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A Motion Planning Algorithm in a Figure Eight Track

Cristian Jardon, Brian Sheppard, Veet Zaveri

TL;DR

A topological approach to robot motion planning is used that relates instabilities in motion planning algorithms to topological features of configuration spaces and shows that the topological complexity of the problem is 3.

Abstract

We design a motion planning algorithm to coordinate the movements of two robots along a figure eight track, in such a way that no collisions occur. We use a topological approach to robot motion planning that relates instabilities in motion planning algorithms to topological features of configuration spaces. The topological complexity of a configuration space is an invariant that measures the complexity of motion planning algorithms. We show that the topological complexity of our problem is 3 and construct an explicit algorithm with three continuous instructions.

A Motion Planning Algorithm in a Figure Eight Track

TL;DR

A topological approach to robot motion planning is used that relates instabilities in motion planning algorithms to topological features of configuration spaces and shows that the topological complexity of the problem is 3.

Abstract

We design a motion planning algorithm to coordinate the movements of two robots along a figure eight track, in such a way that no collisions occur. We use a topological approach to robot motion planning that relates instabilities in motion planning algorithms to topological features of configuration spaces. The topological complexity of a configuration space is an invariant that measures the complexity of motion planning algorithms. We show that the topological complexity of our problem is 3 and construct an explicit algorithm with three continuous instructions.
Paper Structure (23 sections, 4 theorems, 7 equations, 45 figures, 5 algorithms)

This paper contains 23 sections, 4 theorems, 7 equations, 45 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and terminology
  3. Motion planning problem
  4. Basic notions
  5. Configuration space
  6. Configuration space of two robots on a figure eight track
  7. Visualization of the configuration space $X$
  8. Topological Complexity
  9. Homotopy type of $X$
  10. Construction of the Spine
  11. The wedge of seven circles
  12. Calculation of the topological complexity of the configuration space
  13. Algorithm in the configuration space
  14. Algorithm in the chain $C$
  15. Algorithm in $Z$
  16. ...and 8 more sections

Key Result

Theorem 4.1

F1 A continuous motion planning $s$: $X \times X \rightarrow PX$ exists if and only if the configuration space $X$ is contractible.

Figures (45)

  • Figure 1: Physical space $\Gamma$.
  • Figure 2: First and second robot in $\Gamma$
  • Figure 3: Cartesian product $(S^1\vee S^1)\times (S^1\vee S^1)$.
  • Figure 4: $X=(\Gamma \times \Gamma) - D$.
  • Figure 5: Flat representation of the circle.
  • ...and 40 more figures

Theorems & Definitions (5)

  • Theorem 4.1
  • Definition 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Lemma 5.2