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Distributed reconfiguration of 2D lattice-based modular robotic systems

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Abstract

We prove universal reconfiguration (i.e., reconfiguration between any two robotic systems with the same number of modules) of 2-dimensional lattice-based modular robots by means of a distributed algorithm. To the best of our knowledge, this is the first known reconfiguration algorithm that applies in a general setting to a wide variety of particular modular robotic systems, and holds for both square and hexagonal lattice-based 2-dimensional systems. All modules apply the same set of local rules (in a manner similar to cellular automata), and move relative to each other akin to the sliding-cube model. Reconfiguration is carried out while keeping the robot connected at all times. If executed in a synchronous way, any reconfiguration of a robotic system of \(n\) modules is done in \(O(n)\) time steps with \(O(n)\) basic moves per module, using \(O(1)\) force per module, \(O(1)\) size memory and computation per module (except for one module, which needs \(O(n)\) size memory to store the information of the goal shape), and \(O(n)\) communication per module.

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Acknowledgments

The authors wish to explicitly thank the students Reinhard Wallner, Óscar Rodríguez, Sergio Ordóñez and Ángel Rodríguez for their precise work in implementing simulators and simulations. We also wish to thank an anonymous referee for detailed comments that helped to improve the readability of the paper. Suneeta Ramaswami was partially supported by NSF grant CCF-0830589. Ferran Hurtado and Vera Sacristán were partially supported by Projects MTM2012-30951, Gen. Cat. DGR 2009SGR1040, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    Ferran Hurtado, Enrique Molina & Vera Sacristán

  2. Department of Computer Science, Rutgers University, Camden, NJ, USA

    Suneeta Ramaswami

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  1. Ferran Hurtado
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  2. Enrique Molina
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Correspondence to Vera Sacristán.

Appendix

Appendix

1.1 Moves for the 8 sliding square units meta-module

Figure 39 shows how a meta-module made of 8 sliding square units can perform the slide move. Figure 40 shows how the same meta-module can perform the convex transition move without any extra space requirement.

Fig. 39
figure 39

A meta-module of 8 sliding square units performs slide. Only the first half steps are shown, the rest of the move is completed by symmetry (Color figure online)

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Fig. 40
figure 40

A meta-module of 8 sliding square units performs convex transition without extra empty space requirements. Only the first half steps are shown, the rest of the move is completed by symmetry (Color figure online)

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1.2 Moves for the 5 sliding square units meta-module

Figure 41 shows how a meta-module made of 8 sliding square units can perform the slide move. Figure 42 shows how the same meta-module can perform the convex transition move without any extra empty space requirement.

Fig. 41
figure 41

A meta-module of 5 sliding square units performs slide. Only the first half steps are shown, the rest of the move is completed by symmetry (Color figure online)

Full size image
Fig. 42
figure 42

A meta-module of 5 sliding square units performs convex transition without without extra empty space requirements. Only the first half steps are shown, the rest of the move is completed by symmetry (Color figure online)

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1.3 Moves for the rotating square units meta-module

Figure 43 shows how a meta-module made of 8 rotating square units can perform the slide move without any extra empty space requirement. Figure 44 shows how the same meta-module can perform the convex transition move without any extra empty space requirement.

Fig. 43
figure 43

A meta-module of 8 rotating square units performs slide without extra empty space requirements. Only the first half steps are shown, the rest of the move is completed by symmetry (Color figure online)

Full size image
Fig. 44
figure 44

A meta-module of 8 rotating square units performs convex transition without extra empty space requirements. Only the first half steps are shown, the rest of the move is completed by symmetry (Color figure online)

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1.4 Moving meta-modules of hexagonal units

Figure 45 shows how a meta-module made of 6 hexagonal units can change position without any extra empty space requirement, when the units have light extra empty space requirements. Figure 46 shows how a meta-module made of 18 hexagonal units can change position without any extra empty space requirement, when the units have strong extra empty space requirements.

Fig. 45
figure 45

When the hexagonal units have light extra empty space requirements, meta-modules of six units are capable to change position without extra empty space requirements. Only the first half of the atomic moves are shown, the remaining are obtained by symmetry (Color figure online)

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Fig. 46
figure 46

When the hexagonal units have strong extra empty space requirements, meta-modules of eighteen units are capable to change position without extra empty space requirements. Only the first half of the atomic moves are shown, the remaining are obtained by symmetry (Color figure online)

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Hurtado, F., Molina, E., Ramaswami, S. et al. Distributed reconfiguration of 2D lattice-based modular robotic systems. Auton Robot 38, 383–413 (2015). https://doi.org/10.1007/s10514-015-9421-8

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