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NP-completeness of optimal planning problem for modular robots

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Abstract

Self-reconfigurable modular robots (SRM-robots) can autonomously change their shape according to different tasks and work environments, and have received considerable attention recently. Many reshaping/reconfiguration algorithms have been proposed. In this paper, we present a theoretical analysis of computational complexity on a reshape planning for a kind of lattice-type 3D SRM-robots, whose modules are of cubic shape and can move by rotating on the surfaces of other modules. Different from previous NP-completeness study on general chain-type robots (i.e. the motion of any chains and the location of modules can be arbitrary), we consider more practical constraints on modules’ shape (i.e. cubic shape), position (lying in 2D/3D grids) and motion (using orthogonal rotations) in this paper. We formulate the reshape planning problem of SRM-robots with these practical constraints by a (p, q) optimization problem, where p and q characterize two widely used metrics, i.e. the number of disconnecting/reconnecting operations and the number of reshaping steps. Proofs are presented, showing that this optimization problem is NP-complete. Therefore, instead of finding global optimization results, most likely approximation solution can be obtained for the problem instead of seeking polynomial algorithm. We also present the upper and lower bounds for the 2-tuple (p, q), which is useful for evaluating the approximation algorithms in future research.

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Notes

  1. Without loss of generality, we assume that a RC module occupies a space of unit cube.

  2. Noting that each CR movement takes 10 steps (Fig. 3b), 30m steps are required in the reshaping process.

  3. The four steps shown in Fig. 6 actually take six rotation steps; i.e. each of steps 1 and 3 contains two orthogonal rotations.

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Acknowledgements

The authors thank reviewers for their constructive comments that help improve the quality of this paper.

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  1. Zipeng Ye and Minjing Yu are joint first authors.

Authors and Affiliations

  1. Beijing National Research Center for Information Science and Technology, Department of Computer Science and Technology, Tsinghua University, Beijing, China

    Zipeng Ye & Yong-Jin Liu

  2. College of Intelligence and Computing, Tianjin University, Tianjin, China

    Minjing Yu

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  1. Zipeng Ye
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  2. Minjing Yu
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  3. Yong-Jin Liu
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Correspondence to Yong-Jin Liu.

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This work was supported by the Natural Science Foundation of China (61725204, 61521002) and Royal Society-Newton Advanced Fellowship.

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Ye, Z., Yu, M. & Liu, YJ. NP-completeness of optimal planning problem for modular robots. Auton Robot 43, 2261–2270 (2019). https://doi.org/10.1007/s10514-019-09878-9

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