Koopman Operator Based Modeling for Quadrotor Control on $SE(3)$

TL;DR

This paper introduces a systematic method to derive observable functions for Koopman operator modeling of quadrotor dynamics on SE(3), enabling linear control design for complex nonlinear systems.

Contribution

We derive a systematic set of observable functions for quadrotor dynamics on SE(3) and prove their convergence, improving the Koopman operator approach without relying on machine learning or guesswork.

Findings

Finite observable set approximates nonlinear dynamics more accurately with higher dimension

The derived observables converge pointwise to zero

Numerical simulations confirm improved modeling accuracy

Abstract

In this paper, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observable functions (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observable functions that can span the lifted space. Recent methods either use tools from machine learning to learn the observable functions or guess a suitable set of observables that best describes the nonlinear dynamics. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on SE(3). In addition, we prove that the proposed sequence of observable functions converges pointwise to the zero function, which allows us to select only a finite set of observable functions to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.