A Particle Swarm Model for Swarm-based Networked
Sensor S y s t e m s *
B. Anthony Kadrovach and Gary B. Lamont
Department of Electrical and Computer Engineering
Graduate School of Engineering and Management
Air Force Institute of Technology
Wright-Patterson AFB (Dayton), OH 45433, USA
{ B r i a n . K a d r o v a c h . Gmry. Lamont}@aJEi1:. edu
ABSTRACT
avoidartce), c o n n e c t i v i t y ( t h r o u g h cohesion), a n d fault tolerance ( t h r o u g h f o r m a t i o n a d a p t a t i o n ) . However, t h e n e t w o r k
d y n a m i c s chsalenge t r a d i t i o n a l r o u t i n g protocols[8].
G r e a t strides are b e i n g m a d e i n m i n i a t u r i z a t i o n of electronic a n d electro-mechanical systems[17, g]. T h e s e devices,
e q u i p p e d w i t h wireless c o m m u n i c a t i o n s s y s t e m s a n d m u l tiple t y p e s of sensors c a n provide sensor d a t a in n u m e r o u s
e n v i r o n m e n t s i n c l u d i n g those u n s u i t a b l e for t r a d i t i o n a l sensor systems. A large n u m b e r of these devices could work
together llke a s w a r m of insects or a flock of b i r d s to provide high fidelity i n f o r m a t i o n o n a n e a r rea3-time basis.
T h i s p a p e r presents a m o d e l a n d two m e t r i c s t h a t were developed in order to e v a l u a t e s w a r m behavior. T h e s e m e t r i c s
are used as a f o u n d a t i o n for a p r o p o s e d m e t h o d o [ o ~ , for
clessifying s w a r m b e h a v i o r w i t h r e s p e c t to c o m p u t e r network c o m m u n i c a t i o n s .
T h e p a p e r is organized as follows. F i r s t , a b r i e f descript i o n of p a r t i c l e s w a r m i n t e r a c t i o n is given i n S e c t i o n 2. Sect i o n 3 describes t h e m o d e l a n d the v a r i o u s p a r a m e t e r s t h a t
affect s w a r m behavior. T h e n , e v a l u a t i o n m e t h o d s are described a n d a t a x o n o m y is p r o p o s e d for classifyiug different
types of s w a r m s i n Section 4. T h i s speciflc~lly relates the
p a r a m e t e r s described i n Section 3 to the b e h a v i o r of s w a r m
m e m b e r s . Finally, c o n c h i d i n g r e m a r k s w i t h s u g g e s t i o n s for
f u t u r e work are i n c l u d e d i n Section 6.
S w a r m b e h a v i o r as d e m o n s t r a t e d b y flocks of birds, schools
offish, a n d s w a r m s of insects provide a useful m e t h o d for imp l e m e n t i n g a d i s t r i b u t e d n e t w o r k of mobile sensor platforms.
Such m o b i l e sensor s w a r m s y s t e m s are useful for various
search or surveillance activities. S w a r m b e h a v i o r ensures
safe s e p a r a t i o n b e t w e e n s w a r m m e m b e r s while enforcing a
level of cohesion. T h e s e two properties, w h e n considered i n
t h e context of sensors a n d wireless c o m m u n i c a t i o n s , provide
for low r e d u n d a n c y coverage a n d a r o b u s t a n d reliable comm u n i c a t i o n s system. T h i s p a p e r e x a m i n e s particle s w a r m
b e h a v i o r t h r o u g h s i m u l a t i o n w i t h respect to such a sensor
network. A n a l y s i s of swarm b e h a v i o r for v"-~rious p a r a m e t e r
s e t t i n g s i n d i c a t e a classification methodology. T h i s provides
a f o u n d a t i o n for a p r o p o s e d taxonomy.
Keywords
Networking, wireless, p a r t i c l e swarms, d i s t r i b u t e d processing, robotics
1.
INTRODUCTION
S w a r m behavior, as m a n i f e s t e d b y biologicsa organisms,
provides a n ideal m o d e l for developing a s y s t e m for mobile sensor p l a t f o r m s . Flocks of birds, schools of fish, a n d
s w a r m s of insects d e m o n s t r a t e p r o p e r t i e s t h a t are ideal for
networked mobile sensor systems. T h e p r o p e r t i e s of these
biological systems i n c l u d e s w a r m cohesion, as well as particle avoidance a n d a l i g n m e n t . T h e s e properties, w h e n imp l e m e n t e d i n a s w a r m - b a s e d p a r a l l e l / d i s t r i b u t e d processing
sensor n e t w o r k s y s t e m , provide region coverage ( t h r o u g h
2.
PARTICLE SWARMS
T h i s section presents necessary i n f o r m a t i o n for establishing t h e f o u n d a t i o n for i n v e s t i g a t i o n i n t o s w a r m b e h a v i o r for
the p u r p o s e s of s w a r m - b a s e d sensor networks. A collection
of n o t i o n a l sensor s w a r m a p p l i c a t i o n s are described a n d a
m o r e d e t a i l e d d e s c r i p t i o n of a specific s y s t e m is given. Finally, a n i n t r o d u c t o r y discussion of swarms is p r e s e n t e d .
*The views expressed in this article are those of the a u t h o r s
a n d do n o t reflect t h e official policy or p o s i t i o n of t h e U n i t e d
States Air Force, D e p a r t m e n t of Defense, or the U.S. Government.
