=Paper=
{{Paper
|id=Vol-1113/paper2
|storemode=property
|title=A Metric for Hexagonal Grid-World Cooperative Object Recognition Tasks for a Swarm of Agents
|pdfUrl=https://ceur-ws.org/Vol-1113/paper2.pdf
|volume=Vol-1113
|dblpUrl=https://dblp.org/rec/conf/eumas/KingB13
}}
==A Metric for Hexagonal Grid-World Cooperative Object Recognition Tasks for a Swarm of Agents==
https://ceur-ws.org/Vol-1113/paper2.pdf
A Metric for Hexagonal Grid-World Cooperative
Object Recognition Tasks for a Swarm of Agents
David King1 and Philip Breedon1
Nottingham Trent University, Nottingham, NG1 4BU, UK
Abstract. In the future nano-robots could be used in in-vivo medical
applications. By using lattice robots as a unit of measurement tumours
could be assessed as malignant or benign by considering the shape of
their boundaries through cooperative object recognition. With the ob-
jective of moving towards this end goal research has been carried out
using simulations on hexagonal lattices that allow simple agents to coop-
erate to distinguish between two objects’ shapes. This paper determines
a method which gives a value of difference between any two objects con-
structed on a hexagonal lattice. This difference value will provide a metric
for future cooperative object recognition scenarios. The measure of the
difference value was calculated considering varying degrees of knowledge
about sections of the objects’ boundaries. The results were compared to
a simulated task with agents that were homogeneous, anonymous, had
a limited sensor range and no common coordinate system. It was found
that the difference value metric provides a significant correlation to the
time taken to complete a series of cooperative object recognition scenar-
ios.
Key words: multi-agent systems, swarm robotics, cooperative object
recognition.
1 Introduction
With continuing advances in miniaturising electronics there is an increased amount
of interest in nano-scale robots for medical applications [1, 2]. These nano-robots
have suggested applications including isolating harmful cells in the blood stream
[3]; diagnosis of endogenous diseases [4] and antibody delivery systems [5]. Due
to the robots’ individual sizes, a limitation is placed on the robots’ individual
computational and physical capability when compared to larger robots. One ap-
proach that may help overcome this limitation is swarm robotics. Swarm robotics
takes inspiration from the behaviours of social insects (ants, bees and termites)
to provide multi-agent systems methods to complete tasks. Through the agents
local interactions with each other and the environment they inhabit they can
coooperate intelligently [6]. Although specific behaviours have been mimicked
directly: foraging [7], cooperative transport [8] and self-assembly [9], inspiration
alone can be taken from the decentralised approach.
Utilising a swarm robotic approach cooperative object recognition is achiev-
able, where individuals who cannot assess an object alone can cooperate to
identify an object through interacting with it and the other members of the
swarm. In addition to the miniaturisation of robots, this method will potentially
allow for swarms of nano-robots who can identify unwanted or dangerous enti-
ties in human bodies through shape analyses. For example tumours that can be
assessed by their shape to determine if they are malignant or benign [10, 11].
Currently images of tumours are analysed by a radiologists. The system envi-
sioned here would instead have a swarm of nano-bots injected into a person, who
would cooperate to locate and identify tumours, and could potentially be capa-
ble of destroying them. To do this requires that the robots be able to measure
the boundary of the tumour.
Groups of homogeneous lattice robots, who individually are approximations
of regular shapes, can connect together into ordered formations. Two-dimensional
lattice structures can be formed and rearranged by triangular, square and hexag-
onal robots [12, 13, 14, 15, 16, 17]. Three-dimensional lattice structures are
formed by cubical and spherical robots. In some cases these cubical and spher-
ical robots are divided into two parts which can rotate relative to each other.
Structures formed from these types of robot can perform clustered walks [18, 19]
and can replicate themselves [20]. Lattice robots that do not have the capability
provided by having two halves rotate relative to each other have also been demon-
strated to form different structures where the robots’ initial starting structure is
a uniform block. The robots detach from each other at specific points allowing
them to form the required structures [21].
One advantage of lattice based robots, especially where smaller sized and
simpler robots are considered, is their ability to become a unit of measurement.
