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A Short Note on Modeling 2D Taut Ropes with Visibility Decompositions

Adem B. Dalkılıç

TL;DR

This note addresses 2D taut-rope modeling around polygonal obstacles in a constrained space, motivated by haptic feedback for surgical robotics. It introduces Gazan decomposition (GD), a lightweight, online variant of visibility decomposition, to track rope wrapping and unwrapping as the rope ends trace through free space. The main contribution is a linear-time per-step algorithm, $O(n)$ with $n$ obstacle segments, that maintains the rope state $R(t)$ under the small update and single-cut conditions, including a compact pseudocode implementation and a discussion of a collinearity limitation with a linear-time remedy. By extending visibility-decomposition concepts to dynamic rope dynamics, the approach achieves real-time capable updates without heavy precomputation, offering practical impact for real-time haptic sensing and planning in 2D settings. The work situates itself within the broader visibility-decomposition literature, linking classic query-oriented techniques to constructive rope-wrapping models.

Abstract

The problem of modeling ropes arises in many applications, including providing haptic feedback to surgeons who are using surgical robots to realign the distal and proximal ends of split bones. Here, we consider a simplified, 2D variant of the haptic feedback estimation problem and discuss how visibility decompositions greatly simplify the problem. Then, we introduce an efficient, concise algorithm for modeling the dynamics of 2D ropes around polygonal obstacles in O($n$) time, where $n$ is the number of line segment obstacles.

A Short Note on Modeling 2D Taut Ropes with Visibility Decompositions

TL;DR

This note addresses 2D taut-rope modeling around polygonal obstacles in a constrained space, motivated by haptic feedback for surgical robotics. It introduces Gazan decomposition (GD), a lightweight, online variant of visibility decomposition, to track rope wrapping and unwrapping as the rope ends trace through free space. The main contribution is a linear-time per-step algorithm, with obstacle segments, that maintains the rope state under the small update and single-cut conditions, including a compact pseudocode implementation and a discussion of a collinearity limitation with a linear-time remedy. By extending visibility-decomposition concepts to dynamic rope dynamics, the approach achieves real-time capable updates without heavy precomputation, offering practical impact for real-time haptic sensing and planning in 2D settings. The work situates itself within the broader visibility-decomposition literature, linking classic query-oriented techniques to constructive rope-wrapping models.

Abstract

The problem of modeling ropes arises in many applications, including providing haptic feedback to surgeons who are using surgical robots to realign the distal and proximal ends of split bones. Here, we consider a simplified, 2D variant of the haptic feedback estimation problem and discuss how visibility decompositions greatly simplify the problem. Then, we introduce an efficient, concise algorithm for modeling the dynamics of 2D ropes around polygonal obstacles in O() time, where is the number of line segment obstacles.
Paper Structure (4 sections, 2 figures)

This paper contains 4 sections, 2 figures.

Table of Contents

  1. PROBLEM STATEMENT
  2. A SIMPLE SOLUTION
  3. A SMALL LIMITATION
  4. AN OVERVIEW OF RELATED WORK FOR FURTHER READING

Figures (2)

  • Figure 1: Visual depiction of the algorithm. $B$, the fixed end of the rope (filled circle) is shown, the rays of the GD that were computed until an intersection was found are shown (dash-dot lines), the unwrapping ray is shown (dotted), the rope is shown (dashed), and the obstacle line segments are shown (straight) for all $t$ of $A(t)$ (hollow circle).
  • Figure 2: Failure case caused by colinearity. $B$, the fixed end of the rope (filled circle) is shown, the rope is shown (dashed), and the obstacle line segments are shown (straight) for all $t$ of $A(t)$ (hollow circle).