2.1
Network Applications
Self organizing, d i s t r i b u t e d sensor [16, 9] a n d a u t o n o m o u s
c o n t r o l s y s t e m s [13] are n o t new. Researchers a t S a n d i a N~tionaJ L a b o r a t o r i e s have developed a t e c h n i q u e u s i n g s w a r m ing t e c h n i q u e s to i m p r o v e searches for avalanche victims[16].
F u r t h e r m o r e , t h e a u t h o r s i n [9] developed m i n i a t u r e sensor s y s t e m s t h a t have a passive c o m m u n i c a t i o n s capability.
Such s y s t e m s could b e used i n a wide r a n g e of a p p l i c a t i o n s
i n c l u d i n g e x p l o r a t i o n of o t h e r p l a n e t s , search for l a n d mines,
r e p o r t traffic b o t t l e n e c k , or as a n sad to w e a t h e r forecasting. T h e s e particles c a n self-organize a n d r e p o r t d a t a from
m u l t i p l e sensors.
This paper is authored by an employee(s) of the [U.S.] Government and is
in the public domain.
SAC 2002, Madrid, Spain
!-58113-445-2/02/03.
918
direction of travel and peripheral vision. This characterization of t h e neighborhood is problem domain specific. W h e n
discussing autonomous control of uninhabited aerial vehicles, an omnidirectional p r o x i m i t y sensor is often assumed
[14] so t h a t the neighborhood is indeed a circular region with
t h e radius based on t h e sensor characteristics.
These swarm behaviors, when integrated, result in a stable swarm formation (whether flying, floating, rolling, etc.)
where every m e m b e r is at least some m i n i m u m distance from
every other m e m b e r and not any farther than some maxim u m distance (as a result of separation and cohesion). T h e
alignment behavior ensures that the swarm, as a formation,
remains stable even in a d y n am i c environment.
This is illustrated in Figure 2. In this simulation, the first
and third rules (Separation and Cohesion respectively) are
implemented. T h e r a n d o m l y selected initial s t a r t i n g points
are indicated by the asterisks. Figure 2 shows the formation
after a number o[ iterations. T h e edges between particles
indicate that the pair of particles is separated by some maximum distance.
Swarm-based sensor systems are of interest to m a n y activities where surveillance is crucial such as p o t e n t i a l highinterest terrorist targets or even military battlefield inform a t i o n gathering. T h e Joint Battlespace Infosphere ( J B I ) - a
combat information s y s t e m - i s a specific example. T h e J B I
provides access for external users to p o t e n t i a l swarm-based
sensor systems [6]. A more thorough description of swarm
sensors in the J B I is given in [8].
2.2
Particle S w a r m Basics
Particle swarms are used in the optimization and vehicle control domains. In the optimization domain, particle
swarms are used in a population-based, stochastic algorithm
to obtain near o p t i m u m solutions to complex, non-linear optimisation problems[10, 2]. T h e research described in this
paper uses a swarm of sensor platforms (vehicles) and thus
focuses on the control aspects. The control of a large swarm
of vehicles is an ongoing area of intense research [5, 4, 14,
11, 13]. Swarm control issues axe i m p o r t a n t to this research
because it establishes the physical network topology. This is
critical to the development and analysis of the ad-hoc network used for sensor d a t a communications. Further, this
research is devoted to developing a realistic m a t h e m a t i c a l
model of swarm formations in order to characterize swarms
w i t h respect to i m p l e m e n t a t i o n parameters in the context
of a wireless, ad-hoc network communications system.
Reynolds [15] presents the classic swarm control t h e o r y
in the description of his holds model. T h e r e are three basic controlling behaviors t h a t govern m o v e m e n t of particles
within the swarm. These behaviors are presented in Table 1.
.m
...
T a b l e 1: S w a r m P a r t i c l e B e h a v i o r s
Behavior
Descrip~-ion
II
Separation Avoid collisions with nearb~ particles
Alignment Attempt to match velocity with
nearby particles
Cohesion
Attempt to stay close to ~e~rby
particles
4O
-2O
.me
0
20
4O
•
.~
EO
NO
100
F i g u r e 2: S t a t i c S w a r m F o r m a t i o n
T h e emphasis on nearby in Table 1 is i m p o r t a n t . Swarm
behavior is based solely on locally observable p h e n o m e n a
and therefore, particles can only react to swarm particles
t h a t are close in proximity. T h e definition of ,~earb~ is depe nd en t on t h e application and is based on several p ar am eters including speed, maneuverability, and size and n u m b e r
of swarm members. T h e behaviors, summarized from [15],
are described in Table 1 and illustrated in Figure 1.
(a) Separation
(b) Alignment
As Figure 2 demonstrates, the particles form a fairly well
defined, regular formation based only on local interactions.
It is this p r o p e r t y of even dispersal over a region that makes
swarm based sensors advantageous for mobile sensor applications.
A slightly different algorithm is proposed in [1]. T h e b~sic principles of alignment, cohesion, and separation are the
same. However, the alignment rule is applicable regardless of
whether a particle is t o o close or too far. This is illustrated
in E q u a t i o n 1. Cohesion and separation axe encapsulated in
the v a i l , oct vector. T h e sign determines whether particles
are a t t r a c t e d or repulsed. T h e weights ,-- and ~ can be functions of ~.he distance between two particles as well as other
parameters (speed, direction, etc.).