Giplin and Rus [22] show cube agents, Robot Pebbles, which are capable of
duplicating inert shapes with identical resolution to the agents themselves. To
do this the Robot Pebbles must fully surround the shape they are replicating and
pass a signal from agent to agent around the object to determine its shape. This
information is then used to form the replica of that shape from the Robot Pebble
agents themselves. This type of lattice based measurement would be beneficial
where nano-robotic agents neighbour a tumour, or any other object, to determine
any unique distinguishing features in its boundary. One difference from Gilpin
and Rus’s research is that the entire object need not be surrounded as only
distinguishing features need to be identified. This has previously been shown
to be possible on a hexagonal lattice where a swarm of simple agents could
distinguish approximations of hexagons from triangles when cooperating with
each other [23, 24]. As more complex and varied object shapes are considered
on this lattice a metric of difficulty is required to determine a suitable range of
task scenarios. The metric will allow for suitable scenario selection for future
research into training methods for the swarm’s behaviours. Training the swarm
will allow it to adjust to different cooperative object recognition tasks, as well as
give guidance on how much memory the agents will require to complete different
scenarios.
To determine a metric a method is required for comparing two objects’ shapes
similarities to each other whilst considering how much information is currently
known about them. The object shapes in a hexagonal grid-world could be consid-
ered binary images and the arena the background. Shape coding has been used
to store binary images in a range of different ways [25, 26]. Most relevant of these
techniques is chain-coding which maps the relative positions of the neighbouring
pixels at the boundary of a shape on a lattice in a linear order [25]. Usually
chain-coding uses a square lattice but it has also been completed on a hexagonal
lattice [27]. The paths the chain takes to code the shape can follow the pixels
at the inside or the outside of the shape [28, 29]. There are lossy techniques,
where estimations are used to decrease the memory required to store the shape
information. However, lossy techniques are not suitable for this application as
the shapes need to be described accurately. Chain-coding also considers position
and orientation when describing shapes. As the objects are considered identical
despite their location and rotation these two factors are not required for the
cooperative object recognition task metric.
By determining a method of describing the boundary of objects’ shapes on
a hexagonal lattice, without considering their location and rotation, this paper
provides a metric for comparing two object shapes. This metric is then tested
using a series of cooperative object recognition scenario simulations involving a
swarm individually simplistic agents.
2 Simulation Method
In order to understand what the metric will be measuring the simulation platform
first needs describing. This research uses a hexagonal lattice as an arena for
testing cooperative object recognition tasks. Each hexagonal cell can be: an
object cell, which are grouped to form object shapes; a single agent, termed a
hBot; a border, which the hBots cannot pass; or an empty space.
2.1 Object Shapes
Object shapes are constructed by grouping a number of neighbouring hexagonal
object cells together, where the simplest object shape is a single cell. An object
shape can be described without considering rotation or location by the contours
of its boundary region, this is termed the data-chain. If each cell that neighbours
an object shape is given a value determined by how many object cells it touches,
the data-chain is an array of these values arrange in a clockwise manner rep-
resented by the sequence which is first in lexicographical order. All the object
shapes with four object cells (ID0 - ID9) are shown, Fig. 1, with the numbers in
each of their surrounding cells indicating how many object cells they neighbour,
resulting in the data-chains listed.
A scenario is defined by the two object shapes that the hBots must cooperate
to distinguish between. In a scenario one of the object shapes is classed as valid,
this is the object shape the hBots must identify and remove. The other object
shape is invalid; this object shape must not be removed by the hBots. It is the
difference between these two object shapes which determines the difficulty of the
scenario which the metric will determine (section 3.0).
Fig. 1. All ten object shapes (ID0 - ID9) with four object cells with their surrounding
cells indicating the number of object cells they are in contact with which determine
their data-chains.
2.2 hBot Agents
The hBots used in the cooperative object recognition task are homogeneous,
anonymous, have no common coordinate system and are not aware of their posi-
tion relative to the arenas coordinate system. The hBots are modelled with local
sensor and communication, both with a range of one cell. This capability allows
the hBots to determine their current state from their immediate surroundings
and to communicate this state to any hBots that they neighbour, providing a
feedback loop between neighbouring hBots. The current state-level of the hBot
indicates how much knowledge it has about the object’s shape, whilst the state
of the hBot describes this knowledge specifically.
– A hBot at state-level 0 is not in contact with an object shape and therefore
has no knowledge about the object shape, this is state 0.
– A hBot at state-level 1 knows the number of object cells it is neighbouring:
one, two, or three and is equivalent to knowing a single value in a data-
chain. This knowledge is represented by states 1, 2 and 3 respectively. This
value could theoretically be between one and six, however in this research
the object shapes were limited as to only allow situations where the values
one, two and three occur, to initially reduce the total number of states.
