On Controller Design for Systems on Manifolds in Euclidean Space

TL;DR

This paper introduces a novel approach to controller design for systems on manifolds by embedding them into Euclidean space, enabling the use of standard Euclidean control techniques for complex manifold-based systems.

Contribution

The paper proposes a new method that embeds manifold systems into Euclidean space, allowing for global controller design using standard Euclidean techniques, demonstrated on rigid body and quadcopter systems.

Findings

Successfully applied to rigid body tracking

Effective for quadcopter drone control

Enables global controller synthesis in Euclidean space

Abstract

A new method is developed to design controllers in Euclidean space for systems defined on manifolds. The idea is to embed the state-space manifold $M$ of a given control system into some Euclidean space $\mathbb R^n$, extend the system from $M$ to the ambient space $\mathbb R^n$, and modify it outside $M$ to add transversal stability to $M$ in the final dynamics in $\mathbb R^n$. Controllers are designed for the final system in the ambient space $\mathbb R^n$. Then, their restriction to $M$ produces controllers for the original system on $M$. This method has the merit that only one single global Cartesian coordinate system in the ambient space $\mathbb R^n$ is used for controller synthesis, and any controller design method in $\mathbb R^n$, such as the linearization method, can be globally applied for the controller synthesis. The proposed method is successfully applied to the tracking problem for the following two benchmark systems: the fully actuated rigid body system and the quadcopter drone system.