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Hybrid Feedback Control Design for Non-Convex Obstacle Avoidance

Mayur Sawant, Ilia Polushin, Abdelhamid Tayebi

Abstract

We develop an autonomous navigation algorithm for a robot operating in two-dimensional environments containing obstacles, with arbitrary non-convex shapes, which can be in close proximity with each other, as long as there exists at least one safe path connecting the initial and the target location. An instrumental transformation that modifies (virtually) the non-convex obstacles, in a non-conservative manner, is introduced to facilitate the design of the obstacle-avoidance strategy. The proposed navigation approach relies on a hybrid feedback that guarantees global asymptotic stabilization of a target location while ensuring the forward invariance of the modified obstacle-free workspace. The proposed hybrid feedback controller guarantees Zeno-free switching between the move-to-target mode and the obstacle-avoidance mode based on the proximity of the robot with respect to the modified obstacle-occupied workspace. Finally, we provide an algorithmic procedure for the sensor-based implementation of the proposed hybrid controller and validate its effectiveness via some numerical simulations.

Hybrid Feedback Control Design for Non-Convex Obstacle Avoidance

Abstract

We develop an autonomous navigation algorithm for a robot operating in two-dimensional environments containing obstacles, with arbitrary non-convex shapes, which can be in close proximity with each other, as long as there exists at least one safe path connecting the initial and the target location. An instrumental transformation that modifies (virtually) the non-convex obstacles, in a non-conservative manner, is introduced to facilitate the design of the obstacle-avoidance strategy. The proposed navigation approach relies on a hybrid feedback that guarantees global asymptotic stabilization of a target location while ensuring the forward invariance of the modified obstacle-free workspace. The proposed hybrid feedback controller guarantees Zeno-free switching between the move-to-target mode and the obstacle-avoidance mode based on the proximity of the robot with respect to the modified obstacle-occupied workspace. Finally, we provide an algorithmic procedure for the sensor-based implementation of the proposed hybrid controller and validate its effectiveness via some numerical simulations.
Paper Structure (43 sections, 12 theorems, 83 equations, 23 figures, 2 algorithms)

This paper contains 43 sections, 12 theorems, 83 equations, 23 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Projection on a set
  5. Sets with positive reach
  6. Geometric sets
  7. Line
  8. Line segment
  9. Hyperplane
  10. Convex cone
  11. Conic hull boyd2004convex
  12. Tangent cone and Normal cone
  13. Hybrid system framework
  14. Problem Formulation
  15. Obstacle reshaping
  16. ...and 28 more sections

Key Result

Lemma 1

Given a closed set $\mathcal{A}\subset\mathbb{R}^n$, we define the set $\mathcal{G} = \left(\mathcal{A} \oplus \mathcal{B}_{\alpha}^{\circ}(\mathbf{0})\right)^c.$ If $\mathbf{reach}(\mathcal{A}) \geq \alpha$, then $\mathbf{reach}(\mathcal{G}) \geq \alpha$.

Figures (23)

  • Figure 1: Two examples of workspaces that do not satisfy Assumption \ref{['assumption:connected_interior']}.
  • Figure 2: (a) The original obstacle $\mathcal{O}_i\subset\mathbb{R}^2$. (b) Dilation of obstacle $\mathcal{O}_i$ by a structuring element $\mathcal{B}_{\alpha}^{\circ}(\mathbf{0}), \alpha > 0$. (c) Erosion of the dilated obstacle $\mathcal{O}_i\oplus\mathcal{B}_{\alpha}^{\circ}(\mathbf{0})$ by the same structuring element $\mathcal{B}_{\alpha}^{\circ}(\mathbf{0})$.
  • Figure 3: Workspace with two obstacles $\mathcal{O}_{\mathcal{W}} = \mathcal{O}_i\cup\mathcal{O}_j$ such that $d(\mathcal{O}_i, \mathcal{O}_j)<2\alpha$. Left figure shows that the set $\mathcal{O}_{\mathcal{W}}^M$ is not a connected set. Right figure shows that the set $\mathcal{O}_{\mathcal{W}}^M$ is a connected set.
  • Figure 4: Workspace with two obstacle $\mathcal{O}_{\mathcal{W}} = \mathcal{O}_i\cup\mathcal{O}_j$ such that $d(\mathcal{O}_i, \mathcal{O}_j) < 2\alpha$. (a) $\mathcal{O}_{\mathcal{W}}$ does not satisfy Assumption \ref{['Assumption:reach']}. (b) $\mathcal{O}_{\mathcal{W}}$ satisfies Assumption \ref{['Assumption:reach']}.
  • Figure 5: Left figure shows the original obstacle $\mathcal{O}_i\subset\mathbb{R}^2$. Middle figure shows the modified obstacle $\mathcal{O}_{{i}}^M = \mathbf{M}(\mathcal{O}_i, \alpha)$ obtained using \ref{['obstacle_modification_step']}. Right figure shows the convex hull $\text{conv}(\mathcal{O}_i)$ for the obstacle $\mathcal{O}_i$.
  • ...and 18 more figures

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 21 more