather and Hsieh (2011). For335
completesystem:
ness, we briefly summarize the approach here and refercontroller
the
shown in Fig. 3 results in an agent-level stochastic
d
interested reader to Mather and Hsieh (2011) for further deE[N]
AE[N],
dt
control policy,= the
ensemble dynamics of a team of(5)
N robots 39
tails.
P
T
each
control
strategy can be modwhere [A]ijthe
= asame
[A]ii = −
The system state is given by N(t) = [N1 (t), . . . , NM (t)]
. executing
j i and hybrid
(i,j )∈E aij (Mather and
It is important
to note thathybrid
A is a Markov
pro-(pSHS).
Hsieh,a2011).
As the team distributes across the M regions, the rate
in using
eled
polynomial
stochastic
system
cess matrix and thus is negative semidefinite.
This,
coupled
which robots leave a given Vi can be modeled using con340
The advantage
of this approach isPthat it allows the use of mowith the conservation constraint i Ni = N leads to expostant transition rates. For every edge eij ∈ E, we assign a
ment closure
techniques
model
nential stability
of the to
system
giventhe
by time
Eq. (5)evolution
(Klavins, of the
constant aij > 0 such that aij gives the transition probability
2010).of the team across the various cells. This, in turn,
per unit time for a robot from Vi to land in Vj . Different from
distribution
In this work, we note that aij s can be determined experiMather and Hsieh (2011), the aij s are a function of the paenablesmentally
the
analysis
of the stability of the closed-loop en- 39
after the selection of the various parameters in the
rameters c, Tc , and Ta of the individual robot control policy
semble distributed
dynamics.
Thestrategy.
technique
was
control
While the
aij spreviously
can be chosenillustrated
to
is given by Eq. (3), the dynamics of the surrounding fluid,
enableand
the team
of robots
to autonomously
maintain the we
de- briefly
and the inherent noise in the environment. Furthermore,
345
inaij(Mather
Hsieh,
2011).
For completeness,
sired steady-state distribution (Hsieh et al., 2008), extraction
is a macroscopic description of the system and thus a paramsummarize
the approach here and refer the interested reader
of the control parameters from user specified transition rates
eter of the ensemble dynamics rather than the agent-based
to of(Mather
and Hsieh,
2011)
further
details.
is a direction
for future
work.for
Thus,
using the
technique desystem. As such, the macroscopic analysis is a description
T
scribed bystate
Mather
Hsieh
the steady-state behavior of the system and becomes exact as
The system
is and
given
by(2011),
N(t)the=following
[N1 (t),result
. . . ,can
NM (t)] .
stated for our current distributed control strategy:
N approaches infinity.
As the beteam
distributes across the M regions, the rate in
Given G and the set of aij s, we model the ensemble dy350
which robots
a agiven
can with
be modeled
using conTheoremleave
1 Given
team of V
Ni robots
kinematics given
namics as a set of transition rules of the form:
by
Eq.
(1)
and
v
given
by
Eq.
(2),
the
distributed
allocation
f
stant transition rates.
For every edge eij ∈ E, we assign a 40
aij
Ni −→ Nj ∀ eij ∈ E.
(4)
strategy given by Algorithm 1 and Eq. (3), at the ensemble
level is stable and achieves the desired allocation strategy.
www.nonlin-processes-geophys.net/20/657/2013/
Nonlin. Processes Geophys., 20, 657–668, 2013
662
6
K. Mallory et al.: Allocation of swarms in gyre flows
For the details of the model development and the proof, we
refer the interested reader to Mather and Hsieh (2011).
5
Simulation results
We validate the proposed control strategy described by Algorithm (1) and Eq. (3) using three different flow fields:
Fig. 4. Three desired distributions of the team of N = 500 mobile
Fig.resources/robots.
4. Three desired distributions
of pattern
the teamformation,
of N = 500 (b)
mobile
1. the time invariant wind driven multi-gyre model givensensing
(a) A Ring
a Block
sensing
resources/robots.
(a)L-shaped
A ring pattern
formation,
(b) a block
by Eq. (2) with ε = 0;
pattern
formation,
and (c) an
pattern
formation.
Each box
pattern aformation,
an L-shaped
pattern formation.
Each
box of
represents
gyre andand
the(c)number
designates
the desired
number
represents a gyre and the number designates the desired number of
2. the time varying wind driven multi-gyre model given byrobots
contained within each gyre.
robots contained within each gyre.
Eq. (2) for a range of ω 6 = 0 and ε 6 = 0 values; and
6
TEXT: TEXT
3. an experimentally generated flow field using different
1. the time invariant wind driven multi-gyre model given
values of Ta and c in Eq. (3).
by (2) with ε = 0,
We refer to each of these as Cases 1, 2, and 3, respectively.
