Generating Swarm Solution Classes using the Hamiltonian Method of
Swarm Design
M. Li
1
, C. Qiu
2
, J. Park
1
, D. Chan
1
, J. Jeon
1
, J. Na
1
, C. Wong
1
, B. Zhao
1
, E. Chang
1
,
S. Kazadi
1,3
and S. Hettiarachchi
4
1
Swarm Research Group, Jisan Research Institute, 308 S. Palm Ave., Alhambra, CA 91803, U.S.A.
2
University of California Santa Barbara, Santa Barbara, CA 93106, U.S.A.
3
Illinois Mathematics and Science Academy, 1500 Sullivan Rd., Aurora IL 60506, U.S.A.
4
Indiana University Southeast, 4201 Grant Line Rd., New Albany IN, 47150, U.S.A.
Keywords:
Swarm Engineering, Hamiltonian Method of Swarm Design.
Abstract:
We utilize a swarm design methodology that enables us to develop classes of swarm solutions to specific
specifications. The method utilizes metrics devised to evaluate the swarm’s progress – the global variables –
along with the set of available technologies in order to answer varied questions surrounding a swarm design
for the task. These questions include the question of whether or not a swarm is necessary for a given task. The
Jacobian matrix, here identified as the technology matrix, is created from the global variables. This matrix
may be interpreted in a way that allows the identification of classes of technologies required to complete the
task. This approach allows us to create a class of solutions that are all suitable for accomplishing the task. We
demonstrate this capability for accumulation swarms, generating several configurations that can be applied to
complete the task. If the technology required to complete the task either cannot be implemented on a single
agent or is unavailable, it may be possible to utilize a swarm to generate the capability in a distributed way.
We demonstrate this using a gradient-based search task in which a minimal swarm is designed along with two
additional swarms, all of which extend the agents’ capabilities and successfully accomplish the task.
1 INTRODUCTION
Swarm engineering (Kazadi, 2000) as an area of re-
search concerns itself with the translation of a task de-
scription into a design of multiple autonomous agents
whose combined effect accomplishes the desired task.
One method involves utilizing differential equations
derived from state equations to generate behaviors.
The Hamiltonian Method of Swarm Design (HMSD)
is a method for swarm design that begins with func-
tions called global functions defined over the field
of agent measurables (Kazadi and Lee, 2007). This
method utilizes global swarm properties to generate
an abstract phase space. The initial and final system
positions define the task; behaviors implemented by
the agents must change the global properties’ numer-
ical values so as to move the system from the initial
state to the final state. Design of the swarm involves
the analysis of the practical generation of a set of be-
haviors, techniques, and timetables required to move
the system from the initial to the final state.
In this paper, we will examine how the HMSD
can be used to determine the requirements for classes
of swarms that accomplish specific tasks. We begin
with an examination of the way in which the task en-
coded as a function of measurements that individual
agents can accomplish. We continue to a derivation
of the technology matrix which defines the classes of
technologies necessary for the accomplishment of the
task. The technology classes are encoded in terms
of their ability to change the system state as opposed
to a description of the specific technology used. We
demonstrate that, using this approach, we can create
accumulation swarms of varied designs that accom-
plish the accumulation task. Additionally, we use the
source location task to explore the development of
a technological requirement that may or may not be
possible to accomplish on a single agent. In the case
that a single agent-based solution doesn’t exist, the
problem must be solved using a swarm. We derive
three different swarms that can accomplish the task,
demonstrating that in this case, a swarm is absolutely
required in order to accomplish the task.
Li M., Qiu C., Park J., Chan D., Jeon J., Na J., Wong C., Zhao B., Chang E., Kazadi S. and Hettiarachchi S.
Generating Swarm Solution Classes using the Hamiltonian Method of Swarm Design.
DOI: 10.5220/0006122201450152
In Proceedings of the 9th International Conference on Agents and Artificial Intelligence (ICAART 2017), pages 145-152
ISBN: 978-989-758-219-6
Copyright
c
 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
145

