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Dynamic Constrained Stabilization on the $n$-sphere

Mayur Sawant, Abdelhamid Tayebi

Abstract

We consider the constrained stabilization problem of second-order systems evolving on the n-sphere. We propose a control strategy with a constraint proximity-based dynamic damping mechanism that ensures safe and almost global asymptotic stabilization of the target point in the presence of star-shaped constraints on the n-sphere. It is also shown that the proposed approach can be used to deal with the constrained rigid-body attitude stabilization. The effectiveness of the proposed approach is demonstrated through simulation results on the 2-sphere in the presence of star-shaped constraint sets.

Dynamic Constrained Stabilization on the $n$-sphere

Abstract

We consider the constrained stabilization problem of second-order systems evolving on the n-sphere. We propose a control strategy with a constraint proximity-based dynamic damping mechanism that ensures safe and almost global asymptotic stabilization of the target point in the presence of star-shaped constraints on the n-sphere. It is also shown that the proposed approach can be used to deal with the constrained rigid-body attitude stabilization. The effectiveness of the proposed approach is demonstrated through simulation results on the 2-sphere in the presence of star-shaped constraint sets.

Paper Structure

This paper contains 16 sections, 4 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Problem statement
  4. Generic feedback control design
  5. Explicit feedback control design
  6. Unit-quaternion-based constrained attitude stabilization
  7. Simulation results
  8. Conclusion
  9. Examples of separation function
  10. Proof of Lemma \ref{['lemma:safety_lemma']}
  11. Proof of Claim \ref{['claim1:lemmaVTF']}
  12. Proof of Claim \ref{['claim2:lemmaVTF']}
  13. Proof of Claim \ref{['claim3:lemmaVTF']}
  14. Proof of Theorem \ref{['theorem:VTF']}
  15. Proof of Claim \ref{['claim3:theorem']}
  16. ...and 1 more sections

Key Result

Lemma 1

Consider the closed-loop system dynamics_motion_model_on_sphere-proposed_feedback_control_input under Assumption assumption:twice_differentiability. If $d_{\mathcal{U}}(\mathbf{x}(0)) > 0$, then the following statements hold:

Figures (1)

  • Figure 1: Implementation of the closed-loop system \ref{['dynamics_motion_model_on_sphere']}-\ref{['proposed_feedback_control_input']} with $\mathbf{v}_d$ defined in \ref{['ideal_kinematic_planner_example']}. (a) $\mathbf{x}$-trajectories safely converging to $\mathbf{x}_d$, (b) $\|\mathbf{v} - \boldsymbol{\nu}_d(\mathbf{x})\|$ versus time, (c) $\|\mathbf{u}\|$ versus time, (d) $d_{\mathcal{U}}(\mathbf{x})$ versus time.

Theorems & Definitions (6)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Remark 2
  • Proposition 1
  • Lemma 2