(c) Cohesion
F i g u r e 1: S w a r m B e h a v i o r
It should also be noted t h a t only particles within some maximum distance are considered for these calculations. This is
a result of the locally observable phenomena characteristic.
This m a x i m u m distance is another p a r a m e t e r t h a t determines the characteristics of a sw ar m formation. This corn-
In Figure 1, nearby is defined to be the region within the
circle centered on the swarm m e m b e r of interest. In practice,
nearby is often restricted to a section of the circle based on
919
p u t a t i o n a l a p p r o a c h is used t o develop a m o d e l of particle
s w a r m m o v e m e n t s . A m o r e detailed m a t h e m a t i c a l descript i o n of t h es e i n t e r a c t i o n s is given in the n e x t section.
A c o m p r e h e n s i v e review of o t h e r s w a r m algorithms [18,
12, 3] revealed t h a t [1] c a p t u r e s th e essence of s w a r m behaviors and provides a p a r a m e t e r i z e d a l g o r i t h m for b e h a v i o r
analysis.
3.
SWARM MODEL DESCRIPTION
T h e r e are n u m e r o u s c h a r a c te r is t ic s t h a t m u s t b e considered w h e n a t t e m p t i n g to m o d e l p a r t i c l e s w a r m b e h a v i o r as
seen in n at u re- T h e m o d e l fidelity can r a n g e over a s p e c t r u m
t h a t has one end in e x t r e m e s i m p l i c i t y using coarse linear
m o d e l s to the o t h e r end w h e r e e v e r y conceivable n o t i o n is
m o d e l l e d as closely as possible. For obvious reasons, a simplistic m o d e l is insufficient. However, a p p r o a c h i n g t h e o t h e r
e nd o f t h e s p e c t r u m is a r d u o u s at b e s t a n d likely impossible
since all of t h e interactions of n a t u r a l s w a r m s y s te m s are n o t
fully u n d e r s t o o d . T h e p r o p o s e d m o d e l c a p t u r e s the essence
of s w a r m b e h a v i o r while m a i n t a i n i n g tr a c t a b il it y .
T h e m o d e l was i m p l e m e n t e d in M A T L A B TM ( M a t h W o r k s ,
Inc.) initially b e c a u s e of t h e ease and r a p i d i t y of developm e n t . However, it was soon e v id e n t t h a t M A T L A B w o u l d
b e too c u m b e r s o m e for a m o d e l of sufficient fidelity w i t h
a large n u m b e r of n o d e s - o n t h e o r d e r of 1 0 0 - r u n n i n g for a
reasonable s i m u l a t i o n t i m e - o n the order of several t h o u s a n d
t i m e steps. For this reason, t h e m o d e l was p o r t e d to Visual
Cq-q-T~r (Microsoft Corp.). T h i s p r o v i d e d the a d d i t i o n a l
benefit of an effective v i s u a l i z a t i o n e n v i r o n m e n t t h a t was
used t o gain insight into s w a r m behavior.
T h e focus of this section is on t h e m a t h e m a t i c a l m o d e l .
T h e significance of t h e m o d e l ties in b o t h the formulas used
(the a l g o r i t h m ) as wet[ as t h e p a r a m e t e r s . T h e l a t t e r include
m a x i m u m speed and t u r n i n g radius, s e p a r a t i o n d i s t a n c e (radius of c o m f o r t ) , p o p u l a t i o n size, region size, n e i g h b o r h o o d
size, as well a.~ n u m e r o u s w e ig h t in g p a r a m e t e r s which are
discussed later. M o s t of t h e p a r a m e t e r values were chosen e x p e r i m e n t a l l y (i.e. values t h a t m a d e s w a r m f o r m a t i o n
m o v e m e n t stable). T h e goal of this a n d f u t u r e work is to develop t h e r el at i o n s h i p s b e t w e e n the various p a r a m e t e r s and
the continuum of f o r m a t i o n stability.
Figuxe 3 shows a s a m p l e s w a r m s i m u l a t i o n of 15 p a r t i cles b a s e d u p o n this m o r e e x t e n s i v e model. T h e s w a r m
is initially p l a c e d in t h e u p p e r left corner a n d allowed to
m o v e d . T h e j u m b l e of lines r e p r e s e n t t h e tracks of the particles over t h e d u r a t i o n of t h e s i m u l a t i o n as they m o v e from
the u p p e r left corner t o w a r d t h e r ig h t side. T h e o u t e r border represents the edges of the region. T h e inner b o r d e r
defines t h e p o i n t at w h i c h p a r t i c l e m e m b e r s can see t h e border and m u s t confider avoiding it. For n a v i g a t i o n t h r o u g h
t h e region the s w a r m p a r ti c l e s follow a track specified by
re gul a r l y spaced waypoints i n d i c a t e d by t h e black triangles.
W a y p o i n t s are p a r t i c l e s w a r m m e m b e r s t h a t have a fixed
l o c a t i o n (no velocity) and a d i r e c t i o n along t h e i n t e n d e d
route. M o r e on this m e t h o d of n a v i g a t i o n is described later
in this section.