Two metrics are used to compare the three cases. The first is 2. the time varying wind driven multi-gyre model given by
(2) for a range of ω 6= 0 and ε 6= 0 values, and
the mean vehicle control effort to indicate the energy expenditure of each robot. The second is the population root mean
405
3. an experimentally generated flow field using different
square
of the resulting
Fig. 4.error
Three(RMSE)
desired distributions
of therobot
team population
of N = 500 distrimobile
values of Ta and c in (3).
bution
with
respect to the(a)desired
The (b)
RMSE
is
sensing
resources/robots.
A Ring population.
pattern formation,
a Block
(a) No Control
(b) Ring
pattern
formation,
and (c) an of
L-shaped
patternpolicy
formation.
Each box
used
to show
effectiveness
the control
in achieving
We
refer
to
each
of
these
as
Cases
1,
2,
and 3 respectively.
gyre and the number designates the desired number of
therepresents
desired adistribution.
robots
contained
within
each
gyre.
All cases assume a team of N = 500 robots. The robots areTwo metrics are used to compare the three cases. The first is
randomly distributed across the set of M gyres in W. For thethe mean vehicle control effort to indicate the energy expentheoretical
models,
the workspace
W multi-gyre
consists ofmodel
a 4 ×410
4 setditure of each robot. The second is the population root mean
1. the time
invariant
wind driven
given
of gyres,
and
each
V
∈
V
corresponds
to
a
gyre
as
shown
insquare error (RMSE) of the resulting robot population distrii 0,
by (2) with ε =
Fig. 2a. We considered three sets sets of desired distributions,
2. the
timeformation,
varying wind
driven
multi-gyre
model
given bybution with respect to the desired population. The RMSE is
namely
a ring
a block
formation,
and
an L-shaped
(2) for
a rangeinofFig.
ω 6=4.0The
andexperimental
ε 6= 0 values,flow
and data hadused to show effectiveness of the control policy in achieving
formation
as shown
(c) Block
(d) L-Shape
a set3.ofan
3×4
regions. The inner
two cells
whilethe desired distribution.
405
experimentally
generated
flowcomprised
field usingW,
different
C
the complement,
, consisted of the remaining cells.415
This All
cases
assumeofa the
team
N = 500
robots.
The robots
Fig.
5. Histogram
finalof
allocations
in the
time-invariant
values of TW
a and c in (3).
designation of cells helped to isolate the system from bound-are randomly
flow 5.field
for
the of
swarm
of
(a)
passive
robots
exerting
distributed
across
the set
of time-invariant
M
gyresnoinconW. For
Fig.
Histogram
the final
allocations
in the
flow
refer to
of these
Cases
2, and
respectively.
aryWe
effects,
andeach
allowed
the as
robots
to 1,
escape
the3 center
gyresthe theoretical
trols for
and the
robots
exerting
control
forming
the
(b) Ring
pattern
with
models,
the
workspace
W
consists
of
a 4×4
field
swarm
of
(a)
passive
robots
exerting
no
controls
and
metrics are The
useddesired
to compare
the three
cases.
The first is
Tc = 0.8T
a at t = 450, (c) Block pattern with Tc = Ta at t = 450,
inTwo
all directions.
pattern
for this
experimental
robots
exerting
control
forming
the
(b)
ring
pattern
with
T
=
0.8T
c
a
set ofandgyres,
and pattern
each Vwith
corresponds to a gyre as shown
i ∈TV
theset
mean
effort
to indicate
thewithin
energya expen(d) L-shape
c = 0.5Ta at t = 450.
data
wasvehicle
for all control
the agents
to be
contained
single
at t = 450, (c) block pattern with
Tc = Ta at t = 450, and (d) L410
diture
ofof
each
The second
is the population
rootofmean
2(a). We considered three sets sets of desired districell.
Each
the robot.
three cases
was simulated
a minimum
fivein Fig.
shape pattern with Tc = 0.5Ta at t = 450.
square
error
(RMSE)
of
the
resulting
robot
population
distri420
butions, namely a Ring formation, a Block formation, and
times and for a long enough period of time until steady-state
Tc
2
5
8
9
10
Baseline
440
bution
with respect to the desired population. The RMSE isan L-shaped
was
reached.
formation
as5.98
shown3.45
in Fig.
