In this paper, we propose a novel data-driven approach for learning and control of quadrotor UAVs based on the Koopman operator and extended dynamic mode decomposition (EDMD). Building observables for EDMD based on conventional methods like Euler angles or quaternions to represent orientation is known to involve singularities. To address this issue, we employ a set of physics-informed observables based on the underlying topology of the nonlinear system. We use rotation matrices to directly represent the orientation dynamics and obtain a lifted linear representation of the nonlinear quadrotor dynamics in the SE(3) manifold. This EDMD model leads to accurate prediction and can generalize to several validation sets. Further, we design a linear model predictive controller (MPC) based on the proposed EDMD model to track agile reference trajectories. Simulation results show that the proposed MPC controller can run as fast as 100 Hz and is able to track arbitrary reference trajectories with good accuracy. Implementation details can be found in \url{https://github.com/sriram-2502/KoopmanMPC_Quadrotor}

翻译：摘要：本文提出一种基于库普曼算子与扩展动态模态分解（EDMD）的新型数据驱动方法，用于四旋翼无人机的学习与控制。传统方法（如欧拉角或四元数）构建EDMD可观测函数以表示姿态时，存在奇异性问题。为解决该问题，我们采用一组基于非线性系统底层拓扑结构的物理信息可观测函数，直接使用旋转矩阵表示姿态动力学，并在SE(3)流形上获得非线性四旋翼动力学的线性提升表示。该EDMD模型可实现精准预测，并能泛化至多个验证集。进一步，我们基于所提EDMD模型设计了线性模型预测控制器（MPC）以跟踪敏捷参考轨迹。仿真结果表明，该MPC控制器运行频率可达100Hz，且能高精度跟踪任意参考轨迹。实现细节详见\url{https://github.com/sriram-2502/KoopmanMPC_Quadrotor}