T h e g en er al a l g o r i t h m , s h o w n in F i g u r e 4, is described
briefly as follows. For each mobile p a r t i c l e in t h e swarm, a
ne w d i r e c t i o n v e c t o r is calculated. T h i s is done by considering t h e four b o u n d a r i e s a n d t h e four n e a r e s t neighbors. T h e
n u m b e r o f neighbors to consider was chosen by e x p e r i m e n t a t i o n . T o o large a n e i g h b o r h o o d p o p u l a t i o n results in a
form of identity loss. T h i s is c a u s e d by a s t r o n g e r w ei g h t i n g
F i g u r e 3: S a m p l e S w a r m S i m u l a t i o n
of the a l i g n m e n t rule at t h e e x p e n s e of b o t h cohesion and
repulsion.
L o o p Vp¢ E P, i = 1, .., N
Pr o cess b o u n d a r i e s
L o o p Vpj E P ~ , j = 1, ...,N~
Pr o cess n e i g h b o r pj
C a l c u l a t e n ew d i r e c t i o n
end Loop
Move in n ew d i r e c t i o n
end L o o p
Figure
4: G e n e r a l
Swarm
Algorithm
T h e p a r a m e t e r s of F i g u r e 4 are d escr i b ed in T a bl e 2.
T a b l e 2:
Variable
P
N
Pl
Pi
N~
Pi
Swarm Algorithm Variables
Description
The set of mobile particles
The population size (mobile parti-
,-l,.s), IPI
The ira particle in P
The se~. of particles in p~'s neighborhood (includes waypointsl
The number of particles in pi's
neighborhood, IP~[
The Se~ particle in P~
T h e m i n i m u m distance, dm~,~, for all s i m u l a t i o n s de s c r i be d
in this p a p e r were a c c o m p l i s h e d w i t h d m ~ = 60. T h e effect
of t h e neighbors is f u r t h e r l i m i t e d b y the m a x i m u m sight
d i s t a n c e d,~==. T h i s ~alue was set at 4d,,~,~ a nd was also
chosen t h r o u g h e x p e r i m e n t a t i o n . Smatter distarlces cause
t h e s w a r m t o lose cohesion (a s t a b i l i t y effect) white larger
distances seem to increase cohesion b u t l i m i t t h e a m o u n t of
e x p l o r a t i o n by m o r e heavily w e i g h t i n g t h e a l i g n m e n t rule.
T h e process of u p d a t i n g t h e p o s i t i o n of p a r t i c l e p~ generates a t a r g e t d i r e c t i o n v e c t o r t h a t is u p d a t e d w i t h r e s pe c t to
each b o u n d a r y and t h e particles ( i n c l u d i n g way-points) in t h e
n e i g h b o r h o o d ofp~. This is i l l u s t r a t e d in E q u a t i o n 2. In t he
i m p l e m e n t a t i o n , ~t=Tg~= is initialized to 0 a n d t h e n u p d a t e d
920
at each step (i.e. for each b o u n d a r y and each neighbor).
~ . ~ , = F(baundary,
P,)
The vector dot product term provides an a t t r a c t i o n / r e p u l sion mechanism so that particles behind the waypoint are
attracted while particles in front are repulsed (regardless of
particle direction~. This term varies in the range [ - 1 , 1]
since ration and ~
are unit vectors. The c o n s t a n t C'
in Equation 9 was chosen through experimentation to be 10.
T h e weights for attraction and repulsion are calculated
according to Equation I0. The constant D was chosen experimentsJly to be 20. However, the algorithm remained
stable for a fairly wide range of values (2(} -/- 1(}). However,
the value should be sufficiently large so sz to make repulsion outweigh attraction. This ensures that particles do not
collide (at the expense of potentially losing cohesion).
(2)
The boundaries are processed by treating the closest point
on the b o u n d a r y to pi as a particle that must be avoided.
Equations 3 through 5 show how this is implemented.
v.tt..~,
=
Ps -- P,,b E (top, bottom, left, right}
(3)
-A
(4)
1-
The constant A in Equation 4 was chosen to be 3(} by experimentation. Smaller values allow particles to more closely
approach the boundaries. This has the adverse affect of forcing the particles too close together. Larger values result in
an unnecessary limitation of the region of movement. The
vector v , , r o c t is used for a t t r a c t i o n and repulsion. The delineation is made by the sign--positive for attraction and
negative for repulsion.
T h e neighbors of pl (including waypoints) are sorted into
a list according to their distance from p~ with the closest
particle first. A decision tree is used to determine how the
weighting factors are calculated. T h e first step is to determine i r a neighbor is visible (d < d,~,®). If not, the algorithm
simply continues with no change to p~. Next, if the neighbor
particle is a waypoint then the weight is calculated according to the formula given in E q u a t i o n 9. If the neighbor is
not a waypoint, then the calculations are made according
to E q u a t i o n I0 (alignment with attraction, d ~_ d,~i, or
alignment with repulsion, d < dm~,~).
Two calculations are made for each of the three cases described above. These are detailed in Equations 6 and 7.
Additionally, the alignment vector v~I~9, is initialized with
the direction of pj, as shown in E q u a t i o n 8.
I
~0=
wp.,~h f~ ~ .a-d_,.
. . . _ ~ . , . ~2
}
(
d ),
:
d > d..,.
(I0)
A plot of w from Equation I0 is shown in Figure 5. The
z axis has been normalized so that dm~,~ is 1 and dm~ffi is 4.
The values for d > d,n~ are scaled upward by a factor of 8 in
order to make the scale factor effect visible. The i m p o r t a n t
point to note about Figure 5, as pointed out in [1], is that,
at d = d~,~, particles are in a state of equilibrium--i.e.
the weight is zero and particle pj has no effect on p~. These
/
J
-Ig
wp.r,,h
=
B
( c o s 0 + 1)
(6)
-11
'J°,'-°~,
=
P i - P,
(D
Uo,,~..