4. The
experimenRing
Pattern
12.99
3.49
3.66
4.09
used to show effectiveness of the control policy in achieving
tal flow
data
had a set
3 × 4 regions.
inner two
Block
Pattern
- of 11.21
- The12.72
- cells
and
individual robots
follow
fixed -trajectories
navigat5.1the desired
Case I:distribution.
time-invariant flows
L Pattern
- the
30.09
- C when
30.45
- of the
comprised
W,
while
complement,
W
consisted
415
All cases assume a team of N = 500 robots. The robots
ing from one gyre to another. For this baseline case, robots
Table 1. Summary
of the RMSE for each
simulation
pattern
at
aretime-invariant
randomly distributed
the set εof=M0,gyres
W.sFor
cells
helped
totraisolate
travel incells.
straightThis
linesdesignation
at fixed speedsofusing
a simple
PID
For
flows, across
we assume
A =in
0.5,
=remaining
t=450
with
the
time-invariant
flow
field.
models,
the in
workspace
a 425
4 × 4the system
from boundary
effects,
and allowed
the robots
jectory follower
and treat the
surrounding
fluid dynamics
as to
20,theµtheoretical
= 0.005, and
I = 35
Eq. (2). W
Forconsists
the ringofpattern,
of gyres,the
andcase
each when
Vi ∈ Vthe
corresponds
a gyre
as shown
an external
disturbance
source.
The
RMSE
results
for
all
patwesetconsider
actuationtowas
applied
forescape
the center gyres in all directions. The desired pattern 445
in
Fig.
2(a).
We
considered
three
sets
sets
of
desired
distri1 and
6. The
cumulative
ternsexperimental
are summarized
in Table
Tc = f Ta amount of time where f = 0.1, 0.2, . . . , 1.0, andfor this
data
set was
forFig.
all the
agents
to be con420
namely
a Ring
a Block we
formation,
and
control
effort
per
agent
is
shown
in
Fig.
7a.
From
Fig.
6, we
Tabutions,
= 10. For
the block
andformation,
L-shape patterns,
considered
440
We
compared
our
results
to
a
baseline
deterministic
allosimuan L-shaped formation as shown in Fig. 4. The experimen-tained within a single cell. Each of the three cases was
see
that
our
proposed
control
strategy
performs
comparable
the cases when Tc = 0.5Ta and Tc = Ta . The final populastrategy where
thetimes
desiredand
allocation
is pre-computed
a
minimum
of
five
for
a
long
enough
period
tal flow data had a set of 3 × 4 regions. The inner two cellslatedcation
to
baseline robots
case especially
when
Tc = Ta when
= 10 s.
In fact,
tion distribution of the team for the case with
no controls and
andthe
individual
follow fixed
trajectories
navigatcomprised W, while the complement, W C consisted of
430 theof time until steady-state was reached.
even
when
T
<
T
,
our
proposed
strategy
achieves
the dethe cases with controls for each of the patterns are shown
in
ing from onecgyre ato another. For this baseline case, robots
450
remaining cells. This designation of cells helped to isolate
sired
distribution.
The
advantage
of
the
proposed
approach
Fig. 5.
in straight lines at fixed speeds using a simple PID
425
the system from boundary effects, and allowed the robots to5.1 travel
Case
I: Time-Invariant
Flowswhen compared to the
lies
in the
significant energy gains
We compared our results to a baseline deterministic alloescape the center gyres in all directions. The desired pattern 445 trajectory follower and treat the surrounding fluid dynamics
baseline
case,
especially
when
T
Ta ,RMSE
as seenresults
in Fig.for
7. all
We
cation strategy where the desired allocation is pre-computed
as an external disturbance source.c <
The
for this experimental data set was for all the agents to be conFor
time-invariant
flows,
we
assume
ε
=
0,
A
=
0.5,
s
= 20,
patterns
are
summarized
in
Table
1
and
Fig.
6.
The
cumulatained within a single cell. Each of the three cases was simutive
control
effort
per
agent
is
shown
in
Fig.
7.
From
Fig.
6,
I = 35 in (2). For the ring pattern, we conlated a Processes
minimum of
five times20,
and
for a long
enough periodµ = 0.005, and
Nonlin.
Geophys.,
657–668,
2013
www.nonlin-processes-geophys.net/20/657/2013/
that when
our proposed
control strategy
performsfor
compasiderwe
theseecase
the actuation
was applied
Tc = f Ta 455
430
of time until steady-state was reached.
450
rable to the baseline case especially when Tc = Ta = 10 sec.
435
amount
of time where f = 0.1, 0.2, . . . , 1.0, and Ta = 10. For
In fact, even when Tc < Ta , our proposed strategy achieves
5.1 Case I: Time-Invariant Flows
the Block and L-Shape patterns, we considered the cases
F
fl
tr
T
a
T
t=
c
a
in
tr
tr
a
p
ti
w
r
I
th
p
th
7
c
K. Mallory et al.: Allocation of swarms in gyre flows
663
Table 1. Summary of the RMSE for each simulation pattern at t =
450 with the time-invariant flow field. The RMSE for the baseline
case is 4.09.
