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DOI: 10.4230/LIPIcs.SWAT.2026.10

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, and Agnes Totschnig

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

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.

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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)

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@InProceedings{bousquet_et_al:LIPIcs.SWAT.2026.10,
  author =	{Bousquet, Nicolas and Connor, Frank and El Sabeh, Remy and Langevin, Louis-Roy and Mouawad, Amer E. and Nishimura, Naomi and Totschnig, Agnes},
  title =	{{Robotic Arm Rotation: Standing up Is Harder Than You Think}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.10},
  URN =		{urn:nbn:de:0030-drops-260467},
  doi =		{10.4230/LIPIcs.SWAT.2026.10},
  annote =	{Keywords: search, optimization, robotics, robotic arms, parameterized complexity, computational geometry, combinatorial reconfiguration}
}

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Robotic Arm Rotation: Standing up Is Harder Than You Think

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