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Ten Problems in Geobotics

Mikkel Abrahamsen, Dan Halperin

TL;DR

Ten problems bridge robotics and computational geometry, revealing fundamental gaps in motion planning, reconfiguration, and manufacturing-oriented tasks. The survey highlights exact, approximate, and hardness results, linking techniques such as Minkowski sums, caging, and convex covers with practical concerns in assembly and 3D printing. It discusses interpolation between easy and hard instances, minimal-design-modification concepts, and multi-handed assembly as avenues to cope with feasibility and coordination when standard methods fall short. The work also points to potential ML opportunities and emphasizes open questions with broad practical impact for automation and robotics.

Abstract

Robots sense, move and act in the physical world. It is therefore natural that algorithmic problems in robotics and automation have a geometric component, often central to the problem. Below we review ten challenging problems at the intersection of robotics and computational geometry -- let's call this intersection Geobotics. What is common to most of these problems is that the prevalent algorithmic techniques used in robotics do not seem suitable for solving them, or at least do not suggest quality guarantees for the solution. Solving some of them, even partially, can shed light on less well-understood aspects of computation in robotics.

Ten Problems in Geobotics

TL;DR

Ten problems bridge robotics and computational geometry, revealing fundamental gaps in motion planning, reconfiguration, and manufacturing-oriented tasks. The survey highlights exact, approximate, and hardness results, linking techniques such as Minkowski sums, caging, and convex covers with practical concerns in assembly and 3D printing. It discusses interpolation between easy and hard instances, minimal-design-modification concepts, and multi-handed assembly as avenues to cope with feasibility and coordination when standard methods fall short. The work also points to potential ML opportunities and emphasizes open questions with broad practical impact for automation and robotics.

Abstract

Robots sense, move and act in the physical world. It is therefore natural that algorithmic problems in robotics and automation have a geometric component, often central to the problem. Below we review ten challenging problems at the intersection of robotics and computational geometry -- let's call this intersection Geobotics. What is common to most of these problems is that the prevalent algorithmic techniques used in robotics do not seem suitable for solving them, or at least do not suggest quality guarantees for the solution. Solving some of them, even partially, can shed light on less well-understood aspects of computation in robotics.
Paper Structure (12 sections, 4 equations, 13 figures, 1 table)

This paper contains 12 sections, 4 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Shortest coordinated paths for two robots
  3. Can one polyhedron cage another?
  4. Interpolate between easy and hard instances of multi-robot motion planning
  5. Geometric reconfiguration and object rearrangement
  6. Interlocked objects with fewer convex parts
  7. Minimal design modification, or explainability
  8. Remark.
  9. Assembly planning with more than two hands
  10. Compact representation of spatial objects
  11. Partitioning large objects for manufacturing
  12. Convex covering of free configuration space

Figures (13)

  • Figure 1: Two optimal solutions of an instance of Problem 2Discs-Min-Sum in the absence of obstacles, one in orange and one in green. The shaded discs are the unit-disc robots at their start and target configurations, and the dotted circles are of radius $2$ around the start and target configurations. Example following DBLP:journals/corr/KirkpatrickL16.
  • Figure 2: Polygons $A$ and $B$ and the Minkowsky sum $A\oplus (-B)$ on top of $A$. As is seen, $A\oplus (-B)$ has two holes, which correspond to the two "pockets" of $A$ that cage $B$.
  • Figure 3: Left: The setting solved in DBLP:journals/tase/AdlerPSG15: The dark green and orange discs are the start and target positions, respectively. The light colored larger discs of radius $2$ show that the distance between any pair of start and/or target positions is at least $4$. Right: We propose to study the situation where the distance is at least $2+\varepsilon$ for an arbitrary value $\varepsilon>0$.
  • Figure 4: The lifting model demonstrated during the festival Copenhagen Distortion, 2024. Picture by Hao Wu.
  • Figure 5: The Snoeyink-Stolfi construction of 30 convex parts that cannot be taken apart with two hands. Left: An illustration of the actual construction. Middle: A sculpture at UBC inspired by the construction. The left and middle figures are taken from Snoeyink's website https://www.cs.unc.edu/ snoeyink/. Right: The construction printed on a plaster-based 3D printer at Tel Aviv University.
  • ...and 8 more figures