Document Open Access Logo

Robotic Arm Rotation: Standing up Is Harder Than You Think

Authors Nicolas Bousquet , Frank Connor, Remy El Sabeh , Louis-Roy Langevin, Amer E. Mouawad , Naomi Nishimura, Agnes Totschnig

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2026.10.pdf
  • Filesize: 1.2 MB
  • 17 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • search
  • optimization
  • robotics
  • robotic arms
  • parameterized complexity
  • computational geometry
  • combinatorial reconfiguration

Metrics

Abstract

We study motion-planning problems for planar robotic arms that rotate around fixed centers while avoiding collisions. In the SM-RAMP model, each unit-length arm may rotate at most once; the question is whether all arms can be rotated to the vertical position. We resolve an open problem of Bousquet et al. [Bousquet et al., 2026] by proving that SM-RAMP is NP-complete, even in the horizontal-to-vertical setting. Our hardness proof uses a structural analysis of rotation-propagation chains and introduces a combinatorial abstraction of independent interest, the Lighthouse Propagation problem, which we show is itself NP-complete. We then consider the multi-move variant MM-RAMP, where each arm may rotate multiple times among a fixed set of allowed angles (or orientations). We prove that MM-RAMP is PSPACE-complete even when each arm has only a few allowed angles, in sharp contrast with the single-move case. Finally, we give two fixed-parameter tractable algorithms: for MAX-SM-RAMP parameterized by the number k of arms to be made vertical, and for 2A-MM-RAMP (restricted to horizontal and vertical) parameterized by the number 𝓁 of allowed rotations.

Cite As Get BibTex

Nicolas Bousquet, Frank Connor, Remy El Sabeh, Louis-Roy Langevin, Amer E. Mouawad, Naomi Nishimura, and Agnes Totschnig. Robotic Arm Rotation: Standing up Is Harder Than You Think. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.10

Author Details

Nicolas Bousquet
  • CNRS, Université de Montréal CRM - CNRS, Canada
  • Univ. Lyon, CNRS, INSA Lyon, UCBL, LIRIS, UMR5205, F-69621, Lyon, France
Frank Connor
  • McGill University, Montreal, Canada
Remy El Sabeh
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Louis-Roy Langevin
  • McGill University, Montreal, Canada
Amer E. Mouawad
  • American University of Beirut, Lebanon
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Naomi Nishimura
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Agnes Totschnig
  • McGill University, Montreal, Canada

Funding

Remy El Sabeh and Naomi Nishimura: supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
  • Bousquet, Nicolas: supported by the ANR (Agence Nationale de la Recherche) project TOAD (ANR-25-CE48-3716).
  • Totschnig, Agnes: supported by a McCall MacBain Scholarship and partially by FRQNT Grant 332481.

References

  1. Nicolas Bousquet, Remy El Sabeh, Amer E. Mouawad, and Naomi Nishimura. On the complexity of constrained reconfiguration and motion planning. In 51st International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2026), pages 1-15. Springer, 2026. URL: https://doi.org/10.1007/978-3-032-17801-5_1.
  2. Nicolas Bousquet, Felix Hommelsheim, Yusuke Kobayashi, Moritz Mühlenthaler, and Akira Suzuki. Feedback vertex set reconfiguration in planar graphs. Theoretical Computer Science, 979:114188, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114188.
  3. Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the parameterized complexity of reconfiguration problems. Computer Science Review, 53:100663, 2024. URL: https://doi.org/10.1016/J.COSREV.2024.100663.
  4. Ron Breukelaar, Erik D. Demaine, Susan Hohenberger, Hendrik Jan Hoogeboom, Walter A. Kosters, and David Liben-Nowell. Tetris is hard, even to approximate. International Journal of Computational Geometry and Applications, 14(1-2):41-68, 2004. URL: https://doi.org/10.1142/S0218195904001354.
  5. Marcin Brianski, Stefan Felsner, Jedrzej Hodor, and Piotr Micek. Reconfiguring independent sets on interval graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), volume 202 of LIPIcs, pages 23:1-23:14, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.23.
  6. Kevin Buchin, Mart Hagedoorn, Irina Kostitsyna, and Max van Mulken. Dots & boxes is PSPACE-complete. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), volume 202 of LIPIcs, pages 25:1-25:18, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.25.
  7. Petar Ćurković and Bojan Jerbić. Dual-Arm robot motion planning based on cooperative coevolution. In DoCEIS, volume 314 of IFIP Advances in Information and Communication Technology, pages 169-178. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-11628-5_18.
  8. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  9. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0515-9.
  10. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  11. Robert A. Hearn. Games, puzzles, and computation. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2006. URL: https://hdl.handle.net/1721.1/37913.
  12. Robert A. Hearn and Erik D. Demaine. PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci., 343(1-2):72-96, 2005. URL: https://doi.org/10.1016/J.TCS.2005.05.008.
  13. Luu Thi Hue and Nguyen Pham Thuc Anh. Planning the optimal trajectory for a dual-arm robot system using a genetic algorithm considering the controller. In 2022 International Conference on Advanced Technologies for Communications (ATC), pages 292-297, 2022. URL: https://doi.org/10.1109/ATC55345.2022.9942975.
  14. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theoretical Computer Science, 412(12-14):1054-1065, 2011. URL: https://doi.org/10.1016/J.TCS.2010.12.005.
  15. Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, and Marcin Wrochna. Reconfiguration over tree decompositions. In 9th International Symposium on Parameterized and Exact Computation (IPEC 2014), pages 246-257, 2014. URL: https://doi.org/10.1007/978-3-319-13524-3_21.
  16. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/A11040052.
  17. Naoto Ohsaka. Gap amplification for reconfiguration problems. In Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1345-1366, 2024. URL: https://doi.org/10.1137/1.9781611977912.54.
  18. Naoto Ohsaka. Gap preserving reductions between reconfiguration problems. Journal of Computer and System Sciences, 157:103733, 2026. URL: https://doi.org/10.1016/J.JCSS.2025.103733.
  19. Seyed Sina Mirrazavi Salehian, Nadia Figueroa, and Aude Billard. A unified framework for coordinated multi-arm motion planning. International Journal of Robotics Research, 37(10), 2018. URL: https://doi.org/10.1177/0278364918765952.
  20. Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421-444, 1987. URL: https://doi.org/10.1137/0216030.
  21. Jan van den Heuvel. The complexity of change. In Simon R. Blackburn, Stefanie Gerke, and Mark Wildon, editors, Surveys in Combinatorics 2013, volume 409 of London Mathematical Society Lecture Note Series, pages 127-160. Cambridge University Press, 2013. URL: https://doi.org/10.1017/CBO9781139506748.005.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact