Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any t...

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Published in:	Algorithmica Vol. 83; no. 5; pp. 1316 - 1351
Main Authors:	Akitaya, Hugo A., Arkin, Esther M., Damian, Mirela, Demaine, Erik D., Dujmović, Vida, Flatland, Robin, Korman, Matias, Palop, Belen, Parada, Irene, Renssen, André van, Sacristán, Vera
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.05.2021 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Configurations Data Structures and Information Theory Mathematics of Computing Modules Reconfiguration Robots Theory of Computation Pivoting squares Geometric algorithms Modular robots Reconfiguration
ISSN:	0178-4617, 1432-0541
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Abstract	We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses O ( n 2 ) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.
AbstractList	We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses O ( n 2 ) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models. We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses O(n2) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.
Author	Akitaya, Hugo A. Parada, Irene Sacristán, Vera Korman, Matias Arkin, Esther M. Damian, Mirela Flatland, Robin Palop, Belen Renssen, André van Demaine, Erik D. Dujmović, Vida
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References	HemmerlingALabyrinth Problems: Labyrinth-Searching Abilities of Automata, Teubner-Texte zur Mathematik (TTZM)1989BerlinSpringer10.1007/978-3-322-94560-0 ØstergaardEHKassowKBeckRLundHHDesign of the ATRON lattice-based self-reconfigurable robotAuton. Robots200621216518310.1007/s10514-006-8546-1 Nguyen, A., Guibas, L.J., Yim, M.: Controlled module density helps reconfiguration planning. In: Proceedings of 4th International Workshop on Algorithmic Foundations of Robotics (WAFR), pp. 23–36 (2000) StoyKBrandtDChristensenDJSelf-Reconfigurable Robots: An Introduction2010CambridgeMIT Press MurataSYoshidaEKamimuraAKurokawaHTomitaKKokajiSM-TRAN: self-reconfigurable modular robotic systemIEEE/ASME Trans. Mechatron.20027443144110.1109/TMECH.2002.806220 Sung, C., Bern, J., Romanishin, J., Rus, D.: Reconfiguration planning for pivoting cube modular robots. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 1933–1940 (2015) ChennareddySAgrawalAKaruppiahAModular self-reconfigurable robotic systems: a survey on hardware architecturesJ. Robot.201720175013532 An, B.K.: EM-Cube: cube-shaped, self-reconfigurable robots sliding on structure surfaces. In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), pp. 3149–3155 (2008) Ayanian, N., White, P.J., Hálász, Á., Yim, M., Kumar, V.: Stochastic control for self-assembly of XBots. In: Proceedings of ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC-CIE) (2008) Benbernou, N.M.: Geometric algorithms for reconfigurable structures. Ph.D. Thesis, Massachusetts Institute of Technology (2011) YimMShenWSalemiBRusDMollMLipsonHKlavinsEChirikjianGSModular self-reconfigurable robot systemsIEEE Robot. Autom. Mag.2007141435210.1109/MRA.2007.339623 Fitch, R., Butler, Z., Rus, D.: Reconfiguration planning for heterogeneous self-reconfiguring robots. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), vol. 3, pp. 2460–2467 (2003) Unsal, C., Kiliccote, H., Khosla, P.: I(CES)-Cubes: a modular self-reconfigurable bipartite robotic system. In: Proceedings of SPIE Conference on Mobile Robots and Autonomous Systems, vol. 3839, pp. 258–269 (1999) Kurokawa, H., Murata, S., Yoshida, E., Tomita, K., Kokaji, S.: A 3-D self-reconfigurable structure and experiments. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), vol. 2, pp. 860–865 (1998) LarkworthyTRamamoorthySA characterization of the reconfiguration space of self-reconfiguring robotic systemsRobotica2011291738510.1017/S0263574710000718 DumitrescuAPachJPushing squares aroundGraphs Comb.20062213750222100710.1007/s00373-005-0640-1 Murata, S., Kurokawa, H., Kokaji, S.: Self-assembling machine. In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), vol. 1, pp. 441–448 (1994) Abel, Z., Kominers, S.D.: Pushing hypercubes around. CoRR abs/0802.3414 (2008) Chirikjian, G.S.: Kinematics of a metamorphic robotic system. In: Proceedings of IEEE international conference on robotics and automation (ICRA), vol. 1, pp. 449–455 (1994) Rus, D., Vona, M.: A physical implementation of the self-reconfiguring crystalline robot. In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), vol. 2, pp. 1726–1733 (2000) DumitrescuASuzukiIYamashitaMMotion planning for metamorphic systems: feasibility, decidability, and distributed reconfigurationIEEE Trans. Robot.200420340941810.1109/TRA.2004.824936 Salemi, B., Moll, M., Shen, W.M.: SUPERBOT: a deployable, multi-functional, and modular self-reconfigurable robotic system. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3636–3641 (2006) MichailOSkretasGSpirakisPGOn the transformation capability of feasible mechanisms for programmable matterJ. Comput. Syst. Sci.20191021839392606110.1016/j.jcss.2018.12.001 Zykov, V., Chan, A., Lipson, H.: Molecubes: an open-source modular robotic kit. In: IROS-2007 Self-Reconfigurable Robotics Workshop (2007) T Larkworthy (784_CR12) 2011; 29 EH Østergaard (784_CR17) 2006; 21 784_CR24 O Michail (784_CR13) 2019; 102 784_CR14 784_CR21 784_CR11 784_CR22 S Chennareddy (784_CR5) 2017; 2017 784_CR9 K Stoy (784_CR20) 2010 M Yim (784_CR23) 2007; 14 784_CR6 784_CR16 784_CR4 S Murata (784_CR15) 2002; 7 784_CR18 784_CR3 784_CR19 784_CR2 784_CR1 A Dumitrescu (784_CR8) 2004; 20 A Hemmerling (784_CR10) 1989 A Dumitrescu (784_CR7) 2006; 22
