  Continuum soft robots are nonlinear mechanical systems with theoretically
infinite degrees of freedom (DoFs) that exhibit complex behaviors. Achieving
motor intelligence under dynamic conditions necessitates the development of
control-oriented reduced-order models (ROMs), which employ as few DoFs as
possible while still accurately capturing the core characteristics of the
theoretically infinite-dimensional dynamics. However, there is no quantitative
way to measure if the ROM of a soft robot has succeeded in this task. In other
fields, like structural dynamics or flexible link robotics, linear normal modes
are routinely used to this end. Yet, this theory is not applicable to soft
robots due to their nonlinearities. In this work, we propose to use the recent
nonlinear extension in modal theory -- called eigenmanifolds -- as a means to
evaluate control-oriented models for soft robots and compare them. To achieve
this, we propose three similarity metrics relying on the projection of the
nonlinear modes of the system into a task space of interest. We use this
approach to compare quantitatively, for the first time, ROMs of increasing
order generated under the piecewise constant curvature (PCC) hypothesis with a
high-dimensional finite element (FE)-like model of a soft arm. Results show
that by increasing the order of the discretization, the eigenmanifolds of the
PCC model converge to those of the FE model.