– A hBot at state-level 2 knows as much as three individual agents at state-
level 1, as it knows its own state-level 1 state and the states of its neighbours.
This is equivalent to knowing three consecutive values in a data-chain and
includes states 4 – 21.
– A hBot at state-level 3 knows as much as five individual agents at state-
level 1, or three agents at state-level 2. This is equivalent to knowing five
consecutive values in a data-chain and includes states 22 – 264.
As certain states are only achievable when the hBots interact with certain
object shapes, these states can be used by the hBots to distinguish between
object shapes. The total number of states, 265, is determined in part by the
number of object cells the hBots can distinguish between, the way the hBots
move from one state-level to the next and the possible states the hBots can
reach whilst neighbouring each other. A hBot does not distinguish between the
position of its neighbours; when hBots determine their next state they do so by
sending its own state and the states of its neighbours to the state-rule table with
the following format: [own][lowest neighbour][highest neighbour]. This means
that the symmetrical features of the object shape, as described by sub-chains
from the data-chain, will appear identical. For example: where two sets of three
hBots are neighbouring similar parts of object shapes that are mirror images
of each other, Fig. 2. The hBot A neighbours an object shape with the data-
chain {1,1,1,2,2,1,2,1,1,1,3,2} and the hBot B neighbours an object shape with
the data-chain {1,1,1,2,1,2,2,1,1,1,2,3}. The three hBots including hBot A are
on the sub-chain {{1,3,2}} and the three hBots including hBot B are on the
sub-chain {{2,3,1}}. Although these sub-chains are different the resulting state
of hBot A and hBot B at state-level 2 will both be the result of referencing the
state-rule table with values [3][1][2], which returns state 17.
Fig. 2. hBot A and hBot B will both change to the same state even though their
neighbours are the opposite way round. Green is state 1, Blue is state 2, Red is state
3.The resulting states for both hBots A and B is state 17.
For each time-step hBots move with equal probability to any of their neigh-
bouring cells, unless the cell is an object cell or a border cell. If an hBot attempts
to move into a cell containing another hBot it instead remains stationary. When
next to an object cell the hBot generally stays still. However, the hBot has a
probability, determined by its state, of moving away from that cell: 0.01 if it is a
common state to both object shapes or it is only achievable for the valid object
shape and 0.1 if it is only achievable for the invalid object shape. This increases
interaction around the object shapes whilst reducing the chance of stagnation.
If the hBots simply remained stationary, they could be divided amongst multi-
ple object shapes or parts of object shapes without enough neighboring agents
interacting to distinguish between those object shapes.
3 Determining the Difference Values of Object Shape
Pairs
Given a data-chain of an existing object shape it is possible to determine which
states at which state-levels are achievable for a group of hBots by examining
sub-chains, represented by a double curly bracket, of lengths one, three and five.
Definition 1. At sub-chain length one the single number in the sub-chain is
equal to that of the state of the hBot in contact with the object shape at that
same location.
– {{A}} becomes state [A].
At sub-chain length three a conversion is required complicated. The sub-chain
must first be re-ordered and then the relative state of the hBot must be found.
– {{A,B,C}} becomes [B][lowest of A and C][highest of A and C] which returns
the new current state for the hBot at position B.
At sub-chain length five two stages of conversion are required. First three, three
length sub-chains must be found and then the relationship between these three
needs to be found to give the final state.
– {{A,B,C,D,E}} is broken into three {{A,B,C}} , {{B,C,D}} and {{C,D,E}}.
– {{A,B,C}} becomes [B][lowest of A or C][highest of A or C] giving state P.
– {{B,C,D}} becomes [C][lowest of B or D][highest of B or D] giving state Q.
– {{C,D,E}} becomes [D][lowest of C or E][highest of C or E] giving state R.
Now there is a sub-chain {{A,P,Q,R,E}} and Q can be resolved.
– {{P,Q,R}} becomes [Q][lowest of P or R][highest of P or R] which gives the
final state for the hBot at position C.
This method is used as it considers the way in which the hBots interact
with both each other and the object shapes’ boundaries. Sub-chains of lengths,
one, three and five, relate to the information that one, three and five hBots could
gather. Considering each sub-chain in the data-chain, is equivalent to considering
each position that a group of hBots could interact with the boundary of the
object shape.
