The dynamic complexity of robots and mechatronic systems often pertains to the hybrid nature of dynamics, where governing equations consist of heterogenous equations that are switched depending on the state of the system. Legged robots and manipulator robots experience contact-noncontact discrete transitions, causing switching of governing equations. Analysis of these systems have been a challenge due to the lack of a global, unified model that is amenable to analysis of the global behaviors. Composition operator theory has the potential to provide a global, unified representation by converting them to linear dynamical systems in a lifted space. The current work presents a method for encoding nonlinear heterogenous dynamics into a high dimensional space of observables in the form of Koopman operator. First, a new formula is established for representing the Koopman operator in a Hilbert space by using inner products of observable functions and their composition with the governing state transition function. This formula, called Direct Encoding, allows for converting a class of heterogenous systems directly to a global, unified linear model. Unlike prevalent data-driven methods, where results can vary depending on numerical data, the proposed method is globally valid, not requiring numerical simulation of the original dynamics. A simple example validates the theoretical results, and the method is applied to a multi-cable suspension system.

翻译：机器人与机电系统的动力学复杂性常源于其混合动力学特性，其控制方程由根据系统状态切换的异构方程组成。腿式机器人与机械臂在接触-非接触离散过渡过程中会导致控制方程切换。由于缺乏能分析全局行为的统一全局模型，此类系统的分析极具挑战性。算子组合理论通过将系统映射至升维空间中的线性动力系统，为建立全局统一表示提供了可能。本研究提出一种方法，将非线性异构动力学编码为可观测量高维空间中的Koopman算子。首先建立新公式，通过可观测量函数的内积及其与状态转移函数的复合运算，在希尔伯特空间中表示Koopman算子。该公式称为直接编码法，可将一类异构系统直接转化为全局统一的线性模型。与依赖数值数据且结果可能变化的现有数据驱动方法不同，该方法具有全局有效性，无需对原始动力学进行数值仿真。通过简单算例验证了理论结果，并将该方法应用于多缆索悬挂系统。