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Narrow Passage Path Planning using Collision Constraint Interpolation

Minji Lee, Jeongmin Lee, Dongjun Lee

TL;DR

A framework to ensure feasibility throughout the process using a series of subproblems tailored for narrow passage problem is proposed, which begins by decomposing the environment into convex objects and initializing collision constraints with a subset of these objects.

Abstract

Narrow passage path planning is a prevalent problem from industrial to household sites, often facing difficulties in finding feasible paths or requiring excessive computational resources. Given that deep penetration into the environment can cause optimization failure, we propose a framework to ensure feasibility throughout the process using a series of subproblems tailored for narrow passage problem. We begin by decomposing the environment into convex objects and initializing collision constraints with a subset of these objects. By continuously interpolating the collision constraints through the process of sequentially introducing remaining objects, our proposed framework generates subproblems that guide the optimization toward solving the narrow passage problem. Several examples are presented to demonstrate how the proposed framework addresses narrow passage path planning problems.

Narrow Passage Path Planning using Collision Constraint Interpolation

TL;DR

A framework to ensure feasibility throughout the process using a series of subproblems tailored for narrow passage problem is proposed, which begins by decomposing the environment into convex objects and initializing collision constraints with a subset of these objects.

Abstract

Narrow passage path planning is a prevalent problem from industrial to household sites, often facing difficulties in finding feasible paths or requiring excessive computational resources. Given that deep penetration into the environment can cause optimization failure, we propose a framework to ensure feasibility throughout the process using a series of subproblems tailored for narrow passage problem. We begin by decomposing the environment into convex objects and initializing collision constraints with a subset of these objects. By continuously interpolating the collision constraints through the process of sequentially introducing remaining objects, our proposed framework generates subproblems that guide the optimization toward solving the narrow passage problem. Several examples are presented to demonstrate how the proposed framework addresses narrow passage path planning problems.

Paper Structure

This paper contains 12 sections, 1 theorem, 18 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Collision Constraint Interpolation
  3. Signed Distance Function
  4. Smoothed SDF Interpolation between Convex Objects
  5. Homotopy Preserving Collision Constraint Interpolation
  6. Generation of Leaf Set Sequences
  7. Path Planning using Collision Constraint Interpolation
  8. Evaluations
  9. Placing Dishes on the Rack
  10. Extraction of Tool from a Narrow Gap
  11. Nonholonomic Navigation of Maze
  12. Conclusions

Key Result

Proposition 1

If two convex objects $v_1$ and $v_2$ intersect (i.e., $v_1 \cap v_2 \neq \varnothing$), their interpolated object $v^\alpha_{v_1\rightarrow v_2}$ as defined by eq:interpolated_convex is convex and satisfies:

Figures (7)

  • Figure 1: Optimization results for manipulator path planning during tool extraction from a narrow gap, utilizing the proposed collision constraint interpolation framework.
  • Figure 2: Given an environment with convex objects $\mathcal{V}$ colored in gray, a set of green convex objects forms leaf set in (a), and not in (b) or (c). In (b), one of the green object violates Condition 1.1 by intersecting with two objects in $\mathcal{V}$. In (c), the green objects violate Condition 1.2 by intersecting with each other.
  • Figure 3: Object interpolation process using proposed shaping function \ref{['eq:exp_shaping']} with $\eta = 40$ (top row), $\eta=15$ (middle row) and linear interpolation $\eta \rightarrow 0$ (bottom row).
  • Figure 4: Illustration of two sequential processes of gluing leaf sets to the environment.
  • Figure 5: Comparison of homotopy equivalence and the corresponding path planning outcomes when performing interpolation using two distinct formulas.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof