Motion planning for an articulated body in a perfect planar fluid^∗

Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanisms. We use a geometric framework to consider the simplified problem of an articulated two-dimensional body in a potential flow. This paper builds upon the current geometric theory by showing that although the group of Euclidean transformations is non-Abelian, certain tools available for Abelian groups may still be exploited, making use of the semidirect-product structure of this group. In particular, the holonomy in the rotation component may be explicitly computed as a function of the area enclosed by a path in shape space. We use this tool to develop open-loop gaits for an articulated body with two shape variables, using plots of the curvature of the mechanical connection, which relates motion in the shape space to motion of the overall body. Results from numerical computations of the mechanical connection are compared to theoretical results assuming the joints are hydrodynamically decoupled. Finally, we consider a simple method for trajectory tracking in the plane, using a one-parameter family of gaits.