A Maneuvering Model for an Underwater Vehicle Near a Free Surface—Part II: Incorporation of the Free-Surface Memory

Abstract

Using energy-based modeling techniques, we propose a nonlinear, time-dependent, parametric motion model for an underwater vehicle maneuvering near an otherwise undisturbed free surface. By augmenting the system Lagrangian used to derive Kirchhoff's equations for a rigid body moving through an unbounded fluid, we directly incorporate the free surface into the derivation of the equations of motion. This is done using a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">free-surface Lagrangian</i> , which accounts for the instantaneous energy stored within the free surface due to an impulsive vehicle motion as well as fluid memory effects. The system Lagrangian then enables us to derive the six-degree-of-freedom nonlinear equations of motion using the Euler–Lagrange equations. The model structure is similar to standard maneuvering models for surface ships, although additional complexities are present since the hydrodynamic parameters are shown to depend on the vessel position and orientation relative to the free surface. For the proposed model, the vessel motion is unrestricted. This is in contrast to traditional seakeeping models, which use convolution integrals to incorporate memory effects for a vessel, which experiences only small perturbations from steady, forward motion. The proposed motion model is amenable to real-time simulation, design performance analysis, and nonlinear control design. Other important hydrodynamic effects due to viscous flow, for example, may then be incorporated into a robust, nonlinear, closed-loop control system as lumped parameter effects or model uncertainties.