Vector based Control Routines for Swarms of Path Finding Robotic

Devices

Colin Chibaya

Sol Plaatje University, Chapel Road, Kimberley, South Africa

Keywords: XSet, Control Routines, Message Passing, Robotic Devices, Emergent Behaviour.

Abstract: Swarm intelligence systems where robotic devices encoded with primitive actions executed at individual

levels in order to cause swarm level emergent behaviour are appealing to the fields of nanotechnology and

bioinformatics. Interaction between robotic devices allow improved swarm level properties with features

more than the sum of the contributions of the individual robotic devices that form the swarm. However, it is

challenging to pinpoint particular primitive actions which drive robotic devices towards deliberately

engineered emergent behaviour. We propose an XSet model inspired by the behaviours of message passing

agents. The proposed XSet model supports direct device to device interactions in which implicit

communication spaces arise. In this context, an XSet puts together primitive actions, parameters, and meta

information which stipulates when primitive actions are useful to robotic devices. We assess path finding and

path following abilities of message passing robotic devices and compared the measures thereof to the relative

performances of the stigmergic counterparts. Better message passing performances are observed when time

in simulation is sufficiently long, when the population of robotic devices in the swarm is high. Besides giving

a new swarm control model, message passing XSets bring us closer to more generalized swarm control rules.

1 INTRODUCTION

The aptitude to form paths between selected search

points in simulated swarms of robotic devices

emanates from particular control routines that are

collectively designed to cause emergent behaviour

(Chibaya, 2014; Chibaya, 2015; Chibaya, 2019). In

this context, a robotic device is an autonomous

artificial agent designed with a nanite architecture in

mind (Chibaya, 2014). Nanites are tiny electronic

devices assembled at nanometre scale. Control

routines, on the other hand, are computational codes

implemented to characterize robotic devices’

individual-level actions (Negulescu and Barbat,

2004). Emergent behaviour is then non-reducible

phenomena observed in swarms of robotic devices

which cannot be traced back to the individual-level

actions of the robotic devices thereof (Chibaya,

2015).

The design of control routines is, dominantly,

inspired by some phenomena in nature, such as the

map-reading views known from geography (Werfel,

2002; Wehmer et al., 2006), biological metaphors

https://orcid.org/0000-0001-6995-605X

(Chibaya, 2014), geometric views (Ngo et al., 2005),

mathematical matrices (Harris, 2007), or physics

(Spears et al., 2004a; Spears et al., 2004b; Spears et

al., 2005). Successful control routines have been,

commonly, built on stigmergic ant-like colonies

(Chibaya, 2015), artificial bee colonies, or flocking

boids. However, although related emergent

behaviours are plausible, the individual-level actions

of swarm members are blurredly explained. Without

grasping the logic underpinining individual-level

actions of swarm members, simulated swarm

applications will remain naïve. In fact, chances are

high of evolving undesirably hazardous outcomes

from these swarms.

Although they are architecturally autonomous and

naïve, robotic devices’ interactions give rise to

emergent behaviour whose properties are more than

the sum of the contributions of the individual robotic

devices (Chibaya, 2015). Collections of control

routines, together with related parameter values, and

meta information, forms the verbs robotic devices

depend on in order to complete individual-level tasks.

However, what are the white-box design features of

272

Chibaya, C.

Vector based Control Routines for Swarms of Path Finding Robotic Devices.

DOI: 10.5220/0009780802720283

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 272-283

ISBN: 978-989-758-442-8

those control routines included in path finding and

path following robotic devices’ dictionary?

Formalizing the component units of appropriate

sets of control routines with which robotic devices

yield emergent behaviour is an ambitious task

(Chibaya, 2015). The work presented in Chibaya

(2015) proposed the use of an XSet model built on the

characteristics of stigmergic ant-like robotic devices.

In this context, an XSet is an eXtended Set

comprising control routines, parameter values, and

meta information stipulating how and when control

routines are useful to robotic devices (Chibaya, 2014;

Chibaya, 2015). Stigmergic robotic devices interact

indirectly via the environment using virtual

pheromone cues (Chibaya, 2014; Chibaya 2015). The

environment is the shared memory of the swarm.

However, stigmergic XSets do not provide details

regarding how to arrive at generalizable quantities of

pheromone level required by individual robotic

devices at a time. Also, they ignore the effects of

pheromone dissipation to swarm convergence speed

and quality. In fact, the quantities of pheromone

levels used are hard-coded (Chibaya, 2014).

Alternative non-stigmergic XSet models can be

tried for the same task domain. This paper

investigates the design of an XSet model built on the

behaviours of message passing robotic devices

(Chibaya, 2019). The proposed XSet supports

swarms of robotic devices that can directly interact

with one another one-on-one (Chibaya, 2014).

Message passing robotic devices can explicitly share

direction vectors and confidence measures in the

vectors they follow, and use this information to

determine resultant vectors that determine the next

path to follow. Shared vectors, together with related

confidence measures, are used to modify robotic

devices’ perceptions of the direction to the target at

the time. Precisely, robotic devices upgrade or

downgrade their confidence measures depending on

the quality of the information shared amongst

neighbours. With time, the swarms thereof converge

on deterministic paths towards desired targets.

1.1 Statement of the Problem

The particular problem addressed in this paper can be

re-phrased into two questions as follows:

 Which control routines form valid message

passing XSets? In responding to this question,

we investigate the individual actions, the

parameter values, and meta information which

guide message passing robotic devices towards

predictable emergent behaviour. As a case

study and proof of concept, we investigate

those control routines useful for the path

finding and following behaviour in swarms of

robotic devices.

 How do message passing swarms perform

relative to the stigmergic counterpart? In

responding to this question, we administer an

experiment in which we measure speed and

quality of emergence that arise from using the

message passing XSet versus the speed and

quality of emergence yield when a stigmergic

XSet model is used for the same path finding

and path following task. Speed of emergence

evaluates the time it takes a swarm to converge

on a trodden path (Chibaya, 2014). On the other

hand, quality of emergence establishes the

tendencies of robotic devices to

deterministically follow the established paths

(Chibaya, 2014).

While answers to these two questions may not

respond to the very general robotic device control

problem, the message passing XSet model is hoped to

provide features of an alternative approach to the

stigmergic model, which brings us closer to

generalized rules for addressing the broader swarm

control problem.

1.2 Assumptions

We assume message passing robotic devices designed

with abilities to use control routines listed in a

message passing XSet model. The task at hand is to

find and follow paths between selected points situated

in a simulated environment. At any time in

simulation, each robotic device is either searching for

a food-like target or it will be travelling to a nest-like

starting point. Perpetual knowledge of the task at

hand defines a robotic device’s internal state. Internal

state information is kept in robotic devices’ basic

memories, together with the information regarding

the direction vectors being followed, as well as the

robotic device’s confidence measures in those

vectors. Robotic devices are able to interpret the

vectors and confidence measures held in neighbours’

memories by computing their own resultant vectors

which fairly represents the directional views of all the

neighbour robotic devices around. A searching

robotic device uses direction vectors pointing to the

food-like target, while a returning counterpart uses

direction vectors pointing to the starting point.

Robotic devices can relocate towards the direction of

the found resultant vector, detecting targets in each

step, and checking if they should flip between

different internal states when it becomes necessary.

Vector based Control Routines for Swarms of Path Finding Robotic Devices

273

These assumptions connote a message passing XSet

model with a finite cardinality, which supports

robotic devices with a finite number of internal states.

1.3 Overview

Section 2 reviews related works, emphasizing on the

design of robotic device control views in which

formalization of the rules followed is the key problem

addressed. The method we follow in coming up with

the message passing XSet model is presented in

section 3, focusing on the identification and

interpretation of the key control routines required by

message passing robotic devices in computational

terms. The setup of the message passing XSet model

is described in this section. Thereafter, section 4

describes the design of an experiment with which we

evaluate and validate the message passing XSet

model against the stigmergic counterpart. Speeds and

qualities of emergence are measured and reported in

section 5. Section 6 concludes the paper, highlighting

our key observations and the contributions arising.

2 RELATED WORK

Use of XSets to control swarms of robotic devices

towards desired emergent behaviour was first

reported in Chibaya (2014). Subsequent papers then

followed (Chibaya, 2015). However, emphasis has

been on the use of stigmergic XSets which lack

details regarding identification of generalizable views

on the quantities of pheromone levels used by

individual robotic devices in each step. More so, they

ignore the effects of pheromone dissipation to swarm

convergence (Chibaya, 2015). A mathematical model

which captures the quantities of pheromone levels

used by robotic devices, the size of the scene in which

the swarm is deployed, the population of robotic

devices required, the amount of time the swarm

requires in simulation, and pheromone dissipation

factors may, hopefully, lead to more generalizable

views. However, such a mathematical model has not

been found as yet.

Compelling alternatives to the XSet model exist.

Most alternatives, like the stigmergic model, rely on

indirect robotic device interactions coordinated via

the environment (Di Caro et al., 2004; Negulescu and

Barbat, 2004; Bonaneau et al., 1999; Chibaya and

Bangay, 2007; Dorigo, 1992; Dorigo et al., 1999).

Other alternatives are built on cell propagation

theories (Nagpal, 2006), cellular automata (Geer et

al., 2003; Green, 1994; Sanders and Smith, 2009), cell

growth and morphogenesis theories (Nagpal and

Kondacs, 2002), or origami theories (Rothemund,

2006). However, related robotic devices must possess

substantial memory and extra elitist abilities to be

able to handle the computations thereto (Chibaya,

2014). Robotic device orientation is usually based on

hard-marked beacons in the scene (Werfel, 2002),

landmarks (Wehmer et al., 2006), mathematical

models (Ngo et al., 2005), physicomimetic forces

(Spears et al., 2004a; Spears et al., 2004b; Spears et

al., 2005), or some Jacobian matrices (Harris, 2007).

In these cases, robotic devices are able to solve and

convert mathematical equations into directional cues.

In some cases, robotic devices have physically

mounted sensors with which to orientate (Spears et

al., 2004b). However, these elitist features are not

characteristics of the naïve robotic devices we

assume.