2 THE TECHNOLOGY MATRIX
The Hamiltonian Method of swarm design (Kazadi
and Lee, 2007; Kazadi, 2009), involves the writing of
global variables P
i
which are meant to capture the cur-
rent state of the swarm. These are functions of quan-
tities that measure locally measurable aspects of the
system which the individual agents can measure and
manipulate. Mathematically, these are represented by
vectors
−→
s = (s
1
,...,s
N
s
) where N
s
is the number of
local measurables and each s
i
represents the state of
one element of the system. It is clear to see that, with
each agent potentially having K degrees of freedom
and the system itself having more, the number N
s
can
be quite large. Each of the values, s
i
, is assumed to
come from a range S
i
of real numbers, making the
system configuration
−→
s an element of S
1
× ··· × S
N
s
which is itself a subset of R
N
s
.
2.1 The Swarm Design Problem
As each of the P
i
is a function of the local variables,
we write this as
P
i
= f
i

−→
s

. (2.1)
P
i
is meant to capture a global state of the system.
Many examplesmay be developedfor potential global
goals including the location of items that are being
collected, the flow rate of items that are being moved,
construction details, etc. Each of these may be repre-
sented as a potentially complex function of the local
measurables. Each of the functions may have a unique
numerical value corresponding to a specific state of
the system or, at least, a unique numerical value for
the desired system state.
One might imagine writing several functions P
i
describing aspects of the swarm system. A vector
−→
P = (P
1
,... , P
N
P
) results from the combination of the
global properties. The number of global properties
N
P
greatly determines the complexity of the design
problem. We define N
P
as the swarm design dimen-
sion. Therefore, if N
P
= 1 the swarm design prob-
lem is referred to as a 1-d swarm design problem. If
N
P
= 2, the swarm design problem is referred to as a
2-d swarm design problem.
The vector
~
P is an element of the vector space
A = A
1
×···×A
N
P
where each A
i
represents the range
of each P
i
. A is the systemic phase space. Each
element represents a specific set of states. That is,
given a state
−→
P , there is a number of system states
for which the global state is
−→
P . We can define the
system domain I

−→
P

in the following way I

−→
P

=
n
−→
s |
−→
P =

f
1

−→
s

,... , f
N
P

−→
s

o
. If



I

−→
P




= 1
then we say that the system state is well defined.
The goal is to change the state of a system from
an initial state to some defined final state. This is de-
fined in terms of the global properties. Therefore, the
goal is to change
−−→
P
init
to some final desired state
−→
P
fin
.
However, it is clear that both of these states may be
extremely degenerate in the sense that many micro-
scopic configurations may make this happen. We de-
fine the task as either the ordered pair of

−−→
P
init
,
−→
P
fin

or, in the case that the initial state is ambiguous,
−→
−→
P
fin
meaning that all initial states should go to the
final state. That is, the final state
−→
P
fin
is an attractor of
the system under the dynamics we will construct.
As a result, we are interested in the dynamics of
the properties
−→
P . Let us consider now
dP
i
dt
. First, it
is straightforward that, using the Einstein summation
convention,
dP
i
dt
=
∂P
i
∂s
j
ds
j
dt
. (2.2)
this can be rewritten as
dP
i
dt
=
−→
∇P
i
·
˙
−→
s . (2.3)
Defining P as
P =







∂P
1
∂s
1
∂P
1
∂s
2
···
∂P
1
∂s
N
s
∂P
2
∂s
1
∂P
2
∂s
2
···
∂P
2
∂s
N
s
.
.
.
.
.
.
.
.
.
.
.
.
∂P
N
P
∂s
1
∂P
N
P
∂s
2
···
∂P
N
P
∂s
N
s







(2.4)
we can write the time change of the system as
˙
−→
P = P
˙
−→
s . (2.5)
We can then write the goal of the swarm as
−→
P
fin
=
−−→
P
init
+
ˆ
t
f
t
0
P
˙
−→
s dt (2.6)
when the initial and final points are well defined and
as
−→
P
fin
=
−−→
P
init
+
ˆ
∞
t
0
P
˙
−→
s dt (2.7)
if the initial point is not well defined.
2.2 Connecting to the Real World
In the real world, we can describe the swarm engi-
neering problem differently than the somewhat eso-
teric description given above. Generally speaking, the
goal of a swarm is to solve a task. That is, givena task,
we want to determine the following items:
• The technologies that one must use
ICAART 2017 - 9th International Conference on Agents and Artificial Intelligence
146

