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Some geometric and topological data-driven methods in robot motion path planning

Boris Goldfarb

TL;DR

This work will survey some questions, issues, recent work and promising directions in data-driven geometric and topological methods with some emphasis on the use of discrete Morse theory.

Abstract

Motion path planning is an intrinsically geometric problem which is central for design of robot systems. Since the early years of AI, robotics together with computer vision have been the areas of computer science that drove its development. Many questions that arise, such as existence, optimality, and diversity of motion paths in the configuration space that describes feasible robot configurations, are of topological nature. The recent advances in topological data analysis and related metric geometry, topology and combinatorics have provided new tools to address these engineering tasks. We will survey some questions, issues, recent work and promising directions in data-driven geometric and topological methods with some emphasis on the use of discrete Morse theory.

Some geometric and topological data-driven methods in robot motion path planning

TL;DR

This work will survey some questions, issues, recent work and promising directions in data-driven geometric and topological methods with some emphasis on the use of discrete Morse theory.

Abstract

Motion path planning is an intrinsically geometric problem which is central for design of robot systems. Since the early years of AI, robotics together with computer vision have been the areas of computer science that drove its development. Many questions that arise, such as existence, optimality, and diversity of motion paths in the configuration space that describes feasible robot configurations, are of topological nature. The recent advances in topological data analysis and related metric geometry, topology and combinatorics have provided new tools to address these engineering tasks. We will survey some questions, issues, recent work and promising directions in data-driven geometric and topological methods with some emphasis on the use of discrete Morse theory.
Paper Structure (14 sections, 6 theorems, 7 equations, 6 figures)

This paper contains 14 sections, 6 theorems, 7 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Panorama of topological methods in robot motion planning
  3. Topological complexity
  4. Artificial potential vector fields
  5. Topological data analysis
  6. Discrete Morse theory methods
  7. Some preliminaries
  8. Relations between the smooth and discrete Morse theories
  9. Advantages of data-driven applications of discrete Morse theory
  10. Applications of discrete Morse theory
  11. Skeletonization, sphere percolation (Robins et al.)
  12. Density-based modeling (Upadhyay et al.)
  13. Parallelized and distributed planning
  14. A collection of problems

Key Result

Theorem 2.1

Let $M \subset E^n$ be a compact connected $n$-dimensional analytic submanifold with boundary. Suppose we are given a navigation function, that is, an analytic Morse function $\phi \colon M \to [0, 1]$ with a unique minimum in the interior point $p$ of $M$, and with the maximum value attained on all

Figures (6)

  • Figure 1: (a) An illustration of the medial axis as a Morse skeleton and features such as junctions (blue) and ends (green and red). (b) Figure 5 from Robins et al. RDS:15: a visualization of a silica 3D sphere packing with the Morse skeleton of its pore space shown in blue.
  • Figure 2: Figure 4 from Robins et al. RDS:15: illustration of the Morse analysis in a 2D Euclidean distance image (Mt. Gambier limestone); critical cells correspond to the round dots, black for minima, cyan for saddles, and white for maxima. The Morse skeleton is shown in white, and paths between critical cells in cyan. There is duality between the Morse skeleton and the cell partition determined by basins.
  • Figure 3: Standard simple test environments with marked critical 0-cells to serve as goal posts for path planning.
  • Figure 4: From Figure 2 UE:22: protein surface model is shown together with top 10 predicted 1KRN bio-molecule conformations, indicated with different colors.
  • Figure 5: From Figure 5 UE:22: the plots show the total computation time taken (in seconds) by three methods to predict the top 10 IDP docking conformation ensembles around the protein surface model. The green bar shows the method with discrete Morse theory identified goal posts, the other two show results from protein-protein interaction computation servers HawkDock and HADDOCK.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 2.1: Koditschek K:89a, Koditschek/Rimon KR:90KR:91
  • Definition 2.3: Vietoris-Rips complex
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: Forman, Theorem 3.9 rF:98
  • Theorem 3.5: Forman, Theorem 3.4 rF:98
  • Definition 3.6
  • Theorem 3.7: Forman, Theorem 6.2 rF:98
  • ...and 3 more