A few control strategies embrace message passing

views (Trianni and Dorigo, 2005; Raijbhupinder et

al., 2010; Rodriguiz et al., 2007). In these, robotic

devices may hold blocks of textual messages

(Raijbhupinder et al., 2010) to share with other agents

of the swarm. However, processing of textual data is

equally complex. Useful data values are often lost

during text conversion and interpretation. The

message passing model we propose assumes robotic

devices that can share geometric vectors and the

confidence measures associated with using the shared

vectors. These are much easier to interpret. Although

vector arithmetics are popular in machine learning,

network analyses, and spatial data representation,

they are a fairly new angle in resolving the robotic

device control problem. The emphasis we put on

simplicity and specificity in the design of the message

passing XSet gives this strategy a computational

edge. In our view, a message passing XSet will,

potentially, augment the stigmergic counterpart

towards notable developments in swarm intelligence

systems, particularly agent control issues.

3 METHODS

An XSet driven swarm simulator of message passing

robotic devices was developed to solve the path

finding and path following problem on a simulated

environment comprising a food-like target and a nest-

like starting point. The default task of the swarm is to

locate the food-like target, and upon finding it, travel

back to the nest-like starting point. Trips between the

food-like target and the nest-like starting point are

repeated over and over until a set simulation period

lapses.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

274

3.1 Configuration of the XSet

Control routines included in the message passing

XSet are sequentially listed in the order in which they

are used by robotic devices. Meta information include

(a) the alias name of the XSet. Stigmergic XSets were

referred to as stigXSet (Chibaya, 2014). Message

passing XSets are aliased as msgXSet. This alias tells

us the inspiring metaphor on which included control

routines are built. Another meta data is the cardinality

of the XSet. Cardinality tells us how many control

routines are required in each robotic device’s internal

state. Stigmergic robotic devices needed, at most,

four control routines in each internal state (Chibaya,

2014; Chibaya, 2015). From repeated tests, message

passing robotic devices require one extra control

routine in each internal state. The number of internal

states and the amount of memory robotic devices

need are also important meta data. In this case, four

internal states are supported by both stigmergic and

message passing robotic devices. A message passing

XSet can sufficiently drive robotic devices towards

emergent behaviour as long as memory is sufficient

to keep internal state information, direction vectors to

and from the target, and the confidence measures

thereof.

msgXSet (cardinality, states, mem)

{

switch (states)

{

case 0: {MsP, PtV, Nrm, MvP, StS}

case 1: {NOp, NOp, NOp, NOp, StS}

case 2: {MsP, PtV, Nrm, MvP, StS}

case 3: {NOp, NOp, NOp, NOp, StS}

}

Listing 1: Template of a message passing XSet.

We summarize the design of a message passing

XSet in listing 1, where msgXSet is the alias name.

The model accepts three parameters, cardinality,

number of internal states supported, and amount of

memory allocated to each robotic device. In this case,

robotic devices can hold up to 8 memory blocks.

Control routines and related parameter values are

listed between the curly brackets. Each control

routine and its related parameter values are listed after

each switch-case entry. A control routine and its

parameter values are separated by a colon. Parameter

values to a control routine are separated by commas.

Different control routines are, also, separated by

commas. Control routines used in different internal

states are demarcated by curly brackets. A code such

as: (MsP: V

, V

, C) tells a robotic device to share

particular direction vectors and confidence measures.

The vectors shared are used to determine the next

direction to follow. Robotic devices can detect their

proximity to targets using the control routine aliased

as (PtV: V, C). The resultant vectors yield from any

computation are normalized using the control routine

(Nrm: X, Y, Z). Once orientated, robotic devices can

relocate to desired destinations using the control

routine (MvP: X, Y, Z). Should it be necessary at the

time, robotic devices can flip between different

internal states using the control routine (StS: m, n, x).

However, there are moments when robotic devices

have to do nothing. An empty control routine aliased

as (NOp:) is used. The next section describes the

computational interpretation of each of these control

routines.

3.2 Computational Design of Routines

In XSets, control routine names have three letters, e.g.

MsP, PtV, Nrm. These control routines are primarily

used by robotic devices for sharing vectors, detecting

targets, normalizing vectors, making movements, or

flipping between different internal states. This section

discusses the computational interpretation of these

control routines.

We indicated earlier on that message passing

robotic devices require some memory in which to

store internal state information, vectors, and

confidence measures. Keeping internal state

information in memory spells out a robotic device’s

tasks at the time (Panait and Luke, 2004a, 2004b,

2004c). It influences the behaviour of nearby robotic

devices (Parunak, 2005). For example, a robotic

device may flip to another internal state triggered by

a nearby robotic device’s proximity to the target

(Parunak, 2005). In some cases, such flips occur as a

reward for an achievement (Panait and Luke, 2004a).

Robotic devices in the search internal state

communicate with counterparts in the returning

internal state because those robotic devices likely

know where the food-like target is. Robotic devices

in the returning internal state prefer interactions with

members in the searching internal state because those

robotic devices would likely know better about the

direction towards the nest-like starting point. Internal

state changes are repeated every time a target is hit.

Listing 2 interprets the process through which

robotic devices achieve the flipping between different

internal states in computational terms. The mnemonic

(StS: m, n, x) summarizes the control routine for

flipping between internal states. It tells a robotic

device to set its internal state to mode m on condition

Vector based Control Routines for Swarms of Path Finding Robotic Devices

275

bool StS (m , n , x)

{

for-each robotic-device R

if (n is true in domain x)

{

internal state = m

}

Listing 2: Flipping between internal states.

that requirements n are satisfied in domain x. In our

case, m ranges between 0 and 3, representing the four

possible internal states message passing robotic

devices support. Then, n is a set of conditions which

indicates aspects of the simulation to trigger a robotic

device’s interests in changing from one internal state

to another. Inclusion of x sets the domain in which n

is satisfied. For example, (StS:1,0,0) tells a robotic

device to change to internal state case 1, provided that

the target indicators marked as 0 are in quantities

above 0 at the robotic device’s current location.