• The way to deploy the technologies
• The minimal number of agents to use
• The number of teams/groups to assemble
• The number of agents that will be “consumed”, or
lost, during the task
• The time needed to accomplish the task
• The amount of energy required to accomplish the
task
• The supplies required to accomplish the task
These items are, interestingly, either present in the
equations given above in Section 2.1 or capable of be-
ing calculated as a result.
Note that the quantity
˙
−→
s represents the way in
which the local measurables are manipulated. These
local measurables are assumed to be accessible to the
agents; they are things that they can directly mea-
sure and change. This quantity defines how and, ul-
timately, when these measurables must be changed.
While the details of the method of changing the quan-
tity must be developed by an engineer, this provides a
clear design specification.
The quantities
∂P
i
∂s
j
represent the ways in which
changes in the local properties are connected to the
changes in the global properties. For instance, these
may indicate that the value of
ds
j
dt
is coupled with a
value for
∂P
i
∂s
j
that is positive. Therefore, the system
must have some way of determining
∂P
i
∂s
j
. If it is the
case that P
i
must decreased in order for the global goal
to be achieved, this requirement determines a poten-
tial course of action for the swarm:
ds
j
dt
should be the
opposite sign of that of
∂P
i
∂s
j
.
The manner in which
ds
j
dt
couples to changes in
P
i
is determined by the
∂P
i
∂s
j
. In order to couple the
two quantities together correctly, it must be possible
for the agents to determine what
∂P
i
∂s
j
is. This can be
achieved only if there is a mechanism or technology
capable of determining this. The matrix P, therefore,
represents the superset of all of the sensory, commu-
nication, and/or computational technologies that must
be deployed in order to achieve the task. That is, the
non-manipulative requisite technologies or capabili-
ties for the task exist within the matrix P. Yet, under-
standably, these technologies are not represented by
the names and vendors of the technologies to be used.
Rather, they represent the capabilities of the requisite
technology; how that is achieved is up to the discre-
tion of the swarm designer. Note that if any technol-
ogy
∂P
i
∂s
j
does not actually exist, this technology will
either not be used in the solution or will be developed
in order to solve the swarm design problem. Develop-
ing these capabilities is a way of discovering needed
technologies one may not have realized is necessary.
Likewise, the various elements
ds
j
dt
that are
nonzero in order to lead to the desired final system
configuration
−→
P
f
represent the agent-level capabilities
that must be in place in order to change the s
j
values
within the system. These might be lights, grippers,
methods of movement, actuators, internal computa-
tions, or any other method of changing some part of
the system. Therefore the
ds
j
dt
entries represent the
manipulative technologies that must be in place in or-
der to achieve the task.
Together, these two technological requirements
define the superset of all of the technologies and abil-
ities required in order for the task to be achieved. The
system engineer, then, is tasked with identifying or
creating them. A swarm is appropriate in the case that
these abilities can be generated through the interac-
tion of agents when they are not available or possible
with a single agent.
Once we have a set of actuation technologies, or
perhaps if we obtain a list of currently available tech-
nologies, we can determine that the application of any
given technology together with the actuation mecha-
nisms at a given point in the system configuration will
lead to a change in the system configuration ∆
−→
s . This
change will have a concomitant change in the posi-
tion of the system in phase space, which may be ap-
proximated as P∆
−→
s . Therefore, our goal is to deter-
mine the sequence of applications of actuation tech-
nologies that leads the system from the initial point
to the final point. That is, the swarm engineering de-
sign problem consists of determining a set of applica-
tions of actuation technologies and concomitant steps
through phase space

