Characterizing Universal Reconfigurability of Modular Pivoting Robots

Authors Hugo A. Akitaya , Erik D. Demaine , Andrei Gonczi , Dylan H. Hendrickson, Adam Hesterberg, Matias Korman, Oliver Korten, Jayson Lynch , Irene Parada , Vera Sacristán

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Author Details

Hugo A. Akitaya
  • University of Massachusetts Lowell, MA, USA
Erik D. Demaine
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Andrei Gonczi
  • Tufts University, Medford, MA, USA
Dylan H. Hendrickson
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Adam Hesterberg
  • Harvard University, Cambridge, MA, USA
Matias Korman
  • Siemens Electronic Design Automation, Portland, OR, USA
Oliver Korten
  • Columbia University, New York, NY, USA
Jayson Lynch
  • University of Waterloo, ON, Canada
Irene Parada
  • TU Eindhoven, The Netherlands
Vera Sacristán
  • Universitat Politècnica de Catalunya, Barcelona, Spain


This research started at the 34th Bellairs Winter Workshop on Computational Geometry in 2019. We want to thank all participants for the fruitful discussions and a stimulating environment. We would also like to thank a SoCG reviewer for their many contributions that helped improve the presentation of the paper.

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Hugo A. Akitaya, Erik D. Demaine, Andrei Gonczi, Dylan H. Hendrickson, Adam Hesterberg, Matias Korman, Oliver Korten, Jayson Lynch, Irene Parada, and Vera Sacristán. Characterizing Universal Reconfigurability of Modular Pivoting Robots. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We give both efficient algorithms and hardness results for reconfiguring between two connected configurations of modules in the hexagonal grid. The reconfiguration moves that we consider are "pivots", where a hexagonal module rotates around a vertex shared with another module. Following prior work on modular robots, we define two natural sets of hexagon pivoting moves of increasing power: restricted and monkey moves. When we allow both moves, we present the first universal reconfiguration algorithm, which transforms between any two connected configurations using O(n³) monkey moves. This result strongly contrasts the analogous problem for squares, where there are rigid examples that do not have a single pivoting move preserving connectivity. On the other hand, if we only allow restricted moves, we prove that the reconfiguration problem becomes PSPACE-complete. Moreover, we show that, in contrast to hexagons, the reconfiguration problem for pivoting squares is PSPACE-complete regardless of the set of pivoting moves allowed. In the process, we strengthen the reduction framework of Demaine et al. [FUN'18] that we consider of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • reconfiguration
  • geometric algorithm
  • PSPACE-hardness
  • pivoting hexagons
  • pivoting squares
  • modular robots


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