Message passing robotic devices can share

message blocks (Trianni and Dorigo, 2005) of the

format: (x

;𝑓

⃗

;𝑓

;𝑛

⃗

;𝑛

). In these message blocks,

) is the offset of a communicating i

robotic

device. Communication is allowed between a path

finding robotic device and those robotic devices

whose offsets are within the set communication

range. The path finding robotic device self-localizes

relative to nearby robotic devices, placing itself at the

origin of its local coordinate system. The choice of

which vectors are attractive at the time depends on the

path finding robotic device’s internal state. When

searching, robotic devices are attracted to 𝑓

⃗

(food

vectors), weighed by 𝑓

(confidence measures in 𝑓

⃗

The path finding robotic device accumulates the

vectors read from all robotic devices around, each

weighted by its related 𝑓

. A resultant vector is

calculated, which overwrites the path finding robotic

device’s current 𝑓

⃗

. Returning robotic devices are

attracted to 𝑛

⃗

, weighted by 𝑛

. They also find a

resultant vector which points in the likely direction of

the nest-like starting point. Every time a resultant

vector is calculated, either 𝑓

or 𝑛

of the path

finding robotic device is upgraded or downgraded

depending on the quality of the information received.

Below, we show how resultant vectors are calculated.

3.3 Calculation of Resultant Vectors

A similar approach where vectors are shared before

resultant vectors are calculated was used in Ngo et al.

(2005). Resultant vectors summarize nearby robotic

devices’ past experiences (Rodriguez et al., 2007).

These combined experiences upgrade or degrade a

path finding robotic devices’ knowledge and

confidence in the vector it is following.

Upon deployment, robotic devices randomly pick

𝑓

⃗

and initialize 𝑓

to a minimal value possible. The

hope is that 𝑓

would improve as robotic devices gain

more knowledge of the environment through

interactions with other robotic devices. The local

coordinate system created by robotic devices span

over three simulated grid cells in 2D spaces. Figure 1

shows a typical local coordinate system with eight

possible paths a robotic device can follow. The task

of the path finding robotic device is to decide on a

direction to take based on the information shared

from nearby robotic devices.

To find that resultant vector, a message passing

robotic device calculates intersection points and

points of closest approaches between all possible

pairs of vectors read from neighbours. The notion is

that, two vectors representing the knowledge of two

independent robotic devices would intersect at a point

close to the target. That vector which starts from the

origin of the local coordinate system to the

intersection point of the two vectors is a candidate

direction to follow next. In the event of a pair of

vectors not intersecting within the defined scene,

points of closest approach between the selected

direction vectors are determined. Once a set of all

intersection points and points of closest approach is

established, message passing robotic devices use least

squares point estimation to pick a fair direction vector

which represents the knowledge gathered, at the same

time updating or downgrading its own confidence

measure in the new selected direction.

Figure 1: coordinate system for path finding robotic

devices.

Figure 2 shows a set of intersection points and

points of closest approach for an arbitrary case.

However, how do we geometrically find those values

of x and y at which two vectors intersect or where

points closely approach each other? The problem of

determining these coordinates is geometric.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

276

Figure 2: Intersection points and points of closest approach.

If we denote the vector followed by the i

robotic

device as 𝑑

⃗

, and that followed by the j

robotic

device as 𝑑

⃗

. Suppose the offsets of the i

and j

robotic devices relative to the path finding robotic

device are (x

; y

) and (x

; y

) respectively.

Then the points along each line segment

represented by vectors 𝑑

⃗

and 𝑑

⃗

are 𝑑

⃗

+𝑠𝑑



⃗

and

𝑑

⃗

+ 𝑡𝑑



⃗

respectively. Here, 𝑑



⃗

and 𝑑



⃗

are unit

vectors of 𝑑

⃗

and 𝑑

⃗

. Parameters s and t are

magnitudes. Let the points at 𝑑

⃗

+𝑠𝑑



⃗

and 𝑑

⃗

+𝑡𝑑



⃗

be (x

i+1

; y

i+1

) and (x

j+1

; y

j+1

) respectively. The

intersection point of 𝑑

⃗

and 𝑑

⃗

j is therefore (x ; y),

where x and y are computed as shown in equations (1)

and (2). With two points along each vector, we find

the point at which two vectors meet (Wikipedia,

2004) by finding the matrix determinants of the

coordinates of the points identified along each vector.

x





























































(1)

y





























































(2)

When vectors do not intersect, we get a point of

closest approach. If L

=𝑑

⃗

+𝑠𝑑



⃗

and L

= 𝑑

⃗

+𝑡𝑑



⃗

are equations of the line segments along vectors 𝑑

⃗

and 𝑑

⃗

, the point at which these two vectors have a

minimum offset, w=L

- L

, is the point of closest

approach. This point is when w is perpendicular to L

and L

. At this point, 𝑤𝑑



⃗

=0 and 𝑤𝑑



⃗

=0.

