Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method

TL;DR

This paper introduces a novel method to encode heterogeneous robot dynamics into a unified linear model using Koopman operator theory, enabling global analysis without reliance on numerical simulations.

Contribution

The work develops a direct encoding formula for Koopman operators that captures hybrid robot dynamics in a high-dimensional observable space, providing a globally valid, unified linear representation.

Findings

The method accurately models hybrid dynamics without numerical simulation.

Validated on a multi-cable suspension system.

Provides a theoretical foundation for analyzing complex robot behaviors.

Abstract

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.