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Safe and Quasi-Optimal Autonomous Navigation in Environments with Convex Obstacles

Ishak Cheniouni, Soulaimane Berkane, Abdelhamid Tayebi

TL;DR

This work develops a continuous feedback control framework for safe autonomous navigation in environments with convex obstacles, achieving quasi-optimal, collision-free trajectories in $n$-dimensional sphere worlds. The core idea is to iteratively project a nominal destination-driven controller onto obstacle-enclosing cones, yielding locally optimal paths and, in 2D, almost global stability of the target; a sensor-based variant extends these guarantees to unknown 2D convex environments. The authors prove forward invariance of the free space, almost global asymptotic stability of the destination in 2D, and quasi-optimality of the generated trajectories, with extensive simulations and Gazebo experiments validating practical performance. The approach offers real-time applicability by avoiding full map-based planning, using geometric projections and, in the sensor-based case, local sensing to drive safe navigation toward $x_d$.

Abstract

We propose a continuous feedback control strategy that steers a point-mass vehicle safely to a destination, in a quasi-optimal manner, in sphere worlds. The main idea consists in avoiding each obstacle via the shortest path on the cone's surface enclosing the obstacle and moving straight toward the target when the vehicle has a clear line of sight to the target location. In particular, almost global asymptotic stability of the target location is achieved in two-dimensional (2D) environments under a particular assumption on the obstacles configuration. We also propose a reactive (sensor-based) approach, suitable for real-time implementations in a priori unknown 2D environments with sufficiently curved convex obstacles, guaranteeing almost global asymptotic stability of the target location. Simulation results are presented to illustrate the effectiveness of the proposed approach.

Safe and Quasi-Optimal Autonomous Navigation in Environments with Convex Obstacles

TL;DR

This work develops a continuous feedback control framework for safe autonomous navigation in environments with convex obstacles, achieving quasi-optimal, collision-free trajectories in -dimensional sphere worlds. The core idea is to iteratively project a nominal destination-driven controller onto obstacle-enclosing cones, yielding locally optimal paths and, in 2D, almost global stability of the target; a sensor-based variant extends these guarantees to unknown 2D convex environments. The authors prove forward invariance of the free space, almost global asymptotic stability of the destination in 2D, and quasi-optimality of the generated trajectories, with extensive simulations and Gazebo experiments validating practical performance. The approach offers real-time applicability by avoiding full map-based planning, using geometric projections and, in the sensor-based case, local sensing to drive safe navigation toward .

Abstract

We propose a continuous feedback control strategy that steers a point-mass vehicle safely to a destination, in a quasi-optimal manner, in sphere worlds. The main idea consists in avoiding each obstacle via the shortest path on the cone's surface enclosing the obstacle and moving straight toward the target when the vehicle has a clear line of sight to the target location. In particular, almost global asymptotic stability of the target location is achieved in two-dimensional (2D) environments under a particular assumption on the obstacles configuration. We also propose a reactive (sensor-based) approach, suitable for real-time implementations in a priori unknown 2D environments with sufficiently curved convex obstacles, guaranteeing almost global asymptotic stability of the target location. Simulation results are presented to illustrate the effectiveness of the proposed approach.
Paper Structure (20 sections, 13 theorems, 49 equations, 26 figures, 3 tables, 2 algorithms)

This paper contains 20 sections, 13 theorems, 49 equations, 26 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Prior Literature
  4. Contributions
  5. Organization
  6. Notations and Preliminaries
  7. Problem Formulation
  8. Sets Definition and Obstacles Classification
  9. Control Design
  10. Single Obstacle Case
  11. Multiple Obstacles Case
  12. Characterization of the generated trajectories
  13. Safety and Stability Analysis
  14. Sensor-based navigation using a 2D LiDAR range scanner
  15. Convex obstacles
  16. ...and 5 more sections

Key Result

Lemma 1

The solution of the optimization problem min-constraint is unique and is given by where $\xi:\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{N}\to\mathbb{R}^n$ is given by with $\bar{\xi}_i\in\mathcal{V}(c_i-x,\theta_i)$,

Figures (26)

  • Figure 1: 2D representation of the sets in Section IV.
  • Figure 2: Shortest path in a single-obstacle sphere world (blue curve).
  • Figure 3: Successive projections of the control $u_d$ in a two-dimensional sphere world with four obstacles.
  • Figure 4: Quasi-optimal trajectory in 2D workspace.
  • Figure 5: Optimal and quasi-optimal trajectories.
  • ...and 21 more figures

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Lemma 3: Safety
  • proof
  • ...and 20 more