Algorithm 1 shows how 𝑤𝑑



⃗

=0 and 𝑤𝑑



⃗

=0 are

used to solve for s and t, and how we find the point

where w is smallest. Two vectors are parallel when

ac-b

=0. Robotic devices pick a midpoint along the

line segment w as the wanted point of closest

approach. The set of (x ; y) value of intersection points

and points of closest approach for all pairs of vectors

taken from nearby devices is the data from which a

robotic device finds its new path. Hopefully, the

vector found fairly represents the consensus of nearby

robotic devices.

Algorithm 1: Determining the values of s and t.

𝑦 𝑎



𝑥𝑚𝑏



(7)

𝑥𝑦𝑎



𝑥



𝑏



𝑥

(8)

𝑥 𝑎



𝑦𝑚𝑏



(9)

Vector based Control Routines for Swarms of Path Finding Robotic Devices

277

𝑥𝑦𝑎



𝑦



𝑏



𝑦

(10)





∑



∑





∑

x



(11)





∑

𝑥





∑





∑

x



(12)

To pick a direction, least squares point estimation

(Chibaya, 2014) is used on that set of (x ; y)

coordinates. Two least squares regression lines arise,

one for y on x, and another for x on y. The regression

line for y on x is y = a

x + b

, and that for x on y is

x=a

y + b

(Chibaya, 2014). Equations (7) and (8)

determine a

and b

in y = a

x + b

, while equations

(9) and (10) find a

and b

in x = a

y + b

. In both

cases, m is the population of robotic devices around

the path finding robotic device. Equations (11) and

(12) show how we simplify equations (7) and (8) for

and b

. Flipping x and y in these equations gives the

formula for finding a

and b

in equations (9) and

(10). Least squares point estimator finds the point at

which the two regression lines intersect. That point

coincides with the centre of mass (𝑥̅; 𝑦). A vector

starting from the origin of the local coordinates

system to the centre of mass is the resultant vector we

want.

3.4 Updating Confidence Measures

We indicated that each vector has an associated

confidence measure which indicates how well a

robotic device has performed in previous orientation

choices. Confidence measures indicate the robotic

device’s trust in the path followed (Chibaya, 2014).

They reflect the quality of the vectors previously

followed. These are float values between 0 and 1,

where 1 indicates awareness of the direction to the

target and 0 indicates complete ignorance of the scene

features. A robotic device updates its confidence

measure by combining the confidence levels of

nearby robotic devices with its own using equation

(13).

𝑤





𝑡1









𝑤





𝑡





∑







∈



1𝑐𝜆 (13)

If there exist k nearby robotic devices to a path

finding robotic device, then let the vector held in each

of the k robotic devices be denoted as j. Therefore

∑







∈



is the average confidence measure of the k

robotic devices. To minimize the effects of outliers,

we find the spread of the views of the k robotic

devices and denote it as c, which is within the range

0 to 1. To penalize larger c, we use (1-c) as the desired

weight. An average of w

(t) of the path finding robotic

device and w

(t) of the k nearby robotic devices gives

the updated w

(t+1). Some randomness (λ) is added to

allow independence in robotic devices’ actions.

Randomness is especially useful when robotic

devices are isolated from the rest. That way, robotic

devices’ confidence levels would never deplete.

3.5 The Message Passing Routine

Listing 3 presents the message passing routine. The

routine has three parts, one where vectors are shared

in order to create a set of intersection points and

points of closest approach, another part where the

least squares point estimator is used to pick the

vector, and a part where confidence measures are

updated. In the mnemonic: (MsP:v

), v

are

blocks in robotic devices’ memories where

confidence measures are recorded. Then, v

are blocks

in robotic device memories where attractive vectors

are held. The third parameter indicates blocks in

memory in which the resultant vectors are recorded

after they are determined. Robotic devices overwrite

their old vectors in v

by the vectors yield. The routine

tells a robotic device to read vectors held in nearby

devices’ memories, read related confidence

measures, and find a vector to follow.

bool MsP (V

, V

)

{

for-each robotic-device R

for-each robotic-device K

{

pos(K

)= (x

, y

, z

) - D

for-every-other K

{

pos(K

) = (x

, y

, z

)-D

if (D

Ո D

)

{

x = calculated from (1)

y = calculated from (2)

}

else

{

x=random x; y=random y

}

Add (x , y) to array []

}

c = avg (sdvev (x , y))

path-wght=avg(w(t),sum(w(t)

)×c+β

}

Listing 3: The message passing primitive instruction.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

278

bool Nrm (x, y , z)

{

length = x

+ y

+ z

if (length ≠ 0)

{

length:

z =

length

}

else

{

x = random x

y = random y

z = 0.0

path-weight = 0.00001

Nrm (x , y , z)

}

Listing 4: Normalizing vectors.

bool PtV (p

, x)

{

for-each pos-L around device R

{

if (Q(p

) @ L > x)

{

path (R

t+1

) = (L

, L

)

path-weight (R

t+1

) = 1.0

}

Listing 5: Detecting target indicators.

bool MvP (x , y , z)

{

for-each robotic-device R

{

if (R

= (x , y , z))

t+1

= (x+x

, y+y

, z+z

)

}

Listing 6: Robotic device movement.

