Planar propulsion through the manipulation of circulatory flows

The development of thrust forces on the control surfaces of a variety of biomorphic underwater vehicles corresponds to the manipulation of circulatory flows about these surfaces. Lagrangian and Hamiltonian formalism exposes models for underwater vehicle dynamics to analytical tools from geometric mechanics and nonlinear control theory, but the naive introduction of circulation to conservative models for coupled vehicle-fluid systems generically introduces divergent terms to the fluid kinetic energy. We present models for circulation-based propulsion as it pertains to two archetypical underwater vehicle systems. We first present an externally-forced Lagrangian model for a submerged Flettner rotor, comparing gaits predicted by our model to those achieved with a robotic laboratory rotor. We then present and partially analyze a Hamiltonian model for planar carangiform locomotion in which the strength and relative position of a trailing point vortex are controlled to propel a rigid body. The flow around the vortex in this case approximates the flow around an oscillating hydrofoil; we derive a Poisson bracket for this system from the bracket governing the unconstrained interaction of a system of planar vortices using Dirac´s bracket formula.