Elastohydrodynamic instabilities of a soft robotic arm in a viscous fluid

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 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, nonlinear 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 nonlinear equations, using a geometrically exact spectral method, confirm 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.