# Long-term pattern formation and maintenance for battery-powered robots

## Abstract

This paper presents a distributed, energy-aware method for the autonomous deployment and maintenance of battery-powered robots within a known or unknown region in 2D space. Our approach does not rely on a global positioning system and therefore allows for applications in GPS-denied environments such as underwater sensing or underground monitoring. After covering a region, our system maintains a formation and uses an arbitrary number of charging stations to prevent robots from fully discharging. Analyzing the topology of the network formed during robot deployment, we generate virtual recharging trees which the robots use to navigate toward a nearby charging station when needed. All robots that leave the formation are replaced by their neighbors, maximizing the effective coverage provided by the system. We demonstrate the capability of our methods using models, a physics-based simulator, and experiments with real robots.

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## Notes

1. 1.

The original problem of partitioning a graph into a forest is defined as: Given a graph $$G=\{V,E\}$$ and a positive integer $$K \le |V|$$, can the vertices of G be partitioned into k disjoint sets $$V_1, V_2, \ldots , V_k$$, such that for $$1 \le i \le k$$, the subgraph induced by $$V_i$$ contains no circuits.

2. 2.

$$\cup ^z$$ is a zip-union (inspired by Python zip()) operation, which results in a list having members intertwined as if connected with a zipper, for example let $$A = {1, 2, 3}$$ and $$B = {a, b, c}$$, then $$A \cup ^z B = {1, a, 2, b, 3, c}$$.

3. 3.

The degree of a vertex in a graph indicates the number of edges connecting it to other vertices.

4. 4.

To simplify the notation, we define $$p_n=m$$, i.e. the desired number of trees.

5. 5.

A graph without cycles where each vertex has the degree of 2 except the first and last one (Cormen et al. 2011).

6. 6.

The loop (line 5 of Algorithm 4) does not need to go through all the elements in $$R_C$$ on each pass. Since the nodes with the highest degree and centrality are at the top of the list, by default Algorithm 4 takes the first one and declares it a root. This member is removed from $$R_C$$, leaving the next (line 15) with the highest degree and centrality. Given the considered (triangular) graph shapes, it is very rare for the first member of $$R_C$$ to not become a root.

7. 7.

An environment-mediated communication modality is found in insects and some unicellular organisms.

8. 8.
9. 9.

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## Acknowledgements

We thank our former laboratory member Carlo Pinciroli, who greatly influenced this work, and interns Christophe Bedard and Nghia Do for their assistance. This research was supported by the NSERC Strategic Partnership Grant 479149-2015.

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Correspondence to Guannan Li or Giovanni Beltrame.

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## Appendix A: Supplementary materials

### Appendix A: Supplementary materials

Supplementary multimedia content accompanying this paper can be found at: http://mistlab.ca/papers/LongTermPattern/2017/

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Li, G., Svogor, I. & Beltrame, G. Long-term pattern formation and maintenance for battery-powered robots. Swarm Intell 13, 21–57 (2019). https://doi.org/10.1007/s11721-019-00162-1

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### Keywords

• Long-term autonomy
• Energy-aware
• Multi-robot teams
• Graph
• Battery