Tracking Control for an Ellipsoidal Submarine Driven by Kirchhoff´s Laws

In this paper, we study the control of an ellipsoid immersed in an infinite volume of ideal fluid. The dynamics of the uncontrolled body are given by Kirchhoff´s laws. The control system is underactuated: one control is an acceleration along an axis of the ellipsoid and two are angular accelerations around the other two axes. By adopting a backstepping viewpoint, we prove that the position and the attitude of the solid can be forced to approximately follow any given path, using fast-oscillating controls. Moreover, we prove that the controlled mechanical system (which includes the impulses) is completely controllable in an arbitrary small time.