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Minimum-Length Coordinated Motions For Two Convex Centrally-Symmetric Robots

David Kirkpatrick, Paul Liu

TL;DR

This work resolves the minimum-length collision-free coordinated motion problem for two convex centrally-symmetric robots in an obstacle-free plane. It leverages Minkowski sums and Cauchy's surface area formula to transform path-length optimization into a directional integral, yielding an exact characterization: an optimal co-motion exists with at most six pieces, comprised of straight segments or boundary arcs of ${\mathbb A}+{\mathbb B}$, and its length is given by a simple integral depending only on the initial and goal configurations. The authors show that optimal motions can be realized in a decoupled form (only one robot moves at a time) or in a coupled form with monotone orientation, and they introduce a standard-form framework that enables a constructive, verifiable proof of optimality via counter-clockwise tightness. A complete characterization connects the optimal co-motions to tightness properties of the boundary-reach difference and uses corridor-based straight-line tests to identify feasible configurations, while generalizing prior results from discs and squares to arbitrary CCS shapes. The methodology provides a geometric, 2D framework with potential extensions to 3D and to non-centrally-symmetric convex robots, offering both theoretical insight and practical guidelines for optimal coordination in planar robotics.

Abstract

We study the problem of determining coordinated motions, of minimum total length, for two arbitrary convex centrally-symmetric (CCS) robots in an otherwise obstacle-free plane. Using the total path length traced by the two robot centres as a measure of distance, we give an exact characterization of a (not necessarily unique) shortest collision-avoiding motion for all initial and goal configurations of the robots. The individual paths are composed of at most six convex pieces, and their total length can be expressed as a simple integral with a closed form solution depending only on the initial and goal configuration of the robots. The path pieces are either straight segments or segments of the boundary of the Minkowski sum of the two robots (circular arcs, in the special case of disc robots). Furthermore, the paths can be parameterized in such a way that (i) only one robot is moving at any given time (decoupled motion), or (ii) the orientation of the robot configuration changes monotonically.

Minimum-Length Coordinated Motions For Two Convex Centrally-Symmetric Robots

TL;DR

This work resolves the minimum-length collision-free coordinated motion problem for two convex centrally-symmetric robots in an obstacle-free plane. It leverages Minkowski sums and Cauchy's surface area formula to transform path-length optimization into a directional integral, yielding an exact characterization: an optimal co-motion exists with at most six pieces, comprised of straight segments or boundary arcs of , and its length is given by a simple integral depending only on the initial and goal configurations. The authors show that optimal motions can be realized in a decoupled form (only one robot moves at a time) or in a coupled form with monotone orientation, and they introduce a standard-form framework that enables a constructive, verifiable proof of optimality via counter-clockwise tightness. A complete characterization connects the optimal co-motions to tightness properties of the boundary-reach difference and uses corridor-based straight-line tests to identify feasible configurations, while generalizing prior results from discs and squares to arbitrary CCS shapes. The methodology provides a geometric, 2D framework with potential extensions to 3D and to non-centrally-symmetric convex robots, offering both theoretical insight and practical guidelines for optimal coordination in planar robotics.

Abstract

We study the problem of determining coordinated motions, of minimum total length, for two arbitrary convex centrally-symmetric (CCS) robots in an otherwise obstacle-free plane. Using the total path length traced by the two robot centres as a measure of distance, we give an exact characterization of a (not necessarily unique) shortest collision-avoiding motion for all initial and goal configurations of the robots. The individual paths are composed of at most six convex pieces, and their total length can be expressed as a simple integral with a closed form solution depending only on the initial and goal configuration of the robots. The path pieces are either straight segments or segments of the boundary of the Minkowski sum of the two robots (circular arcs, in the special case of disc robots). Furthermore, the paths can be parameterized in such a way that (i) only one robot is moving at any given time (decoupled motion), or (ii) the orientation of the robot configuration changes monotonically.

Paper Structure

This paper contains 29 sections, 9 theorems, 8 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Background
  4. The general approach
  5. Optimal co-motions for two symmetric convex robots
  6. An example of a counter-clockwise optimal co-motion
  7. Certifying non-optimality of certain counter-clockwise co-motions
  8. Optimal co-motions in standard form
  9. Straight-line co-motions
  10. A normal form for initial/goal robot configurations
  11. Standard-form co-motions
  12. The proof of Lemma \ref{['lem:standard']}
  13. Cases with $B_0 \in \text{corr}(A_0,A_1)$
  14. $A_1$ lies above the line forming the upper boundary of $\text{corr}(B_0,B_1)$
  15. $A_1$ lies below the upper tangent through $A_0$ to $({\mathbb A}\!+\!{\mathbb B})[B_1]$
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let ${\mathbb A}$ and ${\mathbb B}$ be two arbitrary convex centrally-symmetric (CCS) robots. For any initial and goal configurations of ${\mathbb A}$ and ${\mathbb B}$, there is a minimum-length collision-free coordinated motion, composed of at most six pieces, taking ${\mathbb A}$ and ${\mathbb B}

Figures (32)

  • Figure 1: Two CCS objects and their Minkowski sums.
  • Figure 2: Minkowski sums as the locus of intersecting placements of CCS objects.
  • Figure 3: Support line $L$ and support point $p$ defining the reach of $S$ in direction $\theta$.
  • Figure 4: The separation of ${\mathbb A}[A]$ from ${\mathbb B}[B]$ in direction $\theta$.
  • Figure 5: The trace of a co-motion from initial configuration (green) to target configuration (red), with intermediate configurations (yellow)
  • ...and 27 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 18 more