∆
−→
s
l

N
steps
l=1
for which the result-
ing change in
−→
P yields
−→
P
f
=
−→
P
i
+
N
steps
∑
l=1
P∆
−→
s
l
. (2.8)
As P may be a function of the state of the system, the
problem may be more generally written as
−→
P
f
=
−→
P
i
+
N
steps
∑
l=1
P

−→
s
l

∆
−→
s
l
. (2.9)
As a result of this sequence of applications, we
can now determine many of the desired quantities.
Clearly, we have determined the technologies to use
as well as their deployment schedule. Each of the
manipulations will require at least one agent. In the
case that more than one agent is required, a team is
Generating Swarm Solution Classes using the Hamiltonian Method of Swarm Design
147

needed. When a manipulation requires the transition
of an agent from an active to an inactive state, this
“consumes” an agent. Summing those for which a
reverse transition is not applied along the pathway in-
dicates how many agents will be consumed. Each of
the transitions requires an amount of energy, supplies,
time, etc. Representing the consumption as γ
l
, we can
write out the consumed quantity as
γ
consumed
=
N
steps
∑
l=1
γ
l
. (2.10)
This expression allows us to calculate what we will
need to accomplish the task along the pathway.
While the determination of this set of applications
of actuation technologies provides a mathematically
rigorous definition of a swarm strategy that will ac-
complish the task, its determination can be nontrivial.
In the remainder of this paper, we will examine two
relatively simple swarms that can be designed using
this approach.
3 ACCUMULATION SWARMS
An accumulation swarm is a swarm that, when ini-
tially organized with agents physically far from one
another, responds by moving the agents toward one
another to eventually form a tightly packed group. A
swarm behaving in this way might be a precursor to a
swarm that then forms a well-ordered formation for a
secondary purpose.
Mathematically, accumulation swarms are defined
as follows: Suppose that each member of the swarm
has a position given by
−→
x
i
. Let the entire swarm have
a set of positions given by
−→
s = (
−→
x
1
,... ,
−→
x
N
). Then we
can define a global property P as
P

−→
s

=
N
∑
i< j
(
−→
x
i
−
−→
x
j
)
2
. (3.1)
P is the dispersion of the group. An accumulation
swarm will have the defining property that
dP
dt
< 0.
Note that
dP
dt
= 2
N
∑
i< j
(
−→
x
i
−
−→
x
j
) ·

d
−→
x
i
dt
−
d
−→
x
j
dt

. (3.2)
This can be simplified to
dP
dt
= 2
N
∑
i
d
−→
x
i
dt
·
∑
j6=i
−→
x
ij
!
(3.3)
where
−→
x
ij
=
−→
x
i
−
−→
x
j
. If we let
−→
x
M,i
=

∑
j6=i
−→
x
ij

then the change in P is given as
dP
dt
= 2
N
∑
i
d
−→
x
i
dt
·
−→
x
M,i
. (3.4)
We may identify
−→
x
M,i
as the center of mass of the re-
maining agents from the point of view of the i
th
agent.
This swarm is defined entirely in terms of its de-
sired behavior by a single global property. As the
swarm requirementsare defined in terms of only a sin-
gle global function, it is a single dimensional swarm.
Examining the form of (3.4), we see that there are
two main terms,
d
−→
x
i
dt
and
−→
x
M,i
. These two terms define
the capabilities of the agents that must be in place in
order for the swarm to perform as desired. That is, in
particular, the swarm must have the ability to move,
and it must have the ability to determine the center of
mass of the system. Therefore, these two basic capa-
bilities define the technological requirements for the
swarm.
In order for the swarm to coalesce, we must have
dP
dt
< 0. (3.5)
I.e., the sum must be negative. The minimal re-
quirement for this to happen the magnitude of the
summed negative values should exceed that of the
summed positive values. There are many ways in
which this can be accomplished including making
each of the terms negative. Making each of the terms
negative requires that the angles between
−→
x
M,i
and
d
−→
x
i
dt
exceed 90
◦
. If this is the case for all agents, the swarm
will always coalesce.
In order to validate these theoretical results, we
simulate a swarm of thirty generalized agents. Each
identical agent has the ability to move and has simu-
lated onboard sensors that report the relative positions
of all other agents. The agents are capable of calcu-
lating relative positions using the coordinates of other
agents. Using this data the agents also can calculate
the center of mass of all agents. All agent movement
is dictated by the center of mass.
The simulation initializes 30 agents with random
coordinates. Each iteration, the agents obtain the rel-
ative positions of other agents. The center of mass
of the set of positions of the other agents is then cal-
culated. The agents then move towards the center of
mass either directly or partially tangentially. When all
agents’ positions are close to or equal to the center of
mass, the simulation stops.
In the first case, the agents move directly towards
the center of mass as they have calculated. In Figure
3.1, all of the agents move towards the center in a
direct path as indicated by the trail each agent leaves.
This accomplishes the task as envisioned.
ICAART 2017 - 9th International Conference on Agents and Artificial Intelligence
148