=
dlrection(pi)
(S)
d Jr: ~mall.-~l}
The wp=~ph factor is used to more strongly weight particles
that are in front of p~ for attraction/repulsion. Particles
that are directly in front (i.e. with respect to current direction of travel) are weighted with one while particles directly
behind are weighted with zero (i.e no contribution). The
vector ~ . t t . ~ is simply the vector pointing from pl to py
and represents attraction if the particles are farther apart
t h a n d,~,. or, negated, represents repulsion. The direction
of a particle is given in degrees with zero along the positive
z axis and increasing in a counter-clockwise fashion. The
function d i r e c t i o n ( ) returns a unit length vector with the
appropriate z and y components. The constant B in Equation 6 was chosen experimentally to be 10. Smaller values
tend to increase instability while larger values limit freedom
of movement.
As stated before, waypoints are fixed particles with a direction along the intended route. W h e n processing a waypoint the weight is calculated according to E q u a t i o n 9.
= Cw~..,~h ( ~ d
) z(--.°,,,.
" ° "-~
'°~' )
Figure 5: P a r t i c l e A t t r a c t i o n W e i g h t
weights and vectors are combined as shown in E q u a t i o n 11 to
determine the contribution made by Pi to the new direction
for ~ .
( v , , w ) j = E~.j~n. + w~attr.ct
[11)
The value for E was chosen experimentally to be 0.I with
an acceptable range of 2.0 to 0.05. The algorithm is sensitive to this value-large v~lues causes the alignment rule to
completely overwhelm attraction/repulsion. This results in
a swarm of particles all moving in the same direction with no
regard to separation distances. However, too small a value
results in a behavior that resembles a stationary swarm of
insects.
The new velocity contributions are combined as shown in
E q u a t i o n 12. The s u m m a t i o n is made over the particles that
are in the neighborhood of particle p~. However, in order to
enstwe cohesion, if particle p~ has no neighbors t h a t meet
the above criteria, the algorithm finds the closest particle
(9)
921
p o p u l a t i o n to efficiently cover a region. B o t h of t h e s e m e t rics focus on d e v i a t i o n a b o u t t h e ideal v ~ u e s . T h e s e m e t rics are s ~ e c t e d b y the various s w a r m p a r a m e t e r s as listed
in T a b l e 3. T h e p a r a m e t e r s in T a b l e 3 c a n b e r e l a t e d t o
~X
Figure
6: P a r t i c l e
Turn
T a b l e 3: S w a r m P a r a m e t e r s
Description
H Parsmet.er
Boundary weight ~Eq. 4)
A
Peripheral vision weight (Eq. 6)
B
IVaypoin~. weight Eq. 9)
C
P~pulsion weighk iEq. 10)
D
Alignment weight (Eq. 11)
E
Speed factor (Eq. 13)
r~ m
Omam
Max turn angle ~Fig. 6)
Max particle speed (Eq. 13)
SvlrLas
Max sight dis~.ance
dmaz
Example
and c a l c u l a t e s t h e n e w d i r e c t i o n for pl t o w a r d it by s e t t i n g
tl~argit ~-
Uattl"act.
~'t-~'9-'
=
~(v.,.,,).~
(12)
J
T h e final step in u p d a t i n g t h e p o s i t i o n o f p~ is to det e r m i n e t h e n ew d i r e c t i o n a n d m o v e pt a c e r t a i n d i s t a n c e
in t h a t direction. As s t a t e d earlier, th e a m o u n t a p a r t i cle c a n t u r n is l i m i t e d b y t h e m a z / m ~ t ~ t~tryt angZe, O,~az.
T h e r e f o r e , t h e n ew d i r e c t i o n of p a r t i c l e pl i8 given b y t h e
m i n i m u m of 0,,a= and Otarget- T h i s iS i l l u s t r a t e d for a left
t u r n in F i g u r e 6. Since A0 > 0,,~= t h e n e w d i r e c t i o n for
p a r t i c l e pi is u p d a t e d t o 01 + 0,,~=. T h e v a l u e for 0 , , , = is
set globally for the s w a r m p o p u l a t i o n . T h e c u r r e n t value is
set at 5 °. L a r g e r values c a u s e i n s t a b i f i t y a n d result in m o r e
of a s w a r m i n g b e h a v i o r (as o p p o s e d to m o r e global s w a r m
formation movement).
O n c e the d i r e c t i o n is d e t e r m i n e d , t h e p a r t i c l e pi is m o v e d
a d i s t a n c e t h a t is p r o p o r t i o n a l t o s o m e f r a c t i o n of t h e m a x i m u m speed. T h e u p d a t e p r o c e s s is d e s c r i b e d b y E q u a tion 13.
J~i' ~- ~ i "~ ¢~aSmazTJvaew
t h e b e h a v i o r s g i v e n in T a b l e 1. P a r a m e t e r s D a n d E rel a t e t o separation and a~ignment r e s p e c t i v e l y w hi t e B a n d
d,na® r e l a t e to cohesion. T h e p a r a m e t e r s 6 , n u a n d s , ~ = are
r e l a t e d t o s w a r m p a r t i c l e c a p a b i l i t i e s ( s p e e d a n d t u r n angle). A d d i t i o n a l p a r a m e t e r s in this a r e a i n c l u d e m a ~ m u m
a c c e l e r a t i o n , i n e r t i a effects, size, etc. V a r i a t i o n of t h e s e pa r a m e t e r s is not. c o n s i d e r e d here. F o r sake o f b r e v i t y a n d
d e m o n s t r a t i o n of t h e two p r o p o s e d m e t r i c s , o n l y v a r i a t i o n
of t h e a l i g n m e n t weight E ( E q u a t i o n 11) is c o n s i d e r e d in
this p a p e r .