Tc
Ring Pattern
Block Pattern
L Pattern
2
5
8
9
10
12.99
–
–
5.98
11.21
30.09
3.45
–
–
3.49
–
–
3.66
12.72
30.45
Fig. 7. C
different
omit the cumulative control effort plots for the other cases
since they are similar to Fig. 7.
In time-invariant flows, we note that for large enough Tc ,
our proposed distributed control strategy performs comparable to the baseline controller both in terms of steady-state
error and convergence time. As Tc decreases, less and less
control effort is exerted and thus it becomes more and more
difficult for the team to achieve the desired allocation. This
is confirmed by both the RMSE results summarized in Table 1 and Fig. 6a–c. Furthermore, while the proposed control strategy does not beat the baseline strategy as seen in
Fig. 6a, it does come extremely close to matching the baseline strategy performance. while requiring much less control
effort as shown in Fig. 7 even at high duty cycles, i.e., when
Tc /Ta > 0.5.
More interestingly, we note that executing the proposed
control strategy at 100 % duty cycle, i.e., when Tc = Ta ,
in time-invariant flows did not always result in better performance. This is true for the cases when Tc = 0.5Ta = 5
for the block and L-shaped patterns shown in Fig. 6b–c.
In these cases, less control effort yielded improved performance. However, further studies are required to determine
the critical value of Tc when less control yields better overall performance. In time-invariant flows, our proposed controller can more accurately match the desired pattern while
using approximately 20 % less effort when compared to the
baseline controller.
5.2
Case II: time-varying flows
For the time-varying, periodic flow, we assume A = 0.5,
s = 20, µ = 0.005, I = 35, and ψ = 0 in Eq. (2). Additionally, we considered the performance of our control strategy
for different values of ω and ε with Ta = 10 and Tc = 8 for
the ring formation and Tc = 5 for the L-shaped formation. In
all these simulations, we use the FTLE ridges obtained for
the time-independent case to define the boundaries of each
Vi . The final population distribution of the team for the case
with no controls and the cases with controls for the ring and
L-shape patterns are shown in Fig. 8. The final population
RMSE for the cases with different ω and ε values for the ring
and L-shape patterns are shown in Fig. 9. These figures show
the average of 10 runs for each ω and ε pair. In each of these
runs, the swarm of mobile sensors were initially randomly
distributed within a 4 × 4 grid cell. Finally, Fig. 10 shows
www.nonlin-processes-geophys.net/20/657/2013/
465
470
475
480
control
difficul
confirm
and Fig
strategy
6(a), it
strategy
fort as
Tc /Ta >
More
control
time-in
mance.
the Blo
In these
mance.
the crit
all perf
troller c
using a
baselin
5.2
485
Fig.6.6.Comparison
Comparison of
of the
the population RMSE in
Fig.
in the
the time-invariant
time-varying
flowfor
forthe
the(a)
(a)Ring
ringformation,
formation, (b)
(b)the
theBlock
block formation,
formation, and
and (c)
(c) the
the
flow
L-shapeformation
formationfor
fordifferent
differentTTcc,,and
andfor
forthe
thePID
PIDcontrol
control baseline
baseline
L-shape
490
controllerininthe
theRing
ring case with time-invariant flows.
controller
flows.
the population RMSE as a function of time for the ring and
L-shape patterns.
495
In time-varying, periodic flows we note that our proposed
control strategy is able to achieve the desired final allocation
even at 80 % duty cycle, i.e., Tc = 0.8Ta . This is supported
by the results shown in Fig. 9. In particular, we note that the
proposed control strategy performs quite well for a range of
ω and ε parameters for both the ring and L-shape patterns.
While the variation in final RMSE values for the ring pattern
is significantly lower than the L-shape pattern, the variations
Nonlin. Processes Geophys., 20, 657–668, 2013
C
For the
s = 20,
we con
differen
Ring fo
all thes
the tim
Vi . The
with no
L-shape
RMSE
Ring an
show th
of these
domly d
664
8
7
(a) No Control
K. Mallory et al.: Allocation of swarms
in gyreTEXT
flows
TEXT:
(b) Ring
(a) Ring
7. Comparison
totalcontrol
control effort
thethe
ringRing
pattern
for for
Fig. 7.Fig.
Comparison
of of
thethetotal
effortforfor
pattern
8
different
T
with
the
baseline
controller
for
time-invariant
flows.
c the baseline controller for time-invariant flows.
different Tc with
TEXT: TEXT
(c) L-Shape
465
500
470
505
475
510
480
Fig. 8. Histogram
the final allocations
flows,more
with pacontrol
effort isof exerted
and thusinitperiodic
becomes
and more
and
ε
=
5,
for
the
swarm
of
(a)
passive
robots
rameters of ω = 5∗π
40
difficult
the team
achieve
thecontrol
desired
allocation.
exerting nofor
controls
and to
robots
exerting
forming
the (b) This is
confirmed
by
both
the
RMSE
results
summarized
in Table 1
Ring pattern with Tc = 0.8Ta at t = 450, and (c) L-Shape pattern
with TFig.
c = 0.5T
a at t = 450.
and
6(a)-6(c).
