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Hybrid Feedback Control for Global and Optimal Safe Navigation

Ishak Cheniouni, Soulaimane Berkane, Abdelhamid Tayebi

Abstract

We propose a hybrid feedback control strategy that safely steers a point-mass robot to a target location optimally from all initial conditions in the n-dimensional Euclidean space with a single spherical obstacle. The robot moves straight to the target when it has a clear line-of-sight to the target location. Otherwise, it engages in an optimal obstacle avoidance maneuver via the shortest path inside the cone enclosing the obstacle and having the robot's position as a vertex. The switching strategy that avoids the undesired equilibria, leading to global asymptotic stability (GAS) of the target location, relies on using two appropriately designed virtual destinations, ensuring control continuity and shortest path generation. Simulation results illustrating the effectiveness of the proposed approach are presented.

Hybrid Feedback Control for Global and Optimal Safe Navigation

Abstract

We propose a hybrid feedback control strategy that safely steers a point-mass robot to a target location optimally from all initial conditions in the n-dimensional Euclidean space with a single spherical obstacle. The robot moves straight to the target when it has a clear line-of-sight to the target location. Otherwise, it engages in an optimal obstacle avoidance maneuver via the shortest path inside the cone enclosing the obstacle and having the robot's position as a vertex. The switching strategy that avoids the undesired equilibria, leading to global asymptotic stability (GAS) of the target location, relies on using two appropriately designed virtual destinations, ensuring control continuity and shortest path generation. Simulation results illustrating the effectiveness of the proposed approach are presented.
Paper Structure (16 sections, 7 theorems, 18 equations, 5 figures, 1 algorithm)

This paper contains 16 sections, 7 theorems, 18 equations, 5 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Notations and Preliminaries
  3. Problem Formulation
  4. Sets Definition
  5. Control Design
  6. Hybrid Control
  7. main result
  8. Numerical simulation
  9. Conclusion
  10. APPENDIX
  11. Proof of Lemma \ref{['lem2']}
  12. Proof of Lemma \ref{['lem3']}
  13. Proof of Theorem \ref{['the1']}
  14. Proof of Lemma \ref{['lem4']}
  15. Proof of Lemma \ref{['lem5']}
  16. ...and 1 more sections

Key Result

Lemma 1

Let $c,a_{-1},a_{1}\in\mathbb{R}^n$ such that $\angle(a_{-1},a_{1})=\psi$ where $\psi\in(0,\pi]$. Let $\varphi_{-1},\varphi_{1}\in[0,\pi]$ such that $\varphi_{-1}+\varphi_{1}<\psi<\pi-(\varphi_{-1}+\varphi_{1})$. Then

Figures (5)

  • Figure 1: Illustration of the different subsets of the free space defined in Section \ref{['section:sets']}.
  • Figure 2: Shadow regions of the virtual destinations.
  • Figure 3: The flow and jump sets for the proposed dynamical hybrid system.
  • Figure 4: Trajectories generated by our proposed hybrid control law in 2D space from nine different initial conditions.
  • Figure 5: Comparison of our proposed hybrid control law with the hybrid control proposed in HybBerkaneECC2019 in 3D space.

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 4 more