For example the data-chain of object shape ID0, {1,1,2,1,2,1,1,2,1,2}, is ex-
amined to find the hBots’ achievable states at state-level 1, 2 and 3 when inter-
acting with the object shape, Table 1. These achievable states are compared to
those of object shape ID1, {1,1,1,2,2,1,1,2,1,1,3}, in Table 2 where the difference
value is found by dividing the number of achievable states at each state-level
that are not present in the other object shape by the length of the data-chain.
The achievable states which are not possible in the alternate object shape are
highlighted bold. This represents the number of positions that a hBot could de-
termine that it is next to one object shape and not the other and the number of
cooperating hBots required to do so.
Table 1. Determining the Achievable States for Object Shape ID0
Data-Chain State-Level 1 State-Level 2 State-Level 3
1 1 [1][1][2] = 5 [5][5][10] = 36
1 1 [1][1][2] = 5 [5][5][10] = 36
2 2 [2][1][1] = 10 [10][5][7] = 104
1 1 [1][2][2] = 7 [7][10][10] = 70
2 2 [2][1][1] = 10 [10][5][7] = 104
1 1 [1][1][2] = 5 [5][5][10] = 36
1 1 [1][1][2] = 5 [5][5][10] = 36
2 2 [2][1][1] = 10 [10][5][7] = 104
1 1 [1][2][2] = 7 [7][10][10] = 70
2 2 [2][1][1] = 10 [10][5][7] = 104
Table 2. Determining the Difference Values of Object Shape ID0 and Object Shape
ID1
State-Level 1: State-Level 2: State-Level 3:
Achievable States Achievable States Achievable States
ID0 ID1 ID0 ID1 ID0 ID1
1 1 5 6 36 52
1 1 5 4 36 26
2 1 10 5 104 32
1 2 7 11 70 112
2 2 10 11 104 112
1 1 5 5 36 37
1 1 5 5 36 36
2 2 10 10 104 103
1 1 7 5 70 40
2 1 10 6 104 57
3 16 184
Difference Values (relative to alternate object shape)
0/10 1/11 2/10 5/11 6/10 10/11
3.1 Results
The difference value for each combination of object shape pair was determined,
excluding object shape ID6. This object shape was not included as it was the only
object shape to include a 4 in its data-chain. The results are shown for achievable
states at state-level 1, 2 and 3 in Fig. 3, left, centre and right, respectively. From
these heat maps it is shown that the higher the state-level the more different the
object shapes appear. It also shows that object shape A’s difference value from
object shape B is not the same as B’s from A. This lack of symmetry is due to the
indiviual features of the boundary of the object shape that sub-chains describe.
If a feature is common to two object shapes that are compared then it cannot be
used to distinguish them. If a feature is unique to one object shape that feature
can be used to distinguish it from the other but not vice-versa. Therefore, it is
possible to have two object shapes where one is easier to distinguish from the
other due to the unique features it has. Finally the heat maps show that object
shape pairs: ID1 and ID2, ID5 and ID7, ID4 and ID8 are indistinguishable, this
is due to them being mirror images of each other.
Fig. 3. Heat-maps of difference values of row object shape relative to column object
shape calculated for state-level 1 (left), state-level 2 (centre), and state-level 3 (right).
4 Cooperative Object Recognition Task
The arena in which the tests were performed was hexagonal with each of the
six sides measuring 21 cells. Within the arena six of each of two types of object
shapes were placed as shown in Fig. 4. One of these object shapes was classed
valid for removal and the other object shape was classed as invalid. Thirty-seven
hBots were placed at the centre of the arena in a tight cluster at time-step zero.
Whenever a hBot reached a state that was achievable for only the valid object
shape, the object shape it was neighbouring would be removed instantly, as if
dissolved. The number of time-steps it took the swarm to remove all of the six
valid object shapes was recorded.
All possible pairs of object shapes with four object cells were tested exclud-
ing object shapes that mirrored each other and object shape ID6. These object
shapes were not included because mirrored object shapes are perceived as iden-
tical in this system and object shape ID6 is the only object shape with a number
4 in its data-chain. Currently the system is only capable of distinguishing object
shapes with numbers 1, 2 and 3 in their data-chains. Each experiment was run
fifty times with both object shapes classed as valid and invalid in turn. Algorithm
1 describes the simulation, hBot and object shape interaction used.