Furthermore, while the proposed control
strategy does not beat the baseline strategy as seen in Fig.
(a)
No Control
(b) Ring
6(a),
it does
come
extremely
to matching
10 shows
the population
RMSE asclose
a function
of time forthe
thebaseline
(b) L-Shape
strategy
performance.
requiring much less control efRing and L-shape
patternswhile
respectively.
fortIn as
shown in periodic
Fig. 7 even
dutyour
cycles,
i.e., when
time-varying,
flows at
wehigh
note that
proposed
Fig.Fig.
9. Final
population
for
9. Final
populationRMSE
RMSEfor
fordifferent
differentvalues
values of
of ω
ω and
and εε for
control
strategy
is
able
to
achieve
the
desired
final
allocation
(a)
the
Ring
formation
and
(b)
the
L-shaped
formation.
T
/T
>
0.5.
(a)
the
ring
formation
and
(b)
the
L-shaped
formation.
(a)
Ring
c
a
even
at 80%
duty cycle, i.e.,
This is supported
c = 0.8T
More
interestingly,
we Tnote
thata .executing
the proposed
by the results shown in Fig. 9. In particular, we note that the
control strategy at 100% duty cycle, i.e., when Tc = Ta , in
proposed control strategy performs quite well for a range of
time-invariant
flows did not always result in better perfor-of 60 s. Figure 11 shows the top view of our experimental
ω and ε parameters for both the Ring and L-shape patterns.
and theto
resulting
obtained
via12.
PIV.
Further
corresponds
a singleflow
gyrefield
as shown
Fig.
The
cells
mance.
is true
forRMSE
the
cases
fortestbed
(c) L-Shape
c = 0.5T
a = 5 cell
While theThis
variation
in final
valueswhen
for theTRing
pattern
details
regarding
the
experimental
testbed
can
be
found
in
of
primary
concern
are
the
central
pair
and
the
remainder
the
Block
and
L-shaped
patterns
shown
in
Fig.
6(b)
and
6(c).
is significantly lower than the L-shape pattern, the variations
Fig. 8. Histogram of the final allocations in periodic flows, with paMichini
et
al.
(2013).
Using
this
data,
we
simulated
a
swarm
boundary
cells
were
not
used
to
avoid
boundary
effects
and
in final
RMSE
allofwithin
10%robots
of perfor5∗πfor
In
these
cases,
less
control
effort
yielded
rameters
ofvalues
ω=
andthe
ε =L-shape
5,
for the are
swarm
(a)improved
passive
Fig.
8. Histogram
of the
final
allocations
in periodic
flows, with
pa- to of
40
500 robots
mobile to
sensors
executing
the control
strategy
given by
allow
escape
the center
gyres in
all directions.
the total
swarm
population.
5·π
exerting
no
controls
and
robots
exerting
control
forming
the
(b)
mance.
However,
further
studies
are
required
to
determine
rameters of ω = 40 and ε = 5, for the swarm of (a) passive robots
Eq.robots
(3). were initially uniformly distributed across the
530
The
Ring pattern
with Tcand
t = 450,
and (c)
L-Shape
a at
exerting
no controls
robots
exerting
control
forming
thepattern
(b) ring over- To determine the appropriate tessellation of the
the critical
value
of T=c 0.8T
when
less control
yields
better
two center cells and all 500 robots were tasked to stay within
with TIII:
= 0.5T
t = 450.
5.3 Case
Experimental
Flows
cwith
pattern
Tca=at0.8T
at
t
=
450,
and
(c)
L-shape
pattern
with
a
workspace,
used
thenoLCS
ridges
all performance. In time-invariant flows, our proposed
conthe
upper centerwe
cell.
When
control
effortobtained
is exertedfor
by the
the
Tc = 0.5Ta at t = 450.
temporal
mean
of
the
velocity
field.
This
resulted
in
the
troller
can
more
accurately
match
the
desired
pattern
while
Using our 0.6m × 0.6m × 0.3m experimental flow tank
robots, the final population distribution achieved by the team
the space
into
a grid the
of 4final
× 3 cells.
Each
10 shows
population
RMSE
as a function
time
the
equipped
with athe
grid
of 4x3
of driving
cylinders,
we for
genshown in Fig.of13(a).
controls,
population
(b)With
L-Shape
using
approximately
20%set
less
effort
whenofcompared
to isthediscretization
12.
