Koopman-Based Rigid-Body Dynamics Modeling and Control: Quadrotor Case Study

A novel and analytical Koopman-Operator-Theory-based methodology is presented to derive rigid-body position and attitude dynamics, with direct applications to underactuated quadrotor and multirotor unmanned aerial vehicles (UAVs). The presented methodology may be used to derive, implement, and test control strategies for generalized robotic platforms and other aerospace systems. Unlike existing data-driven Koopman-based techniques, this formulation is model based and allows for an exact linear model representation of the original nonlinear position and attitude underlying system dynamics. The system model is linear in the autonomous component and state dependent in the control component. The validity range of the finite Koopman model truncation is determined, followed by the controllability and stabilizability analysis. Compared to existing literature formulations, the analytically derived Koopman-based model results in a better approximation of the original dynamics because it uses a more compact truncation of the lifted state space. Further, the system model is derived by using the Koopman approach on the complete system dynamics, without the need for angular velocity dynamic compensation. We show that a truncated subset of the infinite-dimensional model embeds most of the original nonlinear dynamics and can be used to design linear controllers in the lifted space; this corresponds to designing nonlinear controllers in the original state space. A quadrotor UAV is used for implementation and proof-of-concept demonstration purposes. The main advantages of the proposed methodology center around the effective use of linear control strategies for nonlinear plants and for solving the underactuation problem employing a single control loop.