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We provide extra information for the experiments and results in the paper.
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<!-- \section{Appendix}\label{Appendix}
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In this section, we provide extra information required for replicating the results in Section~\ref{sec:Exp} and the simulation codes are available on our repository\footnote{\href{https://git.mistlab.ca/skarthik/local-operations-on-trees}{https://git.mistlab.ca/skarthik/local-operations-on-trees}}. -->
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***
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## Decentralised rules for robots swarms to form Line and Star
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## Decentralised rules for robots to form Line and star
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***
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### 1. Line formation
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* Robots with exactly two neighbors try to straighten the angle between their neighbors by moving towards a direction so as to reduce the obtuse angle.
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The parameters used for line formation experiment are $R_{transfer} = 1m$, *R_{mission} = 1.5m*, $R_{range} = 2.5m$.
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The parameters used for line formation experiment are $R_{transfer} = 1m$, $R_{mission} = 1.5m$, $R_{range} = 2.5m$.
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***
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@@ -40,11 +40,17 @@ We consider a single root star formation. In star topology, all the robots excep
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This table shows the parameters used in the star formation experiments.
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No of robots | $R_{range}$(m) | $R_{transfer}$(m) | $R_{Mission}$ |
In order to verify transformation of trees, we implemented a script in python which verifies the conversion of all possible topologies for a swarm of size 10 to a specific random topology. We also verify that from a random topology it is possible to convert to all other possible topologies.
We also have verified this algorithm in a python script where starting from any random topology we were able to form a star/line for a swarm of size 10 for all the possible $10^8$ combinations.
% Figures~\ref{fig:line_formation} and \ref{fig:star_formation} depict
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% $\lambda_2$, coverage area, and progress of operations. We plot the time
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% evolution of $\lambda_2$ of the graph and the maintained spanning tree, as a connectivity index and a parameter specifying consensus rate.
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% The evolution of the number of nodes with $\text{degree(robot)}=1$ (i.e., having only one
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% neighbor) and $\text{degree(robot)}=2$ is sketched for the line formation and
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% the evolution of the number of nodes with $\text{degree(robot)}=1$ and nodes
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% with $\text{degree(robot)}\geq2$ for the star formation, which is a progress
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% index in each case. The $\lambda_2$ of the manipulated spanning tree and the
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% whole graph examine the connectivity awareness of our method which has to stay
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% greater than zero over the experiment. In the specific case of line, $\lambda_2$
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% of the tree reduces with time and reaches a constant value when the line has
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% been straightened out. Also, $\lambda_2$ of the graph will approach the same
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% value, if $R_{\text{mission}}$ is close to $R_{\text{range}}$, which is the minimum for a
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% given connected graph of $N$ nodes. However, In the case of the star topology,
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% $\lambda_2$ of the tree increases to a constant value of one at the end of the
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% experiment no matter the number of nodes in the system, and the $\lambda_2$ of
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% the graph increases. If $R_{\text{mission}}<R_{\text{range}}/2$ it would have approached an
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% all to all graph which has the maximum $\lambda_2$ for a given connected graph
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% of $N$ nodes. The coverage area has been shown to decrease for the star and to
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% increase for the line case which is showing the trade-off between $\lambda_2$
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% and the coverage area and that is why topology manipulation is needed to provide
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% flexibility. Furthermore, for the star formation, we have plotted the number of
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% nodes with $\text{degree(robot)}=1$ and the number of nodes with
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% $\text{degree(robot)}\geq2$. This is to show that the manipulation operations
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% are changing the topology closer to the star topology with time. The number of
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% nodes with $\text{degree(robot)}=1$ for the star case increases to $N-1$ and
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% there is exactly one node that has $N-1$ neighbors, which is the root. The plots
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% show the evolution of these metrics which is increasing for the number of nodes
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% with $\text{degree(robot)}=1$ and decreasing for the number of nodes with
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% $\text{degree(robot)}\geq2$. For line formation, we have shown a similar metric
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% that reduces to 2 for the number of nodes $\text{degree(robot)}=1$ and increases
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% to $N-2$ for the number of nodes $\text{degree(robot)}=2$ which is the
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% definition of a line topology.
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Figures 3 and 4 depict $\lambda_2$, coverage area, and progress of operations. We plot the time
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evolution of $\lambda_2$ of the graph and the maintained spanning tree, as a connectivity index and a parameter specifying consensus rate.
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The evolution of the number of nodes with $\text{degree(robot)}=1$ (i.e., having only one
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neighbor) and $\text{degree(robot)}=2$ is sketched for the line formation and
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the evolution of the number of nodes with $\text{degree(robot)}=1$ and nodes
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with $\text{degree(robot)}\geq2$ for the star formation, which is a progress
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index in each case. The $\lambda_2$ of the manipulated spanning tree and the
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whole graph examine the connectivity awareness of our method which has to stay
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greater than zero over the experiment. In the specific case of line, $\lambda_2$
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of the tree reduces with time and reaches a constant value when the line has
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been straightened out. Also, $\lambda_2$ of the graph will approach the same
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value, if $R_{\text{mission}}$ is close to $R_{\text{range}}$, which is the minimum for a
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given connected graph of $N$ nodes. However, In the case of the star topology,
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$\lambda_2$ of the tree increases to a constant value of one at the end of the
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experiment no matter the number of nodes in the system, and the $\lambda_2$ of
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the graph increases. If $R_{\text{mission}}<R_{\text{range}}/2$ it would have approached an
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all to all graph which has the maximum $\lambda_2$ for a given connected graph
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of $N$ nodes. The coverage area has been shown to decrease for the star and to
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increase for the line case which is showing the trade-off between $\lambda_2$
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and the coverage area and that is why topology manipulation is needed to provide
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flexibility. Furthermore, for the star formation, we have plotted the number of
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nodes with $\text{degree(robot)}=1$ and the number of nodes with
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$\text{degree(robot)}\geq2$. This is to show that the manipulation operations
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are changing the topology closer to the star topology with time. The number of
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nodes with $\text{degree(robot)}=1$ for the star case increases to $N-1$ and
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there is exactly one node that has $N-1$ neighbors, which is the root. The plots
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show the evolution of these metrics which is increasing for the number of nodes
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with $\text{degree(robot)}=1$ and decreasing for the number of nodes with
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$\text{degree(robot)}\geq2$. For line formation, we have shown a similar metric
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that reduces to 2 for the number of nodes $\text{degree(robot)}=1$ and increases
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to $N-2$ for the number of nodes $\text{degree(robot)}=2$ which is the
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definition of a line topology.
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## Setting up the simlations for Line and Star in Argos3 and Buzz.
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The code for the experiments can be found [here](https://git.mistlab.ca/skarthik/local-operations-on-trees).
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ARGoS3 Simulator ARGoS3 simulator can also be installed from binaries please refer to the official website for more information: https://www.argos-sim.info/
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The instructions below are for installing ARGoS3 from its source.
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Official code repository: https://github.com/ilpincy/argos3
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