[Paper Review] Global Exponential Stabilization on SO(3) via Hybrid Feedback

Soulaimane Berkane, Abdelkader Abdessameud+1 more|arXiv (Cornell University)|Oct 13, 2015

Adaptive Control of Nonlinear Systems24 references3 citations

TL;DR

This paper proposes a hybrid feedback control framework using exp-synergistic potential functions to achieve global exponential stabilization on SO(3), enabling robust attitude control and estimation. The method leverages non-differentiable potential functions with quadratic gradient behavior and synergistic properties at singular points, ensuring exponential convergence to the desired attitude globally.

ABSTRACT

We propose a new hybrid feedback approach for global exponential stabilization on the Special Orthogonal group SO(3), which can be used to solve attitude control and attitude estimation problems. Our approach relies on a new concept of synergistic potential functions on SO(3), coined exp-synergistic, designed for global exponential stability purposes. Roughly speaking, the new synergism concept allows the use of non-differentiable potential functions that enjoy the property of being quadratic with respect to their gradients and being synergistic with respect to the gradients singular points. The proposed approach is used to solve the attitude estimation and attitude tracking problem, leading to global exponential stability results.

Motivation & Objective

• To address the challenge of achieving global exponential stabilization on SO(3) for attitude control and estimation systems.
• To develop a novel class of potential functions—exp-synergistic—that enable exponential convergence despite non-differentiability.
• To design a hybrid feedback control law that ensures global exponential stability using these new potential functions.
• To extend the applicability of synergistic potential functions to non-smooth settings while preserving stability properties.
• To solve the attitude tracking and estimation problems with guaranteed exponential convergence on SO(3).

Proposed method

• Introduces the concept of exp-synergistic potential functions on SO(3), which are non-differentiable but exhibit quadratic behavior in their gradients.
• Defines synergism with respect to gradient singular points, ensuring that the potential function drives the system toward the desired attitude globally.
• Constructs a hybrid feedback control law that switches between different control laws based on the system's state and potential function structure.
• Utilizes the geometric structure of SO(3) to define potential functions that are intrinsic to the rotation group and avoid singularities.
• Employs a Lyapunov-based analysis to prove global exponential stability, leveraging the quadratic gradient property of the potential functions.
• Applies the framework to both attitude tracking and attitude estimation problems, demonstrating its versatility.

Experimental results

Research questions

• RQ1 Can non-differentiable potential functions be used to achieve global exponential stability on SO(3)?
• RQ2 How can synergism be redefined on SO(3) to allow for exponential convergence in the presence of gradient singularities?
• RQ3 What control architecture enables global exponential stabilization using such potential functions?
• RQ4 Can the proposed framework be applied to both attitude tracking and estimation problems?
• RQ5 What are the necessary conditions on the potential function to ensure exponential convergence on SO(3) without requiring smoothness?

Key findings

• The proposed exp-synergistic potential functions allow global exponential stabilization on SO(3) even when the functions are non-differentiable.
• The gradient of the potential function behaves quadratically near singular points, enabling strong convergence properties.
• The hybrid feedback control law ensures exponential convergence to the desired attitude from any initial condition on SO(3).
• The method successfully solves both the attitude tracking and attitude estimation problems with global exponential stability.
• The theoretical framework extends the applicability of synergistic potential functions to non-smooth settings on compact Lie groups.
• The results are derived using a Lyapunov analysis that exploits the quadratic gradient structure and synergistic properties of the potential functions.