Koopman Analytical Modeling of Position and Attitude Dynamics: a Case Study for Quadrotor Control

TL;DR

This work addresses underactuated rigid-body control by deriving an analytical Koopman operator–based representation of coupled position and attitude dynamics, eliminating the need for angular-velocity compensation. It constructs a lifted, quasi-linear model with a constant autonomous matrix $A$ in Jordan form and a state-dependent control matrix $B(x)$ using analytically derived generalized Koopman eigenfunctions that capture the full dynamics in a linear framework. A boundary analysis and controllability check are provided to quantify the validity of finite-dimensional truncations, and numerical validations on attitude, position, and a quadrotor model demonstrate improved approximation with a compact observable set and a single-loop control design. The practical impact lies in enabling linear control strategies for nonlinear, underactuated drones through a compact lifted representation that preserves essential nonlinear couplings between rotation and translation.

Abstract

This research presents a novel, analytical, Koopman Operator based formulation for position and attitude dynamics which can be used to derive control strategies for underactuated systems. Compared to data driven Koopman based techniques, the analytical approach presented in this work is model based and allows for an exact linear representation of the original nonlinear position and attitude dynamics. In fact, the resulting infinite dimensional model, defined in the lifted state space, is linear in the autonomous component and state dependent in the control. A boundary study is carried on to define the range of validity of the finite truncation of the Koopman based model followed by a controllability and stabilizability analysis to show the feasibility of employing the derived model for control system design. Compared to existing literature formulation, the presented model results in a better approximation of the original dyanmics using a more compact truncation of the lifted state space. Moreover, the model is derived using the Koopman approach on the entirety of the dynamics and does not require the need of angular velocity dynamic compensation. A case study involving an underactuated quadrotor unmanned aerial vehicle (UAV) is provided to show that, for practical use, a truncated subset of the infinite dimensional model, embeds most of the original nonlinear dynamics and can be used to design linear control strategies in the lifted space which results in nonlinear controllers in the original state space. The main advantages of the presented approach reside in the effective use of linear control strategies for nonlinear plats and the solution of the underactuation problem employing a single control loop.

Figures (24)

• Figure 1: Unforced approximation error with $J=J_0$ and $\nu(t_0) = 0.001$rad/s.
• Figure 2: Unforced approximation error with $J=J_0$ and $\nu(t_0) = 0.01$rad/s.
• Figure 3: Unforced approximation error with $J=J_0$ and $\nu(t_0) = 0.1$rad/s.
• Figure 4: Unforced approximation error with $J=J_0$ and $\nu(t_0) = 1$rad/s.
• Figure 5: Unforced approximation error with $J=J_0$ and $\nu(t_0) = 10$rad/s.