Nonlinear Modes as a Tool for Comparing the Mathematical Structure of Dynamic Models of Soft Robots

TL;DR

This work introduces eigenmanifolds as a tool to quantitatively compare control-oriented reduced-order models for continuum soft robots, addressing the challenge of infinite degrees of freedom by projecting nonlinear modes into a fixed task space and measuring their similarity. A novel energy-based continuation technique accelerates nonlinear manifold computation, enabling systematic comparison of ROMs derived from different discretization hypotheses. By applying these metrics to piecewise constant curvature (PCC) models and a high-DoF FE-like model, the study shows convergence of PCC eigenmanifolds to the high-fidelity model as discretization increases, with four to five DoFs providing the closest overall match to the rigid baseline in task-space motion. These results establish a practical, geometry-aware framework for validating and selecting ROMs, with potential to inform learning-based model reduction and controller design for soft robotics, where nonlinear modal structure is crucial. The approach is grounded in nonlinear modal theory, using generators parameterized by energy and projections to a fixed dimension, and it highlights that energy-frequency similarity alone may be insufficient to capture cross-model equivalence.

Abstract

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.

Figures (8)

• Figure 1: Schematic of our approach using nonlinear modal analysis and eigenmanifold theory to compare different control-oriented models for the same soft robot. Models differ in assumptions and number of DoFs, making systematic comparison challenging. In the figure, $\sim$ indicates a similar structure, $\nsim$ indicates dissimilarity and a question mark denotes incomparability.
• Figure 2: Schematic representation of the eigenmanifold projection. Given three control-oriented ROMs, their eigenmanifolds $\mathfrak{M}_{1}$, $\mathfrak{M}_{2}$ and $\mathfrak{M}_{3}$ cannot be directly compared in configuration space. To overcome this issue, we introduce a constant dimensional task space $\boldsymbol{h}({\boldsymbol{q}}, \boldsymbol{s})$ in which $\mathfrak{M}_{1}$, $\mathfrak{M}_{2}$ and $\mathfrak{M}_{3}$ are projected and here define similarity measures.
• Figure 3: Stroboscopic plots of the robots in the workspace for the first (a)--(g), second (h)--(n) and third (o)--(u) NMs at $E_{\max} = 1~[J]$ and normalized time instants with respect to the mode period. Each color corresponds to a different model.
• Figure 4: Energy-frequency relationships for the PCC (a)--(e) and FE (f) discretization, respectively. The models show a similar behavior only for the first two modes.
• Figure 5: Energy evolution of the modal Fréchet distance $\boldsymbol{f}(E, L)$ for the $x$ (a)--(e) and $z$ (f)--(j) tip position.