Koopman operator theory has been gaining momentum for model extraction, planning, and control of data-driven robotic systems. The Koopman operator's ability to extract dynamics from data depends heavily on the selection of an appropriate dictionary of lifting functions. In this paper we propose ACD-EDMD, a new method for Analytical Construction of Dictionaries of appropriate lifting functions for a range of data-driven Koopman operator based nonlinear robotic systems. The key insight of this work is that information about fundamental topological spaces of the nonlinear system (such as its configuration space and workspace) can be exploited to steer the construction of Hermite polynomial-based lifting functions. We show that the proposed method leads to dictionaries that are simple to implement while enjoying provable completeness and convergence guarantees when observables are weighted bounded. We evaluate ACD-EDMD using a range of diverse nonlinear robotic systems in both simulated and physical hardware experimentation (a wheeled mobile robot, a two-revolute-joint robotic arm, and a soft robotic leg). Results reveal that our method leads to dictionaries that enable high-accuracy prediction and that can generalize to diverse validation sets. The associated GitHub repository of our algorithm can be accessed at \url{https://github.com/UCR-Robotics/ACD-EDMD}.
翻译:Koopman 操作员理论在数据驱动机器人系统的模型提取、规划和控制方面的势头日益增强。 Koopman 操作员从数据中提取动态的能力在很大程度上取决于选择适当的提升功能字典。 本文中我们提出ACD- EDMD,这是分析一系列数据驱动的库普曼操作员以非线性机器人系统为基础的非线性机器人系统适当提升功能的词典的新方法。 这项工作的关键洞察力是,关于非线性系统基本地形空间(如其配置空间和工作空间)的信息可以被利用来指导赫米特聚氨酯(Hemite)多边基升动功能的构建。 我们显示,在可观测到加权约束时,拟议方法导致易于执行的词典,同时享有可确认的完整性和汇合的保证。 我们用多种非线性机器人系统在模拟和物理硬件实验中(轮式移动机器人、双振动联合机器人臂和软机器人腿)对ACD- EDMD进行评估。 结果显示,我们的方法导致词典的构建,从而能够实现高频/RUMDMD的预测。 通用的演算。