  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.