Fig. 4. The simulated arena with six valid object shapes (ID5), six invalid object shapes
(ID9) and thirty-seven hBots in the centre.
4.1 Comparison of Swarm Results with the Difference Values
The average number of time-steps it took to complete each of the 30 coopera-
tive object recognition tasks was compared to the mean of the three difference
values found for the same two objects, as shown in Fig. 5. These results have a
correlation coefficient of -0.78.
By considering only the difference values at each state-level individually an
indication of how the number of cooperating hBots affect their ability to dis-
tinguish the object shapes. At state-level 1 there was a correlation coefficient
of -0.48. However, this is less than the difference values at state-level 2 and
state-level 3 which have correlation coefficients of -0.73 and -0.84 respectively.
This suggests that the hBots that reach only state-level 1 have a lower effect on
whether or not the object shapes will be distinguished from one another.
5 Discussion
A method was proposed for determining the difference value between two object
shapes. This was done by comparing sub-chains at lengths one, three and five of
the data-chain that represents the object shape. Unlike chain-coding a data-chain
Algorithm 1 The Simulation, hBot and object interaction
repeat
for each hBot i do
if hBot[i] state == a unique valid shape state then
object neighboring hBot[i] is deleted
hBot[i] state = 0
else if hBot[i] state == a unique invalid state then
10% chance of hBot[i] state = 0
else then
1% chance of hBot[i] state = 0
end for
Update contents of all cells
for each hBot i do
hBot[i] senses surroundings
if hBot[i] state == 0 then
hBot[i] moves to random neighboring cell if empty
end for
for each hBot i do
hBot[i] senses surroundings
end for
for each hBot i do
hBot[i] updates current state
end for
Update contents of all cells
Cells displayed
Time-steps++
until all six valid object shapes deleted
Fig. 5. The average difference value of a pair of object shapes compared to the number
of time-steps it took a swarm of hBots to remove the six valid object shapes.
does not have information on its location or orientation which makes it possible
to compare the shapes to each other without considering these two factors. To
measure the accuracy of the found difference values a series of experiments were
run with a range of four cell object shapes. These experiments determined how
long it would take a swarm of hBots to remove six valid object shapes whilst
ignoring six invalid object shapes.
The results showed a significant correlation between the difference values
representative of each of the state-levels with the amount of time-steps it took
the hBots to complete the cooperative object recognition task. As expected,
where only the states achievable by hBots at state-level 1 are considered in the
difference value measurement there was a lower strength correlation. This is
likely due to the hBots being generally unable to distinguish between the object
shapes on their own, with the exception of object shapes ID0 and ID9 which are
the only object shapes without a 3 in their data-chains. Therefore, any other
object shapes can be distinguished from them by a state-level 1 hBot in state 3.
This shows that the difference values give a clear indication of how difficult
it is for a swarm of hBots to complete a cooperative object recognition task
based on the difference between the boundaries of the object shapes. Using this
information it will be possible to derive future task scenarios with a suitable
range of difficulties for comparing different scenarios and techniques for cooper-
ative object recognition. An area of interest is training the hBots to determine
the correct behaviours to distinguish between the object shapes with limited
feedback.
Due to the methods chosen for completing this investigation there were some
limitations to the simulation platform and limitations in the way in which the
agents were modelled, which should be addressed in future research:
– The use of a synchronized grid-world reduced the similarity with physical
systems.
– Using a less common hexagonal lattice, comparisons with existing multi-
agent systems are difficult to make.
– All the agents were modelled with perfect sensors and communication, it
would be interesting to see the effect of miscommunication on the systems
capability to identify objects correctly.
– The system only deals with object shapes that have the values 1, 2 and 3 in
their data-chain, ignoring possible shapes that have the values 4, 5 and 6, to
reduce the state relationships possible.
– The hBots cannot distinguish between hollow shapes and solid shapes, as
they only currently interact with the outer boundary of the object shape
and have no means of communicating to other hBots inside the hollow if
they were there.
– Object shapes that are mirror images of each other appear identical due to
the way the hBots sense their surroundings.
– Currently the number of state-levels would only allow a single hBot to have
the equivalent knowledge of five state-level 1 hBots. This means that certain
shapes currently would not be possible to distinguish from each other.
Overall the difference value metric will be a valuable tool for future studies in
co-operative object recognition research with hexagonal grid-world simulations.
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