A)
B)
C)
Figure 3.1: These figures illustrate the evolution of the coa-
lescing agents when moving directly toward calculated cen-
ters of mass. In this and later figures, the agents are the
triangular-shaped objects in the scene. The black lines il-
lustrate the path taken by the agent connecting to the line.
In the second case, the agents behave identically to
the previous case except that they move in at an angle
of 30 degrees with respect to vector directed at the
calculated center of mass. Figure 3.2 showsthe agents
moving in an indirect path towards the center of mass;
the swarm spirals inward, ultimately still completing
the task.
A)
B)
C)
Figure 3.2: These figures illustrate the evolution of the coa-
lescing agents when moving at an angle of 30
◦
from directly
toward calculated centers of mass.
We again utilize an indirect trajectory but in this
case we assign an offset angle of 60 degrees. As il-
lustrated in Figure 3.3, we again observe the swarm
achieving the task through a longer spiral.
A)
B)
C)
Figure 3.3: These figures illustrate the evolution of the coa-
lescing agents when moving at an angle of 60
◦
from directly
toward calculated centers of mass.
In all cases, the agents accomplish the task by
satisfying the swarm requirement given in equation
(3.5). As these last two cases demonstrate, even when
the agents’ behaviors are significantly perturbed the
swarm is still capable of achieving the global task. In
fact, in this case, it is straightforward to determine
the limit of the perturbation still yielding the desired
global goal.
4 WHEN TO USE A SWARM AND
MINIMAL SWARM SIZE
One of the most important goals of swarm engineer-
ing centers around simply verifying that a swarm is
suitable for the given task. Interestingly, despite the
myriad of studies surrounding the use of swarms, no
systematic approach to the determination of the ap-
propriateness of using a swarm for a given task has
been proposed. In this section, we will examine how
this question can be answered in the context of a sim-
ple swarm. As we shall see, the question is complex
because it requires the consideration of the availabil-
ity of technology, as opposed to simple swarm agent
control algorithms. What makes swarms so interest-
ing is their ability to create, as a group, a capability
Generating Swarm Solution Classes using the Hamiltonian Method of Swarm Design
149

