Elastohydrodynamic instabilities of a soft robotic arm in a viscous fluid

TL;DR

This work develops a geometrically exact, dissipative Cosserat-rod model for a soft robotic arm operating in a viscous fluid under terminal pressure. By formulating the equations of motion on SE(2) with covariant derivatives and linear viscoelastic constitutive laws, the authors reveal a non-Hermitian linear stability problem that exhibits Hopf bifurcations as the follower pressure is varied, followed by re-stabilization at higher pressures. Numerical simulations show stable limit-cycle beating between two critical pressures, while an Euler-Bernoulli-like limit explains how stretch enhances bending stiffness and can suppress instabilities. The framework provides a powerful tool for predicting and controlling viscous dynamics in soft robotics and can be extended to richer 3D geometries and Lie-group settings.

Abstract

The design and control of soft robots operating in fluid environments requires a careful understanding of the interplay between large elastic body deformations and hydrodynamic forces. Here we show that this interplay leads to novel elastohydrodynamic instabilities in a clamped soft robotic arm driven terminally by a constant pressure in a viscous fluid. We model the arm as a Cosserat rod that can stretch, shear and bend. We obtain invariant, geometrically exact, non-linear equations of motion by using Cartan's method of moving frames. Stability to small perturbations of a straight rod is governed by a non-Hermitian linear operator. Eigenanalysis shows that stability is lost through a Hopf bifurcation with the increase of pressure above a first threshold. A surprising return to stability is obtained with further increase of pressure beyond a second threshold. Numerical solutions of the non-linear equations, using a geometrically exact spectral method, confirms stable limit-cycle oscillations between these two pressure thresholds. An asymptotic analysis in the beam limit rationalizes these results analytically. This counterintuitive sequence of bifurcations underscores the subtle nature of the elastohydrodynamic coupling in Cosserat rods and emphasizes their importance for the control of the viscous dynamics of soft robots.

Figures (7)

• Figure 1: The first two roots, $\omega_{1}$ (blue) and $\omega_{2}$ (red), as a function of the strength of the follower force for the different combinations of parameters considered in Table \ref{['tab:parameters']}. The top plots show the real parts (solid lines) and the imaginary parts (dotted lines) of $\omega_{1}$ and $\omega_{2}$. The corresponding plots at the bottom display the locus of $\omega_{1}$ and $\omega_{2}$ as the follower force parameter is swept. In (a), We find an onset of stable oscillations at $\tilde{\mathcal{F}}=20.01$, after which $\omega_{1}$ and $\omega_{2}$ become complex conjugate pairs and $\omega_{2}=\omega_{2}^{*}$. Stability is lost through a Hopf bifurcation, which is observed for $\tilde{\mathcal{F}}_{**}=37.66$ when the real parts of the roots become positive and the roots cross the imaginary axis. A qualitatively similar behavior is observed in (b), where $\tilde{\mathcal{F}}_{*}=14.80$ and $\tilde{\mathcal{F}}_{**}=27.43$. In (c), we observe a qualitatively different behavior, where stability is lost through a Hopf bifurcation at $\tilde{\mathcal{F}}_{**}=49.76$ but subsequently regained for $\tilde{\mathcal{F}}>150$ when the real parts of the roots become negative again and the roots re-enter through the imaginary axis. Finally, in (d), although we find an onset of stable oscillations at $\tilde{\mathcal{F}}_{*}=27.41$, stability is not lost through a Hopf bifurcation and no flutter instabilities are observed.
• Figure 2: Semilog plots of the critical values of the follower force strength as a function of $\tilde{k}_{2}$ and$\tilde{\gamma}_{3}^{-1}$.
• Figure 3: Emergence of limit cycle oscillations of the entire rod (left) and of the top of the rod (right) for $(\tilde{k}_{1},\tilde{k}_{2},\tilde{\gamma}_{3})=(10^{4},10^{2},10^{-2})$ and $\tilde{\mathcal{F}}=50$. Time is measured in units of $\tau_{1}$ in the plots of the tip coordinates. For the visualization of the rod, the color scale is such that later snapshots of the rod are darker. The arrows indicate the configuration-dependent direction of the follower force at each time slice.
• Figure 4: The steady-state limit cycle amplitudes of the coordinates of the tip of the rod as a function of the strength of the follower force.
• Figure 5: The first eigenmode $(0,Y_{1},\Theta_{1})$ and its associated deformations for $(\tilde{k}_{1},\tilde{k}_{2},\tilde{\gamma}_{3})=(10^{4},10^{2},10^{-2})$ and $\tilde{\mathcal{F}}=10$, before the onset of oscillations.