Controlling non-controllable scallops

Abstract

A swimmer embedded on an inertialess fluid must perform a non-reciprocal motion to swim forward. The archetypal demonstration of this unique motion-constraint was introduced by Purcell with the so-called “scallop theorem”. Scallop here is a minimal mathematical model of a swimmer composed by two arms connected via a hinge whose periodic motion (of opening and closing its arms) is not sufficient to achieve net displacement. Any source of asymmetry in the motion or in the forces/torques experienced by such a scallop will break the time-reversibility imposed by the Stokes linearity and lead to subsequent propulsion of the scallop. However, little is known about the controllability of time-reversible scalloping systems. Here, we consider two individually non-controllable scallops swimming together. Under a suitable geometric assumption on the configuration of the system, it is proved that controllability can be achieved as a consequence of their hydrodynamic interaction. A detailed analysis of the control system of equations is carried out analytically by means of geometric control theory. We obtain an analytic expression for the controlled displacement after a prescribed sequence of controls as a function of the phase difference of the two scallops. Numerical validation of the theoretical results is presented with model predictions in further agreement with the literature.