that their constituent members cannot. We shall see
in this example that, when technologyis not available,
the swarm can make up the ability. In such a case, a
swarm is required. Additionally, we can determine
how many agents must be in this swarm in order for
the task to be achieved.
We consider a task quite similar to plume track-
ing (Spears et al., 2009) or odor source identification
swarms. We suppose that in a space there exists a
source of some desirable thing (light, food, odorant,
etc.), and that the dispersion of the substance is de-
fined by some real function of position S(
−→
x ). The
goal is to find a “good" location in the sense that it is
locally optimal (Kazadi et al., 2015).
As indicated in Sections 2 and 3, we begin to de-
sign this swarm by defining a global property of the
swarm.
P =
N
s
∑
i=1
S(~x
i
) (4.1)
where S is the position-dependent average measure-
ment of the target,
−→
x
i
is the position of the i
th
agent,
and N
s
is the number of agents in the swarm. The time
derivative is
dP
dt
=
N
s
∑
i=1
−→
∇S·
˙
−→
x
i
. (4.2)
In order to move the swarm to an optimum, we must
have that
dP
dt
=
N
s
∑
i=1
−→
∇S·
˙
−→
x
i
≥ 0. (4.3)
Equations (4.2) and (4.3) indicate both the technolo-
gies and their functional requirement. Firstly, two
technologies are needed for this process to work. The
swarm agents must have a method of evaluating the
quantity
−→
∇S, and the agents must have the ability to
move, as indicated by
˙
−→
x
i
. Technologies that give us
these capabilities can be used to generate the desired
motion. Secondly, the motion given by
˙
−→
x
i
and the
quantity
−→
∇S must be oriented in such a way that the
sum of positive products at most equals that of neg-
ative products. Given these requirements, the swarm
will move toward the optimum.
As we’ve indicated above, swarms are indicated
for a task if their capability exceeds that of the indi-
vidual agents. Focusing on the technology that gener-
ates
−→
∇S, we can ask whether or not individual agents
have access to a technology with ability to determine
−→
∇S. There are two possibilities:
1. such a technology exists and can be integrated
with the agent design; or
2. such a technology does not exist.
In the first case, the gradient sensor can be integrated
on a single agent. The problem will be solved by that
single agent; a swarm is not needed – the agent can
find its way to the optimum of S.
In the second case, there are again two possibili-
ties:
1. A single agent, by moving around and sam-
pling the local area can generate a local gradient,
thereby making the information available; or
2. Such a movement and integration is unavailable
or impractical.
Again, in the first case, the integration of this techno-
logical capability solves the problem and only a single
agent is required. In the second case, a single agent
has no way of obtaining this information. Therefore,
if we mean to get the data needed to solve the prob-
lem, we need at least two agents. As a result, a swarm
with a minimal size of two (2) is required.
We can design swarms that accomplish the overall
task with the limitations imposed by the last case. We
describe three strategies for accomplishing the gradi-
ent ascent and demonstrate their capabilities below.
Swarm of Two Agents. Two agents represent the
smallest group that can accomplish this task in the
event of the restrictions given above. Our swarm ac-
quires and processes the local gradient by coopera-
tion. One agent initially remains stationary while the
second agent orbits the first in a circular orbit, sam-
pling the value of S(
−→
x ) at points along its path
. The
second agent executes more than one full orbit, mea-
suring the intensity of the field as it goes. After its
first time around, the second agent stops at the loca-
tion of the highest intensity in the first orbit when it
re-encounters it during the second orbit. The direc-
tion indicated by the two positions of the agents when
the second one stops is approximately that of the lo-
cal gradient. Once the second agent stops, the agents
switch roles and the process restarts. After multiple
iterations, the swarm finds and stays at the area with
the global intensity peak.
We illustrate in Figure 4.1 the performance of the
two-agent swarm below:
Figure 4.1: The trial of the two-agent swarm in a radiation
field modeled by functions one through three.
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The swarm finds the highest intensity, overcoming
the limitations that would hobble a single agent.
Swarm of Three Agents. We construct a simu-
lated swarm comprising three identical agents. Each
of these agents is capable of making an intensity
measurement at their location. Additionally they
can determine the relative range and bearing of the
other agents. The agents’ utilize a modified physi-
comimetic control algorithm (Gordon et al., 1999;
Spears and Gordon, 2007; Spears et al., 2004) in
which the force function is given by
−→
F = 5arctan

