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Safe Bubble Cover for Motion Planning on Distance Fields

Ki Myung Brian Lee, Zhirui Dai, Cedric Le Gentil, Lan Wu, Nikolay Atanasov, Teresa Vidal-Calleja

TL;DR

Inspired by sampling-based planning algorithms, three algorithms for constructing a safe bubble cover of free space are presented, named bubble roadmap (BRM), rapidly exploring bubble graph (RBG), and expansive bubble graph (EBG), and shows that safe bubbles can be obtained from any Lipschitz-continuous safety constraint.

Abstract

We consider the problem of planning collision-free trajectories on distance fields. Our key observation is that querying a distance field at one configuration reveals a region of safe space whose radius is given by the distance value, obviating the need for additional collision checking within the safe region. We refer to such regions as safe bubbles, and show that safe bubbles can be obtained from any Lipschitz-continuous safety constraint. Inspired by sampling-based planning algorithms, we present three algorithms for constructing a safe bubble cover of free space, named bubble roadmap (BRM), rapidly exploring bubble graph (RBG), and expansive bubble graph (EBG). The bubble sampling algorithms are combined with a hierarchical planning method that first computes a discrete path of bubbles, followed by a continuous path within the bubbles computed via convex optimization. Experimental results show that the bubble-based methods yield up to 5- 10 times cost reduction relative to conventional baselines while simultaneously reducing computational efforts by orders of magnitude.

Safe Bubble Cover for Motion Planning on Distance Fields

TL;DR

Inspired by sampling-based planning algorithms, three algorithms for constructing a safe bubble cover of free space are presented, named bubble roadmap (BRM), rapidly exploring bubble graph (RBG), and expansive bubble graph (EBG), and shows that safe bubbles can be obtained from any Lipschitz-continuous safety constraint.

Abstract

We consider the problem of planning collision-free trajectories on distance fields. Our key observation is that querying a distance field at one configuration reveals a region of safe space whose radius is given by the distance value, obviating the need for additional collision checking within the safe region. We refer to such regions as safe bubbles, and show that safe bubbles can be obtained from any Lipschitz-continuous safety constraint. Inspired by sampling-based planning algorithms, we present three algorithms for constructing a safe bubble cover of free space, named bubble roadmap (BRM), rapidly exploring bubble graph (RBG), and expansive bubble graph (EBG). The bubble sampling algorithms are combined with a hierarchical planning method that first computes a discrete path of bubbles, followed by a continuous path within the bubbles computed via convex optimization. Experimental results show that the bubble-based methods yield up to 5- 10 times cost reduction relative to conventional baselines while simultaneously reducing computational efforts by orders of magnitude.
Paper Structure (16 sections, 2 theorems, 12 equations, 11 figures, 3 algorithms)

This paper contains 16 sections, 2 theorems, 12 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Constructing a Bubble Cover
  5. Bubbles for Safe Space Representation
  6. Bubble Roadmap
  7. Rapidly Exploring Bubble Graph
  8. Expansive Bubble Graph
  9. Planning in Bubble Cover
  10. Discrete Planning of Bubble Path
  11. Continuous Planning
  12. Extension to Unknown Environments
  13. Evaluation
  14. Planning Shortest Distance Paths
  15. Comparison of Bubble Sampling Algorithms
  16. ...and 1 more sections

Key Result

theorem thmcountertheorem

Consider a constraint $l(\mathbf{y}) \geq 0$, where $l:\mathbb{R}^m \rightarrow \mathbb{R}$ is Lipschitz continuous with constant $L$. Then, for any $\mathbf{y}$ such that $l(\mathbf{y}) \geq 0$, all points $\mathbf{y}'$ in ball $B(\mathbf{y}, \frac{l(\mathbf{y})}{L})$ centered at $\mathbf{y}$ with

Figures (11)

  • Figure 1: Left: We construct 'safe bubbles' from a distance field representation of the environment, whose radii are given by the distance to obstacles. Middle: We present three algorithms for sampling safe bubbles (cyan), and a hierarchical planning method that first computes a bubble path (red) and then a continuous trajectory (dashed line) within the bubble path. Right: Our approach scales effectively to higher dimensions.
  • Figure 2: Illustration of the BRM algorithm with varying number of samples (red crosses) and bubbles (cyan). Not all random samples lead to a valid bubble due to footprint and minimum radius requirements. The free space is filled as the number of samples increases (from (a) to (c)).
  • Figure 3: Illustration of the RBG algorithm. Similar to RRT, RBG promotes even spatial coverage by sampling and steering towards random points (red crosses). The center of a new bubble (red diamond) is set at the perimeter of the nearest bubble (blue). Repeating this process from a) to b), we obtain a safe bubble cover of the free space with limited variation in radii.
  • Figure 4: Illustration of EBG with $N_{explore}=4$. a) New bubbles (red) are expanded from current (blue). b) A new bubble from a) is considered, but discarded because of overlap (red cross) c) Repeating this process fills the free space with confirmed bubbles (cyan).
  • Figure 5: An illustration of EBG modified for static unknown environments. Before the goal (black diamond) is in fully expanded bubbles (green), the planned trajectory (black solid line) points to the closest fully expanded bubble. As frontier bubbles (yellow) become visible, more fully expanded bubbles (green) appear.
  • ...and 6 more figures

Theorems & Definitions (5)

  • definition thmcounterdefinition: rigatos2015nonlinear
  • theorem thmcountertheorem: Safe bubble
  • proof
  • theorem thmcountertheorem: Random bubbles do not contain each other
  • proof