Magnitudes of vectors are variable. They dictate

robotic devices’ movement steps. Normalizing these

vectors reset the magnitudes to 1, defining unit

robotic device steps. Listing 4 summarizes the

semantics for normalizing a vector. The control

routine receives the x, y, and z components of a vector

and use these values to find the length of the vector.

This length is non-zero when a robotic device has

neighbours. Components of the vector are divided by

this length. Isolated robotic devices yield vectors of

magnitude 0, after which the robotic devices follows

randomly picked direction vectors with depleted

confidence measures. Random vectors are recursively

normalized.

It is critical that robotic devices detect targets

(Cavalcanti and Freitas, 2005) in order to trigger

internal state changes. Points where key objects exist

in the simulation environment are marked by target

indicators (Cavalcanti and Freitas, 2005). Target

indicators are virtual chemicals set when the

environment is launched. A method is required with

which message passing robotic devices can detect

target indicators and appropriately interpret this into

vector information. This is not the first time vectors

have been used to interpret pheromone levels

(Chibaya, 2014). A magnitude of zero indicate that

the robotic device is not yet on target. Listing 5

describes target detection and conversion of related

information to vectors. It accepts two parameters, one

passing the ID of the target indicator and another

setting the lowest level of target indicators to trigger

changes in robotic devices’ actions. If detected, the

robotic device overwrites its vector by a vector

pointing to the location of the target indicators. This

upgrades confidence measures to the highest measure

possible, indicating absolute trust in the vector to the

target.

Movement is the robotic device’s last task in each

cycle. Listing 6 interprets the movement policies run

after orientation. Three parameters indicating the

offset of the chosen destination are received into the

routine. In these offset values, the z component is

always 0 as we operate in 2D.

Meta information stipulates each XSet’s precise

cardinality. However, there are moments when

robotic devices require fewer control routines than

stipulated. (NOp:) tells robotic devices to do nothing

when it becomes necessary. This is merely an empty

routine.

4 EXPERIMENTS SETUP

An experiment is administered in which to assess path

finding and path following properties in swarms

controlled by message passing and stigmergic XSets

(Chibaya, 2014).

The platform on which robotic devices are

assessed for path finding and path following abilities

is called an environment (Chibaya, 2014). In

computational terms, an environment is a grid of rows

and columns which intersect to form cells which we

regard as locations. A location is practically a tuple

which stores information such as target indicators.

For experimental purposes, we assume a 1000 × 1000

grid environment. Target and the starting point are

hard coded at particular locations to allow repeatable

tests.

Vector based Control Routines for Swarms of Path Finding Robotic Devices

279

Robotic devices are allowed to run for, as a case

study, 10,000 steps in which to score performances.

However, simulation time can be changed without

compromising the desired outcomes. Each swarm

consists of, as a case study, 5000 robotic devices.

A performance metric measures the extent to

which emergent behaviour is manifest as a result of

robotic devices using the control routines listed in an

XSet. Two performance metrics are of interest in this

study. First, quality of emergence measures

adherence of robotic devices to the schedule,

evaluating robotic devices’ engagement with the task

at hand (Werfel et al., 2006). On the other hand, speed

of emergence evaluates timeliness. Combined, the

two metrics yield an index of merit of the XSet used.

To measure speed, the time it takes the first

robotic device to find the target is determined. Time

is measured in iterations. Then, the time it takes the

last robotic device in the swarm to also find the target

is found. The time gap between these two time

intervals is the speed of emergence towards the target.

Speed of emergence towards the starting point is the

time gap between the times taken by the first and last

robotic device to complete round trips. The average

between speed of emergence towards the target and

speed of emergence towards the starting point is the

overall speed of emergence of the swarm. Ten

repeated simulations administered, before centrally

placed speed measures, are reported.

To determine quality of emergence, we set a time

frame in which successful trips of robotic devices in

each direction are counted and recorded. Ten repeated

tests are also administered in order to achieve

smoothened and centrally placed performance trends.

The index of merit is determined by scaling speed

and quality measures so that they lie in the range [0;

1]. Equation (14) shows how the average of the scaled

metrics is found, giving an index of merit of an XSet.

Index of merit = 0.5×((1-

) +

) (14)

The hypothesis which drove the experiment

administered is: message passing XSets are a useful

alternative for controlling swarms of robotic devices

towards emergent behaviour. The null hypothesis

thereto is: there aren’t significant differences in

performance between swarms coordinated using

message passing and stigmergic XSets. The

dependent variables in this experiment are XSets’

indices of merits. Two independent variables are: the

time taken in simulation and the control levels at

which measures of emergence are extracted. In this

case, performance measures are extracted at intervals

of 1,000 iterations. All other variables are controlled,

including the population of robotic devices, size of

the environment, the positions of the starting point

and the food-like target, the cardinalities of XSets,

number of internal states, and the sizes of the memory

blocks supported by robotic devices. Listing 7

summarizes the design of the experiment, showing

the procedure followed.

Listing 7: Experiment design.

5 RESULTS

The key results reported are speeds of emergence,

qualities of emergence, and indices of merits of

message passing and stigmergic XSets. In these

results, measures of central tendencies and dispersion

are important. Correlations extracted at a 99% level

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

280

of confidence are also key. The sample size we use is

statistically small, so two-tailed correlation tests are

administered because we cannot tell, in advance, the

sign of the correlation coefficients we will get. The

critical correlation value we use is 0.765 when alpha

is 0.01. Absolute values of the correlation coefficient

above 0.765 indicate sufficient evidence to accept the

hypothesis that there are significant relationships

between the speeds and qualities of emergence yield.

