Offline models for autonomous robots often fail under time-varying dynamics outside their training distribution. Koopman operator theory offers a linear representation of nonlinear dynamics via lifting, but its transition to real-time recursive estimation may suffer numerical vulnerabilities: covariance windup under low excitation when using exponential forgetting, and vanishing gain without forgetting. This paper introduces a Covariance-Regulated Recursive Koopman Learning (CR-RKL) framework with two complementary strategies--error dead-zone gating and constant-trace normalization--each independently capable of preventing covariance explosion and parameter freezing, with the latter additionally preserving the geometric structure of uncertainty. Validated on a non-holonomic differential-drive robot with wheel slip and Stribeck friction and on a 26-gram butterfly-inspired flapping-wing micro aerial vehicle, CR-RKL achieves numerically stable and accurate online modeling, and when embedded in model predictive control, it maintains reliable tracking performance under uncertain, time-varying dynamics.

翻译：针对自主机器人的离线模型在时变动态超出训练分布时往往失效的问题。库普曼算子理论通过提升映射提供非线性动态的线性表示，但其向实时递归估计的转化可能面临数值脆弱性：采用指数遗忘时低激励条件下的协方差膨胀，以及无遗忘时的增益消失。本文提出协方差调控递归库普曼学习（CR-RKL）框架，包含两种互补策略——误差死区门控与常迹归一化——各自独立地防止协方差爆炸与参数冻结，其中后者还能保持不确定性的几何结构。通过具有车轮滑移和斯特里贝克摩擦的非完整差动驱动机器人，以及26克仿蝴蝶扑翼微型飞行器进行验证，CR-RKL实现了数值稳定且精确的在线建模；当嵌入模型预测控制时，它在不确定时变动态下仍能保持可靠的跟踪性能。