Connectivity c a n b e m e a s u r e d b y e x a m i n i n g t h e d i s t a n c e
b e t w e e n p a r t i c l e s (specifically t h e d i s t a n c e b e t w e e n n e a r e s t
n ei g h b o r s) . T h i s m e t r i c m e a s u r e s t h e a b i l i t y o f t h e s w a r m
f o r m a t i o n t o m a i n t m n cohesion. A t e a c h t i m e s t e p t h e
d i s t a n c e b e t w e e n e a c h p a r t i c l e a n d its n e a r e s t n e i g h b o r is
r eco r d ed . T h e m i n i m u m an d m a x i m u m v a l u e s in t h i s ~ e c t o r
provide the connectivity information. Plots of the m a x i m u m
s e p a r a t i o n and s e p a r a t i o n d e v i a t i o n - - u s i n g a s e c o n d o r d e r
s t a t i s t i c a b o u t t h e ideal d i s t a n c e - - - f o r E = 0.05 (solid line)
a n d E : 0.50 ( d o t t e d line) are s h o w n in F i g u r e 7 (a) a n d (b)
respectively. T h e i d eal s e p a r a t i o n d i s t a n c e , as n o t e d before,
is 60. Clearly, t h e s w a r m for E = 0.50 m a i n t a i n e d a m o r e
(13)
T h e v e c t o r e , ~ is a u n i t v e c t o r in t h e d i r e c t i o n as specified
above. T h i s v e c t o r is scaled b y t h e m a x i m u m speed, s . . . .
a n d a p a r a m e t e r a , . T h i s p a r a m e t e r is e q u a l to t h e inverse
of t h e n u m b e r of p a r t i c l e s in t h e n e i g h b o r h o o d of pi. T h e
t h o u g h t is t h a t p a r t i c l e s in a m o r e cra~ded n e i g h b o r h o o d
n e e d to m o v e m o r e c a u t i o u s l y (i.e. m o r e slowly).
T h e n u m e r o u s p a r a m e t e r s u s e d in t h e s w a r m m o d e l det e r m i n e t h e b e h a v i o r o f t h e s w a r m f o r m a t i o n . T h e effect
of t he se p a r a m c t e r s an d a m e t h o d for e v a l u a t i n g s w a r m beh a v i o r is d e s c r i b e d in t h e n e x t section.
140
,
~ ~
.
.
.
...............
.
~
.
.
.
......
~
4.
.
-
........ ...........................
..~
S W A R M EVALUATION M E A S U R E S
T h e p r e c e d i n g m o d e l d e s c r i p t i o n p r o v i d e s the f o u n d a t i o n
for analysis of s w a r m b e h a v i o r in t h e c o n t e x t of s w a r m - b a s e d
sensor networks. T h e p r o p e r t y of c o h e s i o n ( c o n n e c t i v i t y ) is
i m p o r t a n t to m a i n t a i n i n g reliable c o m m u n i c a t i o n s while t h e
p r o p e r t y of a v o i d a n c e ( r e d u c t i o n of sensor o v e r l a p ) is i m p o r t a n t t o sensor efficiency. M e t r i c s are p r o p o s e d to m e a s u r e
t h e s e p r o p e r t i e s an d thus p r o v i d e a classification m e c h a n i s m
for c a t e g o r i z i n g s w a r m s y s t e m s .
O t h e r research has b e e n d o n e t o e s t a b l i s h m e a s u r e s of
s w a r m p e r f o r m a n c e . A m e t h o d u s in g th e ideal gem law is
p r o p o s e d in [7]. However, t h e m e t r i c p r o p o s e d is used to
m e a s u r e a different b e h a v i o r - - - n a m e l y t h e t i m e it takes t o
e s c a p e an enclosed region. T h e m e t r i c s p r o p o s e d in this
p a p e r m e a s u r e connectivi~ and coverage e~cienc~/. Connectivi~ is a m e a s u r e of p a r t i c l e s e p a r a t i o n d i s t a n c e (which
is i m p o r t a n t to n e t w o r k c o m m u n i c a t i o n s c o n n e c t i v i t y ) w h i l e
coverage e ~ c i e n c y r e l a t e s th e a b i l i t y o f a g i v e n sensor s w a r m
ZK
.
.
.
.
•
•
•
•
•
,a
IQ
~l
Figure
7: C o n n e c t i v i t y
stable formation (with respect to reducing the m a x i m u m
d i s t a n c e an d t h u s i m p r o v i n g c o n n e c t i v i t y ) t h ~ t h e s w a r m
w i t h t h e lower v al u e o f E . I m p r o v e m e n t s in f o r m a t i o n stab i l i t y for larger values o f E t h a n 0.50 are negligible. T h i s is
922
i m p o r t a n t since it allows one to conclude that~ for t h e given
veJues of all the other parameters, increasing E beyond 0.50
provides no improvement in stability. As noted above, increasing E reduces the effect of a tt r a c ti o n /r e p u ls i on . This
has the undesired effect of decreasing the ability to explore.