References_xml	– reference: Murata, S., Kurokawa, H., Kokaji, S.: Self-assembling machine. In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), vol. 1, pp. 441–448 (1994) – reference: MurataSYoshidaEKamimuraAKurokawaHTomitaKKokajiSM-TRAN: self-reconfigurable modular robotic systemIEEE/ASME Trans. Mechatron.20027443144110.1109/TMECH.2002.806220 – reference: DumitrescuASuzukiIYamashitaMMotion planning for metamorphic systems: feasibility, decidability, and distributed reconfigurationIEEE Trans. Robot.200420340941810.1109/TRA.2004.824936 – reference: Salemi, B., Moll, M., Shen, W.M.: SUPERBOT: a deployable, multi-functional, and modular self-reconfigurable robotic system. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3636–3641 (2006) – reference: Benbernou, N.M.: Geometric algorithms for reconfigurable structures. Ph.D. Thesis, Massachusetts Institute of Technology (2011) – reference: Zykov, V., Chan, A., Lipson, H.: Molecubes: an open-source modular robotic kit. In: IROS-2007 Self-Reconfigurable Robotics Workshop (2007) – reference: MichailOSkretasGSpirakisPGOn the transformation capability of feasible mechanisms for programmable matterJ. Comput. Syst. Sci.20191021839392606110.1016/j.jcss.2018.12.001 – reference: HemmerlingALabyrinth Problems: Labyrinth-Searching Abilities of Automata, Teubner-Texte zur Mathematik (TTZM)1989BerlinSpringer10.1007/978-3-322-94560-0 – reference: Sung, C., Bern, J., Romanishin, J., Rus, D.: Reconfiguration planning for pivoting cube modular robots. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 1933–1940 (2015) – reference: Abel, Z., Kominers, S.D.: Pushing hypercubes around. CoRR abs/0802.3414 (2008) – reference: Kurokawa, H., Murata, S., Yoshida, E., Tomita, K., Kokaji, S.: A 3-D self-reconfigurable structure and experiments. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), vol. 2, pp. 860–865 (1998) – reference: Rus, D., Vona, M.: A physical implementation of the self-reconfiguring crystalline robot. In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), vol. 2, pp. 1726–1733 (2000) – reference: Ayanian, N., White, P.J., Hálász, Á., Yim, M., Kumar, V.: Stochastic control for self-assembly of XBots. In: Proceedings of ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC-CIE) (2008) – reference: DumitrescuAPachJPushing squares aroundGraphs Comb.20062213750222100710.1007/s00373-005-0640-1 – reference: LarkworthyTRamamoorthySA characterization of the reconfiguration space of self-reconfiguring robotic systemsRobotica2011291738510.1017/S0263574710000718 – reference: ØstergaardEHKassowKBeckRLundHHDesign of the ATRON lattice-based self-reconfigurable robotAuton. Robots200621216518310.1007/s10514-006-8546-1 – reference: An, B.K.: EM-Cube: cube-shaped, self-reconfigurable robots sliding on structure surfaces. In: Proceedings of IEEE International Conference on Robotics and Automation (ICRA), pp. 3149–3155 (2008) – reference: Fitch, R., Butler, Z., Rus, D.: Reconfiguration planning for heterogeneous self-reconfiguring robots. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), vol. 3, pp. 2460–2467 (2003) – reference: ChennareddySAgrawalAKaruppiahAModular self-reconfigurable robotic systems: a survey on hardware architecturesJ. Robot.201720175013532 – reference: StoyKBrandtDChristensenDJSelf-Reconfigurable Robots: An Introduction2010CambridgeMIT Press – reference: Nguyen, A., Guibas, L.J., Yim, M.: Controlled module density helps reconfiguration planning. In: Proceedings of 4th International Workshop on Algorithmic Foundations of Robotics (WAFR), pp. 23–36 (2000) – reference: Unsal, C., Kiliccote, H., Khosla, P.: I(CES)-Cubes: a modular self-reconfigurable bipartite robotic system. In: Proceedings of SPIE Conference on Mobile Robots and Autonomous Systems, vol. 3839, pp. 258–269 (1999) – reference: YimMShenWSalemiBRusDMollMLipsonHKlavinsEChirikjianGSModular self-reconfigurable robot systemsIEEE Robot. Autom. 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Robot. – volume: 21 start-page: 165 issue: 2 year: 2006 ident: 784_CR17 publication-title: Auton. Robots doi: 10.1007/s10514-006-8546-1 – ident: 784_CR21 doi: 10.1109/ICRA.2015.7139451 – volume: 29 start-page: 73 issue: 1 year: 2011 ident: 784_CR12 publication-title: Robotica doi: 10.1017/S0263574710000718 – volume-title: Self-Reconfigurable Robots: An Introduction year: 2010 ident: 784_CR20 – ident: 784_CR6 – volume: 22 start-page: 37 issue: 1 year: 2006 ident: 784_CR7 publication-title: Graphs Comb. doi: 10.1007/s00373-005-0640-1 – volume-title: Labyrinth Problems: Labyrinth-Searching Abilities of Automata, Teubner-Texte zur Mathematik (TTZM) year: 1989 ident: 784_CR10 doi: 10.1007/978-3-322-94560-0 – volume: 102 start-page: 18 year: 2019 ident: 784_CR13 publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2018.12.001 – volume: 20 start-page: 409 issue: 3 year: 2004 ident: 784_CR8 publication-title: IEEE Trans. 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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Configurations Data Structures and Information Theory Mathematics of Computing Modules Reconfiguration Robots Theory of Computation
Title	Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers
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