G
r
2

ˆr+
−→
∇I



−→
∇I



. (4.4)
We assign G to 1200. When this simulation is run,
three agents are randomlyplaced in the arena. Each it-
eration each agent obtains the intensity measurements
from other agents and uses these to calculate the force
as in (4.4). The agent then moves in the direction of
the combined force. To calculate the gradient, we use
a first order linear approximation defined by
I(x, y) = ax + by+ c (4.5)
where x and y are the coordinates of each agent, and a
and b are undetermined coefficients. We can calculate
the local gradient
~
∇(I) = (a,b). (4.6)
In Figure 4.2, the performance of the three-agent
swarm are displayed. It is worth noting that, con-
strained by the method we implemented in the al-
gorithm, the swarm does not always move perpen-
dicularly to the contour lines. The swarm linearly
approximates its local gradient vector. The further
apart the agents are, the less accurate the estimation
is. Nonetheless, once in the correct configuration, the
swarm is able to find the peak intensity. One crucial
advantage of this three-agent design, comparing to the
two-agent swarm, is that the former does not require
designated memory space for each agent, for the col-
lection of all three agents is capable of computing the
gradient instantaneously from their relative positions
and their current intensity readings.
Figure 4.2: The trial of the three-agent swarm in a radiation
field modeled by two functions.
Large Physicomimetic Swarms. We illustrate in
Figure 4.3 the performance of a multi-agent swarm.
As in the three-agent swarm, the agents in the multi-
agent swarm do not require internal memory; the
agents compute the local intensity gradient vector in-
stantaneously which is again a linear approximation.
For the multi-agent swarm to find the peaking inten-
sity for the three functions is not extremely challeng-
ing given that swarm consists of 20 agents and the
ability of the swarm controlled by physicomimetics
to perform well in noisy fields (Hettiarachchi et al.,
2008). It is highly improbable in these three scenarios
that the swarm would get stuck in a local minimum;
there are multiple neighboring agents which are capa-
ble of obtaining the intensity measurements required
to compute their force vectors. Moreover, there are
no other obstacles in the field. The swarm easily finds
the peaks of the first two functions causing the agents
to surround the peak. In the third function the swarm
faces difficulty due to the flat trough like peak. Since
the swarm stretches out in the trough, it becomes chal-
lenging for the agents in the back of the formation to
compute intensity. Given adequate time, the swarm is
capable of reaching the peak.
Figure 4.3: The trial of the large physicomimetic swarm in
a radiation field modeled by two functions.
In all three cases, the swarm finds the highest in-
tensity, overcoming the limitations that would hobble
a single agent.
5 CONCLUSION AND FUTURE
WORK
This paper examined the HMSD. Global functions of
the local measurables were used to gemerate a state
space describing the swarm’s state. Next, a set of
swarm requirements was developed that generating
swarm classes. This approach equates to a mathemat-
ical proof that the swarm’s task will be achieved, if
sometimes indirectly.
Within the equations generated from the global
functions are mathematical descriptions of the varied
technologies that one must have in order to achieve
the task. The technology matrix is made up of ex-
pressions that describe the function of the technology
Generating Swarm Solution Classes using the Hamiltonian Method of Swarm Design
151

on one or more aspects of the phase space describing
the system. It is sufficient to enable the identification
of solutions or the determination that such equipment
does not yet exist.
In our estimation, the completion of a task is not
dependent on a swarm if all requisite technologies can
be deployed on a single agent. However, if the tech-
nologies cannot be implemented on a single agent,
then a swarm is required.
We illustrated these principles using two classes
of swarms: accumulation swarms and gradient ascent
swarms. All developed gradient ascent swarms which
achieved the task. It was demonstrated that mini-
malist stigmergic swarms consisting of two agents
could achieve the gradient ascent task, as could larger
physicomimetic swarms, as long as they implemented
the technological and behavioral requirement that
emerged, even when they had to do it cooperatively.
These developments enable us to approach the
problem of swarm engineering from a different point
of view than has generally been employed. The global
properties amount to metrics on the state space of the
system and their time derivatives enable the identifi-
cation of technologies that are related to the achieve-
ment of the goal. The design problem, then, is re-
duced to developing a method of applying these tech-
nologies sequentially so that the system will move in
state space from its initial state to a predetermined
one. As a result, this approach reduces the swarm
design problem to a multidimensional search through
technology space, guided by the movement through
phase space.
We have focused in this study on designing low di-
mensional swarms with simple design requirements.
We turn in future work to more complex swarms
whose design requirements are multidimensional as
opposed to single dimensional.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the efforts and
assistance of several students of the Illinois Math-
ematics and Science Academy. These students are
Katherine Bezugla, Ankit Agrawal, and Rohit Mitta-
palli. These students assisted in some of the simula-
tion programming for Section 4 in this paper.
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