The
cells
cell
corresponds
to
a
single
gyre
as
shown
Fig.
Ring
and
L-shape
patterns
respectively.
RMSE values
for the flow
L-shape
10 % 535
of distribution is shown in Fig. 13(b). The control strategy
eratedina final
time-invariant
multi-gyre
fieldare
to all
usewithin
in simulabaseline
Incontroller.
time-varying, periodic flows we note that our proposed
of
primary
concern
are
the
central
pair
and
the
remainder
Fig. 9. Final population RMSE for different values of ω and ε for
500
515
485
520
525
490
the totalimage
swarmvelocimetry
population. (PIV) was used to extract the
tion. Particle
was applied assuming Tc /Ta = 0.8. The final RMSE for
control strategy is able to achieve the desired final allocation
(a)boundary
the Ring formation
andnot
(b) the
L-shaped
formation.
cells were
used
to avoid
boundary effects and
surface
flows
at
7.5
Hz
resulting
in
a
39×39
grid
of
velocity
different
values of c in (3) and Ta is shown in Fig. 14(a) and
even Case
atII:
80%
duty
cycle,
i.e.,
T
=
0.8T
.
This
is
supported
5.2 5.3
Case
Time-Varying
Flows
to
allow
robots
to
escape
the
center
gyres
in all directions.
c
a
III: experimental flows
measurements.
The
data was
collected
for a total
of 60that
sec.
RMSE
as a function
of time uniformly
for different
values ofacross
c and the
Ta
by the results
shown
in Fig.
9. In particular,
we note
the
The robots
were initially
distributed
FigureUsing
11
shows
the
top
view
of
our
experimental
testbed
are
shown
in
Fig.
14(b).
proposed
quite
well
a flow
range tank
of = 0.5,
ourcontrol
0.6 mstrategy
× 0.6
mperforms
× 0.3 mflow,
experimental
two center cells and all 500 robots were tasked to stay within
For the
time-varying,
periodic
we for
assume
A
and
flow
field
obtained
viaofPIV.
Further
details we
540
The
results
obtained
using nothecontrol
experimental
flow field
505 the
ω resulting
and ε parameters
forof
both
the
Ring
and
L-shape
patterns.
equipped
with
a
grid
4
×
3
set
driving
cylinders,
the upper
center
cell. When
effort is exerted
by
sregarding
= 20,
µ
=
0.005,
I
=
35,
and
ψ
=
0
in
(2).
Additionally,
cell
corresponds
to
a
single
gyre
as
shown
Fig.
12.
The
cells to
While
in final
RMSE
forfield
theinRing
thetheexperimental
testbed
canvalues
beflow
found
that thethe
proposed
control strategy
has achieved
the potential
generated
avariation
time-invariant
multi-gyre
to(Michini
usepattern
in sim- shows
the robots,
final population
distribution
by the
of effective
primary concern
are flows.
the central
pair and
the remainder
we
considered
thethis
performance
of (PIV)
our
control
for
is
significantly
thanvelocimetry
the L-shape
pattern,
theused
variations
et al.,
2013).
Using
data,
we
simulated
a swarm
ofstrategy
500
in realistic
However,
the
perforulation.
Particlelower
image
was
to ex- be
team is shown
in Fig.
13a.
With
controls,
theresulting
final
population
boundary
cells
were
not
used
to
avoid
boundary
effects
and
in
final
RMSE
values
for
the
L-shape
are
all
within
10%
of
different
values
of ω
and
ε7.5
with
Ta = 10given
T×c(3).
= 8grid
for mance
thedistribution
mobiletract
sensors
executing
theatcontrol
by
will require
good
matching
between
thestrategy
amountwas
of
the surface
flows
Hz strategy
resulting
inand
a 39
39
is
shown
in
Fig.
13b.
The
control
to allow robots to escape the center gyres in all directions.
the
total swarm
population.
To of
determine
the
tessellation
of
the
control
effort
a
vehicle
can
realistically
exert,
the
frequency
Ring
formation
and
Tappropriate
=
5
for
the
L-shaped
formation.
In
velocity
measurements.
The
data
was
collected
for
a
total
applied
assuming
T
/T
=
0.8.
The
final
RMSE
for
different
c
c a uniformly distributed across the
530
The robots were initially
workspace,
we used the
LCS
ridges
obtained
forobtained
the 545 in
which
the
auctions
occur
within
a cell,
and
scales
all
these
simulations,
we
use
the
FTLE
ridges
for
two
center
cells
and
all
500 robots
were
tasked
to the
staytime
within
510
5.3 Case III: Experimental Flows
temporal
mean
of
the
velocity
field.
This
resulted
in
the
of
the
environmental
dynamics
as
shown
in
in
Figs.
