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Global Stabilization of Antipodal Points on n-Sphere with Application to Attitude Tracking

Xin Tong, Shing Shin Cheng

Abstract

Existing approaches to robust global asymptotic stabilization of a pair of antipodal points on unit $n$-sphere $\mathbb{S}^n$ typically involve the non-centrally synergistic hybrid controllers for attitude tracking on unit quaternion space. However, when switching faults occur due to parameter errors, the non-centrally synergistic property can lead to the unwinding problem or in some cases, destabilize the desired set. In this work, a hybrid controller is first proposed based on a novel centrally synergistic family of potential functions on $\mathbb{S}^n$, which is generated from a basic potential function through angular warping. The synergistic parameter can be explicitly expressed if the warping angle has a positive lower bound at the undesired critical points of the family. Next, the proposed approach induces a new quaternion-based controller for global attitude tracking. It has three advantageous features over existing synergistic designs: 1) it is consistent, i.e., free from the ambiguity of unit quaternion representation; 2) it is switching-fault-tolerant, i.e., the desired closed-loop equilibria remain asymptotically stable even when the switching mechanism does not work; 3) it relaxes the assumption on the parameter of the basic potential function in literature. Comprehensive simulation confirms the high robustness of the proposed centrally synergistic approach compared with existing non-centrally synergistic approaches.

Global Stabilization of Antipodal Points on n-Sphere with Application to Attitude Tracking

Abstract

Existing approaches to robust global asymptotic stabilization of a pair of antipodal points on unit -sphere typically involve the non-centrally synergistic hybrid controllers for attitude tracking on unit quaternion space. However, when switching faults occur due to parameter errors, the non-centrally synergistic property can lead to the unwinding problem or in some cases, destabilize the desired set. In this work, a hybrid controller is first proposed based on a novel centrally synergistic family of potential functions on , which is generated from a basic potential function through angular warping. The synergistic parameter can be explicitly expressed if the warping angle has a positive lower bound at the undesired critical points of the family. Next, the proposed approach induces a new quaternion-based controller for global attitude tracking. It has three advantageous features over existing synergistic designs: 1) it is consistent, i.e., free from the ambiguity of unit quaternion representation; 2) it is switching-fault-tolerant, i.e., the desired closed-loop equilibria remain asymptotically stable even when the switching mechanism does not work; 3) it relaxes the assumption on the parameter of the basic potential function in literature. Comprehensive simulation confirms the high robustness of the proposed centrally synergistic approach compared with existing non-centrally synergistic approaches.
Paper Structure (18 sections, 10 theorems, 43 equations, 4 figures)

This paper contains 18 sections, 10 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Notations and Lemmas
  4. Problem Formulation
  5. Main Results
  6. Continuous Feedback Control
  7. Synergistic Control
  8. Construction of a Centrally Synergistic Family on n-Sphere
  9. Application to Attitude Tracking
  10. Centrally Synergistic Potential Functions on 3-Sphere
  11. Global Hybrid Attitude Tracking
  12. Simulations
  13. Conclusion
  14. Acknowledgment
  15. Proof of Lemma \ref{['lem:nec_cond_synFunc']}
  16. ...and 3 more sections

Key Result

Lemma 1

Let $n \geq 3$ and $S \in \mathbb{R}^{n \times n}$ be skew-symmetric such that $S^3 = - a^2 S$ for some $a > 0$. Then, for all $\phi \in \mathbb{R}$, $e^{S \phi} \in \mathbb{R}^{n \times n}$ is an orthogonal matrix and $e^{S \phi} = I + a^{-1} \sin ( a \phi ) S + a^{-2} \bigl( 1 - \cos ( a \phi ) \

Figures (4)

  • Figure 1: Discontinuous attitude measurement: the scalar part of $Q_n$.
  • Figure 2: Scenario A: global attitude tracking by CS Hybrid controller.
  • Figure 3: Scenario B: hybrid behavior of CS Hybrid controller.
  • Figure 4: Scenario C: quaternion measurement sensitivity with switching faults (shadow zone).

Theorems & Definitions (22)

  • Lemma 1: Bernstein2018
  • Lemma 2
  • proof
  • Definition 1: Mayhew2013Mayhew2013b
  • Remark 1
  • Lemma 3
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 12 more