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.
Similar content being viewed by others
Notes
Without loss of generality, we assume that a RC module occupies a space of unit cube.
Noting that each CR movement takes 10 steps (Fig. 3b), 30m steps are required in the reshaping process.
The four steps shown in Fig. 6 actually take six rotation steps; i.e. each of steps 1 and 3 contains two orthogonal rotations.
References
Ahmadzadeh, H., & Masehian, E. (2015). Modular robotic systems. Artificial Intelligence, 223(C), 27–64.
Asadpour, M., Sproewitz, A., Billard, A., Dillenbourg, P., & Ijspeert, A. J. (2008). Graph signature for self-reconfiguration planning. In IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 863–869).
Chirikjian, G., Pamecha, A., & Ebert-Uphoff, I. (1996). Evaluating efficiency of self-reconfiguration in a class of modular robots. Journal of Field Robotics, 13(5), 317–338.
Cormen, T. H., Leiserson, C. E., & Rivest, R. L. (1990). Introduction to algorithms. Cambridge: The MIT Press.
Davey, J., Kwok, N., & Yim, M. (2012). Emulating self-reconfigurable robots-design of the SMORES system. In IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 4464–4469).
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: WH Freeman.
Gorbenko, A., & Popv, V. (2012). Programming for modular reconfigurable robots. Programming and Computer Software, 38(1), 13–23.
Hou, F., & Shen, W. M. (2010). On the complexity of optimal reconfiguration planning for modular reconfigurable robots. In Proceedings of the IEEE international conference on robotics and automation (ICRA) (pp. 2791–2796).
Hou, F., & Shen, W. M. (2014). Graph-based optimal reconfiguration planning for self-reconfigurable robots. Robotics and Autonomous Systems, 62(7), 1047–1059.
Jing, G., Tosun, T., Yim, M., & Kress-Gazit, H. (2017). An end-to-end system for accomplishing tasks with modular robots: Perspectives for the AI community. In Proceedings of the twenty-sixth international joint conference on artificial intelligence, IJCAI-17 (pp. 4879–4883).
Kurokawa, H., Kamimura, A., Yoshida, E., Tomita, K., Kokaji, S., & Murata, S. (2003). M-TRAN II: Metamorphosis from a four-legged walker to a caterpillar. In IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 2454–2459).
Kurokawa, H., Tomita, K., Kamimura, A., Kokaji, S., Hasuo, T., & Murata, S. (2008). Distributed self-reconfiguration of M-TRAN III modular robotic system. The International Journal of Robotics Research, 27(3–4), 373–386.
Liu, Y. J., Yu, M., Ye, Z., & Wang, C. C. (2018). Path planning for self-reconfigurable modular robots: A survey. Scientia Sinica Informationis, 48(2), 143–176.
Murata, S., Yoshida, E., Kamimura, A., Kurokawa, H., Tomita, K., & Kokaji, S. (2002). M-TRAN: Self-reconfigurable modular robotic system. IEEE/ASME Transactions on Mechatronics, 7(4), 431–441.
Pamecha, A., Ebert-Uphoff, I., & Chirikjian, G. S. (1997). Useful metrics for modular robot motion planning. IEEE Transactions on Robotics and Automation, 13(4), 531–545.
Romanishin, J. W., Gilpin, K., Claici, S., & Rus, D. (2015). 3D M-blocks: Self-reconfiguring robots capable of locomotion via pivoting in three dimensions. In Proceedings of the IEEE international conference on robotics and automation (ICRA) (pp. 1925–1932).
Romanishin, J. W., Gilpin, K., & Rus, D. (2013). M-blocks: Momentum-driven, magnetic modular robots. In IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 4288–4295).
Salemi, B., Moll, M., & Shen, W. M. (2006). SUPERBOT: A deployable, multi-functional, and modular self-reconfigurable robotic system. In IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 3636–3641).
Stoy, K., & Brandt, D. (2013). Efficient enumeration of modular robot configurations and shapes. In IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 4296–4301).
Stoy, K., Brandt, D., & Christensen, D. J. (2010). Self-reconfigurable robots: An introduction. Cambridge: MIT Press.
Sung, C., Bern, J., Romanishin, J., & Rus, D. (2015). Reconfiguration planning for pivoting cube modular robots. In Proceedings of the IEEE international conference on robotics and automation (ICRA) (pp. 1933–1940).
Yu, M., Liu, Y. J., & Wang, C. C. (2017). EasySRRobot: An easy-to-build self-reconfigurable robot with optimized design. In IEEE international conference on robotics and biomimetics (IEEE-ROBIO) (pp. 1094–1099).
Yu, M., Liu, Y. J., Zhang, Y., Zhao, G., Yu, C., & Shi, Y. (2019). Interactions with reconfigurable modular robots enhance spatial reasoning performance. IEEE Transactions on Cognitive and Developmental Systems. https://doi.org/10.1109/TCDS.2019.2914162.
Yu, M., Ye, Z., Liu, Y. J., He, Y., & Wang, C. C. L. (2019). Lineup: Computing chain-based physical transformation. ACM Transactions on Graphics, 38(1), 11:1–11:16.
Acknowledgements
The authors thank reviewers for their constructive comments that help improve the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the Natural Science Foundation of China (61725204, 61521002) and Royal Society-Newton Advanced Fellowship.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10514-019-09878-9