Strict adherence to the m i n i m u m separation distance results
in a formation t h a t moves with a high degree of rigidity and
high degree of stable connectivity whereas a m o r e loose adherence to m i n i m u m separation distance results in a more
fluid formation (i.e. b e t t e r able to adapt to changing envir o n m e n t a l conditions) and less stable connectivity.
T h e connectivity measure can be related to the t h er m o d y n a m i c principle of temperature[19]. Particles with higher
t e m p e r a t u r e would cause a higher variance in the connectivity measure. Further work needs to be done to investigate
the potential parallel between the principles of t h e r m o d y namics and particle swarm behavior.
Coverage e~cienc# is related to the ability of a sensor
swarm to maximize coverage for a given number of sensors.
For a large number of particle swarm sensors, the region
of coverage can be approximated by a rectangle with an
area proportional to the number of sensors. Furthermore,
the most efficient arrangement of these sensors in two dimensions is a triangular lattice [8]. A bounding box can
be used to ap p ro x i m a t e the area of the swarm formation
and, assuming a fixed perimeter, the most efficient shape is
that of a square. Therefore, the sensor coverage efficiency
is related to the ratio of the bounding box dimensions so
t h a t walues closer to unity are more efficient. P l o t s of this
metric for three cases are shown in Figure 8. T h e base-
II E
T a b l e 4" C o v e r a g e E f f i c i e n c y V a r i a n c e
Unguided Guided
0.05
0.1607
0.4487
0.10
0.25
0.50
1.00
0.4342
0.3317
0.2847
0.257'1
along a diagonal, the efficiency metric would be very close
to unity but the coverage would n o t be optimal. E x a m i n a tion of the swarm formations over t i m e for the test cases
presented above revealed t h a t the formations were indeed
roughly square-like. F u r t h e r work could be done in order to
relate this metric to b o u n d i n g box area and the number of
swarm particles.
5.
CLASSIFICATION CATEGORIES
A starting point for suggested categories for classifying
sensor swarm behaviors include (type-of) birds, fish, insects,
and other animals. As noted before, for increased weighting
of the alignment rule ( p a r a m e t e r E in E q u a t i o n 11), more
rigid formations t h a t m a i n t a i n a smaller deviation a bout
the ideal m i n i m u m distance result. This t y p e of formation
is more like a school of fish m o v i n g in a n o n- t hr e a t e ni ng
environment. Reducing E results in a formation t h a t is
more chaotic (greater deviation a b o u t the ideal separation
distance) but is more adaptable to e n v i r o n m e n t a l conditions
(e.g. a swarm of insects).
Observe t h a t the t a x o n o m y is being developed in the context of networked sensor systems and hence concentrates
on communications aspects. As opposed to the t a x o n o m y
presented in [3], the work here is concerned with link establishment and duration for wireless communications. Results
indicate t h a t there are numerous types of swarm formations
as shown in Figure 9. T h e vertical axis represents the scale
s
i
1
rd J
i
s
l.O
0.0539
0.0648
0.0431
0.0353
Behavior
1.4
~,~t~_
r
l l.ll
Ordered Ch~lic
] ii: i':. . . . . . . .
O.i
0,0
a
I
,
I
I
i
i
i
I
e, i
F i g u r e 8: C o v e r a g e E f f i c i e n c y
line case, indicated by the dashed line near unity, is for an
unguided swarm, i.e. no waypoints and E ---- 0.50. T h e
worst efficiency, represented by the other dashed line, is obtained for a guided swarm with E = 0.05 T h e efficiency
for a guided swarm with E = 1.00 is represented by the
solid line. T h e measure of the deviation of this efficiency is
calculated using a second-order statistic about unity. T h e
values of this metric for various configurations are shown
in Table 4. Clearly, as E increases, the coverage efficiency
increases (smaller variance). It should be noted here that
this metric must be used carefully. T h e potential fallacy
lies in assuming that the formation is tightly packed in the
bounding box. However, if the formation was strung out
F i g u r e 9: S w a r m
Classification
of the b e h a v i o r - w h e t h e r global, i.e. the entire swarm formation, or regional. T h e lateral axis represents the a m o u n t
of order in t h e s w a r m - o r d e r e d like a school of fish or chaotic
like a cloud of insects. T h e d e p t h axis represents the degree
of coupling between p a r t i c l e s - t i g h t l y or loosely coupled in
the sense of sharing environment i n f o r m a t i o n t h r o u g h some
form of communication. Several examples serve to illustrate
the classification scheme. A single, large school of fish is an
example of a swarm in t h e / G l o b a l , Ordered, Loose] class. A
colony of ants foraging in-widely s c a t t e r e d groups might be
categorized as a [Regional, Chaotic, Tight] swarm. Finally~
923
a p a c k of wolves c o u l d b e classified as a / R e g i o n a l , Ordered,
Tight] s w a r m f o r m a t i o n .
Figuxe 9 shows a s h a r p d e m a r k a t i o n Between t h e different
regions. However, in r e a l i t y t h e r e is a c o n t i n u u m on w h i c h
s w a r m f o r m a t i o n s m a y exist. Differing o r d e r e d - c h a o t i c beh a v i o r c a n be o b t a i n e d b y v a r y i n g the p a r a m e t e r E and t h e
neighborhood size. W h i l e the o t h e r p a r a m e t e r s of T a b l e 3 axe
n o t a d d r e s s e d specifically, their affect c a n b e d e s c r i b e d from
a n i n t u i t i v e s t a n d p o i n t . F o r instance, d e c r e a s i n g the p e r i p h eral weight, BI a n d sight distance, d ~ = , results in a collection of s w a r m s a c t i n g a l m o s t i n d e p e n d e n t l y ( t h e [Global,
Ordered, Loose] class in F i g u r e 9).
