Modelling, Controllability and Gait Design for a Spherical Flexible Swimmer

Abstract: This paper discusses modelling, controllability and gait design for a spherical flexible swimmer. We first present a kinematic model of a low Reynolds number spherical flexible swimming mechanism with periodic surface deformations in the radial and azimuthal directions. The model is then converted to a finite dimensional driftless, affine-in-control principal kinematic form by representing the surface deformations as a linear combination of finitely many Legendre polynomials. A controllability analysis is then done for this swimmer to conclude that the swimmer is locally controllable on $\mathbb{R}^3$ for certain combinations of the Legendre polynomials. The rates of the coefficients of the polynomials are considered as the control inputs for surface deformation. Finally, the Abelian nature of the structure group of the swimmer's configuration space is exploited to synthesize a curvature based gait for the spherical flexile swimmer and a rigid-link swimmer.