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Sublevel Set Approximation in The Hausdorff and Volume Metric with Application to Path Planning and Obstacle Avoidance

Morgan Jones

TL;DR

This article shows that if a sequence of functions converges strictly from above/below to a function, then these functions yield a sequence sublevel sets that converge to the sublevel set of <inline-formula><tex-math notation="LaTeX">$V$</tex-math></inline-formula> with respect to the Hausdorff metric.

Abstract

Under what circumstances does the ``closeness" of two functions imply the ``closeness" of their respective sublevel sets? In this paper, we answer this question by showing that if a sequence of functions converges strictly from above/below to a function, $V$, in the $L^\infty$ (or $L^1$) norm then these functions yield a sequence sublevel sets that converge to the sublevel set of $V$ with respect to the Hausdorff metric (or volume metric). Based on these theoretical results we propose Sum-of-Squares (SOS) numerical schemes for the optimal outer/inner polynomial sublevel set approximation of various sets, including intersections and unions of semialgebraic sets, Minkowski sums, Pontryagin differences and discrete points. We present several numerical examples demonstrating the usefulness of our proposed algorithm including approximating sets of discrete points to solve machine learning one-class classification problems and approximating Minkowski sums to construct C-spaces for computing optimal collision-free paths for Dubin's car.

Sublevel Set Approximation in The Hausdorff and Volume Metric with Application to Path Planning and Obstacle Avoidance

TL;DR

This article shows that if a sequence of functions converges strictly from above/below to a function, then these functions yield a sequence sublevel sets that converge to the sublevel set of <inline-formula><tex-math notation="LaTeX"></tex-math></inline-formula> with respect to the Hausdorff metric.

Abstract

Under what circumstances does the ``closeness" of two functions imply the ``closeness" of their respective sublevel sets? In this paper, we answer this question by showing that if a sequence of functions converges strictly from above/below to a function, , in the (or ) norm then these functions yield a sequence sublevel sets that converge to the sublevel set of with respect to the Hausdorff metric (or volume metric). Based on these theoretical results we propose Sum-of-Squares (SOS) numerical schemes for the optimal outer/inner polynomial sublevel set approximation of various sets, including intersections and unions of semialgebraic sets, Minkowski sums, Pontryagin differences and discrete points. We present several numerical examples demonstrating the usefulness of our proposed algorithm including approximating sets of discrete points to solve machine learning one-class classification problems and approximating Minkowski sums to construct C-spaces for computing optimal collision-free paths for Dubin's car.
Paper Structure (23 sections, 24 theorems, 119 equations, 9 figures)

This paper contains 23 sections, 24 theorems, 119 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Application to Path Planning and Obstacle Avoidance
  3. Notation
  4. Sublevel Set Approximation
  5. Sublevel Set Approximation In The Hausdorff Metric
  6. Sublevel Set Approximation in The Volume Metric
  7. Numerical Sublevel Set Approximation
  8. An optimization problem for $L^\infty$ approximation
  9. An optimization problem for $L^1$ approximation
  10. SOS implementation of $L^\infty$ and $L^1$ approximation
  11. Approximation of Semialgebraic Sets
  12. Approximation of Unions of Semialgebraic Sets
  13. Approximation of Minkowski Sums
  14. Approximation of Pontryagin Differences
  15. Approximation of Discrete Points
  16. ...and 8 more sections

Key Result

Theorem 1

Consider a compact set $\Lambda \subset \mathbb{R}^n$, a function $V : \Lambda \to \mathbb{R}$, and a family of functions $\{J_d \}_{d \in \mathbb{N}}$ that satisfies the following properties: Then for all $\gamma \in \mathbb{R}$ we have that,

Figures (9)

  • Figure 1: Plot associated with Example \ref{['ex: Approximation of unions of semialgebraic sets']} showing the approximation of the union of semialgebraic sets, $X_1 \cup X_2 \cup X_3$ given in Eq. \ref{['numerical union of semialg']}, shown as the green region.
  • Figure 2: Plot associated with Example \ref{['ex: mink and pont']} showing approximations of $X_1 \oplus X_2$ and $X_1 \ominus X_2$, where $X_1$ and $X_2$ are given in Eq. \ref{['mink and pont numerical']}.
  • Figure 3: Plot associated with Example \ref{['ex: ROA']} showing the approximation of the ROA of the single machine infinite bus system from trajectory data.
  • Figure 4: Plot associated with Example \ref{['ex: lorenz']} showing various angles of our approximation the Lorenz attractor from trajectory data.
  • Figure 5: Plot associated with Example \ref{['ex: path planning']} showing collision free trajectories of Dubin's car in both the workspace and C-space.
  • ...and 4 more figures

Theorems & Definitions (49)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Lemma 1: Semialgebraic sets can be written as a single sublevel set
  • proof
  • Lemma 2: Unions of sublevel sets in single sublevel set form
  • proof
  • Definition 1: Minkowski Sum
  • ...and 39 more