T h i s classification scheme provides a f o u n d a t i o n for evalu a t i n g s w a r m s of sensor p a r t i c l e s in the c o n t e x t of n e t w o r k
c o m m u n i c a t i o n s . Wireless, a d - h o c c o m m u n i c a t i o n s p r o t o cols e m p l o y e d in these s y s t e m s c a n b e e v a l u a t e d a n d o p t i m i z e d a c c o r d i n g to specific use in e i t h e r a s t a t i c or d y n a m i c
sense. D y n a m i c p r o t o c o l o p t i m i z a t i o n is i m p o r t a n t since
p a r t i c l e s w a r m s can a d a p t to the e n v i r o n m e n t a n d thus fall
i n t o a different classification c a t e g o r y over t i m e .
6.
of M e m p h i s , A p r i l 2001.
www.msci.memphis.edu/~franklin/coord.html.
[5] V. G a z i a n d K. M. P a s s i n o . S t a b i l i t y of
one-dimensional discrete-time asynchronous swarm. In
Proc. Of the I E E E lnt'l Syrup. On Intelligent
Con~.rol/IEEE Conf. On Control Applications, M e x i c o
City, Mexico, 2001.
[6] K. G. H i l l m s n , J. P. H a n n a , a n d M. J. W a l t e r .
M o d e l i n g t h e J o i n t B a t t l e s p a c e Infosphere. I n
Proceedings of the 6th Intl. Command and Control
Research and Technologll S~rtposium, A n n a p o l i s , M D ,
J u n e 2001.
[7] S. D. J a n t z , K. L. Doty, J. A. BagneU, a n d I. R.
Z a p a t a . K i n e t i c s of r o b o t i c s : T h e d e v e l o p m e n t of
u n i v e r s a l m e t r i c s in r o b o t i c s w a r m s . I n Proceedings of
the 1997 Florida Conference on Recent Advances in
Robotics, A p r i l 1997.
[81 B. A. K a d r o v a c h and G. B. L a m o n t . Design a n d
a n a l y s i s of s w a r m - b a s e d sensor s y s t e m s . I n
Proceedin9~ of the Midwest Sgmposium on Circuits
and Systems, F a l r b o r n , OH, A u g u s t 2001[9] J. M. K a h n , R. H- K a t z , a n d K . S. J. P i s t e r . N e x t
c e n t u r y challenges: M o b i l e n e t w o r k i n g for s m a r t d u s t .
I n ProceedingJ of ff~e A C M / I E E E Intl. Con]. on
CONCLUSIONS
T h i s p a p e r e x a m i n e s a sensor s w a r m n e t w o r k s y s t e m v i a
s i m n i a t i o n a n d classification. Careful a t t e n t i o n was given
to i n c o r p o r a t i n g p a r t i c l e b e h a v i o r s in o r d e r to r e a l i s t i c a l l y
m o d e l such biological p h e n o m e n a as cohesion, repulsion,
avoidaatce, p e r i p h e r a l vision, d i r e c t i o n of travel, etc. Connectivitll and coverage e ~ c i e n c y m e t r i c s are p r o p o s e d t o
e v a l u a t e s w a r m b e h a v i o r s in t h e c o n t e x t of sensor n e t w o r k
s y s t e m s . T h e s e m e t r i c s showed a c o r r e l a t i o n b e t w e e n the
a l i g n m e n t weight a n d c o n n e c t i v i t y a n d coverage efficiency.
A d d i t i o n a l w o r k m u s t still b e a c c o m p l i s h e d to i n v e s t i g a t e
p o t e n t i a l c o r r e l a t i o n s b e t w e e n the o t h e r p a r a r n e t e r s . O t h e r
areas of f u t u r e work m i g h t i n c l u d e i n v e s t i g a t i o n of t h e p o t e n t i a l p a r a l l e l b e t w e e n t h e principles of t h e r m o d y n a m i c a n d
p a r t i c l e swaxm b e h a v i o r . I n v e s t i g a t i o n of t h e s e principles
m a y l e a d to a m o r e d e t a i l e d m o d e l of s w a r m b e h a v i o r a n d a
m o r e c o m p l e t e u n d e r s t a n d i n g of the p a r a m e t e r i n t e r a c t i o n s A d d i t i o n ~ l y , w o r k needs to b e done to m o r e f o r m a l l y r e l a t e
t h e coverage el~ciency m e t r i c to t h e s w a r m h o u n d i n g b o x
a r e a ~nd the n u m b e r of s w a r m particles.
7.
Mobile Computing and Networking (MobiCora 99),
[10]
[11]
[12]
[13]
[14]
ACKNOWLEDGMENTS
T h e a u t h o r s would like to t h a n k the A i r Force R e s e a r c h
L a b o r a t o r y ~nd specifically, Dr. R o b E w i n g for hie i n s i g h t
a n d suggestions in d e v e l o p i n g p a r t i c l e s w a r m e v a l u a t i o n m e t rics mad m e t h o d s .
8.
[15]
[16]
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