14(a)
the upper center
cell. When no control effort is exerted by the and
the time-independent
case to define
the boundaries
of each
Nonlin. Processes Geophys.,
20, 657–668,
2013
www.nonlin-processes-geophys.net/20/657/2013/
discretization
of
the
space
into
a
grid
of
4
×
3
cells.
Each
14(b).
This
is
an
area for future
investigation.
Using our 0.6m × 0.6m × 0.3m experimental flow tank
robots, the final population
distribution
achieved by the team
Vi . The final population distribution of the team for the case
equipped with a grid of 4x3 set of driving cylinders, we genis shown in Fig. 13(a). With controls, the final population
with no
controls
and themulti-gyre
cases with
for
Ring and
erated
a time-invariant
flowcontrols
field to use
in the
simuladistribution is shown in Fig. 13(b). The control strategy
535
9
K. TEXT
Mallory et al.: Allocation of swarms in gyre flows
TEXT:
665
(a)
(a)
Fig. 10. Comparison of RMSE over time for select ω and ε pairs for
Fig. 10. Comparison of RMSE over time for select ω and ε pairs for
the (a)the
Ring
and (b) L-shaped patterns in periodic flows.
(a) ring and (b) L-shaped patterns in periodic flows.
n of RMSE over time for select ω and ε pairs for
L-shapedvalues
patterns
in periodic
flows.
of c in Eq.
and Ta isOutlook
shown in Fig. 14a and RMSE
6 Conclusions
and(3)Future
as a function of time for different values of c and Ta are
(b)
in Fig. 14b.
In thisshown
work,
we presented the development of a distributed
The results obtained using the experimental flow field
and
Future
Outlook
550
hybrid
control
forcontrol
a teamstrategy
of robots
to potential
maintain
Fig. 11. (a) Experimental setup of flow tank with 12 drive
shows
that strategy
the proposed
has the
to a desired spatial
distribution
a stochastic
geophysical
fluid enders. (b) Flow field
be effective
in realistic in
flows.
However, the
resulting perfor(b)for image (a) obtained via particle im
mance
will
require
good
matching
between
the
amount
of
locimetry (PIV).
vironment.
We assumed robots
a map of the workspace
presented
the development
of a have
distributed
control
effort
a setting
vehicle can
realistically
exert,some
the frequency
which
in
the
fluid
is
akin
to
having
estimate
of
ategy for ina which
teamthe
ofauctions
robotsoccur
to maintain
a deFig. 11. (a) Experimental setup of flow tank with 12 driven cylinwithin a cell, and the timescales
Fig. 11. (a) Experimental setup of flow tank with 12 driven cylinthe
global
fluid
dynamics.
This
can
be
byand
knowing
bution inofa the
stochastic
geophysical
fluidachieved
enenvironmental
dynamics as shown
in
Fig. 14a ders.
b. (b) ders.
Flow field for image (a) obtained via particle image ve555
the locations
of
the
material
lines within the flow field that (b) Flow field for image (a) obtained via particle image veThis
is
an
area
for
future
investigation.
locimetrylocimetry
(PIV).(PIV).
sumed robots have a map of the workspace
separate regions with distinct dynamics. Using this knowl-
setting
to having
some estimate
of
edge,isweakin
leverage
the surrounding
fluid dynamics
and inher6This
Conclusions
and
future
outlook
ynamics.
can benoise
achieved
by knowing
ent environmental
to synthesize
energy efficient conwhich in the fluid setting is akin to having some estimate of
he material
linestowithin
the
flow
fieldallocation
that
trol strategies
achieve
a distributed
of the teamthe global fluid dynamics. This can be achieved by knowIn this work,
we
presented
the development
of a distributed
560
todistinct
specific
regions
in theUsing
workspace.
Our to
initial
results
with
this
knowling the locations of the material lines within the flow field
hybriddynamics.
control strategy
for
a team
of robots
maintain
a de-show
that separate regions with distinct dynamics. Using this
sired
spatial
distribution
in
a
stochastic
geophysical
fluid
enthat
using
such
a
strategy
can
yield
similar
performance
as
the surrounding fluid dynamics and inhervironment.
We
assumed
robots
have
a
map
of
the
workspace
deterministic
approaches
thatefficient
do not explicitly
account forknowledge, we leverage the surrounding fluid dynamics and
l noise
to synthesize
energy
conthe impact of the fluid dynamics while reducing the control
achieve
a distributed allocation of the team
Nonlin. Processes Geophys., 20, 657–668, 2013
effort www.nonlin-processes-geophys.net/20/657/2013/
required by the team.
s565in theFor
workspace.
Our
initial
results
show
future work we are interested in using actual ocean
strategy
can
similar
performance
as allocation stratflow data toyield
further
evaluate
our distributed
(LCS
ders. (b) Flow field for image (a) obtained via particle image velocimetry (PIV).
