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Constrained Stabilization on the n-Sphere with Conic and Star-shaped Constraints

Mayur Sawant, Abdelhamid Tayebi

Abstract

The problem of constrained stabilization on the n-sphere under star-shaped constraints is considered. We propose a control strategy that allows to almost globally steer the state to a desired location while avoiding star-shaped constraints on the n-sphere. Depending on the state's proximity to the unsafe regions, the state is either guided towards the target location along the geodesic connecting the target to the state or steered towards the antipode of a predefined point lying in the interior of the nearest unsafe region. We prove that the target location is almost globally asymptotically stable under the proposed continuous, time-invariant feedback control law. Nontrivial simulation results on the 2-sphere and the 3-sphere demonstrate the effectiveness of the theoretical results.

Constrained Stabilization on the n-Sphere with Conic and Star-shaped Constraints

Abstract

The problem of constrained stabilization on the n-sphere under star-shaped constraints is considered. We propose a control strategy that allows to almost globally steer the state to a desired location while avoiding star-shaped constraints on the n-sphere. Depending on the state's proximity to the unsafe regions, the state is either guided towards the target location along the geodesic connecting the target to the state or steered towards the antipode of a predefined point lying in the interior of the nearest unsafe region. We prove that the target location is almost globally asymptotically stable under the proposed continuous, time-invariant feedback control law. Nontrivial simulation results on the 2-sphere and the 3-sphere demonstrate the effectiveness of the theoretical results.

Paper Structure

This paper contains 33 sections, 10 theorems, 95 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Problem formulation
  4. Constrained stabilization under conic constraints
  5. Constrained stabilization under star-shaped constraints
  6. Safety and stability analysis
  7. Application to Constrained Attitude Stabilization
  8. Reduced attitude control
  9. Full attitude control using the unit-quaternion
  10. Simulation results
  11. Geometric construction of a star-shaped set on the -sphere
  12. Constrained stabilization on
  13. Constrained stabilization on -sphere
  14. Conclusion
  15. Proof of Lemma \ref{['lemma:reverse_geodesic_always_stay_outside']}
  16. ...and 18 more sections

Key Result

Lemma 1

Let $\mathcal{A}$ be a star-shaped set on $\mathbb{S}^n$. Then, for every $\mathbf{g}\in\sigma(\mathcal{A})\cap\mathcal{A}^{\circ}$ and for all $\mathbf{x}\in\partial\mathcal{A}$, one has

Figures (12)

  • Figure 1: Illustration of (a) a star-shaped set and (b) a gs-convex set on $\mathbb{S}^n$.
  • Figure 2: Conic constraints \ref{['definition:conic_constraints']}.
  • Figure 3: $\mathbf{x}$-trajectories under the negative gradient-based control law \ref{['negative_gradient_control_law']}. (a) Case with an undesired unstable equilibrium. (b) Case with an undesired stable equilibrium.
  • Figure 4: Representation of $-\mathbf{P}(\mathbf{x})\mathbf{g}_i$ for $\mathbf{x}\in\partial\mathcal{U}_i$.
  • Figure 5: Illustration of mutually exclusive sets $\mathcal{R}_i$, where $i\in\mathbb{I}_a$.
  • ...and 7 more figures

Theorems & Definitions (16)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 2
  • Remark 3: Continuous control input
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 6 more