Figure 3 compares the indices of merits yielded

from using the two XSets. The x- axis is marked by

the control levels. The y-axis shows the average

scaled indices of merits thereof. The speeds and

qualities of emergence found satisfy Kolmogorov-

Smirnoff tests for normality with a statistical p value

above 0.05. A correlation coefficient of 0.91 is

yielded when speed and quality of emergence in the

stigmergic category are compared, implying a strong

correlation between speed and quality of emergence

in that model. The message passing counterpart

yielded a correlation coefficient of 0.86 when the

same metrics are compared. Chances are therefore

slim that any generalized views we deduce from these

results would significantly differ from the trends and

relationships shown.

Figure 3: Comparisons between indices of merits.

We have sufficient evidence to accept the

hypothesis that the two XSets are alternatives to one

another, both achieving plausible outcomes. We

make the following additional observations:

 The stigmergic XSet reaches a turning point in

performances with time in simulation, after

which swarm performances deplete. A turning

point in performances occurs when the

environment gets saturated with pheromone

levels which reverts robotic devices to a

random wandering mode. We noted in

(Chibaya, 2015) that pheromone dissipation

could be, to some extent, the remedy to this

flaw. On the contrary a message passing XSet

improves related swarm performances in

Sigmoid-like patterns. Related performances

gradually improve until robotic devices

converge on deterministic paths. Thus, while

extended time in simulation is detrimental to

stigmergic swarm performances, it allows

message passing swarms to build vector fields

with deterministic cues to the targets. Swarm

performances are therefore a function of time

in simulation in both cases.

 Stigmergic control routines come in specific

sequences in order to yield good performances.

Similarly, message passing control routines are

executed sequentially. The configuration of the

two XSets is similar, connoting possibilities of

a generalized XSet.

 Message passing and stigmergic XSets both

influence swarm behaviour. They are, both,

useful swarm drivers. This is a milestone in the

study of robotic systems.

The results provide sufficient evidence to support

the view that message passing XSets are an

alternative to stigmergic XSets. No significant

differences are observed between their performances,

with both XSets showing causal properties.

6 CONCLUSION

A new XSet for controlling swarms of robotic devices

towards desired formations is proposed. The new

XSet is inspired by the behaviours of message passing

agents. Controlling swarms of robotic devices

requires us to provide five meta data during the

configuration of the XSet. We need to provide the

alias name of the XSet, cardinality, number of

internal states supported, amount of memory, and the

separators used to demarcate the XSet’s entries. Six

routines form the vocabulary of message passing

XSets as follows:

 (MsP:) – a control routine with which robotic

devices share vectors, finding resultant vectors

to follow next, updating their confidence

measures, and orientating in each step.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

12345678910

Indices of merits

Stigmergic Xset

Message passing Xset

Random wandering Xset

Vector based Control Routines for Swarms of Path Finding Robotic Devices

281

 (PtV:) – a control routine for detecting target

indicators, updating confidence measures, and

orientating robotic devices towards the targets.

 (MvP:) –a routine for relocating robotic

devices.

 (Nrm:) – a routine for normalizing vectors.

 (StS:)– a control routine which allows robotic

devices to conditionally flip between different

internal states.

 (NOp:)– telling robotic devices to do nothing.

An experiment to evaluate the message passing

XSet for causal properties showed that this is an

alternative swarm control protocol to the stigmergic

version. Four contributions emanate from this work:

 The design of the message passing XSet adds

to new developments towards practical use of

robotic device based swarm intelligent

systems.

 The control routines used are creative, adding

relevant content to the robotic device control

and programming problem.

 The metrics used to measure the performances

of XSets are innovative. These metrics can be

useful in verifying other forms of emergent

behaviours, opening up new research avenues

in areas related to quantification of emergency.

 The statistical tests applied during validation of

the XSets and tests for normality on the results

are also innovative. Similar statistical tests may

inspire the development of more scientific and

deductive outcomes with positivism angles.

Although the general robotic device programming

problem is not resolved, this work brings us closer to

such generalization. It provides a baseline upon

which further investigations may arise. More so, the

work strengthens the foundation set when the

stigmergic XSets were identified. Importantly, the

investigations undertaken may soon inspire the

development of more generic control routines with

which robotic devices, in general, would engineer

predicable object assembly.

ACKNOWLEDGMENTS

We acknowledge support from the department of

Computer Science at Sol Plaatje University, for

allowing us time to work on this article, the brotherly

advices, and financial support, without which this

work would not have been a success. However,

professor Shaun Bangay remains our most inspiring

mentor ever.

REFERENCES

Chibaya, C. 2019. A Message Passing XSet for Path

Finding Robotic Devices. In the proceedings of the 1

International Multidisciplinary Information

Technology and Engineering Conference.

Chibaya, C. 2014. An investigation into XSets of primitive

behaviours for emergent behaviour in stigmergic and

message passing ant-like agents. A PhD thesis

submitted at Rhodes University. South Africa.