666
K. Mallory et al.: Allocation of swarms in gyre flows
590
595
600
(a)
10
TEXT: TEXT
605
12. FTLE
thetemporal
temporal mean
of the
veFig. 12.Fig.
FTLE
fieldfield
forforthe
mean
of experimental
the experimental
velocity data. The field is discretized into a grid of 4 × 3 cells whose
locity data. The field is discretized into a grid of 4 × 3 cells whose
et al., 2009; Lolla et al., 2012), and realistic 2D and 3D unboundaries are shown in black.
boundaries are shown in black.
bounded flow models provided by the Navy Coastal Ocean
Model (NCOM) database. Particularly, we are interested in
610
extending our strategy to non-periodic, time-varying flows.
In addition, we are currently developing an experimental
testbed capable of generating complex 2D flows in a controlled laboratory setting. The objective is to be able to eval580
uate the proposed control strategy using experimentally gen615
erated flow field data whose dynamics are similar to real(b)since our proposed strategy reistic ocean flows. Finally,
(a)
(b)
quires robots to have some estimate of the global fluid dyFig.Fig.
14.14.
(a) (a)
Final
RMSE
forfor
different
values
of of
c and
TaTusing
the
Final
RMSE
different
values
c and
a using
namics,
another
immediate
direction
forconstant
future
work
is to deexperimental
flow
field.
T
/T
=
0.8
is
kept
throughout.
c
a
the
experimental
flow
field.
T
/T
=
0.8
is
kept
constant
througha
Fig. 13. Population distribution for a swarm of 500 mobile sensors
585
termine
howtime
welltime
one
canc cestimate
fluid dynamics
given
RMSE
over
select
andc and
Ta the
parameters
ononananexper(b) RMSE
over for
for select
Ta parameters
ex- 620
over
period
of 60 sec
(a) withfor
noa controls,
and (b) (b)out.
Fig.a13.
Population
distribution
swarm ofi.e.,
500passive,
mobile sensors
knowledge
of
the
locations
of
Lagrangian
coherent
structures
perimental
duty
cycleTcT/T
=0.8
0.8 is
flow flow
field.field.
TheThe
duty
cycle
is kept
keptconstant
constant
aa=
c /T
with
= 0.8T
. controls, i.e., passive, and (b) with imental
overcontrols
a periodwith
of 60Tsc (a)
withano
(LCS) in the flow field.
throughout.
throughout.
575
controls with Tc = 0.8Ta .
inherent environmental noise to synthesize energy efficient
control strategies to achieve a distributed allocation of the
team to specific regions in the workspace. Our initial results
show that using such a strategy can yield similar performance
as deterministic approaches that do not explicitly account for
the impact of the fluid dynamics while reducing the control
effort required by the team.
For future work we are interested in using actual ocean
flow data to further evaluate our distributed allocation strategy in the presence of jets and eddies (Rogerson et al.,
1999; Miller et al., 2002; Kuznetsov et al., 2002; Mancho
et al., 2008; Branicki et al., 2011; Mendoza and Mancho,
2012). We also are interested in using more complicated
flow models including a bounded single-layer PDE ocean
model (Forgoston et al., 2011), a multi-layer PDE ocean
model (Wang et al., 2009; (a)
Lolla et al., 2012), and realistic 2-D and 3-D unbounded flow models provided by the
Nonlin. Processes Geophys., 20, 657–668, 2013
590
595
600
605
Navy
Coastal Ocean KM
Model
database.
Particularly,
Acknowledgements.
and(NCOM)
MAH were
supported
by the Of625
we
in extending
strategy
to non-periodic, EF
ficeare
ofinterested
Naval Research
(ONR) our
Award
No. N000141211019.
time-varying
flows.
In addition,
we are
currently
developing
was supported
by the
U.S. Naval
Research
Laboratory
(NRL)
No. N0017310-2-C007.
IBS
was supported
ONR
anAward
experimental
testbed capable of
generating
complexby2-D
grant in
N0001412WX20083
and the
NRL Base
Research Program
flows
a controlled laboratory
setting.
The objective
is to
authors control
additionally
acknowledge
beN0001412WX30002.
able to evaluate theThe
proposed
strategy
using ex-support by the ICMAT
Severo
Ochoa
SEV-2011-0087.
perimentally
generated
flow
fieldproject
data whose
dynamics are
similar to realistic ocean flows. Finally, since our proposed
strategy requires robots to have some estimate of the global
fluid
dynamics, another immediate direction for future work
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Acknowledgements. K. Mallory and M. A. Hsieh were supported by the Office of Naval Research (ONR) Award No.
N000141211019. E. Forgoston was supported by the US Naval
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Edited by: S. Wiggins
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