Chibaya, C. 2015. An XSet based protocol for coordinating

the behaviour of stigmergic ant-like robotic devices. In

the ACM International Conference Proceeding Series

and the SAICSIT’ 2015. South Africa.

Geer, P, McLaughlin, H,W, Unsworth, K. 2003. Cellular

lines: an introduction. In Discrete mathematics and

theoretical computer science.

Green, D, G.. 1994. Emergent behaviour in biological

systems. Complexity international.

Sanders, J, W, Smith, G. 2009. Refining emergent

properties. International Institute for software

technology. Electronic Notes in Theoretical Computer

Science.

Di Caro, G, Ducatelle, F, Gambardella, L,M. 2004.

AntHocNet: An adaptive nature-inspired algorithm for

routing in mobile ad hoc networks. Technical report

Dalle Molle Institute.

Negulescu, S,C, Barbat, B, E. 2004. Enhancing the

effectiveness of simple multi-agent systems through

stigmergic coordination. In the 4

symposium on

engineering of intelligent systems.

Bonaneau, E, Dorigo, M, Theraukaz, G. 1999. Swarm

Intelligence: From natural to artificial systems. Oxford.

Chibaya, C, Bangay, S. 2007. A probabilistic movement

model for shortest path formation in virtual ant-like

agents. In the ACM International Conference

Proceeding Series, and In SAICSIT 2007 on IT research

in developing countries.

Dorigo, M. 1992. Optimization, learning and natural

algorithms. A PhD thesis submitted to the Dipartimento

di Electronica, Politecnico di Milano. Milan, Italy.

Dorigo, M, Di Caro, G, Gambardella, L,M. 1999. Ant

algorithms for discrete optimization. In the Proceedings

of Artificial Life.

Nagpal, R. 2006. Self-Organizing shape and pattern: From

cells to robots. IEEE Intelligent Systems.

Nagpal, R, Kondacs, A, Chang, C. 2002. Programming

methodology for biologically-inspired self-assembling

system. In the Proceedings of the American Association

for Artificial Intelligence.

Rothemund, P, W, K. 2006. Folding DNA to create nano-

scale shapes and patterns. Computation and neural

systems and computer science.

Werfel, J. 2002. Autonomous multi-agent construction.

Amorphous projects. On-line on http://groups.csall.mit.

edu/mac/projects/amorphous.6.978/nal_nal.pdf.

Wehmer, R, Boyer, M, Loertscher, F, Sommer, S, Menzi.

2006. Ant navigation: one-way routes rather than maps.

Current biology. Elsevier Science Ltd.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

282

Ngo, V, T, Nguyen, A,D, Ha, H. 2005. Integration of

Planning and Control in Robotic Formations. In the

Proceedings of the 2005 Australian Conference on

robotics and Automation.

Spears, W, M, Spears, D,F, Zarzhitsky, D, Heil, R. 2004a.

Physicomimetics for mobile robot formations. In the 3

International conference on autonomous agents and

multi agent systems.

Spears, W, M, Spears, D, F, Hamann, J, C, Heil, R. 2004b.

Distributed, physics-based control of swarms of

vehicles. Autonomous robots.

Spears, W, M, Spears, D, F, Zarzhitsky, D. 2005.

Physicomimetics positioning methodology for

distributed autonomous systems. In the Proceedings of

the government microcircuit applications and critical

technology conference, Intelligent Technologies.

Harris, D, M, J. 2007. Direct motion of a parallel-linkage

robot through the Jacobian. 12

IFToMM World

Congress, France.

Trianni, V, Dorigo, M. 2005, Self-Organisation and

communication in groups of simulated and physical

robots. Technical report: Université Libre de Bruxelles.

Biological Cybernetics.

Raijbhupinder, S,D, Harwinder, S, S, Amarpreet, S, G.

2010. Load Balancing of Ant Based Algorithm in

MANET. International journal of computer science

and technology.

Rodriguez, S, Salazar, R, McMahon, T, Amato, N, M.

2007. Roadmap based group behaviour: generation and

evaluation. Technical report: Parasol Lab, Computer

Science.

Panait, L, Luke, L. 2004a. A pheromone-based utility

model for collaborative foraging. In the 3

International Joint Conference on Autonomous Agents

and Multi-Agent Systems.

Panait, L, Luke, L. 2004b. Ant foraging revisited. In the 9

International conference on simulation and synthesis of

living systems.

Panait, L, Luke, L. 2004c. Learning Ant Foraging

Behaviours. In the 9

International Conference on the

Simulation and Synthesis of Living Systems.

Parunak, H, V, D. 2005. A survey of environments and

mechanisms for human-human stigmergy. In the 2

International conference on environments for Multi

Agent Systems.

Wikipedia. 2004. Line-line intersection. Wikipedia.

Cavalcanti, A, Freitas, R, A. 2005. Nanorobotics control

design: A collective behaviour approach for medicine.

IEEE Transactions on NanoBioScience.

Werfel, J, Nagpal, N, Seung, H, S. 2006. Ant-hills built to

order: automating construction with artificial swarms.

A PhD thesis submitted to the MIT.

Gengan, D, Schoeman, M, A, Van der Poll John, A. 2014.

An ant-based mobile agent approach to resource

discovery in grid computing. In the proceedings of the

SAICSIT 2014.

Vector based Control Routines for Swarms of Path Finding Robotic Devices

283